Applications of Orlicz Spaces (Pure and Applied Mathematics)

0. therefore, ^(^) = 0 for 0 < f < a and $(v) = Jjj1'1 ^(s)ds = 0 if 0 < |v < a, i.e., ^ ^ II. A contradiction. By Proposition 3 (iii), $ is an AT-function. Finally, \I> is also an JV-function, since the complementary function of an JV-function is an AT-function as pointed out in Section 1.1. D Now we give some examples, which are Young functions but not AT-functions. Example 5. Let $i(u) = jj e*dt = e\u\ — 1, then its complementary Young function is given by

— v

v v\ > I.

By Definition 2, $1 G / n II n /// but $1 ^ IV since <^i(0) = 1, and ^i G / n /// n IV but #1 ^ //. Note also that for any K > 2, *!(2t;) - 0 - KVi(v) if 0 < v < J, and lim ^ - 1, i.e., $1 ^ V2(0). A basic property for AT-complementary pair ($, \l/) is that $ G V 2 (0)($ G V 2 (oo), or $ G V 2 ) iff * G A 2 (0)(# G A2(oo), or \I> G A 2 ). This example shows that the Young pair does not satisfy the above assertions. We shall give a classification of these growth conditions for Young functions (see Definition 8 below). Example 6. Let $2(«) = ^ if w < 1; = \u - \ if \u\ > 1. Then its complementary Young function \l>2 is given by ^ r 2 (v) = -^j- if |v| < 1; = +00 if \v > 1. By Definition 2, $2 G / n // n IV but $2 g ///,

30

/. Introduction and Background Material

and #2 e II n III n IV but *2 i I- Note that for all K > 0, # 2 (2v) = oo = KV2(v) if |v| > 1, and Jim - 1, i.e., $2 £ V2(oo). t—yoo

Example 7. Let $3(n) = \u , then * 3 (v) = 0 if v\ < 1;= +00 if v\ > 1, which follows from (1) and (2). We see that $3 G I n II but $2 £ /// u IV, and W 3 ^ I U II but ^3 G III n IV. Note that ^^ = 2 for all u > 0, i.e., $3 ^ V 2 (0) U V 2 (oo) but $3 G A 2 . Let us introduce the following: Definition 8. Let $ be a Young function. (i) $ G A 2 (oo) means $ G I and limsup AJy < oo, u—-too

(ii) <£ G A 2 (0) means $ G II and limsup ^vv < oo, u—>Q

(iii) $ G A 2 means $ G I n II and sup ^vv < oo, •u>0

^ '

(iv) $ G V 2 (oo) means $ G I and lim inf ^Ly v > 2, (v) $ G V 2 (0) means $ G II and lim inf J, v > 2, and finally u->0

*W

(vi) $ G V 2 means $ G I Pi II and inf v? v wv > 2. w>0

v ;

Prom this definition, we find that <3> G A 2 iff $ G A 2 (oo)nA 2 (0), and $ G V 2 iff $ G V 2 (oo) n V 2 (0). Let ($;, ^i), i = 1, 2,3 be given as in Examples 5-7. Then $! G A 2 (0)\V 2 (0), $1 G V 2 (oo)\A 2 (oo), *i G A 2 (oo)\V 2 (oo), and tyl £ A 2 (0) U V 2 (0). For i = 2,3,$i G A 2 but ^i ^ V 2 . Thus, the basic property for JV-functions, recalled in Example 5, need not be true for the Young pair. Remark 9. For a Young function $, Maligranda [2, p. 98] defined the A 2 condition of $ somewhat differently. However, our Definition 8 (i)-(iii) is seen to be the same as his definition. By Proposition 4 and Definition 8 we have the following: Corollary 10. Let ($, ^) be a Young pair. If $ G A 2 and $ G A 2; then $ and \£ are N-functions. If (ft, S, fj,) is an abstract measure space and $ is a Young function, we let L® and I/*) denote, as before, for measurable functions:

{/ (. Jn

= < f : ft -> R, |/||$ = sup { \ fgdfj, : py(g} < 1 > <

1.3 Notes on Young functions and general measures

31

where ^ is the complementary Young function of $, and L<*> = (/ : ft -» R, ||/||w = inf ik > 0 : / $ (() dp < 1\ < 00} . I I 7n \ f c / J J It is verified that || • ||$ and || • ||($) are norms if a.e. equal functions are identified and then L® = L^ as sets, and these are Banach spaces. We again call them Orlicz spaces. Same proofs hold, as may be seen from our earlier volume [1]. We recall the result of Wu et al. [1], stating that for an TV-function <&, 0 ^ / G L* =>• the existence of a k* > 0 such that 11/11* =

[l + P*(**/)].

(3)

If $ = $3 ($3(14) = |u|), which is a Young (not an N-) function, then (3) cannot hold unless k* = oo and -&rp$>3(k*f) is taken p®3(f). As another example, let $ be a continuous Young function that vanishes only at the origin and suppose that the complementary Young function ^ of $ has the same properties (by Proposition 4, $ is an TV-function), then it is shown that ||/||$ = ||/||($) iff / = 0 (cf. Rao [5], Lemma 1). Since for $3 of the above Young function, which is not an TV-function, this result is not applicable (it cannot hold since, in fact, the equality is valid for L 1 ). One of the important features of Orlicz spaces is that the two norms || • ||$ and || • ||($) are equivalent and rarely equal, in contrast to the Lebesgue spaces. The spaces L® and L^ are isomorphic. In some ways this is a weaker property, but it is compensated by the existence of a rich collection of isomorphically equivalent Orlicz spaces. This is exemplified by the following result of considerable importance in applications. Theorem 11. Let $ be a continuous Young function such that $(u) = 0 iffu — 0, i.e., e I nil. Then there exists a strictly convex continuous Young function $1 such that L*(JJ,) = L*1^), on any measure space (fi, £, /Lt), and their norms are equivalent. Thus the spaces are isomorphic (but not isometric). Moreover, z/<& is ^-regular, so is $1. Proof. This result was proved in our earlier volume ([1], p. 153), but we include a simple construction for completeness. Thus we have

r\u\ = \ V(t)dt. ./o

(4)

32

/. Introduction and Background Material

Define a new function (f>i(t) = 0 and set /• / Joo


(5)

Since 0 and strictly increasing, $1 is strictly convex. Moreover, it is clear that $(u) < $i(u) < 2$(w),u € R Hence / e L*(jz) iff / € L*i(yi) and ||/||w < ||/||(4>l) < 2||/||w. Thus, the spaces are isomorphic. The A2-regularity is immediate from construction. D In the above reference an uncountable collection of distinct Young functions generating equivalent norms is given. Note in particular, if $(w) = w|, then $1 constructed above is strictly convex and A2regular, and hence even though L^(IJL) = Ll(p,} is not strictly convex, its isomorphically equivalent (collection) L^l^(n) is strictly convex (cf. our [1], p. 272). On the other hand, 4>(u) = \u\ and its (Young) complementary function ^ are not ./NT-functions, but are necessary to consider these "boundary" functions if the Orlicz space theory is to generalize the Lebesgue theory. Moreover, L1 and L°° are some of the most important concrete Banach spaces which are also algebras. These spaces cannot be included in Orlicz spaces based on iV-functions alone. Let us start with the following: Definition 12. A vector space X is called a commutative Banach algebra if (X, \\ • ||) is a Banach space and for each or, y in X a product xy is defined and the latter satisfies xy = yx as well as the inequality \\xy\\ ^ 11^ II I \y I 5 so that multiplication is a continuous operation in the norm topology of X. As examples, (i) X = Co(R), the space of continuous functions vanishing at ±00, with uniform norm; (ii) L°°(R), under pointwise multiplication; (iii) X = L 1 (M), the Lebesgue space of integrable functions on (R, S, //) with (j, as Lebesgue measure and multiplication as convolution, i.e., for /, g E -^1(R), the convolution / * g is given by (/ * g)(x) = f f(x- y)g(y)dn(y),

x e R.

(6)

JR

These are commutative, and if X — B(y) ,the space of all bounded linear operators on a Banach space 3^ with operator norm, then it is a noncommutative Banach algebra. A change of variables, and translation invariance of ^ shows that f *g = g* f m (6), and ||/*
1.3 Notes on Young functions and general measures

33

Proposition 13. Let $ be a Young function andL*(R) be the corresponding Orlicz space on the Lebesgue measure triple (R, £, //) as above. 7/L*(R) is closed under convolution, in the sense that /, g € I/*(R) implies f * g is defined as (6) and f * g G L^(R), £/&en i/iere exists a constant Q < K < oo such that \\f * 9\\* < K\\f\\*\\9\\*>

/,0€L*(R).

(7)

On i/ie o£/ier /mnd, i/ $ € / D /I, and z/ there is an absolute constant K such that (7) holds for all simple functions f and g, then (7) again obtains for all /, g € L*(R); so L*(R) is closed under convolution. Proof. It was already noted that f*g = g*f whenever it is defined, and by hypothesis, Z/*(R) is closed under convolution as multiplication, so that it is an algebra. Then the mapping / h-» Tf defined as Tf(g) — f*g on the spaces T : L*(R) -> #(L*(R)), is a homomorphism, since for any h € L*(R) one has

and the left side is also (/ * g) * h = Tf*g(h), so that TfTg = Tf*g = Tg*f = TgTf, and /•->•/ (the complex conjugate) is an "involution". Consequently, by the abstract theory the spectral radius of T/, namely ||Ty||sp = lim |T?||n exists, and satisfies ||T/||ap < J^"||/||$ for some J n—>CXD 0 < K < co (cf., Loomis [1], Lemma on p. 92). Moreover, ||T/|| = | \Tf \\sp in this case. Hence, we have II7/H = sup{||:Z>G/)||* : ||^||« < 1} < K\\f\\*,

(8)

which implies (7), since

In the reverse direction, suppose (7) holds for simple functions /, g. Then f*g G L*(R) and 0 < K < oo is an absolute constant. It suffices to consider 0 < /, g since for the general case / = fi — /2, g = g\ — gi implies / * g = fi * gi - /2 * g\ - f\ * #2 + h * 92 and the result will follow, by linearity, for all /, g in L*(R) (and then similarly for complex functions). Now we can find simple 0 < fn,gn in L$(R) such that fn t /, <7n t 9- Let no be arbitrarily fixed, and from (6), /n0 * 9n t /n0 * 9i a-e- and hence by the monotone property of the

34

/. Introduction and Background Material

norm: ||/no * gn\\$ < #||/n 0 ||*ll0ll* yields, on letting n -> oo, that 11/n0 *0II* < #ll/noll*H0ll* < #11/11*11011*- Since fno *# = g*fno, this implies, on letting no —> oo, that (7) holds. D Remark. The continuity of $, in the statement is used in approximating f , g in L*(R) by simple function sequences. If &(u) — 0 for 0 < \u\ < a , a > 0 , o r < I > ^ / , the simple function (pointwise) approximation has to be replaced by uniform sequences, and the argument runs into trouble. The question arises whether there are Young functions $ € / n // for which (7) holds. In fact, if $(w) = |it|, then L*(R) = Ll(R) and the result is classical (with K — 1 in (7)). However, by Theorem 11 there are strictly convex continuous Young functions 3>i equivalent to $ here so that (7) holds for some K > 0 (not necessarily = 1). We now present a sufficient condition on $, for which $ e / D // and (7) again holds for some K > 0. Proposition 14. Let 3> be a Young function such that $ G / D // and = $+(0) > 0. Then L*(R) is closed under convolution; in fact, ll/*0ll*<^yll/ll*yi*, so i/ia^ JC = —T

/,0€L*(R),

(9)

m

Proof. First observe that (since
|u| > 0,

(10)

so for any 0 ^ g € L*(R) we have, with /^ as Lebesgue measure as before, with (10): '

i i ^ ••• i ^ i y 0 d^ ^ TTT^T / $ TTTH

\ i ^ A ^ < ^7^'

/ii\ I11)

by definition of the gauge norm. It follows that \\g\\i < —T^yll^H*. Hence Z/*(R) C L X (R) and the embedding is continuous since v?(0) > 0. Next note that for any /,g 6 L*(R),/ * g exists in But we have, with ^ as the complementary Young function of

(f *

g)(x)h(x)dij,(x)

1.4 Interpolation results on Orlicz spaces

35

= sup \ I L I / f(x - y)g(y)dn(y)\h(x)dn(x) J I JR JR

:\\h\\w < 1 \ )

< sup{ / \g(y)\\L I \f(xUK 7R

Taking tf=^y > 0, we get (7). D In addition to $(w) = |u| (so
1.4 Interpolation results on Orlicz spaces We first recall the classical interpolation theorem due to M. Riesz and refined by G.O. Thorin. It states: if 1 < p\ < p2 < oo, I < qi < q2 < oo consider an intermediate pair ps,
ps

PI

P2 qs

qi

q-2

the corresponding Lebesgue spaces Lpi (n},Lqi (p,) on (fi, E, //), i = 1,2. Now if T : Lpi (//) —> Lqi (/u), i = 1, 2, is a linear operator defined on both the spaces satisfying ||T/||g. < Ci\\f\\pi,i = 1,2, / € (LP1 nL? 2 )^), then T is also defined on Lps(n) into L 9s (/Li), for all 0 < s < 1, and satisfies \\Tf\\qs
0<S<1

(2)

for any / e Lps (//). This result has a complete generalization to a large class of Orlicz spaces, which have important applications. For this it is necessary to define intermediate ./V-functions and the corresponding

1

36

/. Introduction and Background Material

spaces, generalizing (1) and (2), as follows. Let ($1,^2) be a pair of TV-functions and define $s by the equation:

$7l(u) = [ $ i l ( u ) ] l - s [ $ 2 l ( u ) } s , 0 < s < l,n > 0.

(3)

If p~l + q~l = l,z — 1,2, then it is immediately seen from (1) that p~l + q~l = 1 for all 0 < s < 1. If \I/i is complementary to $j, i = 1, 2, and tys is defined analogously to (3), then (3> s ,\l> s ) is not necessarily a complementary pair. The following result implies a useful substitute with which the desired extension can be formulated and proved. Lemma 1. Let (j, \I/i),? — 1,2 be two pairs of complementary Nfunctions, and let (<& s , ^ s ) be the intermediate functions, as defined by (3). Then both $s and \&s are also N -functions, and if^/^ is complementary to 3>s, we have

^t(v) < ^s(v) < tf+(2u), v>0.

(4)

Moreover, the Orlicz spaces L*s(n) and L^^ (/z) on (SI, S,/Lt) have the same elements as sets, are topologically equivalent in the sense that

and again relation (5) holds if \\ • |(#) is replaced by || • ||^. Proof. Since ^1 is concave, one can verify, with the classical Holder inequality for sequences, that $~ x is concave and that $s is an Affunction. The result is also true for Young functions with simple modifications, and then (4) and (5) obtain without much difficulty. The details are spelled out in Rao and Ren [1], pp. 223-224 and will not be reproduced. The analog of (5) for Orlicz norms were not given, but from Theorem 2.16, there exist &i,&2 > 0 for g G L* s (/z) and 2g <E L** (/^) such that

1+ and

II2PII.J =

1.4 Interpolation results on Orlicz spaces

37

Then by (4) we obtain:

1

i+ 1+

as asserted.

D

We now present an Orlicz space extension of the Riesz-Thorin theorem for a generalization and applications below. Theorem 2. Let 3>j,Qi,z = 1,2 be N-functions, (Q,E,/^), (Q', £', z/) be a-finite measure triples. If3>s,Qs are the intermediate N-functions, as in (3), consider the (complex) Orlicz spaces M^i C L $i (//), and Lfi^v^i — 1,2 and similarly M* S ,L^ S (^). Suppose that a linear transformation T : M^i —>• L(^i(v) , i= 1,2 is defined on the bounded functions, satisfying, for some constants GI > 0,

Then T is also defined on all functions of M*s into L®s(v}, for each 0 < s < 1 and satisfies

I|r/||(o.)
(7)

max(||T/||Q+, ||r/||Q.) < cJ-^ll/H*.,/ € M*v

(8)

and [In (6)-(8), the operators are first defined on simple or bounded functions and then they have unique extensions to all of M$i or M®s without affecting the bounds. We use the same symbol T after the extensions also for simplicity.] The proof based on the three-line theorem of complex analysis is an Orlicz space extension of that given by the classical procedure for the Lebesgue spaces, and the detailed argument is given in Rao and Ren [1], pp. 227-229, and will be omitted here. The result is also true for the real case, except that on the right side of (7) and (8) the constant is to be doubled, i.e. 2c]~sC2For the packing problem and other applications in the following chapters, a special multilinear formulation of the above result is needed. First let us introduce a concept in the following:

/. Introduction and Background Material

38

Definition 3. Let $ — ($1, . . . , <$n) be an n-tuple of TV-functions $j, and (Oj, Sj, fj,j) be cr-finite spaces, j = 1, . . . , n. For each p > 1, and ntuple of weights A = (Ai, . . . , A n ), Xj > 0, consider the tensor product space: = {f = (/i, . . . , / „ ) : fj € M*> with norm ||

<

<

defined for f G x n

p

Ml 1 JyU&j)

"

i p

,7

i/ j

1— < rp < oo (9)

j?/''

nax||/j II (*j) j

»

if

p=QG

or the norm || • ||$)P defined in the same way as (9) in which || • ||($.) is replaced by the Orlicz norm | • U^., 1 < j' < n. If ^j is the complementary iV-function of $j, denote by x M*-7'^-), || • ||(*))9, and\\ • \\ytq for .7=1 the same weights A = ( A i , . . . , A n ) and q — p/(p — 1). For convenience we let Af*(A) = ( x M**r(/^-), || • ||$ip) and similarly M*(A), where we j=i set ^ = ( ^ i , . . . , ^ n ) with the complementary A^-function n-tuple. It is easily verified that the weighted norms are given by:

=

SU

P

= sup

(10)

and similarly

||f ||$)P = sup

1.4 Interpolation results on Orlicz spaces

39

It is also not difficult to see that M ( A ) is a Banach space under either of these (equivalent) norms. In the same way one finds that L p ( \ ) ( L p ( X ) ) is a Banach space which is a (finite) weighted tensor product of L®i(nj)(L®i(nj}} with the Lp-norms. We can now present Cleaver's [1] formulation of the interpolation theorem for these weighted tensor product spaces as follows: Theorem 4. Let fy = ($a, . . . , $im), Q{ = (Qn, . . . , Q in ), * = 1,2 be m,n-tuples of N-functions, I < ri,r2,£i,£2 < oo, A = ( A i , . . . ,A m ) and suppose rj = (771, . . . , r/n] be given positive numbers. Next let <E»S = ($ s i,... , <& sm ), Q3 = (Qsi,... , Qsn) be the associated intermediate N-functions, 1 rs

1 —s ri

s l r-2 ' ts

l —s ti

s £2'

~

~

and (Qj,Ej,/^j), (fi^.,Sj,,i/fc),j < m, and k < n be a -finite spaces. If T : M^ (A) -> L^^(ry),z = 1,2 is a bounded linear operator with bounds K\,K-2 on complex spaces, so that \\Tf\\(Qi}tti < ^||f||(*i)>Pi , f € M ( X ) , i = 1,2, then T is also defined on Mrs and one has the bound

(A) into L[

(13)

(rj), for all 0 < s < 1,

),t.
(14)

Similarly for the Orlicz norms, one gets

max(||rf||g.,t., ||rf|| Q + jt- ) < Kt-sKI\\f\\*s,ra •

(15)

The same result holds for real spaces, except that the bounds in (14), (15) have to be replaced by Except for notational complexity, the proof is a simple modification of that of Theorem 2. The details are again found in Rao and Ren [1], pp. 232-239, and will not be reproduced here. We specialize the above results when $1 = $, a given TV-function but < & 2 s — x2. We start with a useful observation:

40

/. Introduction and Background Material

Lemma 5. Let $Q(U) = u 2 ,$ an arbitrary N -function and $ s ,0 < s < 1 be the inverse of the following: $-l(u) = [$-1(w)]1-s[$-1(u)]s = [-l(u)]l-su*,u > 0.

(16)

Then $s € A 2 n V 2 , 0 < s < 1 (but not, of course, for s = 0). Proof. Recall that f < a$ < 0$ < I (from Section 1.1), and by (16) we have

Hence for 0 < s < 1,

and f $-i( ) 1 , _ / 1V 1 a<s?s = inf < ——T u : 0 < u < oo > = (0;$)N l I —-p I > —. Consequently, $s € A2 as well as $s € V2, by Theorem 1.7. n Remark. The above result holds if 3?o(u) = \u Po, 1 < po < oo> and in (16) replace 142 by u^o . The following result is an Orlicz space analog of one of the classical Clarkson inequalities for the Lp-spaces; and it will be used in Chapter 2 to estimate bounds of nonsquare as well as von Neumann-Jordan constants. Theorem 6. Let $ be an N-function and $s be the inverse of that of (16), 0 < s < 1. If (fi, E,//) is a a-finite space and L* s (^) is the corresponding Orlicz space, then we have for /, g G L*s (/x):

and (ll/ + 9\\ls + ll/ - Pill.) * < 2t (\\f\\£ + yl*^)

2

.

(18)

1.4 Interpolation results on Orlicz spaces

41

Proof. Consider the tensor product space of M*(//) with itself and with unit weights. More explicitly, let <3>i = ($, $) be the 2-vector of JV-functions, A = (1,1), 1 < r\ < oo, and set = {(/,
(19)

where

To further identify with the proceeding theorem, take Qi = ($, <&) = $1 and Q2 = $2 = ($o> ^o) where $ 0 (w) = u2. Set ri = 1, r2 = *2 = 2, and ti = +00. Next define the (linear) operator T : M®* —> L^ (p,) by the equation: We then have

,*! = max(il/ + pll(*)» ll/ Hence i^i = 1 and since || • ||($0) = || • IJ2, we find

Thus ^2 — \/2- Let rs and ts be given by

1 _ 1 —s rs ~ f i

s

I _ I —s

' r-2, ' r~ ts ~ ~~*ti

s

^" 12 T~

so that with the particular values of ri, r^ t\, t% chosen above,

rs = ;«. = -, 2 —s s

(20)

and from Theorem 4, T : M r ( f s) -> L(£S\IJL) is well-defined, where Qs = <|s = ($ s ,$ s ), with $s as in (16). Then (14) gives:

42

/. Introduction and Background Material

since K\~*K% = 2 f . It follows from (20) that

\

-J / 1

a

»

\ —o /

\ ~ & /

(22)

I

and

from which (17) follows since 3>s G A 2 by Lemma 5. The argument for the Orlicz norm is identical giving (18). D We can deduce Clarkson's inequalities from this result. The original proof of these is less revealing as one uses a manipulation of the conjugate exponents (p, q). (See Hewitt and Stromberg [1], p. 225227. However, the inequality (25) below is somewhat different from the latter.) The (7-finiteness of (JL is no restriction here. Corollary 7. (Clarkson [1]) Suppose that 1 < p < oo,
(24)

and (\\f + 9\pp + \\f- 011?)* < 2* ( H / I I 2 + W|)*, 2 < p < oo.

(25)

Proof. If 1 < p < 2, let 1 < a < p < 2,$(u) = |w| a ,$ 0 (w) = u2 and s = * I ° . Then 0 < s < 1 and ^ w = w^ or $ a u = u p . Hence II ' ll(* s ) = II ' lip, and since lim f = -^ = g; lim^ = ^ we get (24) a4,l

"

a.4,1

^

from (17). Similarly let 2 < p < b < oo, $(u) = |w|6, $ 0 (w) = ^2 and s = ^Ef}Then 0 < s < 1 and 4> s (n) = w p, lim -s = p. lim ^^ = 9- and hence 2 ~

6t°o

6t°o

(25) follows from (17). D We also can obtain the modulus of convexity (and of smoothness) for a large class of uniformly convex (and smooth) Orlicz spaces from (17) and (18). First recall that the modulus of convexity (smoothness) of a Banach space X is defined as: 6 ( X , e) = inf l - i \f + g\\ : f , g e S(X), \\f - g\\ - e < 2

1.4 Interpolation results on Orlicz spaces

43

and (p(X, r) = sup {1(||/ + 0|| + ||/ - 0||) - 1 : / € S(X), \\g\\ = r}} . \ lz J/

where S(X) = {x € X : \\x\\ = 1} is the unit sphere of X. The space X is uniformly convex (smooth) if 6(X,e) > 0 for every 0 < £ < 2 (if \\m^p(X,r) = 0). It can be verified that the uniform smoothness condition here is equivalent to uniform Frechet differentiability of 5 (AT). Specializing X to a certain subclass of Orlicz spaces, we can obtain from the above: Corollary 8. Let $, (fi, S,/z) and $ s ,0 < s < 1 be as in Theorem 6. If Xs = £<*•>(/*) [or = L*'0/)7, then we have forQ<e<2, e)>l-(2-e)*

(26)

and

p(XatT)<(l + T&)*r-l, r > 0 . (27) Hence Xs, 0 < s < 1 is both uniformly convex and smooth simultaneously. Proof. Consider X8 = L^*'^(fj), first. Thus if ||/ - g||($s) — e, then (17) implies for f , g £ Xs, Consequently,

Taking infimum of /, g e S(X8) on the left, we get (26) and also its uniform convexity if 0 < e < 2. On the other hand, if ||/||($8) = 1 and IMI(* 8 ) = Ti then since | > 2,

1

«(!!/ +0II(*.) + ||/-0||(*.) Z

1•

1

a

2

< o(ll/ + 0ll(* ) + !!/-g\\h )) Z ^ ' a

2

2i

by (17),

44

/. Introduction and Background Material

Consequently 1

2

2-s

Taking the sup on the left over / € S(XS] and ||^||($s) = T, we get (27). Since the right side above has a vanishing derivative at T = 0, this also implies that Xs is uniformly smooth. Similarly (18) provides the same conclusion with the Orlicz norm. D Remark. It is known for a long time that between any two Lp spaces one can construct an Orlicz space L* where the TV-function $ is not A2-regular so that the space is not reflexive (cf., e.g., Krasnoselskii and Rutickii [1], pp. 28-29). On the other hand, the above corollary implies that for any TV-function $ one can find uniformly convex and smooth Orlicz spaces, namely L* s (//),0 < s < 1, between the given L*(//) and the Hilbert function space L2(/j,). This shows the vastness of the class of Orlicz spaces in comparison with the Lebesgue classes. The interpolation Theorem 6, a useful specialization of the Orlicz space extension of the Riesz-Thorin convexity theorem given as Theorem 2, reveals these noteworthy geometrical facts. We next consider weighted tensor product spaces and the resulting interpolation inequalities, by another specialization of Theorems 2 and 4. These will be used for work in Chapters 3 and 5 in obtaining results on normal structure coefficients and packing theorems. The result is essentially due to Cleaver ([1], p. 201) for the Orlicz norms. Theorem 9. Let $, $ s ,0 < s < I and L®s(p,} be as in Corollary 8 on a a-finite space (fi, £,/-f). Then for any finite collection fi € n

ZA $S '(//), 1 < i < n, Ci > 0, £ Ci = I and c = max (1 — Ci), we have i=l

l
2 J^IIA - E^/jllfr.) < cf~ 2 . £ ^CiCjUfi - /,-llfft.).

(28)

The same inequality holds for Orlicz norms also. Proof. We have included the proof of an analogous inequality in our companion volume ([1], p. 240). However, (28) is somewhat different from the earlier one, and we add the essential detail for completeness, with the gauge norm. Consider the vectors, , . . . ,cic n , c 2 c 3 ,... ,c 2 c n ,...c n _ic n )

1.4 Interpolation results on Orlicz spaces

45

where A has n'n2~ ' = m components, 77 has n components and each of $i,2 has m components, with 3>o(w) = u2. Similarly set Q\ = ($, . . . , $ ) , $2 = ($o» • • • ? $o) each with n components. If fa is a simple function, I < i < n, define f = (f1, . . . , /n-1) /

= ( / ! » • • • > /n-l) ~ (/I - /2> /I - /3> • • • > /I - /n),

/

—

(/I ) /2 > • • • > fn-2)= V/2 ~~ /3s • • • > /2 ~~ /n)?

andfinally,/"-1 = /J1'1 - /n-i - In so that /] = /« - /i+J-, 1 < t < n — I j j > i. Thus f has also m components. We define a linear operator on the weighted tensor product spaces introduced for Theorem 4 above first on simple (vector) functions f just defined: * M r , )

,

( n - *i = oo)

T : M r (f*>(A) -> Mt(2Q2)(77)

,

(r2 = *2 = 2)

by the expressions

where

n— i

i—l

fc=i Observe that (from (9)): ll f ll(*i),ri = max{ll/j-ill(*) : 1 < « < n - l,n > j > «}

It is now necessary to compare the bounds for [(Tf ||(Q.) )t ., i = 1,2 with the norm bounds in the range spaces. Although this is an algebraic simplification, it is somewhat long and tedious. We now present the result, omitting the algebra for the reader to verify. It is given by: II" H(gi),*i = H rf ll(Qi),oo < c||f ||(»l)|00 = c||f ||(»l)iri

(29)

where c = max (1 — c,-), and 3> l<j
\\Tf\\iQM

= ||Tf ||(0a),2 < ||f||(*2),2 = Hf|l(* 2 ),r 2 .

(30)

46

/. Introduction and Background Material

The omitted computations are analogous to those given in Wells and Williams ([1], pp. 81-82). Let rs,ts be denned by 1 1 —s s 1 1 —s s - = + - ; -. = —_ + -;

rs

n

r2

ts

ti

ti

. . (31)

so that rs — ts — -s^<s<\. We can now apply Cleaver's extension given as Theorem 4, to conclude that (k\ = c,k2 = 1) ,t.
(32)

Thus T is also defined as a bounded linear operator on Mra (A) on all its simple functions into the (complete metric) space M^s'(n) and has an extension (same bound) onto the whole space for 0 < s < 1, since such functions are dense in Mrs (A) by Lemma 5. Substituting the values rs and ts from (31) into (32) it reduces to (28). Since no explicit forms of the gauge norms are used in this computation, the same work can be carried through for the Orlicz norms, and (28) holds for it also. D Taking 3>(u) = u p , 1 < p < oo, the above result becomes the Lpinequality due to Williams, Wells and Hayden ([1], Thm. 2(c)-(d)), (see also Wells and Williams [1, Thm 15.1) which we state for a comparison. Corollary 10. Let fi G Lp(n], I < p < oo, 1 < i < n, on a cr-finite n

space (17, E, IJL). Then for any GJ > 0, S GJ — 1, one has:

where c = max (1 — Ci), and j3 > -£y if 1 < p < 2, or (3 > p if l
P

2 < p < oo.

The verification is entirely similar to that of Corollary 7, and is left to the reader. We also need another inequality for use in later applications. It was proved in our companion volume ([1], p. 240) for the Orlicz norm, but as noted above it holds for the gauge norm as well. We now state the result.

1.4 Interpolation results on Orlicz spaces

47

Theorem 11. (Cleaver [2]) Consider an N-function $, a a-finite space (Q, S, IA) and the intermediate N-function 3>s between $ and $o :u *-* |n|2. Then for any fi € L^'^(M)>ci > 0, S c» = 1,1 < i < n, one /ias i=l

V^ ^,11 f, - f,ll 2rs. <: 9r>::¥rfi \ " ,...11 f,\\ ?-: »,j=i t=i

(34)

&e same inequality holds if \\ • \\(&s) is replaced by \\ • ||$s, where c = max (1 — Ci) and 0 < s < 1. Again (34) reduces to the result of Hayden and Wells ([!]) by taking 4>(u) = |u|p, 1 < p < oo, and we state it for a comparison. Corollary 12. Let fi € LP(/JL) on a a -finite (fi, E, fj,}, I < p < oo and 1 < i < n. Then for any C{ > 0, S Ci = 1, one has i—l

(35)

c = max (1 — Ci), and ! < / 3 < p z / l < p < 2 , o r l < / 3 l
if2
Remark. The weights GI > 0 are taken as S c^ = 1 in all the above i=l

work only for convenience. If S Ci = a, we can replace c$ by ^ and c z=l

will have to be replace by max(o; — c») = c so that c2"^ is replaced by a /3-l£2-/3

Bibliographical notes. As stated at the beginning, this chapter contains a summary of the results from the general theory of Orlicz spaces, and we concentrated mostly on TV-functions which will be adequate for many applications. We included the details of the results which are not readily available in our companion volume [1]. Here we treated the quantitative indices at some length because of their use in the following geometric applications, but the analysis is also important in obtaining some finer aspects of Orlicz spaces with the Young functions. The generalized Clarkson inequalities, as applications of the interpolation theory, given in the text are adapted from the recent work due to Ren [3], which when specialized to the Lp-spaces have a simpler

48

/. Introduction and Background Material

proof of the classical work. [However, the classical work did not use interpolation theory and hence involved longer computations and special techniques.] We also included some aspects of the (general) Young functions which are not TV-functions in such studies as the convolution algebras. More on these will be given in the last chapter. Most of the references and due credits to the work are already pointed out in the text at appropriate places. We now proceed to the stated theoretical and other applications.

Chapter II Nonsquare and von Neumann-Jordan Constants

The following work is devoted to calculating the nonsquare and von Neumann-Jordan constants, defined for any Banach space, specialized to Orlicz spaces based on JV-functions. The generalized concepts allow a finer analysis of the geometric properties of Orlicz spaces using the quantitative indices introduced and detailed in Chapter I. Thus the first section contains the abstract concepts. Sections 2 and 3 are devoted to obtaining the lower and upper bounds with the respective concepts. Finally Section 4 treats the von Neumann-Jordan constants of Orlicz spaces and gives exact values for certain intermediate spaces.

2.1 Introduction We first recall various concepts and immediate consequences related to an arbitrary Banach space X and find specialized conditions for them when X is an Orlicz space. The latter work will be useful for concrete applications. As usual, let S(X] = {x 6 X : \\x\\ = 1} denote the unit sphere of X. Following James [1], X is termed uniformly nonsquare if there is a constant 0 < c < 1 such that for x, y € S(X) \\x + y\\<2-2c, or \\x - y\\ < 2 - 2c.

(1)

It is known that (cf. James [1], or Beauzamy [1], pp. 256-261) a uniformly nonsquare X is reflexive. In view of this property, it is useful to introduce the following concept, as was done by Gao and Lau [1]:

49

50

//. Nonsquare and von Neumann-Jordan Constants

Definition 1. For a Banach space X, the parameter J(X) is termed a nonsquare constant (in the sense of James), where J(X) = sup{mm(||a; - y||, \\x + y||) : x, y 6 S ( X ) } .

(2)

From (1) and (2) it is easy to deduce that (cf. Gao and Lau [1]) X is uniformly nonsquare iff J(X) < 2. On the other hand, James [1] has already shown that the following holds (in this terminology): Proposition 2. If X is a uniformly convex Banach space, then J(X) < 2, (so it is uniformly nonsquare). Proof. Recall that the modulus of convexity of a Banach space X is defined as: 6(X, 8} = inf{l - hx + y|| : x,y € S ( X } , \\x - y\\ >e},e> 0, Zt

and X is uniformly convex if 6(X,e) 0 < eo < 2, and if ||a: — y\\ < £Q, then

> 0 for 0 < e < 2. Thus fix

x + y||, \\x - y||) < \\x - y\\ < £0.

(3)

If, on the other hand \x — y\\ > £Q, then by hypothesis 6(X, EQ) > 0?so

, \\x - y||) < ||x + y|| < 2(1 - 6(X, e0)).

(4)

From (2) - (4), it follows that J(X) < 2. D Remark. Gao and Lau [2] noted that for any Banach space X, J(X) = supjs > 0 : 6(X,e) < 1 - f } holds. On the other hand, Schaffer [1] termed X uniformly nonsquare if there exists a > 1 such that max(||a; + y||, ||o: - y||) > a,

x, y E S ( X ) ,

and introduced a nonsquare constant g(X) as: g(X) = inflmaxdlo: + y||, \\x - y||) : x, y But then Gao and Lau [1] have shown that if dim(A') > 2, then 1 < g(X) < \/2 < J(X) < 2 and g(X)J(X) = 2. Hence g(X) > 1 iff J(X) < 2 so that X is uniformly nonsquare in the sense of James. Thus both definitions of James and Schaffer are equivalent.

2.1. Introduction

51

We now consider another geometric parameter, based on a classical result of Jordan and von Neumann [1], introduced by Clarkson [2]. Definition 3. The von Neumann- Jordan constant, CNJ(X) of a Banach space X is the smallest c > 0 such that for all x, y € X — {0},

It is clear that 1 < CNJ(X) < 2, and by the Jordan- von Neumann Theorem, CNJ(X) = 1 iff X is a Hilbert space. The following simple result is useful (cf. Kato and Takahashi [1]). Lemma 4. For any Banach space X, one has J(X)2 < 2cNJ(X).

(6)

Proof. In fact, x,y €. S(X) implies

< \\x + y\\2 + \\x - y\\2 <<2cNJ(X)(\\x\\2 + \\y\\2)

which implies (6) because of (2). D The following, as yet unpublished result due to T.F. Wang (communicated to Ren) has interest in this connection: Theorem 5. For a Banach space X, J(X) < 2 iff CNJ(X) < 2. Proof. If cNJ(X) < 2, then by (6), J(X) < 2. So only the converse needs a proof. Thus from the left side of (5) it is easy to see 2 MrP4-7/II 4- \\r - ?/ll 2 + =*.ȣ*. CNJ(X) = sup{ z ! a pfn L

vll 'll

'

)

11*11 • (7)

If J(X] < 2, then we can find a 0 < J < 1 such that (see (2)): sup{min(||z + 7/11, ||z - y||) : y, z € S ( X ) } <2-6.

(8)

52

//. Nonsquare and von Neumann-Jordan Constants

If 1 = \\y\\ > \\x\\ > 1 - f , then from (8) we have H^ + y\\ < 2 - 5, (the same inequality holds if y is replaced by —y). But we also have

< 2 - 6 + (1 Hence from (7)

If 1 = \\y\\ and 0 < \\x\\ < 1 — f , one has a bound for the left side of (9) as: (\\r\\ + L1^2

/-i _ || xT ||N2

UFII 2+ )_ _ 2 _ v 1

IHI

ll ll/

i

<

2 _ -(i _ ||a;||)2

Prom (7), (9) and (10) we get 1 + (I - £

as claimed. D A consequence of Proposition 2 and the above result is the: Corollary 6. (Kato and Takahashi [1]) If X is uniformly convex, then cNJ(X) < 2. A Banach space y is finitely representable in X if for any a > 1 and each finite dimensional subspace 3>i C 3^, there is an isomorphism T from y\ into X such that for all x G y\ one has: - l II^H l r l l 2; < \\-L \\TT\\ < n\\r\\ X\\ ^ 0,\\X\\.

A Banach space X is super-reflexive if no nonreflexive Banach space is finitely representable in X (cf. James [2]). Such spaces are characterized by Enflo [1] (cf. also Pisier [1]) as:

2.2. Lower bounds for J(L<*)) and cNJ(L*)

53

Lemma 7. A Banach space is super-reflexive iff it admits an equivalent norm under which the space is uniformly convex. For a proof of this result, see Beauzamy ([1], pp. 273-290). Prom Theorem 5 and the above, one has the interesting: Corollary 8. Let c?fj(X} and J(X} be the infima of von NeumannJordan and of nonsquare constants of equivalent norms of X. Then the each of the following assertions implies the other two: (i) X is super-reflexive, (ii) CNJ(X] < 2, (in) J(X] < 2. The following relations for function spaces Lp on a cr-finite (17, S, p,) and 1 < p < oo are known: (Clarkson [2]):

cNJ(Lp) = max(2t~ 1 ,2 1 ~t).

(11)

The relation also holds if p = -f-oo, and similarly for the following: (Gao and Lau [1]):

J(LP) = max(2p, 2 1 ~p);

(12)

(Kato and Miyazaki [1]): cNJ(W^(R)) = CNJ(LP), for the Sobolev space W* (R) based on LP(R) on the Lebesgue line (R, 5, n). In the following sections we establish (11) and (12)-(12') as consequences of the corresponding results for Orlicz spaces based on subsets of the Lebesgue line, in lieu of the general (17, S, p,). 2.2 Lower bounds for J(L<*>) and cNJ(L*) In this and the following sections, we obtain lower and upper bounds for the nonsquare and von Neumann-Jordan constants for a fairly large class of Orlicz spaces. Then in Section 4 we obtain exact values of these constants for a certain class of reflexive Orlicz spaces, namely the special interpolation spaces considered in the last chapter which however properly contain the corresponding Lebesgue spaces. Thus we start with an N-function $ and the £*(//) (with both gauge and Orlicz norms) on a diffuse cr-finite or counting (17, S, /u) often taking 17 = [0,1] or £1 = R+ or (17 = N) and n as Lebesgue (or a counting) measure to simplify writing. (In the last case we write (?, the sequence space.) Also we use the same notations for quantitative indices of Section I.I for a complementary pair of TV-functions ($, \I>), namely, a$, /3$, a, /?$, Q^, /?£, and similarly for #. For an TV-function, we can present the following

54

//. Nonsquare and von Neumann-Jordan

Constants

Theorem 1. Using the above notations we have: (i) (1) (2) (3) Similarly,

(ii) max(|a;2, 20J) < CNJ(L^[Qt 1}), , 2(^)2) < cNJ(L^(R+)),

(4) (5) (6)

Proof, (i) To establish (1), by definition of a$, there exist 1 < un | oo, such that lim & -i/ 9 "un \ — a$. Hence for any £ > 0, we can find uno n->cx> *

(,^ n;

(= WQ > 1 (say)) such that e.

/(7)

Choose a pair of disjoint measurable sets Gi,G-2 in [0,1] satisfying v(Gi) = z~,i = 1,2, and set /; = $- 1 (2u 0 )XG i ,« = 1,2. Then ||/i|| ( *)=:landby(7) II f { II — If I I / I - /2||(*) — I/I

Hence J(L<*>[0, lj) > mindlA - /2||w, ||A + /2||w) > - - . (8) Similarly, by definition of /?$, given e > 0, there exists a VQ > 1, such that

Again choose disjoint measurable Ei in [0,1] satisfying fj,(Ei) = ^~. If we set 51 = $~ 1 (^o)(X£; 1 + XE2),92 = fc^MCxsi - X£? 2 ), then (9) bi + 02||* > 2/3$ - e.

(10)

55

2.2. Lower bounds for J(L<*>) and cNJ(L*)

Hence «/(£<*> [0, 1]) > 2/3$ - e. This and (8) imply (1) since e > 0 is arbitrary. The proof of (2) is entirely similar. Regarding (3), by definition of a$, for given 0 < £ < 1, find UQ such that 0 < UQ <

and -i


* + £ wmcn means

)$- 1 (2n 0 )]>u 0 .

(11)

Let &o > 0 be the smallest integer such that &o < ~ < &o + 1> choose c > 0 such that 2koUQ + <3>(c) = 1. Define a pair of (vector) functions of finite disjoint supports (of &o + 1 points) as:

Then p<s>(fi) = 1 and hence ||/i||($) = l,i = 1,2. By (11) we get 1-6

= p$

l-e

2k0u0 l-2tt 0 > -- > —- > 1 — £

I —£

Hence

Next by definition of /3$, for a given 0 < e < 1 there is a VQ satisfying 0 < v0 < § and £-i°\ > )0J — , or equivalently

Again let ko be the smallest integer satisfying ko < ^ < ko + I and choose £ > 0 such that 2kov0 + <£(£) = 1. Now define two functions (vectors) of finite supports (of 2&o + 1 points) as:

, t, o, o, . . . )

A= h=

W, • • • , -

(v0), 0, t, 0, 0,

56

//. Nonsquare and von Neumann-Jordan Constants

Then ||/»||($) = 1 and p$>(fi) = l,i = 1,2, and with (13) we get as before

Since e > 0 is arbitrary, (13) and (14) imply (3). (ii) By Lemma 1.4 for any Banach space #, we have \J(X)2 < CNJ(X).

(15)

Hence (4) - (6) can be deduced from (i). For instance, since (a$)-1 > 1, and 2/3$ > 1, we get max I —2"5 2^| 1 = - I max( —, 1

with X = L(*)[0, 1], which is (4). D The corresponding result with Orlicz norms can be given as: Theorem 2. With the same notation on L®(R+), $ an N -function, with fy as its complementary function, we have the following inequalities: (i) max (2 fa, —} < J(L*[0, 1]),

(16)

max (2fa, —} < J(L*(R + )),

(17)

V

a*/

max

(ii) max f2^|, -V) < cNJ(L*[Q, 1]), \ toiy / 2 < CjVj(i*(R)), max f 2(^)2, r^)

V

2(o!^) y

.

(19) (20)

(21)

2.2. Lower bounds for J(L(*>) and cNJ(L*)

57

Proof. To illustrate the techniques, distinguishing from the preceding, we establish (17) and (18). Then (ii) follows with (15) as in the previous case from (i), and (16) is similar. So consider, for (17), the definition of P*. For given 0 < e < 1, we can find 0 < VQ < oo such that -l

. { (22)

5 e ^ - -

By the nonatomicity of //, there exist disjoint measurable sets Gi C R+ such that n(Gi) = £.. Then /; = 9-*v$VQ)XGi,i = 1,2 have unit (Orlicz) norms and with (22) ll/i - Ml* = ll/i + /all* = ^n

> 2/3* - e,

which implies that 2/3* <J(L*(R+)).

(23)

Next by definition of a*, for any 0 < £ < 1, we can find no > 0 such that

Choose again measurable disjoint sets Ei C R+ such that 2^,* = 1,2. If we let /i = g_?j?no) (x^ +X^ 2 ) and /2 = ^-i? Xf; 2 ), then the /j have unit norms and with (24), II /1-/2 f f \\9 II = II/1+/2 f -j- f II« =- -;— ^-^no)

ay + e

Thus (25) implies (a*)"1 < J(L*(R)) which together with (23) establishes (17). Regarding (18), from definition of (3§, for 0 < e < I we can select 2 > vn I- 0 such that for all n > 1 Jf-wfol > /3» - I-

(26)

Again let kn be the smallest integer such that kn < -^- < kn + l. Since

^ t °o as i; | 0, we get > *

W

> 2fc ra ^- 1 (4-)»

(27)

58

and

//. Nonsquare and von Neumann-Jordan Constants

( , n+1)>P - 1 ( F ^_) > r^)>^- 1 (.L).

(28)

Let

Then 6n -> 0 and cn < 2^'~ 1 (2^-) -> 0 as n -> oo. Let n0 > 1 be chosen such that bncn < e for n > no- Define the (sequences or) functions /i, /2 of disjoint finite supports, by taking /i with the initial ^-components as 6 no , the rest zeros and ji having the second knocomponents as bno and the rest zeros, so ||/i||$ = bnokno$~l(j^-) = 1, i = 1, 2. Then by (27), (28) and dropping the suffix no for simplicity (so 6 for 6 no , k for kno etc.)

-c}> (2/3° - e)6A:*- 1 () - be K

This shows that 24 < J(t*).

(29)

Similarly, from the definition of a^, for given 0 < e < 1, we find > un J, 0 such that for n > 1 , (30)

Let fcn < 2^: < fcn + 1, *„ = pfcn*-1^)]'1 and sn = (fcn + i)*"1!^!) ~ ^n*"^^-)- Tnen in I 0, sn I 0 as n ^ oo, so that there is n0 such that £ n s n < |, n > no- We also have 2^nSn <

<

— < - — ~ .

(31)

2.3. Upper bounds for J(L<*>) and J(L*)

59

Define /i , /2 of finite supports (vectors) such that the first 2fc components of /i consist of t(= tno > 0) with k = kno components, and the rest zeros, and of /2 with t for the first k and —t for the next k components, and zeros elsewhere. Then ||/i||* = i2/u^ f "" 1 (^) — l,i = 1,2, and by (30) and (31),

= 2t[(k + i J t f - ^ - . ) - *], (8 = >2t

.

1-e

which with 29 This shows -^( ) ? proves (18). D a * < J(O, Example 3. Let Lp e {£p[0, 1], L"(R+), ^p}, K p < oo. Then

max(2p,2 1 -p)< J(Lp), 1

1

max(2p- ,2 ~p) < c NJ (L^).

(32) (33)

Indeed, let $(u) = u|p so that a<j> = /3$ = a$ = y5$ — a$ = /3| = 2~ p . Since L" = L<*> with || • ||w = || • ||p, we obtain (32) and (33) from Theorem 1. Actually there is equality in each of (32) and (33) as shown in Corollary 4.2 later. Remark 4. In view of Proposition 1.2 of James' and Theorem 1.5 we deduce that J(X] = CNJ(%) = 2 for any nonreflexive Banach space X. Thus if L<*>[0, 1] is nonreflexive (i.e., $ £ (A2 n V 2 )(oo)), then fa = 1 or a$ = ^. Hence (1) and (15) imply = m a x — ,2/3*

<J

[0,l] < \2cjvj(L(*)[0,l]) < 2,

and the above result is obtained for Orlicz spaces. Also, Hudzik [1], Wang and Chen [1] proved that uniform nonsquareness (of Schaffer) coincides with reflexivity for these Orlicz spaces, (cf. also Grzaslewicz, Hudzik and Orlicz [1]).

2.3 Upper bounds for J(L^) and «/(£*) We begin with reflexive Orlicz sequence spaces with the gauge and Orlicz norms.

60

//. Nonsquare and von Neumann-Jordan Constants

Theorem 1. For an N-function $ e A 2 (0) n V 2 (0), we have (i) «/(£(*>) < 2, and (ii) J(t*) < 2. Proof. We establish (i) in steps, for clarity. 1. For $ as in the statement we assert that there is 0 < 6 < | satisfying <&(—) < (

£)4>(u),

u < $~1(1).

(1)

In fact, since $ € V 2 (0), there are UQ > 0 and 0 < 61 < | such that $(f) < (\ - 5i)$(u} for 0 < \u\ < u0 (cf. Cor. 1.1.4). We can suppose UQ < $-1(l). By continuity of u •->• A^/ < | on the closed [WG, $~1(1)], we can find a 0 < 82 < \ such that $(f) < (\ — ^ 2 )$(u) on it. Taking 6 = min($i, £2) we get (1). 2. For 0 < 5 < \ chosen in (1), we get, with $ e A 2 (0) n V 2 (0),|n|, v <$-l(l),

For, by (1), if iu> > 0, then

and by convexity

which gives (2) by addition. Since $(•) is symmetric, the same result holds even if uv < 0. 3. For any fi e £&\ ||/t||(*) = l,i = 1,2, we have p®(fi) = 1 since $ G A 2 (0), and *(£(&)) < 1,A: > 1. So by (2)

fc=i

2 00

fc=l 2-26.

2.3. Upper bounds for J(L<*>) and J(L*)

61

Hence . mm

[ / f i + /2\ 9 //i — / 2 \ 1 ^ 1 5x P* —^— h/ * —^— < 1~ -

/Q\ (3)

4. Since $ <E A 2 (0), there is a k0 > jzj(> 1) such that fco$(w) for all |u| < S-Hl). Let 7 = ^^, and e0 = 1 - ^, so that 0 < 7 < 1 and J < £Q < 1. Hence by the A2(0)-condition, we have

<

-7

Foranyy e^*) with/o*(flf) < 1-5 we get ^-^l) > ^^(l-S) > \g(k)\ and (4) implies

-

1

oo

®(9(k^ - 1?

or

It follows from (3) and (5) that, for ||/i||($) = 1, /I -I" /2 n

n /I — /2

This implies J(^*)) < 2 - 2e0 < 2. The proof for (ii) is similar.

D

The corresponding result for Orlicz spaces L*(£7), n being Lebesgue measure, is given by: Theorem 2. For each N -function $ G (A2 n V 2 )(cxD) one has (i) J(L*[0,1])<2, and Proof. We establish (i). Consider the constants A, B defined by A = mf{kf >l:kf = l + p*(fc//), / € Since fe/ > 1, it is clear that 1 < A < B < oo, the set of fc/'s above being nonempty. In fact, by Young's inequality and the A2-condition one has sup

j kfgd»

: p*(g) < 1 1 < i[l + p*(*/)] < oo,

(6)

62

//. Nonsquare and von Neumann-Jordan Constants

for any k > 0. To see that A > 1, suppose on the contrary that A = I. Then there exist fn e 5(L*) and kfn 4- 1 such that kfn — l = p$>(kfnfn}<> 0 = lim (kfn - 1) = lim n— —>oo

p*(kfnfn)

—

/ n ), by convexity, n ) > c0 > 0,

(7)

since the modular topology and norm topology are equivalent for L* when $ € A2(oo). The contradiction in (7) shows that A > I. We now claim that B < oo, assuming $ 6 V2(oo) and p<(ti) < oo. Indeed, this condition implies lim

-r.

r:

IMI*-4oo \\g\\*

(oj

= +OO,

(see Rao and Ren [1], p. 174, where the result is stated for the equivalent gauge norm || • ||($)). Now if B — oo, we get from definition the existence of fn e 5(L*) and kn —>• oo, such that n)

.

(9)

Setting gn = knfn so that ||pn||* = &n||/n||<£ = kn -> oo, one deduces from (8) and (9), the following contradiction: 1= lim

kn

+

\\gn\

= O + oo.

Thus B < oo must hold. Another proof was given in Chen ([1], p. 21). Now let uQ = fc-^f (1 - A~l))/B > 0 where 1 < A < B < oo defined above. For /1? /2 G 5(L*), choose fei, &2 € [A, 5] such that 1 - ||/i||* = f [1 +-P*(*i/i)],» = 1,2. /Cj

(10)

Observe that k^k^ 6 [j^g, ifsl' * = ^ 2 anc* ^ € V 2 (oo) implies the existence of 0 < 6 < 1 such that

2.3. Upper bounds for J(L<*)) and J(L*)

63

But by the convexity of <£, for 0 < AI < A2 < 1, one has AI

~

A2

and hence, writing A2 — -Q and AI >

g, (11) implies ,

M>tlo.

(12)

We now assert that for any ki e [-4,5], max(|wi|, (1*2!) > MO, , (MI - M 2 )

< (2 - 6) --*(*i«i) + -$(fc 2 H 2 )

. (13)

To prove this, by symmetry considerations, we may assume MI, u2 > 0. Suppose MI > MQ, so that ^MI > MI > MO (since ki > 1) and (12) implies i

2

(14) and by convexity of 3>, k2

Adding (14) and (15) and dividing by fc*^|2 , we get (13). The same holds, by symmetry, if M2 > MQ in the above, i.e., (13) always obtains. Let again /; e 5(L*), satisfying (10) for i = 1,2 (ife. > 1). If Ef. = {u e [0, 1] : \fi(u)\ > M0}, i = 1, 2 one has with (9): "-i

> i-

"-i

since jn(n) = 1, i = l,2.

(16)

64

//. Nonsquare and von Neumann-Jordan Constants

Adding the expressions for i = 1, 2 in (16), one gets

/ [-

JEflUEf2

+

U

(17)

^-T

Integrating (13) over Efl U Ef2 (with Ui — /i), we find -f

< (2 - 6) f JE

(18)

On the other hand if max(|ui|, \U2\) < UQ, the convexity of $ gives

. N i . + ^2 +

N

K2

-U2)

Using this and integrating as in (18) on (Efl U Ef2)c results in the inequality:

/WK*S <2

(19)

2.3. Upper bounds for J(LW) and J(L*)

65

Adding (18) and (19) and using (17) we get

= 2|^L + -^i + ^-^ < P* ( T^- ( A + /2; «2

-6 f JE

< 4-5(1-). Hence + /2||», ||/i - /a||#) < 2 -

(1 -

).

(20)

Consequently, «/(£*( [0, 1])) < 2 since 5 > 0 and A > 1. A similar argument establishes (ii). D Using the same method as in the theorem, we can prove the result for L (R + ). Here in A2 and V 2 , UQ = 0, and the proof slightly simplifies. Thus we present the result leaving the detail to the reader: $

Theorem 3. // the N-function $ e A 2 n V2, then (i) J(£<*>(R+)) < 2, and (ii) J(L*(R+)) < 2. By Theorems 1, 2, 3 and 1.5, we can state Theorem 4. Let $ be an N-function, and consider

T/ien A$ is reflexive iff J(A"$) < 2, or iff CNJ(X<^) < 2. To see that dim(A$) = oo is essential for the above conclusions through J(A'), consider i\ = {a = (a!,...a n ) : ||a||i = E"=1 |aj|}. Then i is n-dimensional and so is reflexive. But J = c/vj = 2.

66

//. Nonsquare and von Neumann-Jordan Constants

In fact, i f e i - (1,0,... ,0),e 2 = ( 0 , 1 , 0 . . . ,0), then ||ei||i = ||e2||i = 1, and ||ei + e%\\i = ||ei - e 2 ||i = 2. Thus J(^i) = 2. So the assertions of Theorem 4 do not hold for arbitrary reflexive X. Example 5. Let $(ar) = x2. Then £<*> = I2, and (5 = \ satisfies (1) and £0 = (1 - ^) holds for (4). By Theorem 1, J(l2) < \/3. However, cNJ(£2) = I and J(i2} < ^2cNJ(l2) = >/2. A sharp upper bound for nonsquare constant of (general) reflexive Orlicz spaces in the sense of James is still not known. Such a bound can be obtained for a class of reflexive Orlicz spaces, and this will be investigated in the next section.

2.4 Sharp bounds for cNJ(L&')) and cNJ(L*') The class of spaces to be considered are the intermediate Orlicz spaces between L 2 (fi) and an L®(£1) for an TV-function <£. The desired result is given by the following: Theorem 1. Let $ be an N -function and 4>s be the inverse of ^~l(u} = [$-l(u)}l-sK l(u)]s,

u > 0, 0 < s < 1,

(1)

with $o(w) = u2 . 7/(O, S,/i) is a a -finite space, then (i) for Z/* S )(Q) on (Q, S,//,), with gauge norm,

and J(V*'>(to))<21-*,

(2)

(M,) £/ie same results hold for L®s (fi) tyzi/i Orlicz norm also. Proof, (i) By Theorem 1.4.8, for any /, g e L^ S ^(Q) we have

0

2 1 f _1_ «l! « I/ + 0 M ( * . )

I +

1 s 2 2 s 2_s I f f,\\ s \n < ^ ' O o / ' l | / ' | | 2 - s J _ l l n l l 2 ~ s ^~2~~ I/ ~ 0 I I ( * 8 ) J ^ ^ Vll/llf*.) + H0ll(*a)J '

/"^^ WJ

and using the Holder inequality for sequences, namely »l,g=-£T,

(4)

f - g\\2(^a), n = 2,p - ^,q = ±, in (4), one gets ll/ + 011?*.) + ||/ - 0||?*.) < 21-s[(||/ + 011?*.))" + (||/ - 0||?*.))•]' + \\9\\*}2-S-

(5)

2.4- Sharp bounds for cNJ(L^^) and cNJ(L*°)

67

Similarly, letting p — j5| and q = 1 — s, 0 < s < 1, (4) gives

--

''

(

*'V

+

x

(*.))^' Prom (5) and (6), we deduce that for / ^ 0 ^ g a.e.,

(6)

Thus the first of (2) holds. But since J(X) < ^1cNJ(X} for any Banach space, the second of (2) also holds. (ii) Next using Theorem 1.4.8 for the Orlicz norm, we get the same result in exactly the same way. D As a consequence of an earlier computation and the above theorem, we have the following precise result for the Lp-spaces: Corollary 2. Let Lp € {LP([Q, I]),LP(!+),^}, 1< p < oo. Then J(LP) = max(2p , 2 1 ~ p ) . (7) Proof. By Example 2.3, it suffices to show < in both cases of (7). Now let 1 < a < p < 2, and take $(u) = |u|a, and s = fjfEfj- < 1. Then $s(u] = |u|p, and since lim(l — s) = - — l,lim(l — | ) = -, we get (i)

cNJ(Lp) = max(2?- 1 ,2 1 -t);

aj.1

(tt)

"

o4-l

the desired inequality from (2). Next let 2 < p < b < oo, and take $(u) = \u\b, and s = 2 )^~%| < 1. Then $s(u) = \u\p again, and since lim (1 — s) = 1 —p- and lim (1 — f ) = 1 —p-, we get the desired 6—>-oo

b-+oo

bound from (2) immediately. Consequently (7) follows from these two statements. D Using Theorems 1, and 2.1 as well as Lemma 1.4, we can deduce the following general statement: Corollary 3. Let $ be an arbitrary N-function and S be as the inverse o/^J 1 as in (1), 0 < s < 1. Then we obtain max — , 2 / 3 * s < J(L<*->[0,1]) < i2c^j(L(*-)[0, 1]) < 2 x max -£-,2/3j

< J(^*-') < V2c J vj(^ ( ^ ) ) < 21-*,

(9)

68

//. Nonsquare and von Neumann- Jordan Constants

and

max We illustrate these inequalities in the following: Example 4. Let $(w) = u\2p + 2\u\p, I < p < oo. Then $s of (1) is given by Consequently the a$s , f3$>s indices are seen to be

and a<£s = a^^(3^a = /3$s, so that Corollary 3 gives: 1 2i-f ~t ^> JT / /,r«>,i \ ^ J "

r

5"/

1 < P < 2,

,i/ 2 < p < oo,

,»/ 1 < P < | , 2 —

The corresponding inequalities (or bounds) for the CATJ are obtained as:

«7 2 < p < oo, , s

I r- -^~

«/ 1 < P < |

, i/ | < p < o o ,

and [0,1]) > 2

1

~ s - , 1 < p < oo.

It may be observed that the limit values of CNJ as p —> oo or p —> 1 are all equal and the value is 21-s, (except for the last one) as p —> 1. With Corollary 3, exact values of J-and c^v-constants can be found for a class of reflexive Orlicz spaces with 4> s . We again present the results for the gauge and Orlicz norms separately.

2.4- Sharp bounds for cNJ(L^^} and cNJ(L*°)

69

Theorem 5. Let $ be an N -function and $5 be the related intermediate N -function defined by (1), 0 < s < 1 so that $s € A 2 n V 2 . Then we have the stronger statements: (i) $ £ A2(oo) n V2(oo) => ) = 2 x -t ; CJVJ (L<*-)([0,1])) - 21-,

(11)

A 2 (o)nv 2 (<))=*• j(/(*.)) = 2 1 -*;c A rj(*<*->) = 21-8,

(12)

A2 n V2 =»• ) - 21-*;

CjV j(^*-)(R+))

- 21-8.

(13)

Proof, (i) If $ ^ A2(oo) n V 2 (oo), then either (3$ = 1 or a$ = \ by Theorem 1.1.7. Also from (1), one has ,tt>0.

(14)

Now if ;0$ = 1, this gives a*. = (a*.)1"^^)* ^ y1'1 ' V

since a$

^ 2'

2'

Hence /

max(

1

i

a.

,2/3$J = 21"'.

/

x

(15)

If a$ = ^, from (14) one finds 1

_,_.

,

since /3$ < 1.

Thus (15) holds again. Also (11) follows from (15) and Corollary 3. Parts (ii) and (iii) are similarly established. D To demonstrate the corresponding result for the Orlicz norm we present the following analog of Corollary 3, which is obtained from Theorems 2.2., 1 (ii) and Lemma 1.4:

70

//. Nonsquare and von Neumann-Jordan Constants

Corollary 6. Let $, $s be as in (1), 0 < s < 1, and \P+ be the complementary function to $s. Then we have max) 2V.—1- ) < J(L*'[0,1]) < \/2cNj(L*'[0,l])

< 21-*,

(16) max 2/3°s+ , -o\ V

Q; 4. / *7/

< J(r*) < j2cNJ(l*-) < 21-*, "

(17)

and

max f 2V.— I <J r (^* s (K + ))

(18)

We can now state the analog of Theorem 5 for the Orlicz norm. Theorem 7. Let $,$ s ,0 < s < I , be as in (1). Then the following assertions hold: [again $s E A 2 D V 2 •/ ^ if3> £ A 2 (oo) n V 2 (oo), then ^[0,l])-2 1 - s ;

(19)

(ii) if$i A 2 (0) n V 2 (0), then J(^) = 2 1 -t,

c JVJ (^) = 21-;

(20)

cNJ(L*'(B+)) = 2l~s.

(21)

A 2 n V 2 , i/ien = 2 x -f ,

Proo/. (i) By Theorem 1.1.15, 2/3^+ = -1- and ^— = 2/3$s, so that s

*s

^

the left sides of (8) and (16) are equal: maxf — ,2/3*.) = m a x ( 2 / 3 ^s , a V«* s / V *

(22)

Hence (i) follows from the proof of Theorem 5(i). Similarly, we can verify (ii) and (iii). D We end this chapter with the following illustration.

2.4- Sharp bounds for cNJ(L^^) and cNJ(L**)

71

Example 8. Let $ be a function defined as the inverse of $~l(u) = [log(l + u)]*tu*,u > 0, 1

[0, i]) = J(L<*)(R+)) - 2* 0, 1]) = cNJ(L^(R+)) = V2 J(L*[0, 1]) - J(L*(R+)) - 2* +

0, 1]) = CJVJ (L*(R )) - x/2

(23) (24) (25) (26)

and >>

[ 24

{

2

' p , 2 < p < oo,

(27)

!_ I 2

:_i'1
2 2 p,2 < p < oo.

(28)

These are verified by taking $i(w) = exp(|u p) — 1, $o(w) = u 2 , so that u)]^u^,u> 0,0 < s < 1. Then $(n) — $1 (w), and c$1 = 00,0^ — p. From Theorem 1.1.2, we note that $1 i A 2 (oo),$i ^ A 2 but $1 e A 2 (0) n V 2 (0). Hence (23) and (24) follow from Theorem 5(z) and (Hi). Similarly (25) and (26) follow from Theorem 7(i] and (Hi). For the remaining two, consider \&(v) =
fl-s

°

°

si

1

1— 2

and 4= 1 —<-^B- = I4 - -ir, so by Theorem 1.1.8 C\j/ ^p a»=/3»=7»=2-(*

+

i>,

and

a J = / 3 S = 7 * = 2*-*. These relations imply (27) and (28) at once.

1

72

//. Nonsquare and von Neumann-Jordan Constants

Bibliographical notes. From the classical Jordan-von Neumann [1] characterization of a Hilbert space by the parallelogram identity ||/ + 9\\2 + ll/ - 9\\2 = 2(||/||2 + IMI 2 ),/, g 6 X, Clarkson [2] introduced the constant CNJ(X) of a Banach space X to measure its "departure" from Hilbert space. Based on the work of James [1], Gao and Lau [1] introduced the nonsquare constant J ( X ) for a Banach space X. The relationship between these constants is given in Theorem 1.5, due to T. F. Wang (communicated to the second author in October 1997 and included here with his permission). The work of Section 2.3 is generally known (cf., e.g., Chen [1]) since the two definitions of uniform nonsquareness, due to R. C. James and J. J. Schaffer, are equivalent. Here the results are presented with a somewhat different type of arguments. Sections 2.2 and 2.4 are adapted from Ren [4]. The computations of the coefficients CNj(L®"(£i)) are included for comparison.

Chapter III Normal Structure and WCS Coefficients

This chapter is also devoted to some geometric properties of Orlicz spaces, specialized from the general Banach spaces, to be used in concrete applications. The results include normal structure and WCS coefficients of Orlicz spaces, and their relations with the quantitative indices of these spaces considered in the preceding chapter, using both the gauge and Orlicz norms. The work refines and extends the corresponding results of earlier authors. 3.1 Introduction The concept of normal structure was introduced by Brodskii and Milman[l] to describe the centers of convex sets and fixed points of isometrics in normed linear spaces. Later Kirk[l] has shown that a weakly compact convex subset of a normed linear space with normal structure has the fixed point property for nonexpansive mappings. This led to an extended study of Banach spaces with normal structure. Then Gillespie and Williams[l] introduced the uniform normal structure concept and soon after Bynum[l] and Maluta[l] defined certain other geometric coefficients for these spaces and analyzed some of the interrelations between them. We recall these concepts and then present the relevant abstract results to specialize and refine them in the context of Orlicz spaces. Definition 1. The normal structure coefficient X, is defined as:

N(X) of a Banach space

N(X) = inf \ -7-77- : A C X a bounded closed convex set, d(A) > 0 1 , (1) (r(A) } where d(A) = sup{||:r—y\\ : x, y e .4} is the diameter, and r(A) is the relative Chebyshev radius of A, namely, r(A) = inf {sup{||z - y\\ : x e A} : y € co(A)} ,

73

74

III. Normal Structure and WCS

Coefficients

co(A] being the closed convex hull of A. Since 1 < N(X) < 2, one says that X has a uniform normal structure provided N(X] > I . Some basic properties of N(X] can be presented as follows: Theorem 2. (i)(Maluta[lj) If X is nonreflexive, then N(X) — 1, and if X is infinite dimensional, then N(X] < \/2. (ii)(Amir[l]) If X is reflexive, then

N(X) = inf { 4-4 :AcX, finite, d(A) >0}. (r(A) J

(2)

Proof. To get a feeling for these concepts, we sketch a proof of the results. (i) (After Amir[l]). It is known that in a nonreflexive Banach space X, there exists, for each E > 0, a sequence {xn, n > 1} such that for each m and y' e CG{XI, • • • ,xm},y" e co{xi,i > m+ 1} one has 1 — 5 < ||y' — y"|| < H-£, (cf., e.g., Milman and Milman[l]). If A = co{xn,n > 1}, then d(A) < 1+ £ and r(A) > 1 - e. Hence, 1 < N(X) < ^ so that ]V(A') = 1 since E > 0 is arbitrary. Regarding the second inequality, let d(X, y) — infJIlTHIlT" 1 !! : T : X —>• y, an isomorphism onto }, the "Banach-Mazur distance " between a pair of Banach spaces. Then -xN(X)
(3)

On the other hand, by a known result of Dvoretzky's, if dim X = +00 for each e > 0 and integer n > 0 one can find a subspace y C X such that y is isomorphic to the Euclidean space /^ of dimension n, and d(y, / „ ) < ! + e. Thus taking y = ye here and noting that N(Pn} = y^±^(cf. Routledgefl]), we get from (3) 2(n

Since e and n are arbitrary, this implies N(X] < \/2(ii) Let C C X be a, closed convex set with d(C) — 1. Given an a < r(C'), one gets f| B(x: a) n C = 0. Since a closed ball B(x, a) and C in a reflexive zee

3.1 Introduction.

75

X are weakly compact, then a finite subset A C C can be found such that P| B(x, a] n co(A) C D B(x, a) n C = 0, and a < r(A). x£A

x£A

Letting a /* r(C), this implies (2). D The next result will be used in the following section, and it is due to Dominguez Benavidesfl]. Theorem 3. If X is a reflexive rotund (— strictly convex ) space, and A C X is a finite subset, then there is a set B C A such that (i) r(B] > r(A), and (ii) ||6 — x\\ = r(B) for every x E B, where b is the relative Chebyshev center of B with respect to co(B)(cf. Definition 1). Proof. We include a brief argument for this result also. Thus let y0 be the relative Chebyshev center of A and consider AI — {x E A : \\yQ — x\\ — r(A)}. We claim that r(Ai) > r(A). For otherwise, we can find an e > 0, such that 1 1 7/0 — x\\ + £ < r(A] if x E A\Ai. Let y\ be a similar center of A\ and take 0 < A < 1 to satisfy A||y0 — Ui\\ < f • Then for a; 6 AI we get by supposition \\x - yo + A(y 0 - yi)||

< A||x -

Vl\\

+ (1 - A)||x - y0||

If x € A\Ai, then \\x — yQ + A(y 0 - T/I)|| < r(A) - | < r(A), so that for every x G ^4, ||a; — yo + A(T/O ~~ 2/i)|| < r (^)- This contradicts the minimality of yo, since y0 + A(yo — yi) E co(A). Since A is finite there is a B C A, minimal in the nonempty subsets of A, satisfying (i}. It also must satisfy ( i i ) , for otherwise one can find a BI, a proper subset of B, satisfying (i), by the last argument contradicting the minimality again. D Remark 4.

(i) For a Banach space X, Maluta[l] defined

d(A)

^

*, a s i n ( l )

1 /

=

It is termed a "self-Jung constant" in Amir[l]. A brief account of Jung constants of Banach spaces is given in Section 4.1. (ii) The bounded sequence coefficient, BS(X], is defined as: A(tx.\\ -rr-TT : {^» i > 1} is a bounded

76

III. Normal Structure and WCS

Coefficients

nonconverging sequence in X } , introduced in Bynum[l], has been shown to be just N(X], ( i.e., BS(X] = N(X}} by Lim[lj. Here A({xi}} — lim sup{||xj — Xj\\ : i, j > n}, the asymptotic diameter of {xi, i > I } . We present a few other results, without proofs, that will be analyzed in the context of Orlicz spaces later on. Recall that X is said to have the fixed point property if for each closed, bounded convex set 0 / K C X and mapping T : K ->• K satisfying \\Tnx - Tny\\ < j\\x - y\\,x,y € K,n > 1, then T has a fixed point in K. Here T is also called a 7—Lipschitz mapping. Theorem 5. (Casini and Maluta[l]) // X satisfies N(X] > 1, then it has the fixed point property for 7—Lipschitz mappings, satisfying 7 < JN(X}. Recall that X has Schur property iff weak and strong convergences are equal (as in the case of the classical i1}. Definition 6. (Bynum[lj) For a Banach space X without the Schur property, the weak convergent sequence (WCS) coefficient is defined as: ( A(fr.\} WCS(X) = inf I Y V( : {xt, i > 1} C X is weakly but I

r

(\xij)

not strongly convergent sequence },

(4)

where A({xi}} and r ( { x i } ) are the asymptotic diameter and the relative Chebyshev radius with respect to W({xi}}. It is seen that 1 < WCS(X) < 2, and X is said to have weakly uniformly normal structure when WCS(X] > 1. Theorem 7. (Zhang[lj) If X does not have the Schur property, then WCS(X] = inf {A({xi}} : x{ G S(X),Xi ->• 0 weakly} .

(5)

Here S(X] is the unit sphere of X, as usual. Another relation is given by Theorem 8. (Bynum[lj) If X is a reflexive space without the Schur property, then I < N(X] < WCS(X] < 2. (6) The list of results needed in the following work will include the next three theorems.

3.1 Introduction.

77

Theorem 9. (Dominguez Benavides, Lopez Acedo and Xu[l]) Let X be a space with WCS(X] > l,C C X a weakly compact convex set, T : C —> C a nonexpansive mapping (i.e., \\Tx — Ty\\ < \\x-y\\ ). Then T has a fixed point. To proceed further, another concept due to Maluta[l], will be useful. Definition 10. For a Banach space X, define the coefficient (lim sup d(xn+\, co{xi}") = sup { n^°° _ , , - , , : {xi} C X

is a bounded nonconstant sequence } , where d(A, B) is the distance between the sets A and B and d(A) the diameter of A Malutafl] proved that D(X] = 0 iff dim A' < oo; \ < D(X) < 1 if dim X = oo, and D(X] = 1 for nonreflexive X. Theorem 11. (Xu[l], Prus[l]) // X is an infinite dimensional reflexive Banach space, then D(X] = (WCS(X}}-\ (7) This answers a question posed in Maluta[l]. Some examples with numerical values of the above noted coefficients, have been given by Bynum[l] and Dominguez Benavidesfl]. Let £1 = [0,1] or [0, oo) with Lebesgue measure (or a counting measure with f2 = N ) and L p (f2), lp be the corresponding Lebesgue spaces with 1 < p < oo, then = WCS(Lp(ti)} = min(2 1 -p, 2p) = N ( l p ) .

(8)

= 2'.

(9)

and

Both (8) and (9) will be deduced from the Orlicz space results given in Sections 2-4 below. Moreover, the work of Section 4 shows that there are classes of Orlicz spaces for which strict inequalities obtain for (8) and (9) with both the gauge and Orlicz norms already in the sequence spaces. As a final item of this section, we include the following result due to Chen and Sun[l], and Wang and Shi[l] for some Orlicz spaces.

78

III. Normal Structure and WCS

Coefficients

Theorem 12. Let Q = [0, 1] or [0, oo) with Lebesgue measure (or the sequence spaces (fi = TV) with counting measure) as above, and <3> be an N— function. Then one has (i) L<*)(fi) is reflexive iff N(L-W(Sl)) > 1, (ii) L*(fl) is reflexive iffN(L*(tt}} > 1. The details of (i) may be found in Chen ([1], pp. 107-1 13). Some other results on geometric constants may be found in Prus[2], Xu[2], AyerbeDominguez Benavidesfl], as well as Dominguez Benavides[2]. As noted above, we turn to finding explicit bounds on Orlicz spaces using the indices introduced in the previous chapters. 3.2 WCS and N— coefficients of Orlicz function spaces We again concentrate on pairs of complementary N— functions ($,\I/), and intervals Q = [0, 1] or R+ — [0, oo) with Lebesgue measure. The results admit extensions to a a— finite space (0, S, n) having the Rademacher property, i.e., the space on which a system of Rademacher functions (recalled below) exists. Using the notations introduced before, we can present: Theorem 1. Let $ be an N— function and let 0;$, /3$, a$, (3$ be the quantitative indices of Section 1.1. Then one has (i)

WCS(M^[Q,1}) <2a<j,,

(1)

])<-i-; P*

(2)

and $ G V2(oo) implies

(3) and $ 6 V2 implies WCS(M^(R+)} < i-.

(4)

P$>

Proof, (i) (1) is established by using a property of Rademacher functions Rn, which are defined as follows. For each a > 1, let l - l

1=1

'

2rlfl

I

2 a

' "

3.2 WCS and N-Coefficients

of Orlicz Function Spaces.

79

Here a > 1 will be chosen suitably so that Rn : [0, ^] —)• {—1,1} is in L2[0, £] and the sequence {Rn,n > 1} has interesting properties. [ In fact one may define Rn more briefly as Rn(i) = sgn(sin 2 " +1 — ) , V a} and they can be regarded as independent random variables on the probability space ((0, ^), ^f )• We use the more explicit form given above.] Now a > 1 is chosen as follows. By definitions of a$ there exist 1 < Uk /* oo such that lim $-i(2u\ ~ Q:*' Hence for any e > 0, one can find k0(= k0(e}) > I such that

and we take a = Uk0 in the above. If /n = $ 1(o)Rn which is a simple function, one has p$(/n) = 1 for all n > 1. However, (M^[0, 1])* = L*[0, 1] C Ll[Q, 1], so that by a known property of Rademacher functions (cf., e.g., Guerre-Delabriere[l], p. 27) one has for each g € L*[0, 1], lim !_

Jo" fn(t)g(t)dt = 0, so that fn —> 0 weakly. Moreover, by the orthogonality of Rn and (5), for m ^ n

so that Thus (1) holds, £ > 0 being arbitrary. For (2), since $ G V2(oo)[or equivalently \I> e A2(oo) ], for any e > 0 from definition of /3$ , we can find 1 < Vk /* oo such that

so that v*, A > 1 .

(6) 00

By considering a subsequence, if necessary, it may be assumed that Z) oo k < *=i 1, and let Gk C [0, 1] be disjoint measurable sets such that ^(Gk) = ^~ where

80

III. Normal Structure and WCS

Coefficients

\JL is the Lebesgue measure. If we let fk = $~l(1vk}xGki then ||/fc||($) = 1 and by (6), for m ^ n P*((£* - e ) ( / m - / n ) ) < 1, so that ||/m — fn\ ($) < (fa— z}~~1 and hence A ( { f n } ) < (/3$ —e)" 1 . But since ^ G A 2 (oo), for each g G I/*[0,1], || 0 as n —>• oo. Consequently, by the Holder inequality, $ fn(i)XGn9(t)dt -)• 0 for each 0 G L*[0,1], i.e., /n -> 0 weakly. Hence by Theorem 1.7 we get (2). (ii) The proof of (3) is entirely similar to that of (1), and so we consider (4) when $ G V2. Then for any £ > 0, there is 0 < VQ < oo such that > •#* SP - e

(7) V /

The simple function /„ = 3>~l(2vQ)xEn, En - [^, ^), satisfies ||/n||($) = 1, and since ^ G A2 so that L^(R+] = M*(JR + ), one has ||pX£ n ll* ~* ® for En C [^-, oo), by the dominated convergence theorem. Thus fR+ fngdt —> 0 as n —>• oo (by the Holder inequality) for each g G L^(R+). So /n —>• 0 weakly. However for m ^ n, from (7), we also have

Hence >!({/„}) < sup ||/n - /m||($) < (/^, - e)"1, showing that (4) holds, m^n

since £ > 0 is arbitrary. D Corollary 2. For an N— function $, and f2 = [0, 1] or .R+, one has: (i) if L^(£l) is nonreflexive, then WCS(LW(ty) = WCS(MW(to)) = 1;

(8)

(ii) z/Z/*)(Q) Z5 reflexive, then

min(2o;$, ==-),

for ft = R+.

(9)

(in) For any N-function $, : \/2.

(10)

3.2 WCS and N- Coefficients of Orlicz Function Spaces.

81

Proof, (i) In this case, the iV-function $ ^ V2(oo) or $ e V2(oo)\A2(oo) so that either 0;$ = | or 0;$ > | but /?$ = !. Then by Theorem l(i), WCS(MW[0, 1]) < 1. Also by the proof of Theorem l(i), we see that Z/*)[0, 1] lacks Schur property, and the definition of WCS— coefficient implies 1 < WCS(LW[Q,I]) < WCS(MW[Q,1]), so that (8) holds for fi = [0,1]. The proof is similar for J7 = R+. (ii) If Z,<*)(n) is reflexive, then M<*)(ft) = Z,(*)(ft) and (9) is a consequence of Theorem 1 . (iii) It is sufficient to show that min(2o;$, j-} < \/2- If this is not true, then 2a$ > \/2 as well as /?$ < -T=. But this contradicts the fact that Q!$ < ^$. Similarly (10) is proved if fi = R+. D The above assertions are not valid for Orlicz sequence spaces. For instance, if $(u) = |u|p, 1 < p < 2, then WCS(l^) = 2? > >/2. We now study the case of Orlicz norm which is more involved. Theorem 3. Let ($,^) be an N — complementary pair and o;$,/3^,a^,^ be quantitative indices introduced earlier. Then one has: (i) Q,l})<±-, (11) and $ e V2(oo) implies WCS(M*[Q,1]) <2ay,

(12)

WCS(M*(R+)} < ^-, P*

(13)

(")

and $ e V2 implies Proof, that

WCS(M*(R+)) < 2av. (14) (i) By definition of $p, for any 0 < s < |, we can find a w0 > 1 such

Consider the Rademacher sequence {Ri,i > 1} on [0, £-] as in the proof of (1), and define ft = ^ U o ) ^,i> 1. Then ft e M*[0, l]°and u

o

n

ii

-i

• ^ -,

82

III. Normal Structure and WCS

Coefficients

Also as in the preceding theorem, it is seen that fi —> 0 weakly. By (15) for II f. _ f.\\ \\Jl

_ _

Jj\\$

—

^jy_ 1

This establishes (11) since e > 0 is arbitrary. Regarding (12), $ E V 2 (cxD) <^> ^ G A2(oo), and hence there is a VQ > 0 and K > 2 such that for v > vn

(16) \ ^/

^

\Z/

and

2vt/)(2v) < *(4v) < X 2 ^(t;),

(17)

ip being the left derivative of ^, as usual. Now for 0 < e < ^, since | < ay < 1, we have by (16) and (17), for v > VQ .?_ < ^(2)

<

vil>[(a9 + e)v]

<

-r,--, <

By definition of ay, there exist ^(VQ] < un /* oo such that n—>oo lim w. T ._i/o \ = ^"nj a^. For this e > 0, we can find n0 = n 0 (e) > 1 such that for n > n0, [^" 1 (w n )/^~ 1 (2u n )] < a$ + e. Let ^ = ) I r ~ 1 (2u no+ i) so that v0 < Vi /* oo, and

Since the Vi satisfy (18), we can extract a subsequence, denoted by the same symbols, such that lim

l

*

=b

(20)

and f - b - TNow for all large enough i, (20) may be written as:

'-^-
(21)

(22)

3.2 WCS and N—Coefficients

of Orlicz Function Spaces.

83

Since Vi /* oo, we can assume that £) ^TT < 1 and (22) holds for alH > 1. (Vt) i>l

Choose disjoint Borel sets Gj C [0,1] with n(Gi) = -J^y and define ,_

fi

1.

-

Then ||/i||$ = 1, and since ty € A 2 (oo), /z(Gi) —>• 0 as i —>• oo, we have for each g € (M$[0,l])* = £<*>[(), 1], II^XcJIw -> 0 as i -» oo. And then by Holder's inequality, /Q ftgdt —> 0 or fi —> 0 weakly as i —>• oo. But from (22) and (19) we have for i ^ j,

= 1+

+ ^T-T$

j

= (a* + g)

H

Since by (21), (6 — ^ ) > ^ — ^ = K:) the above computation gives:

and hence A({/j}) < 2(o;^ + e)(l + IKe). Since AT is a fixed constant and £ > 0 is arbitrary, this establishes (12). (ii) The proof of (13) is similar to that of (11), and so let us consider (14) when $ e V2 or ^ e A 2 . Now by definition of a$ for £ > 0 there is a VQ > 0 such that e.

(23)

84

III. Normal Structure and WCS

Coefficients

Let d = [g, ^), and define /, = ^rf^yXG, so that ||/;||$ = 1 and for each g 6 L^(R+)(= M^(R+)) one has > 0, i -» oo,

so that ||<7XGiH(*) —> 0 and hence by Holder's inequality ->• 0,

i.e., fi —> 0 weakly. But by (23), we also have for i ^ j

This implies (14) since £ > 0 is arbitrary. D An immediate consequence, similar to Corollary 2, is Corollary 4. lei $ 6e an N—function and Q = [0,1] or .R+, w£/i // as the Lebesgue measure, as before . Then (i) L*(fi) is nonreflexive => ^^^(^^(Q)) (ii) Z/$(il) is reflexive => ( ^ being complementary to

(24) n(J-,2o*), M^

Q-.R+,

(iii) For any N— function $, one /ias

We also record the following Theorem 5. Let <3> be an N— function and Q be as in Corollary 4- Then (i) L<*)(fl) w re/Zea:it;e i^ WCS(LW(ty) > 1; (ii) L*(Q) is reflexive iff WCS(L*(ty} > 1. Proof. Note that 1 < N(X) < WCS(X] < 2 if X is a reflexive Banach space and does not have the Schur property (cf. Theorem 1.8). Hence (i) follows from Corollary 2(i) and Theorem 1.12(i), whereas (ii) follows form

3.2 WCS and N— Coefficients ofOrlicz Function Spaces.

85

Corollary 4(i) and Theorem 1.12(ii). D We remark that Theorem 5 is not valid for Orlicz spaces l^ and /* (see Section 3 below). The relation between WCS(LW(ty) and Maluta's £>(£<*> (fi)) can be seen from the following result. Theorem 6. For an N— function $ one always has: (25)

and D(L*(ty) = [WCS(L*(ty)]-1,

(26)

$

where the L (J7) spaces are as in Corollary 4Proof. If L<*)(fi) is nonreflexive, then WCS(L^(Q,}} = 1, by Corollary 2(i). Thus (25) follows since D(X) = 1 for any nonreflexive X. But if L<*>(ft) is reflexive, then (25) is a consequence of Theorem 1.11. Similarly (26) is established. D It may be noted again that Theorem 6 is not valid for /$ and l^ (see Section 4). In general it is not easy to find the exact values of N(L^(ty), WCS(L(*>(n)), JV(L $ (Q)) or WCS(L*(ty) for all reflexive Orlicz spaces with either the gauge or Orlicz norms. However, we can do this for a class of reflexive Orlicz spaces defined by certain intermediate N— functions. Theorem 7. Let $ be an arbitrary N— function, 3>o(w) = u2 and 3>s be the intermediate N— function of$Q,$, i.e., the inverse of $~l where < l,u>0. Then for a a— finite measure space (fi, E, /^), the normal structure satisfy: 2* < m i n W ( L ( * ' > ( n ) ) , W ( L * ' ( n ) ) .

(27) coefficients (28)

Proof. It was seen before (cf. Corollary 1.4.10) that both Z/ $ *)(fJ) and L®° (17) are uniformly convex, hence reflexive, so that we can invoke Theorems 1.2(ii) and 1.3. Thus let A C L^')(fi) be a finite set. Then by Theorem 1.3 there exist a subset B — {/j, 1 < i < n} of A such that r(B) > r(A) n

and a point 6 = E Cjfj

n

€ co(B),Cj

> 0, E GJ = I satisfying conditions

86

III. Normal Structure and WCS

Coefficients

\\b — /i||($) = r(B], 1 < i < n. Using Theorem 1.4.12, we get n

n

0

7i

TI

0

c

j\\fi - fjllfV-p

(29)

where c = max(l — c,-) < 1. It then follows from(29) that J 3
~

=

2

so that f > 2f. Similarly using Theorem 1.4.12, the corresponding inequality for the Orlicz norm is obtained. This establishes (28). D From this we immediately deduce the following: Corollary 8. (Dominguez Benavidesfl]) Let I < p < oo, and (7 = [0, 1] or R+ Then 1 ~ p , 2p). (30) Proof. IP

Let <& p (u) = \u p. Then a$p = /3$p = a$p = /3$ = 2 ~ p , since p ), and by Corollary 2(ii) we get min(2 1 ~p,2?).

For the second equality, let 1 < a < p < 2 and (u) = |w|a, $o(u) — w 2 , and 0 < s = f5 < 1. Then $ a (w) = |w| p and by (28) one obtains

Since lim | = 1 — -, this implies

3.2 WCS and N— Coefficients of Orlicz Function Spaces.

87

On the other hand, for 2 < p < b < oo, let $(u) = \u\b,s = 2|=£. Then 0 < s < 1 and ^~l(u) = UP, lim f = -, so that b—too

2p
P

2
Combining this inequality with the previous one, (30) follows upon using Theorem 1.8, since N(X) < WCS(X] for reflexive X without the Schur property. D Remark 9. Since the counting measure is u— finite, the above proof and Theorem 7 imply that for 1 < p < oo min(2 1 ~p,2p) < N ( l p ) .

(31)

We shall show later (cf. Section 4) that there is equality in (31). As an application of Theorems 7 and 1.5, we have: Corollary 10. Let (fi, E, /^) be a a— finite space and $s be as in Theorem 7. Then for each 0 < s < 1, the Orlicz spaces 1/^(0) and L $3 (f2) have the fixed point property for uniformly 'j—Lipschitzian mappings with 7 < 2t. We next consider the classes for which exact values of these coefficients can be obtained when the measure space is on [0, 1] or R+ with Lebesgue measure. Let us start with: Theorem 11. Let $ be an N— function and $s be as in Theorem 7, 0 < 5 < 1. Then we have: (i) $ <£ A2(oo) H V2(oo) =» 0, 1]) = WCS(L&')[Q, 1]) = 25;

(32)

N(L^(R+}) = WCS(L(*°\R+)) = 2f .

(33)

(ii) $ 0 A 2 H V2 =>

Proof.

From Theorem 7 and Corollary 2(ii), one has for £1 = [0, 1] or R+,

2 < N(L*°ty] < WCS(L*a(ty) < <

(34)

III. Normal Structure and WCS

88

Coefficients

Since

\fu

u > 0,

one has

and similarly,

A2(oo) D V2(oo) =>•/?$ = 1 or 0;$ = | (cf.

Thus (i) becomes, since Theorem 1.1.7), we get mn

= 2*.

For (ii), $ £ A 2 n V2 => /?$ = 1 or a$ = |, so that

min 2a*,,=— V

=25.

P

Thus both (32) and (33) are true. D In the same way we can prove the following for the Orlicz norm. Theorem 12. Let <E> and 3>s be as in the preceding theorem. Then: (i) $ g A 2 (oo)nV 2 (oo) =*> [0, 1]) - ^C5(L$S [0, 1]) = 2t ;

(35)

N(L*°(R+)) = WCS(L**(R+}) - 2f .

(36)

(ii) $ ^ A 2 n V2 =>•

Proo/. From Theorem 7 and Corollary 4(ii), one has for fi = [0, 1] or R+,

(37)

3.2 WCS and N- Coefficients of Orlicz Function Spaces.

89

where ^+ is the complementary TV— function of $s. (i) By Theorem 1.1.15 we have 2<x*.P9+ = 1 = 2a*+/9*..

(38)

Hence WCS(L^[Q, 1]) and WCS(L*'[Q, 1]) have the same upper bound by (34), (37) and (38). The desired result is immediate from this and Theorem (ii) Since 2o;$a/3^+ = 1 = 2a^,+/3^s, the result follows from (34), (37) and Theorem ll(ii), in which Q = R+.O We now present a couple of examples illustrating these results. Example 13. Let Q(u) be the inverse of Q~1(u) = {exp[log(l + u)]i - l}5 us u > 0. If Q = [0, 1] or R+, then we get tf(L«>(n)) - WCS(LW(ty) = 2*,

(39)

and = 2.

(40)

Indeed, it is known that (cf. e.g., Krasnoselskii and Rutickii[l], p. 34)

is an TV-function. Since $ £ A2(oo) and Q -1 (u) = [^ > 7 1 ( w )]s=i) (39) follows from Theorem 11, and (40) from Theorem 12. Example 14. Let $ p (u) = |u|2p + 2|w| p , 1 < p < oo. Then l-i

— 1) p w 2 , 0 < s

Clearly, $ £ A 2 fl V2- But we can not invoke Theorems 11 and 12 here. However, the geometric coefficients can be estimated by using (34) and (37), obtaining 2* < #(£<*••*>[0,1]) <

90

III. Normal Structure and WCS

Coefficients

2* _

~^~

! ' p < oo,

and the same estimates hold for L $s>p (f2). It is observed that there are equalities here as p —>• oo for Q = [0,1] or R+ but as p -> 1 for fJ = /2+. 3.3 WCS coefficients of Orlicz sequence spaces We note as before, the closed separable subspace m* = {f G /* : p$(A/) < oo,VA > 0} by m^,ra* for the same set with gauge and Orlicz norms. We start with the gauge norm. Theorem 1. Let $ be an N—function.

Then

P<j>

\P<3>

(1)

P$ /

where

and Q< — _ Qliri . t. _ i o . . . I> . p sup J< ,._\2kL $ 1/n . / c — 1 , 2 ,

f^oj o^

Proof. The left half of (1) is essentially due to Dominguez Benavides and Rodriguez ([1], Thm. 2 ). To prove it, suppose the contrary so that (4)

Then there is a sequence {/n € m^\n > 1} such that ||/n||($) — 1,/n —>• 0 weakly and limsup||/ n - /m||(*) < (/^*)~ 1 - Taking a subsequence (denoted m,n—>oo

by the same symbols ) of these functions of disjoint supports, we have l/n - /m||(*) <

T<

for

all 0 < u <

.

(5)

3.3 WCS coefficients of Orlicz Sequence Spaces.

91

Since p*(/n) = 1 = p*(/m), we get max{|$(|/n(z)|), |$(|/ m (z)l)} < |forall i > 1 and from (5)

i,||/n -

,,

/m||(*)

.

(6)

and similarly

*Thus from (6) and (7) for ra 7^ n, one has Jn

J

1=1

This contradiction proves the original assertion of the left half of (1). Regarding the right half, we first show WCS(m<») < ± P$

(8)

By definition of /3$ (cf. Sec. 1.1.), there exist 1/2 > Uj \ 0, such that lim *-i(2u\ = P%- Hence for 0 < e < 1/2, we can find j0(= jo(e)) such that 2uj < e and $ _ 1 i 7 ^l > /?$ — £, j > jo- Setting u0 — UJQ for convenience, we have -e)$-1(2uo)] 0 satisfying 2fc 0 wo + $(
(10)

92

HI. Normal Structure and WCS

Coefficients

Consider the (kQ + 1)-vectors XQ = ($"1(2u0), • • • , $~l(2uQ), tQ) and ZQ = (0,0, • • • , 0) and define fa = (Z0, • • • , Z0, X0, ZQ, • • •), i > 1, where X0 is in ith position. Then /; 6 m^ and ||/j |($) = 1 since p$(fi) = l,i > 1. For any

1 so that

lim

= 0.

Hence /; —> 0 weakly by Theorem 1.2.15. But (9) and (10) imply

1+e

l+E

1 l+E 1 l+E

Hence

1+g

implying (8). Finally, we prove

(11) For any natural number k > 1, consider the k— vectors ** = ^'(^(l, 1, ' ' ' , 1) and Zk = (Q,Q,--, 0) K

with dimX fc = dimZk = k, and define fi ~ (Zk, ' ' ' i Zk, Xk, Zk, Zk, • • •),

where Xk is in ith position. Then ||/i||($) = p<s>(fi) = l,i > 1 and for £ > 0

*(f A) =

3.3 WCS coefficients of Orlicz Sequence Spaces.

93

which implies that /, -> 0 weakly. But for i ^ j, we have

^

V2fc>

Since /c is arbitrary, one has W) < A({ft}) < inf

: A; = 1,2, • • • = - - , l^

V 2 jfc><

J

P$

which is (11). D Example 2. (Bynum[l]) VFCS(P) = 2?,1 < p < oo. This follows from Theorem 1 if $p(u) = \u\p so that m<*"> = IP and ^p = /?£p = ^$p = 2~p, since 5*_V("\ = 2~i,' 0 < u < oo. Corollary 3. For an N— function $, we ^; i/$ 0 A 2 (0), then WCS(l^) = WCS(m^} = 1; (b) »/$ e A 2 (0), 1. Proo/. (a) If $ ^ A 2 (0), then ^g = 1 by Theorem 1.1.7. The right side of (1) implies

i.e., WCS(lW) = WCS(mW) = I . (b) If $ € A 2 (0), then l^) = m^ and /?g < 1. Since g-"^ < 1 for every w0 € (0, |], we see that ^$ < 1 and the left side of (1) implies 1 < WCS(lW). D For further study, we need the following result, from Yan[l]. Lemma 4. For an N— function $(u) = /J (j)(t)dt,

define

for t > 0, u > 0 anrf c > 1. ZTien F$(t) w increasing (decreasing) on (O,^"1^)] z/ anc? on/?/ if for any c > 1, G$(c, w) zs increasing ( decreasing ) on (0, M], where UQ > 0.

94

III Normal Structure and WCS

Coefficients

Proof. We first assume that $'(i) = (£). Then for a fixed c > 1, the derivative of G$(c, u) at u e (0, ^] is

^Ur<j>^C, r f U), — nil

t* l-fc

,

L__. ,

.

Q^^lr";/) ^x: \ {sLi I \

Let ti = $~l(u) and t2 = ^(cu) for w e (0, ^ ], then 0 < *i < t2 < $~l and

which proves the lemma. The general result can be proved by using some elementary lemmas, which are as follows: (A). A function /, with f'± existing, is increasing iff both f'+ > 0 and f- > 0 . (B). Let F£\t] = ^ and F^(t) = ^. Then F^(t] is increasing (fy\

(decreasing) iff F^, '(t) is increasing (decreasing) (cf. Yanfl]). D In view of Corollary 3, we deal only with $ € A 2 (0), which implies that F$(t) = t-j$- is finite on (0, $-1(l)]. Now we can prove the following. Theorem 5. For $ e A 2 (0), we have: (i) if F$(t) = l-j±± is increasing on (0, ^"^(l)], then ;

(12)

(ii)z/F$(t) is decreasing on (0, $~1(1)], then WCS(l(}) = 2<^,

(13)

where C^ = limF<Sfv(t). x * t-»o Proof. (i) By Lemma 4, the hypothesis implies that G$(u] = G$(2,w) =

3.3 WCS coefficients of Orlicz Sequence Spaces.

95

$-1$) is increasing on (0> f]> so that

which proves (12) by Theorem 1. (ii) The condition shows that C$ exists and

which implies (13) again by Theorem 1. D We illustrate the result for several TV— functions. Example 6. If $(u) = e\u\ - \u\ - 1, so V(v) = (1 + H) log(l + \v\) - \v\, then 3 ^

« 1.3364

(14)

\2/

and

WCS(l(*}) = \/2. In fact, for each t > 0, r

,.

t*'(t)

t(e* - 1) J-t-l

and

which implies (14) by Theorem 5(i). On the other hand, v;

(1 + 1 ) log(l + t) - t

and C^w = limF^(t) (0)/ n V 2v(0). Also, v / = 2, i.e., * <E A 2 v y t->0

(15)

96

III. Normal Structure and WCS Coefficients

for t > 0, since v 2 ^ dlJL \

uy

f

/"lH~^

< (\Ji/

\

/ /

\ /*1"{~* /77/ \

dw)/ \^i/

9 /

=

~~9~ 1 i1r+it' u /

by the CBS-inequality. Hence (15) follows from Theorem 5(ii). Example 7. Let

\II\P ilog(e+ u\)h '

2
defining an N— function (cf. Sec. 1.1). Then WCS(l(*'}) = 2?.

(16)

Indeed, we have F*,(t)=pp (e + t) log(e + t}' So C£ = limF&tt) = p. Hence (16) follows from Theorem 5(ii). ^=P

*

.Q

^P \

/

-*

\

/

\

/

Example 8. Let $ r (u) = e|u|r - 1,1 < r < oo. Then

Jog 3-log 2

(17)

In fact, the function

is increasing on (0,oo) D (0,1/2] and ot\r = @$T = limG f $ r (w) = 2 >-, so that $r G A 2 (0) n V 2 (0). Hence

and (17) follows from Theorem 1.

3.3 WCS coefficients of Orlicz Sequence Spaces.

97

Example 9. Consider the A^—function $V(w) = |w| p log r (l + |u|),

r > 0 and 1 < p < oo.

Then >) = 25*.

(18)

Again for t > 0, we have "—'

t$'r(i) *„,(*)

rt ' (l + t)log(l+t)

and so Cg P' = lim.FV, r (*) = P + r > 1, i.e., $P)T. G A 2 (0) n V 2 (0). Now t—}Q T

d

v

MX _

^'[^gl1 + * ) - * ]

^ n

^n

Hence, (18) follows from Theorem 5(ii). Example 10. Let

Then WCS(l(Mp)) = 2r.

(19)

p

Indeed, the function FMp(t) = M ,} is decreasing on (0, M^"^!)] = (0, 1] and CM = \imFMp(t) = p. Hence (19) follows from Theorem 5(ii). It should be noted that for some TV— functions we can not apply Theorem 5. To illustrate this we give the following example, due to Yan [1]. Example 11. Let

Then WC5(/ W )=43.

(20) -1

In fact the function t !->• F$(t) = ^^- is not monotone on (0, $ (l)], so that Theorem 5 can not be applied for this $. By a simple computation we can verify that

i = ^ and

= 0 = 2~^,

III. Normal Structure and WCS

98

Coefficients

which proves (20) by Theorem 1. We now consider several analogs for the Orlicz norm. Theorem 1'.

Let (, ^) be a pair of complementary N—functions.

Then

where

u

sup

-

(22)

with i/j being the (left) derivative of ty, and o = inf

(23)

Proof. The proof is similar to that of Theorem 1, but the details are computationally somewhat different, and so we include the essentials. Thus consider the left half of (21) and suppose it is false. Then we must have WCS(m*) <~ = inf

:0< u<

Hence there is a sequence {fn,n > 1} C m$, ||/n| $ = 1, such that /„ —>• 0 weakly, and limsup ||/n — /m||$ < (/J^)"1. ^ere we can a^so assume that the m,n— ¥00

f^s have disjoint supports and for n ^ m l l f n ~ /m||* <

0 < U <

T <

(24)

Then by Theorem 1.2.16 there is a k(= k(m, n)} > 1 such that for n ^ m In

1

I /n

k(fn-f

J /

and l/n -

= E*

(25) m *

/„(«)

+

3.3 WCS coefficients of Orlicz Sequence Spaces.

99

Consequently, for alH > 1 max < T——n

, ——m

> <

But ||/n — /m||$ < 2, and from the above we conclude

This and (24) give the pair of inequalities k\fn(i)\

_!/!

. l

and a similar inequality by interchanging m and n. Hence from (25) and (26), we get exactly as in Theorem 1 (following (6) and (7)): ^ * I II f _ =l \ll/n

This contradiction proves the left half of (21). Regarding the right half, we first show WCS(m*) < 2c4

(27)

By definition of o§, (cf. Sec. 1.1), there exist | > vn \ 0 satisfying a^ = lim a/-uo v So, given E > 0, there is no > 1 such that n—too * \£Vn> fy~^(v ) —-T-—^
n>n0.

(28)

If kn is the integral part of ^-, then ^-q^- < 2vn < -£-, and since ^—^ /* oo as i; \ 0, we get from (28):

100

III. Normal Structure and WCS Coefficients

Let

so that cn —>• 0 and dn —> 0 as n —>• oo. Choose HI > n0 such that cn HI. Letting &o = kni, CQ = cni and do = dni, we define /$, ? > 1 as /i = (Z0, • • • , ^o, ^o, ^o, • • •)>

^o in ith place,

where XQ — (do, dQ, • • • , dQ), ZQ = (0, 0, • • • , 0) are both k0— vectors. Then fi e m$, ||/i||$ = dok0V-l(±) = 1. Since (m $ )* = /<*) and im < sup -p*(£fi)

= k0 im ——5- = 0,

fi->Q weakly by Theorem 1.2.15. But by (29), for i

- +

Hence ^({/i}) < 2(aJ +e)(l +e), proving (27). Next we assert ^^^(m*) < 2<.

(30)

For any integer k > 1, let Zk = (0, 0, • - - , 0) and Xk = k^-l()

(1,1,-", 1)

be &— vectors, and define fi = (Zk, • • • , Zk, Xk, Zk, Zk, • • •),

Xk in ith position.

3.3 WCS coefficients of Orlicz Sequence Spaces.

101

Then fi e m$, ||/i||$ = 1, i > 1 and fa -> 0 weakly, since lim

t^o

A- I" i)! = l i m ^

£ \

1

-0.

Hence = lim sup{||/i-

which implies (30), since A; is arbitrary. Finally, the right half of (21) follows from (27) and (30). D Similar to Corollary 3, one has the Corollary 3'. For an N— function $, we have:

(a) if® (£ A 2 (0), then WCS(l*) = WCS(m*} = 1; (b) if$e A 2 (0), then WCS(l*) > 1. Proof, (a) If $ ^ A 2 (0), then /3g = 1 by Theorem 1.1.7. The right side of (21) and Theorem 1.1.15 imply 1 < WCS(l*) < WCS(m*} < 2<4 = -i- = 1, P$ which proves (a). (b) If $ 6 A 2 (0), then /* = m$ and ^g < 1. Since |^Jy < 1 for every w0 e (0, i$(2^[*-1(l)])], we see that ^| < 1, and the left side of (21) implies 1 < WCS(l*}. D For further use, we have the following also from Yan [1]. It can be deduced from a general result given in Lemma 5.2.15 later. Lemma 12. Let $(w) = /Ju' (t)dt, \&(i>) = /J"' tl>(s}ds be a pair of complementary N— functions and define ^) =

-

M>0.

(3!)

Then F$(t) is increasing (decreasing) on (0, ^[^"^(l)]] if and only if Fy(s} is decreasing (increasing) on (0, vJ/"1^)].

102

III. Normal Structure and WCS

Coefficients

The analog of Theorem 5 is as follows: Theorem 5'. For an N— function <3> € A 2 (0), with ^ being its complementary function, we have: (i) ifF^(t) = f - is increasing on (0, 2?/;[*-1(l)]], then WCS(,*) <

;

(32,

(ii) if F$(t) is decreasing on (0, 2'0[^~1(1)]], then (33)

where C$ = limF$(i). Proof, (i) By Lemma 4, the hypothesis implies that the function G$(u) = $-i(2u) *s a^so mcreasing on (0? 2^(^[^ r ~ 1 (-'-)])]' smce 2V ; [^ r ^ 1 (l)] — ^>"1{<^) >. Hence by definition of (3$ in (22), one has (34)

On the other hand, it follows from Lemma 12 that F\f,(s) decreasing on (0, ^"^l)], which implies 2^~ 1 f i ) min(2al, 2oi) = 2c4 - ^_ ^.

= ^$is v(S)

(35)

Hence (32) follows from (34), (35) and Theorem 1', since I* = m* if $ € A 2 (0). (ii) The hypothesis implies that C$ exists, and by Lemma 4, the function °M = | is decreasing on (0, i$(2^[*- 1 (l)])]- Hence (36)

But 1a%l3l - 1 (cf. Theorem 1.1.15). From this and (36), we get -

= 24 = 2^,

(37)

3.3 WCS coefficients of Orlicz Sequence Spaces.

103

which proves (33) by Theorem 1'. D Example 6'. Let $,# be as in Example 6. So -0(s) = log(l + s),s > 0, tf-^l) = e - 1 and ^-l(l}] = 1. By Theorem 5' we have 1.28299 «

z

2

2^>~ 1 (M < WCS(l*) < « 1.34498

(38)

and J*) = \/2.

(39)

Problem 13. Find the value of WCS (I®), where the iV— function $ is as in Example 6'. Applying Theorem 5'(ii), we have the following Example 7'. Let $p be as in Example 7, then

) = 2?.

(40)

Example 9'. Let p)r be as in Example 9, then WCS(l*p>T) =

p+

\/2.

(41)

Example 10'. Let Mp be as in Example 10, then

} = 2p.

(42)

Remark 14. For the coefficient D(X) of X (cf. Theorem 1.11) defined by Maluta, we have D(/P) = 2~p (Example 2), _ill and D(/W) = D(/*) =

=

(Examples 6 and 6'),

(Examples 7 and 7'), (Examples), D(l(*"
(Examples 9 and 9'),

104

III. Normal Structure and WCS ) = 2~p

Coefficients

(Examples 10 and 10')

and

£>(/(*)) = 2-1

(Example 11).

These examples supplement the work in Cui, Hudzik and Zhu[l]. Here is a brief application of these results to some fixed point theorems on sequence spaces. Corollary 15. Consider an N— function $ G A 2 (0). Then every nonexpansive mapping T from a weakly sequentially compact convex subset C of /($)(/$) inj.Q g nas a fixea point. Proof. This follows from Corollary 3(3') and Theorem 1.9. D In what follows, we present the WCS values for a class of intermediate functions &s considered several times before, without using the interpolation theorem. Theorem 16. Let $ be an N— function, QQ(U) = u2 and $s be the usual intermediate function, i.e., the inverse of 0<s0.

(43)

Then we have 2f < J- < WCS(l^} < mm (-L, -L] P$s \P*. P*. /

(44)

^ < ^r < WCS(l**} < min(2<s+ , 2a'v+) P$3 "

(45)

and

where ty+ is the complementary function to $s. In particular, if 3> £ A 2 (0), then = 2f = WCS(l*s] = [D(l*')]~l.

(46)

Proof. Since <E>5 G A 2 nV2 for any $, regarding (44) and (45), from Theorems 1 and 1' we only need to show .,/ft ) < 2 ~ f .

(47)

105

3.3 WCS coefficients of Orlicz Sequence Spaces. Indeed, (43) implies that for u > 0

(48)

which proves (47) and hence (44) and (45) also. Now if $ g A 2 (0), then /3g = 1 and by (48)

From Theorem 1.1.15 and the above, we get -

n

1

(49)

= 2*.

Finally, (46) follows from (44), (45), (49) and Theorem 1.11. D Example 17.

Consider the iV—function Q, which is the inverse of

e2 < u < oo

(§)*(«)',

with

= 0. Then we assert that -1 = WCS(l(Q}] = $2 = WCS(1Q] = [D(1Q]}-1.

(50)

Indeed this Q can be verified as the intermediate function <E>S for s = | where 0< I 4p TO ^

11Uu

,

— ?/Lt I <" Q <* ^ ^^ r*~i ^f^J,

with $(0) = 0, (see. e.g.,Kaminska [2], p.304). Then Cg = lim^f^ = oo and so $ 0 A 2 (0). The assertion (50) follows from Theorem 16. We now specialize Theorem 16 for $ € A 2 (0). In what follows we deal with the gauge norm and omit the discussion on the Orlicz norm. Theorem 18.

For $ € A 2 (0),$ 0 (w) = u2 let 3>~l be as in Theorem 16.

106

III. Normal Structure and WCS

Coefficients

We have the following assertions: t) = |^| is increasing on (0, ^"^(l)], then 1-s

(51)

= 22

is decreasing on (0, $~ 1 (1)], then l = 2 f+ ^,

(52)

where Cg = limF$(t). Proo/. Since $s e A 2 (0) n V 2 (0), the left sides of both (51) and (52) are true by Theorem 1.11. (i) The hypothesis, Lemma 4 and (48), imply that both G$(u) = $-1$) and Gs>3 (u) — <J>,Ii,\\ are increasing on (0, ^1. From Theorem 5(i) one has ^ ' o \ i 2J \ ' S (2u) ,

^'(l)

proving (51).

__ i_ (ii) The given condition implies that C$ exists, /?^ = 2 c* and that G$ 3 (u) is decreasing on (0, |], so that

which proves (52) by (44) in Theorem 16. n We conclude this section with the following: Example 19. Let $ r (u) = e\u\r - 1, 1 < r < oo. Then [log(l 4- u)]r and

Since $r e A 2 (0) n V 2 (0) and the function F$r(i) = ^ (0,oo), it follows from Theorem 18(i) that

is increasing on

3.4 More on Orlicz sequence spaces

107

We now analyze the sequence spaces in more detail. 3.4 More on Orlicz sequence spaces We consider relationships between the coefficients N(l^), WCS(l^} and £>(/<*>) as well as those between N(l*}, WCS(l*} and £>(/*). In view of Theorems 3.1 and 3.1' and the discussion in Section 1, we have immediately the following conclusions: (A) If $ ^ A 2 (0), so that l^ and I® are not separable (and not reflexive), then £>(/(*)) = N(lW) = WCS(l(*}} = 1 = WCS(l*) = N(l*) = D(l*). (B) If $ 6 A 2 (0)\V 2 (0), so l^ and J* are separable but not reflexive, then (with the notations of Theorems 3.1 and 3.1') min ~

<2

and

= l < < WCS(l*) < min 2aJ, 2^ < 2. (C) If $ e A 2 (0)nV(0), so that 1^ and /* are reflexive (and separable), then by Theorems 1.12, 1.8 and 1.11 one has: 2,

(1)

and

< WCS(l*} = -

< 2.

(2)

It may be noted that the first two inequalities in both (1) and (2) between N(-) and WCS(-) are trivial in view of Theorem 1.8. However, it is not easy to find general reflexive Banach spaces for which these inequalities are strict. We now present a class of reflexive Orlicz sequence spaces for which this property is verified. We first establish the following:

108

III. Normal Structure and WCS

Theorem 1. we have

Coefficients

Let (Q,^) be a pair of complementary N— functions. Then < min (2^-0- 1 ,

(3)

<min(-^,2c4J,

(4)

and further m&x{N(lW),N(l*)}

< min (2<4,-^-) .

(5)

Proof. First consider the bound with 2a$ in (3). By definition, given 0 < £ < 1, there is an n0 > 1/e and u0(= M 0 (e,n 0 )) > 0 such that uQ2n° < s and : <4 + e.

(6)

Let A; be the integral part of (u 0 2 n °)~ 1 , and choose £ > 0 such that ku02n° + $(t) = 1. Fixing this A; and t > 0, let /j = (1, • • • , 1) be a vector of 1's of size A;2 n °~ z , and define a set of elements of l^ as: ^ = (0,4-4 ••',/;,-40, ( ) , • • • ) ,

where the number of non-zero blocks for /j is 2l (each of size k2n°~'1 and the blocks are alternating in signs ), i = l , 2 , - - - , n o . Let fa be defined as fa = tei+$~1(uQ)zi, where e,\ = (1,0,0, • • •) is the first unit coordinate vector. Since p$(fa) = 1, i < n0, we have /i||($) = 1 and Ano - {fa, 1 < i < nQ} is a set of unit vectors. By (6) and the definition of A; we get for i ^ j

H f . _ r I,

=

**-lM < w-lM

2(

o+£)_

Hence d(^no)<2(aS>+5).

(7)

But if / - (a1; 02, • • •) £ / W and g = (blt b2l • • -) e /*, then {/, 0) = g 0^64 »=i is well-defined. Let & = ***,_"9i^OJ^, 1 < z < n0 and note that -—^ < -—^ t/1 ^2

3.4 More on Orlicz sequence spaces

109

for Vi > v-2 > 0,

by definition of A;. With /; defined earlier, we get (fi,gj) — 0 for i ^ j and (fi,9i) = u0k2no > 1 - u02no >!-£. Observing that ||/||($) = sup{|{/,0)| : ~

no

ID

IMI* < 1}) we find for / = £) Cjfj, £ c3•, = 1 and GJ > 0, > (fi - f , 9 i ) = (!-

Consequently, r(A no ) > (1 - e) max (1 - a : 1 < i < n 0 ) Thus (7) and (8) give

This establishes the first inequality of (3) since e is arbitrary. Regarding the second part of (3), if l^ is nonreflexive, then N(l^) = 1 (cf. Theorem 1.2(i)) and since /3$ < 1, the desired inequality is true. If /^ is reflexive, so l^ = m^\ by Theorems 1.8 and 3.1 one has < WCSl

<

This and the preceding inequality imply (3). For (4), we proceed similarly. Consider its first half. By definition of /3§, given 0 < £ < 1/2, there is a sequence Vj \ 0 such that ^i^l > /?£ — e for j > jo > 1- Fix n > l/e and choose v G {vj : j > jQ} such that v2n < e. If k is the integral part of (v2n)~l, then k
Let

a - k1n^-1 (-?-} andJ/
110

III. Normal Structure and WCS

Coefficients

where zi, • • • , zn are as in the first part of the proof of (3). Then ||/j||$ = 1 and if i a

\

n

~ /

av

Hence if Bn — {/j : 1 < i < n] we get

Let #z = ^"^w)^, 1 < z < n, so that p
a

av

av

(11)

00

Again if / is a convex combination, / = Z c^/j, then by (11) j=i ll/i - 711* > {/i - 7,^) = (1 - *) > (1 - e)(l - Ci)^^and

^"Hf) / 1\ $~l(v] r(Bn) > ^-^(1 - e) (l - -) > (1 - ,)2^-^. av \ nj av

(12)

It follows from (9), (10) and (12) that

d(Bn) r(Bn]

V-i(2v)

1

(1 -

Since £ is arbitrary, this proves the first half of (4). However, if /* is nonreflexive, then N(l®) = 1 and since 1 < 2o;^, the second part of (4) holds in this case. If /* is reflexive, then N(l*) < WCS(l*} < 1a% by Theorem 1.8 and 3.1'. Thus (4) holds in all cases. Finally, (5) follows from (3) and (4) since 2a|/5° = 2<4/3g - 1 (cf. Theorem 1.1.15). D Example 2. (Dominguez Benavides[l]) We have 1< p < oo.

(13)

3.4 More on Orlicz sequence spaces

111

Indeed, by Remark 2.9, it suffices to verify that N ( l p ) does not exceed the right side of (13). But this is immediate from (3) since for $ P (M) = \u\p, lp = We now present a class of reflexive Orlicz spaces l^ for which WCS(l^}, and give examples. Theorem 3. For an N- function $ € A 2 (0) n V 2 (0), if 2<*J < i, P$

(14)

where

as in Theorem 3.1, then we have 7V(/ W ) < WCS(l(V).

(15)

In particular, (15) holds if C^ = l i m f - exists and

where : 0 < t <*-'(!).

(17)

Proof. Since /3$ < fl® for every TV— function <$, the hypothesis (14), (3) and Theorem 3.1 imply mn

so that (15) holds. In particular, if C$ exists, then by Corollary 1.1.9, a$ = _ i 0$ — 7$ ~ 2 c* and 1 — ^ o - < - ^ - < A - b y (16). Hence we get N(l(9)) < min 2

<%,2

=2

< 2 .

(18)

112

III. Normal Structure and WCS

Coefficients

To complete the proof it suffices to show that v (19) >

2^ —< J-.

By (17), f

< B* for 0 < t < ^~l(l).

ttQ
^~l(l), we have

Letting ^ = $~l(u) and *2 = $ -1 (2w) for 0 < u < 1/2, we get 0 < ti < t2 < S-Hl) and __^(ta) / — —•- r

or

which proves (19). Finally, (15) follows from (18) and (19). D Example 4. Let $ r (w) = e |u|r - 1. If 1 < r < 1 + C, where

0

[log(log2)

~ log(los3 ~ log2 )^ « °-7736'

then < WCS(l(*r)).

(20)

In fact, $r 6 A 2 (0) n V(0) (cf. Example 3.8), a^r = 2~r and _ _ *' ~ $-*(!) ~ (

log 2

'

Hence 2a^r < (P*,)'1 for 1 < r < 1 + C, and (20) follows from Theorem 3. Example 5.

Consider the N— function defined by:

3.4 More on Orlicz sequence spaces

113

For this function, W ( J * ) < To see this, note that Cg = lim^fj^ = f, so that $ 6 A 2 (0) n V 2 (0) by Theorem 1.1.2 and a$ = /3$ = 2~3. Hence from Theorem 1 and Example 3.11 we get

1 < N(l&) < 2o& = 23 < 43 = WCS(lw).

We also present a class of spaces /* for which N(l*) < WCS(l*) holds, as follows. Theorem 6. For an N- function $ e A 2 (0) n V 2 (0), if

24 < -L p$

(21)

where

& =sup

: <

° "-

as in Theorem 3.1', then we have < WCS(l*).

(22)

In particular, (22) holds if C$ = lim t-^r exists and

where B$ = sup <

: 0 < t < 2'0[\I>~1(1)] I .

(24)

Proo/. Since /3g < ^|, (22) follows from (5), (21) and Theorem 3.1': N(l*) < min 24, = 24 < - < \ p$ / p$

In particular, if C£ exists, then oj = ^g = 2~^* and by (23)

(25)

114

III. Normal Structure and WCS

Coefficients

Similar to (19), by definition of B$ in (24), we can verify 2^ < J_.

(26)

P$

Hence from (25) and (26) it results that N(l*) < mm (V~%, 2%) = 2^% < 2^ < J- < WCS(l*), \ J P$ as desired. D Example 7. Let $ p) r(w) = |w| p log r (l + |w|), 1 < p < oo and r > 0, as in Examples 3.9 and 3.9'. If p + r < 2, then C| r = p + r < 2 so it follows from Theorems 3 and 6 that max

It is nontrivial to find exact values of TV(-) for l^ or ^$ in general. Here we present such values for a class of intermediate functions <$,, considered many times before. In what follows, a < {b, c] stands for the two inequalities a < b and a < c. Theorem 8. Let $ be an N— function, $Q(U) = u2 and $s be the usual intermediate function, i.e., the inverse of

0 < s < l,u > 0.

(27)

Then we have 2* < 7V(/^)),7V(^) < min (tel.,--

.

(28)

In particular, if 3> £ A 2 (0) n V 2 (0) we get = 2! =N(l*').

(29)

Proo/. Since $5 e A 2 n V 2 , 0 < s < 1 (cf. Lemma 1.4.7), (28) follows from Theorem 2.7 (with // as counting measure) and Theorem 1. Now if $ g A 2 (0) H V 2 (0), then /?£ = 1 or a^ = 1/2. Since

3.4 More on Orlicz sequence spaces

115

we have -jp = 2* < 2<43 or 2alt = 2* < -^-. Hence if $ 0 A 2 (0) n V 2 (0) we find mm

^ o M ( S '^J =

which proves (29) by (28). D We illustrate Theorem 8 by the following: Example 9. Let $ r (u) = e\u\r - 1, 1 < r < oo. Then $r e A 2 (0) n V 2 (0) and Since by (28) we get

{

21-f-171, l < r < 2 2f +i T^,

2 < r < oo.

It should be noted that limM/(^)) = r—>1 limWf/*') = 2* = r—>oo lim N(l(^} = r—^cxo lim M/*').

r—>1

Example 10.

Let Q be the inverse (A 7 "—function) of

{

(—logu) 2 w? ;

0 < u < e~ 2

x

P c / ^ ^i (- *w?u7// yl 2^ j

P c.

^ _\ ?/ u/ <^" ^s. rv^i *^\^

with Q-1(0) = 0. Then we assert that = 24 = 7V(/S).

(30)

116

III. Normal Structure and WCS Coefficients

Indeed, one verifies that Q is the intermediate function $s at s — |, where $ is given as the inverse of [$~1(0) = 0 and] — wlogw,

0 < u <e u < oo.

Moreover, one finds

so that $ g V 2 (0) by Theorem 1.1.7. Since Q = [$s]a=i, we get (30) from Theorem 8. Bibliographical notes. Wang and Shi [1], and Chen and Sun [1] studied uniformly normal structures of Orlicz spaces for the Orlicz and gauge norms respectively. Inequality (12) in Theorem 2.3 is proved by T, F. Wang (communicated to Ren). The WCS coefficients of Orlicz sequence spaces satisfying A2(0)— condition were considered by Dominguez Benavides and Rodriguez[l]. Most of the results with gauge norm are taken from Ren ([2], [4]) and Yan [1]. The corresponding results for the Orlicz norm have not appeared in print.

Chapter IV Jung Constants of Orlicz Spaces

In studying the geometric properties of Banach spaces, W. Jung [1] introduced in 1901 certain invariants, for Banach spaces, with values in [|, 1], and calculated their values for finite dimensional Euclidean spaces. Later several authors found the exact values for certain Lf and other spaces. In fact some workers in the subject think that "estimation of Jung constants is one of the directions of research of the geometric theory of normed spaces". This chapter, like the preceding ones, is devoted to results on Jung constants for Orlicz spaces under both the Orlicz and gauge norms. Also for a class of intermediate Orlicz spaces exact values are calculated, and for general Orlicz spaces lower and upper estimates are obtained.

4.1 Introduction to Jung constants This parameter of a Banach space X is defined in terms of the ratio of the Chebyshev radius and the diameter of a set in X. Recall that for a set A c X, the diameter of A is d(A] = sup{ \\x — y\\:x,y£A} and if A, B C X are sets then the relative Chebyshev radius r(A, B) = inf{sup[||rr — z\\ : x e A] : z 6 B}, and the (absolute) Chebyshev radius of A is r(A, X}. By definition r(A,B) — r(cd(A),B] and r ( A , X ] = r ( c o ( A ) , X ) , where co(A) is the closed convex hull of A. The following concept for a normed linear space was introduced by Jung [1] in 1901 and it was termed by Griinbaum [1], the 'Jung constant': Definition 1. The Jung constant of a normed linear space X, JC(X) is defined as: JC(X) = sup I r „,, ( d(A)

: A C X, bounded, d(A) > 01. }

(1)

The definition implies \ < JC(X] < 1, and in some works (cf., e.g., Amir

117

118

IV Jung Constants of Orlicz Spaces

[1], Franchetti and Semenov [1]) 2JC(X) is termed the Jung constant. Jung [1] showed that

where I2n is the Euclidean n-space. He also showed for an arbitrary ndimensional normed linear space Xn, JC(Xn] = \ iffXn = /£°. Also Bohnenblust [1] established that JC(Xn) < ^~ for any n-dimensional normed linear space Xn. For some infinite dimensional spaces the following results are available: JC(l2} = -± (Routledge [1]), JC(L°°[0, 1]) = JC(l°°) = \ and JC(Ll[Q,l]) = JC(l1} = JC(cQ) = JC(C(T}} = 1 where T is a compact Hausdorff space, without isolated points (cf., Berdyshev [1], Papini [1] and Amir [1]). Davis [1] showed that JC(X) = | iff X is a B\— space, i.e., a space which is norm one complemented in every super space (cf., Franchetti [1]). Obtaining lower and upper bounds respectively by Berdyshev [1] and Pichugov [1], the following result was deduced: JC(L p [-7r,7rj) = m a x ( 2 ^ ~ 1 , 2 " p ) , 1 < p < oo.

(3)

For LP(0,) with Jl — [0, 1] or R+, I < p < oo, (3) will be reproved by using Orlicz space results in Section 3. Indeed computing exact values JC(X] for different normed linear spaces X is a nontrivial task. We start with the following general result: Proposition 2 (Amir [1]). If X — Y* , the dual space of a normed linear space Y, then JC(X] = sup

f :AcX,

finite, d(A) > O .

(4)

Proof. Let A C X be a bounded set with d(A) = 1. Then the Chebyshev radius r(A, X] < d(A] = 1 and so for each 0 < a < r(A, X] the closed balls {B(x,a),x e A] cover A where B(x,a] = {y e X : \\y — x\\ < a} and at least a pair of balls must be disjoint. In particular f| B(x, a) — 0. But x&A

by the Alaoglu-Birkhoff-Kakutani theorem each B ( x , a ] is weak*-compact (since X = Y*} and hence there is a finite subset A\ = {x±, • • • ,xn} C A such that f| B(XI,O) = 0. This means each r ( A , X ] can be approximated x^Ai

by r(^4i, X] since a, (0 < a < r(A, X}} is arbitrary. So using this in (1), one gets (4). D

4.1 Introduction to Jung constants

119

A few more general results on Jung's constants will be included in this section and use them to improve their bounds for Orlicz spaces in the rest of this chapter. The following result is also from Amir [1]: Theorem 3. Let X be a dual space and Xa C X, a G D, be a net of linear manifolds directed ('< ') by inclusion. If each Xa is the range of a norm 1 linear projection Pa and \J Xa = X , then a&D

JC(X] = sup JC(XQ) = lim JC(Xa). a<ED

(5)

"€£>

Proof. If P : X —> Y is a projection with range F, \\P\\ = 1, then for A C y, x e X we have r(A, Px] < r(A,x) and taking the infimum over X, one gets r(A, Y) < r(A, X } . There is equality here since A C Y and the 'inf is already attained in Y. From (1), this yields JC(Y) < J C ( X ] , and hence JC(Xa] < JC(Xp) < JC(X] < 1 for all a < ft in D. Hence by Proposition 2, given e > 0, one can find a convex set A generated by a finite number of Xi G X,\ < i
Since U-^a is dense in X, we can find (by directedness of D) an a in D and x\ £ Xa such that \\Xi — x'j\\ < e, 1 < i < n. If A' = co{x'i : I < i < n}, then A' C Xa and

r(A',Xa) d(A')

r(A,X}-E d(A) + e '

(

'

Since e > 0 is arbitrary, from (6) and (7) we get immediately (5):

as asserted. D The above result is useful in obtaining lower bounds for the Jung constants of Orlicz spaces. To obtain the corresponding upper bounds, the following result, a slightly restricted version due to Pichugov [1], will be useful. Theorem 4. If X is a separable rotund dual Banach space, and {xi,i > 1} is a dense denumerable set, let Xn = sp{xi, 1 < i < n}. Then JC(X] < lim inf JC(Xn).

(8)

120

IV Jung Constants of Orlicz Spaces

Proof. Let A C X be a bounded set and Xn be as given. Since Xn is rotund, for each x G A, there is a unique y G Xn closest to a: and let Pn : x —> y be the correspondence. Let An = Pn(A) and Pn is a (nonlinear) mapping. Given E > 0, there is a zn 6 A^n such that (by definition of r(A, X}} \\zn-x\\
Vrrein.

(9)

Then {zn,n > 1} is a bounded set in X(— Y*} so that it is weak* -compact. By extracting a subsequence we get znk —>• ZQ 6 X in the weak* -topology. Hence \\y - z0\\ < lim inf||7/ - znk\\, y 6 A. (10) n

k

too

Since X is also separable, for each y G A, an integer no can be found satisfying y <E j4 nfc , nfc > n 0 . Then for nfc > n0,

< 25 + JC(Xnk)d(Ank) < 2e + JC(Xnk}[d(A} + 2e}.

(11)

Hence (8) is a consequence of (10) and (11). D The following result is also from Pichugov [1] which gives some useful properties of the Chebyshev center and radius. Theorem 5. Let Xn be an n- dimensional Banach space, A C Xn be a compact convex set with r(A,Xn) as its Chebyshev radius. Then a point 2/o ^ Xn is the Chebyshev center of A iff there is an HQ < n + 1 such that there are x,- e A, /< € X*, ||/i|| = 1, and (x{ - y 0 ,/;} = ||a;» - 2/o|| and no

Ci > 0, "52 Ci = 1, X] Q/i = 0, /or which \\Xi — yo\\ = r(A, Xn), 1=1 1=1 Moreover, when these conditions hold, we have n)

1=1 j=i

d /or eac/i fixed 1 < A < CXD; t/ie number no

A=

CiC-ar. -

i = 1,

(12)

4.1 Introduction to Jung constants

121

satisfies

MI A• jj\\

_ _ n+1'

(13)

The proof of this result uses some properties of the subdifferential of the convex functional y \—> F(y) = max{||rr—t/||,y G Xn} and then (12) and (13) are deduced with a straight forward computation, via Jensen's inequality. We omit the detail, and refer the reader to Pichugov's [1] paper. In some cases, the subset An+\ = co{xi, l
H -H

is a similar one of order 2(n +1). It is suspected, but not yet established, that there exists Hadamard matrix of order n + l i f n + l = 4m for some integer ra. Such matrices have been constructed for many values of ra, in particular for 1 < m < 67 (cf., e.g., Golomb and Baumert [1]). If n+ 1 = 2k, then JFf( n+1 ) X ( n+1 ) exists for all k > 2. An Hadamard matrix #(n+1)x(n+1) is said to be in normal form if its first row and column consist of Is. We denote by -fir nx (n+i) by removing the first row of #( n +i)x(n+i)- The following example #4 X4 illustrates its use for Jung's constants. Example 7. Consider an JV-function 3> e A 2 (oo),f2 — [0,1] and ^ is Lebesgue measure. If n + 1 = 4, then

1 1 1 1 1 1-1-1 1 - 1 1-1 1-1-1 1

1 #3x4 —

1 - 1 - 1

1 - 1

1 - 1

1 - 1 - 1

1

For fixed u > 1, divide the subinterval [0, £] C [0,1] into G; = [^,

122

IV Jung Constants of Orlicz Spaces

1, 2, 3. Let \Gl be the indicator function of G{ and let a — $~l(u}. If (/I, /2, /3, .A) =ft(X'G,, XG,, XG 3 )#3x4,

so that

and /i(U d] = £ and Mi (^ = 1 (±.\ 1=1

\'

/

'-

V

one

gets ||/,||(ci,) = 1, and for

/

i ^ j , l
_

^

Since $ € A 2 (oo), (Z,W[0,1])* = ^*[0,1] where * is the complementary 4

A^-function of <£, we have with ()i — ^fl,cl = \,Y. Q,9i — 0, ||.9i||* = 1, and i=l

(/i ~ 0, ^j) =

T1 / fiQidfJ,

=

By Theorem 5, the set ,44 = co{/z, 1 < i < 4} has '0' as its Chebyshev center in X3 = sp{XG,, 1 < i < 3} C L™ [o, £] . Thus r(A 4 , X3) = 1, and

In general, if n + 1 = 2 m ,?r?, > 2, for w > 1 we divide [0, -] into (7^ =

with ^ = ^j/i,Cj = ^-, and a = ^"^u) in the above construction; An+i c °{/n 1 < i < ^ + 1} will have '0' as its Chebyshev center in Xn = s l
d(An +l

4.1 Introduction to Jung constants

123

This construction, and the result, will be used in the next section. The following lemma employs a similar argument and is recorded here for reference. It is due to Ivanov and Pichugov [1]. n

Lemma 8. Let Xn be the space of elements f = Y^ f ( i ) e i with \e^ 1 < i < n} i=\ n

as basis unit vectors and X£ of elements g = Y^ g(i}(!>i with {;, 1 < i < n} 1=1 as biorthogonal basis and for which both Xn and X* respectively use the l\ and I™-norms. Let An+i = co{fi, 1 < i < n+ 1} and ( / ! ) • • • > /n+i) = (GI, • • • , en)Hnx(n+i),

where ^( n +i)x(n+i) 2S a normalized Hadamard matrix (if it exists). Then the Chebyshev center of An+\ is the origin '0', and \ fi\\ = r(An+i,Xn) for all I < i < n + 1, ||/i - fj\\ = d(An+1),i + j. Proof. As in the preceding example, we verify the conditions of Theorem 5. Thus one only has to note that in each row of the Hadamard matrix (except the first removed one) there are as many 1's as there are — 1's, and 80 c i = ^+1 for all 1 < i < n + 1. n We also include the following result for later use. Lemma 9. Let A be a nonempty set in the closed unit ball of a Banach space X. If'there is a ZQ G X,r(A,zo) < 2, then \\ZQ\\ < 3. Proof. If ||z0|| > 3, then one has r(A, zQ) = sup{||x - z0\\ : x e A} > sup{||,2o|| - \\x\\ : x e A} > 2, a contradiction. So ||2;o|| < 3 holds. D We finally include Remark 10. Recall that Maluta [1] defined a constant N(X] for a Banach space X as:

f r(A A) } N ( X ) = sup < ,' : A C X bounded closed convex, d(A) > 0 > . (14) I d(A) J We term N ( X ) the self-Jung constant of X [2N(X) is called by that name in Amir [1], p.4]. In view of the concept of normed structure coefficient N ( X ) (cf. Remark 3.1.4), we have N(X] = [N(X)]-1. By a known result of Garkavi and Klee (cf. also Klee [1]), for any bounded closed convex A C

124

IV Jung Constants of Orlicz Spaces

X,r(A,X] = r(A,A) iff X is an inner product space or dim(A') < 2. Since r(A, X] < r(A, A), it follows that for every Banach space X,

- < JC(X] < N(X] < 1.

(15)

Zj

This inequality will be used in the following sections. 4.2 Lower Bounds of JC(L^) and JC(L*) We start with Orlicz spaces with the gauge norm. Theorem 1. Let
(1)

and if also <£ G A2(oo), then -<JC(L<*>[0,1]).

(2)

Proof. By definition of /#$ (cf. Sec. 1.1), there exists a sequence 2 < ^ /* oo such that lim =/?». fc^oo$-1(2i;fc) ^

(3) v '

But for any 0 < e < |, there is a v0 G {vk '• k > 1} satisfying /?.-, and *-i( 2wb ) > 5*^1. £

(4)

Choose an integer n0 > 1 such that 2v0 — 1 < n0 < 2v0, s o l < ^ < l + ^ ri-i 2, and put ^ = [, ), /, = 3>-l(2v0)XEi, 1 < i < n0. Then \-2vo and by (4) for i \\fi ~ fj\\(*) = If >1 = {fi : 1 < i < n0}, then by (5) d(A) < j^-£. Let r0 = r(A, L^[0, 1]). Then there is a 0 € L(*)[0, 1], such that ll/z ~ 0||(#) < r(A, g) < r0 + ,

1 < i < n0.

(6)

4.2 Lower bounds ofJC(L^)

and JC(L*}.

125

Our aim is to find an estimate to r0 so that ^y bounds JC(L*[0, 1]) from no

below. Let E = VEi = [0, ^) C [0, 1], and 9l = gXE. Then \ft -

9l\(t)

=

|/i - 9\XE < \fi ~ 9\ in [0, 1], and hence r(A,gi)
(7)

Now define Fj = {t e Ei : gi(t) < $~l(2v0)} and consider no

92 =

diXF +

Then ^2 < ( ^ > ~ 1 (2^o) and |/j - ^|(t) < \ft - 9l\(t), t <E E for 1 < i < n0, so that r(A,00"1(2^o), |/< - 9z\(t) < \fi - 92\(t), and

r(A,93}
(9)

no

Finally, let gQ = gfyxe,-, where 6,, = ^y fE. g^. Then for any RI > \\fi — #3 1|($)> with Jensen's inequality, by definition of the gauge norm we get

1

>

fi ~9o

126

IV Jung Constants of Orlicz Spaces

So \\fi - 00||(*) < Ri and \fl - 00||(*) < ||/t -tfallc*),1 < * < n0. Thus r(A, <7o) < r ( j 4, ^3)- Since TO < r(^4, 0) = 1, the inequalities (6) -(9) imply r(A,9o) < r 0 + | < 2 .

(10)

If we let 0 < \i = $-iLv x < 1, Aio = min{A; : 1 < i < n0}, then

1=1

By Lemma 1.9 and the choice of VQ above, we get > « \\n( II > A, \ o > *)>

- x

—,

V "o /

so that Aio < §. Thus by (10), r

o+2 >

r

(^'^o) =max{||/i -^o||(*) : 1 < i < n0} > \\(fio -

= ll(l-Ai 0 )/ i o || ( *) = l - A i o > l - | . Then ro > 1 — £ and from Definition 1.1 we have

which establishes (1) since e > 0 is arbitrary. Now we prove (2). Since $ 6 A 2 (oo), L^[0, 1] is a separable dual (of M*[0, 1]) space. By definition of a$, we can find 1 < Uk /* oo satisfying lim ^-1/9 u ^ = Q!$ and then, for given £ > 0, there is UQ € {u^, k > 1} such ;->oo * (* k) A;->oo that

Let D be the set of integers for which #(n+i)x(n+i) exists, so that D D {n+1 = 2 m , m > 2} implying that D is infinite. If n + 1 G D, divide [0, ^] C [0, 1] into n-parts: d = fo, -^-) , G2 = [LJnu -, -^-), • • •,' Gnn = fI2nuo ^ 1 , ' -V and let ^ L i"o / o ' nuo / ' uo / ' ^n = sp{XGk :l
4.2 Lower bounds ofJC(L^) if HI < ni+i, and

U n+l&D

and JC(L*).

127

Xn is dense in L^[0, U— ]. Consider the mapping °

Pn--9*—> g bjXGj, bj = ^y IG. gdfji. Then Pn on £,(*) [o, ^] is linear, and by Jensen's inequality for ||^||($) = 1,

so ||Pn|| < 1. Actually equality holds since g — X\QI ' JLIJ =>• PnQ = 9- Hence by Theorem 1.3, we deduce that for n + 1 G D UQ

-}}> JC(Xn). woJ/

(12)

To show (a^)"1 bounds the right side of (12) from below, let n0 + 1 G -D be large enough so that ^ < e. Deleting the first row of the Hadamard matrix to have # no x(n 0 +i)> define Ano+1 = co{fi : 1 < i < n0 + 1} C Xno, where (/I, ' ' ' , /n 0 +l) = ^~ 1 (wo)(XGi, ' ' ' , XG n o )^n 0 x(n 0 +l)-

Then we find ||/i||($) = 1, 1 < i < n0 + 1, and with (11) for i ^ j

implying £).

(13)

The Chebyshev center of AIO+I is 0 G Xno (cf. Example 1.7), and so by (13)

It remains to show that in the left side of (12), we can replace [0, ^-] by the unit interval / = [0,1]. Let ys = L^[0,s] and Qs be defined as Qsg = 9X[o,s}, g € L($)[0,1]. Then Qsg e ya, \\QS\\ = 1 and since yai C y,2 for 0 < si < 52 < 1, U 34; is dense in £^[0,1]. By Theorem 1.3 again JC(L<*>[0,1]) > JC(ys), 0 < s < 1. In particular if s = ^, we get the

128

IV Jung Constants of Orlicz Spaces

desired assertion, and hence, with (12) and (14) we see that (2) holds. D It is noted that | = JC(l°°) < JC(cQ} — 1 where c0 is the separable sequence subspace of l°°, by Sec. 4.1. However, we show below that for an JV-function <E>, JC(L(*)[0, 1]) = JC(M^[0, 1]), for the separable subspace M(*)[0, 1], after observing the following: Corollary 2. For any N-function 3>, we have ),

(15)

]).

(16)

and if $ 6 A 2 (oo), then

Proof. oo

We consider (15). Let vn /* oo be a sequence satisfying (3) and

7

£ 2V < 1- Next choose a disjoint sequence of Borel sets G; C [0,1] such i=l

*

that n(Gi) = ^ and let fi = $~l(vi)XGi- Thus ||/i||(*) = *-$£) , and ll/t - /jll(*) = 1 for z ^ j. If {/j : i > 1} = A, then d(A) — 1. Since elements of M^[0, 1] have absolutely continuous norms and p(Gi) —>• 0 as i —>• oo, we get lim ||/oXGj|(*) = 0 for /0 e M^[0, 1}. Also, using (3) one has r(A/o) = sup{||/ -/o||(*):/ 6 A}

But /0 G M($)[0, 1] is arbitrary. Hence , 1]) = inf{r(A, /0) : /„ € M^fO, 1]} > and this implies (15) since d(A) — 1. If $ e A 2 (oo), then M^[Q, 1] = L<*>[0, 1] so that (2) implies (16). D Corollary 3. If $ & A2(oo) n V 2 (oo) ; then , 1]) = JC(MW[0, 1]) - 1.

(17)

4.2 Lower bounds ofJC(L^)

and JC(L*}.

129

On the other hand, for an N -function $, Z/*)[0, 1] is reflexive iff JC(Z,<*>[0, !])
(18)

and in general, ^ < JC(L^[Q, 1]). Proof. If $ <£ A2(oo) or $ € A2(oo)\V2(oo) so that 0$ = 1 or $ e A2(oo) but 0:$ = | (cf. Theorem 1.1.7), then (17) follows from Theorem 1 and Corollary 2. If L($)[0, 1] is reflexive, then by Theorem 3.1.12 (and Remark 1.10) we get ,/£(£<*> [0,1]) < W(L<*>[0,1]) < 1. On the other hand, this inequality implies by (17), G A2(oo) n V2(oo) so that Z/*)[0, 1] is reflexive. Finally, if the last inequality is not true, then both /3$ < 4- and 4- > 2^ or a$ > 4= > /5$, which can not happen since 0;$ < /?$ always, n We now state the analogs of the preceding results if the unit interval [0, 1] is replaced by R+. The proofs being quite similar (as seen in the preceding work), they will be omitted. Theorem 4. For an N -function $>, one has h < JC(L^(R+}),

(19)

and if$£ A 2j it is true that (20)

The corresponding Corollaries can be stated as follows: Corollary 5. For each N-function $, fa < JC(M^(R+}},

(21)

and if$ G A2, then (22)

Corollary 6. // $ g A2 n V2, then JC(L^(R+)) = JC(M^(R+)) = 1. On the other hand, ^ < JC(L^(R+}) = JC(M^(R+}) always, and LW(R+) is reflexive iffJC(L^(R+)) < 1. We shall now present the analogs for the Orlicz norm, and the details are included since the argument is somewhat different. Theorem 7. For an N-function $, with ^ as its complementary, one has 0,1]).

(23)

130

IV Jung Constants of Orlicz Spaces

If, moreover, $ G A2(oo); then /3y < JC(L

[0,1]).

(24)

Proof. As before, by definition of ay, we can find 2 < Vk /* oo such that lim WM-if?\*vk)\= a$i implying that for any e > 0, there is a VQ G {vk '• k > 1} fc-400 —• satisfying ~

T~T

^

/

-*-"

f\,

1

£~

T" *\ CCA]/ "T" C 5

Q f\ /"I

dilU.

1

,

,

*"«^

^

,

, .

I O Cx \

\£j\Jj

Consider the integer HQ such that 2^o — 1 < HQ < 2^o, or equivalently i <1 - — <^-
(26)

Let A = {fi : I < i < n0}, where fi = lTj Jy° .YE-l with Ei — Lf^-, ^- ), then ^ ^fo / ^^o ^^o / ||/i| $ = 1, and for i ^ j'by (25) we have

so that d(A) < 2(a;$ + e). If r0 = r(^,L*[0, 1]), and £• > 0 fixed above, we can find a g 6 L*[0, 1] satisfying max{||/i - ^||$ : 1 < i < n0} = r(A, g) < r0 + -. We need to find a lower estimate to r 0 . We proceed exactly as in Theorem no

r

\

l(see (6) and (7)). Thus, let9l - gXE, E - U ^ = [O, ^) C [0, 1]. Then r(A,gi) < r(A,g] and let #2 be defined as: -

~

9l

where Fx = {* G E, : Pl (t) < ^rf^y}- Thus £2 < ^^y and |/z |/» - 9 i \ ( t ) , t£E,so that r(A, 9-2) < r(A, g^. Next let F - {i e E : g2(t) > 0}, and #3 = ^XF- Then it is seen that |/j — gz\(t) < \fi — g2\(t], t € E, and

4.2 Lower bounds ofJC(L^)

and JC(L9).

131

no

r(A,g3) < r(A,gz). Hence if gQ = £ biXE^ with bi = -^!Ei9zdn, then as in the proof of (1), if ||/j — g$\\$, ^ 0, there is a ki > 0 such that for i < nQ

Using the Jensen inequality we have the lower estimate for this as:

ll/«-fcll* > f 2^o

)^,lV*-1(2.o)

- # 3 } dp, **

> Hence r(^4, #0) < r (A#a) < r0 + | < 2. Let Aj = ^H min{Aj : 1 < z < n0}. Then 0 < Aj < 1 and

It follows from these estimates that (cf. also Lemma 1.9) 6A,;

Thus Aio < |, and we have £•

^0 + 77 > ^(-^j Po) ^ 11/in ~~ Po||$ ^ Z

This shows that rn > 1 — £ and then 0.11) >

1-e + e)'

and A,n =

132

IV Jung Constants of Orlicz Spaces

Since e > 0 is arbitrary, (23) follows, and we turn to (24). If e > 0, fc->00 * ^ > there is UQ € {WA;, & > 1} such that \ k

-> P$ R-r — — c. r

c

( )7\ \4 I)

Let D be the set of integers n + 1 > 1 for which H(n+i)X(n+i) exists; and it contains infinite set of integers. Choose n0 G D, with ^- < e, and disjoint intervals (7j = ^"IT' rT^T )' * —n°' so ^na^ ^(Gi) — ^T- ^et v4no+1 = co{/j : 1 < i < n0 + 1} in which (/l>/2, ' ' ' > / n 0 + l ) — ^~T7

r(XGi> ' ' •)XGn0)^nox(no+l)-

So ||/i||$ = 1 and for i =£ j n /•

r ii

2wo

by (27). Hence d(^ no+ i) < j^- As seen in Example 1.7, AIO+I nas its Chebyshev center at "0" in Xno = sp{xd,l < i < n0} C L* [o, ^]. So r(Ano+l,Xno) = I and JC(Xno) > —t— > ^^. «(Aio+l)

(28)

1+ £

It remains to verify that

jc (L*[O, i]) > jc V(L* L[o,UQ\J —]}> JC(xno). V

However,

U n+le£> r

7

Xn is dense in L* Lm, U—o Jj since $ e A 2 (oo), and the mapping -I

Tl

Pn from L* [0, ^j to Xn defined by Png = E bjXGj, fy = ^-y /G. gdp as _7 — 1

4.2 Lower bounds ofJC(L^)

and JC(L*}.

133

in Theorem 1, shows that it is (clearly linear and ) contractive. The last is seen from the computation: if 0 ^ g € L$ m, ^1 with \\g\\ = £[1 + p(kg)}, then

k \\9\\9-

As in the earlier result, it is possible to find a g with equality. Hence \\Pn\\ = 1. By Theorem 1.3, this gives

JC (l* [o, -1) > sup JC(Xn] > JC(Xno). \

L

UQ]/

n+l£D

That the bound increases if [0, ^1 is replaced by [0,1] is also the same as in the previous result, so that (24) follows, n We now present the analogs of Corollaries 2 and 3 for gauge norms, which are proved with a similar argument. Corollary 8. For an N-function <J> with \t as its complementary, M$[0,1] C ), 1] as before, then 1

<JC(M*[0,lj),

(29)

and if moreover, $ e A2(oo), then fa < JC(M*[Q, 1]).

(30)

Corollary 9. If $ g A2(oo) n V2(oo), then JC(L*[0,1]) = JC(M*[0,1]) = 1.

(31)

On the other hand, L*[Q, 1] is reflexive iff JC(L*(Q, !])
(32)

134

IV Jung Constants of Orlicz Spaces

and in general, JC*(I/*[0,1]) > -^. We next present the corresponding versions of the results if the unit interval [0,1] is replaced by R+, The proofs being simple modifications, as in the case of gauge norms, they are again omitted. Theorem 10. For any N-junction & with \I> as its complementary, we have

and if $ G A 2; then

— < JC(L*(R+}),

(33)

fa < JC(L*(R+)).

(34)

The Corollaries of this result then are as follows: Corollary 11. For the usual subspace M®(R+) of L®(R+], we have: 1

;*+)),

(35)

ZUfty

and if & G A 2 , then

fa < JC(M^(R+}}.

(36)

Corollary 12. $ £ A2 n V 2; then JC(L*(R+)} = JC(M*(R+}} = 1, andL*(R+} is reflexive iff JC(L*(R+)) < 1. In general, ± < JC(L*(R+)). Combining all the preceding results with Theorem 1.1.15, we may state the following: Theorem 13. For a pair (<$, \I/) of complementary N-functions: (a) $ G A 2 (oo) n V 2 (oo) => max f — , p9} < min{JC(L(*}[0, ll), JC(L*[0,1])}; \2o;<j,

/

(37)

(b) $ G A2 n V2 =>•

/ I

- \

Proof. Since the hypotheses imply $ € A 2 (cxD)( or A 2 ), the results follow from Theorems 1, 7, 4 and 10 and the fact that (cf. Theorem 1.1.15) ay fa and 1a$fa — 1 = 2a^^$. n

4.3 Bounds for JC(L^) and JC(L*°)

135

4.3 Bounds for JC(H*>)) and JC(L*°) We now consider a class of intermediate Orlicz spaces and obtain the corresponding bounds, which improve the preceding work. Theorem 1. Let $ be an N-function and $o(u) = u2. If $s> 0 < s < I , is the intermediate N -function with \&+ as its complementary function, then 2~*,

(1)

2~f ,

(2)

and

where ft = [0,1] or#+. Proo/. By Theorem 3.2.7 and Remark 1.10 we find < 2- ,

and similarly 2-.

Hence the two inequalities imply (1). As for (2), by Lemma 1.4.7, $s e A2 n V 2 so L*^(fi) is a reflexive ( hence dual) separable space. Let {^ : i > 1} be a countable dense set in L*s (£7), and Xn = sp{ 0, no

2 Cj = 1, that satisfy the conditions of Theorem 1.5 relative to a Chebyshev 1=1 center / € Xn. Taking A = ^37, 0 < s < 1, it follows from Theorems 1.5 and 1.4.12 that

Hence

r(A,Xn] d(A)

~

Vn + 1

136

IV Jung Constants of Orlicz Spaces

This gives JC(Xn}

< 2-f .

Letting n —> oo, we get by Theorem 1.4

Similarly we can prove These two inequalities imply (2).D We deduce Pichugov's [1] result from this as follows. Corollary 2. // 1 < p < oo, q = -p-^, and £1 as above, then JC(Lp(ty) = JC(Lq(ti)} = max(2-«,2~i).

(3)

Proof. Let $ p (u) = |U|P, so that $p € A2 n V2 and L p (fi) = L<**>(fi). Note that a$p = /3$p = a$p = ^$p = 2~?, (4) by Theorem 2.13 one half of (3) holds: max(2p~ 1 ,2~p) < JC(Lp(fy).

(5)

For the other half, from Remark 1.10 and Corollary 3.2.8 we have JC(jy(0)) < „,,* = -1-^i-rr = max(2i- 1 , 2~J). (6) ^(^ p (^)) min(2 -?, 2?) Thus (5) and (6) imply (3).D Corollary 3. For an intermediate $s, as in Theorem 1, we have maxf-i-.O < {JC(L^[Qtl]),JC(L*'[0,l})} y/Q;$s J

< 2~i,

(7)

< 2~f.

(8)

and if^l — R+ , then max

-,^ < {JC(L^(R+)),JC(L*°(R+))}

Proof. This follows from Theorem 2.13 and the above theorem. D Example 4. Let 1 < p < oo and $ p (u) = |w|2p + 2|w| p . Then ,

0 < s < 1.

4.3 Bounds for JC(Z,<*'>) and JC(L*>)

137

Now one finds _

fj/Jh

Q

Z7^ Oih

~~~

1 • ___

IllTl

S

\ /

f ) ~*>

"~* / &p

7!

£

and BA. = Ba>. as well as

Hence Corollary 3 implies 2~* > {JC(L<*')[0,1]), JC(L*'[0,1])} > 2^-*, and

2-

( 2 ZP

2

,

p < oo.

For a class of intermediate Orlicz spaces considered in this section, exact values of Jung constants can be found: Theorem 5. For an N-function <3>, let 3>s be the associated intermediate function, 0 < s < 1. Then we have: $ $ A 2 (oo)nV 2 (oo) =*• JC(L<*'>[0, 1]) - JC(L*'[0, 1]) = 2-*,

(9)

JC(L(^(R+}} = JC(L*>(R+)) = 2-f.

(10)

$ 0 A2 n V2 =>• Proo/ By Theorem 1.1.7, | < a$ < /3$ = 1 when $ g A2(oo), and

so that

138

IV Jung Constants of Orlicz Spaces

Hence max f^"?/^) — 2 ~ 2 . If $ 0 V 2 (oo), then | = 01$, < /3$ < 1 so that 2^7 = 2"f > A*, implying (9). 1 If $ 0 A 2" n V 2"i, then max (T^ -, 3$ } = 2~i,i so that (10) follows from this i\ 2a<j> ' i^^s J \ I and Corollary 3 (cf. (8)).D Now Corollary 3 is complemented by the following: Corollary 6. Under the same conditions of Corollary 3, the inequalities there hold if LP>\ti) and L**(ft) are replaced by L^(fi) and L*< (ft). We illustrate Theorem 5: Example 7. Let 1 < p < oo and let Q be the inverse of s

Q~l(u) = [log(l 4- u)]&u*, u > 0. Then , 1]) = JC(LQ[0, 1]) = 2-4

(11)

and

JC(L( Q H^ + )) - JC(LQ(R+)) = 2 - 4 .

(12)

p

To see this, if $(u) = (exp(|w ) - 1), then

Thus Q - [$sL = i. Since C$ = lim | = oo, $ ^ A 2 (oo) and $ ^ A 2 . Hence (11) and (12) follow from (9) and (10) respectively. It should be observed that, under the given conditions, Theorem 5 states that

where 17 = [0, 1] or R+. Franchetti and Semenov [1] obtained lower bounds for certain rearrangement invariant spaces, and upper bounds for other types of spaces. The Orlicz sequence spaces are investigated recently by T. Zhang [1] for the values of Jung's constants and they will be discussed in the next section. 4.4 Bounds for JCl™

and

As noted before, JC(l°°) — | < JC(c0) = 1 although c0 is a separable subspace of /°°, a dual space. Here we show that for any TV-function <£,

4.4 Bounds for JC(l^} and JC(l*}

139

= JC(mW), which is nontrivial if $ g A 2 (0). First we establish: Theorem 1. For any N-function $, one gets /3° < min { JC(mW), JC(/ ($) )} < 1 and JC(m<*>) = JC(/(*>) = 1 if® £ A 2 (0). Proof. By definition of /?£ we can find w^ \

(1)

0 such that lim t_ir f c | == /3£ Jfc-KX> *

VwfcJ

and hence for any 0 < e < \ there is u0 G {u^ : A; > 1} satisfying

Let k0 be the integer part of ^, so that w0 < ^ and ^ < ^^ . Hence

e e , ., r. 2w0 < - < -- and 1 — 2u0 > 2 l+e l+e Now let OQ = "J1"1 (j^J and define constant fco-vectors ho = (ao,ao> • • • > a o); go = (0,0, • • • 0 ) . Set

fi = (go, • • • , 9o, h0, g0, • • •), i = 1, 2, • • • with /i0 in the i

t/l

position. It is convenient to consider the infinite vector

where the first "1" occurs on the mth slot and ends at the nth, 1 < m < n. Thus fi = aox [m, n] with m— I + (i — l)ko and n = iko. For an element g = (<7(1), #(2), • • •) G m/*) or /(*) with pointwise multiplication, we get 9X [m, n] = (0, 0, • • • , 0, g(m),g(m + ! ) , - • • , g(n), 0, 0, • • •)• Consider the infinite set A — {fi : i > 1}. Then ||/j||($) = 1 and for i ^ j > 1, we have

140

IV Jung Constants of Orlicz Spaces

by (2). This implies d(A) < j^-£. If g = (0(1),0(2),.--) <E m**>, let P^g = (0(1), • • • ,0(n), 0, • • •), so that ||0 — .Pn0||($) —>• 0 as n —>• oo, implying that {Png : n > 1} is a Cauchy sequence with limit g in m^ (not true for g & l^ / m^). Thus, _

__

„, ,, v ,,

,,

P(j_i)fc 0 0||($) -> 0 as i —> oo

and

r(A,g)

= sup ||/j — 0||($) > limsup ||/i — 0||($) i>l

i—>oo

> lim sup || (fi — 0)x[l + (i — I ) k 0 , i k 0 ] ||($) z—>oo i—>oo

i

It follows therefore that

We conclude from (3) and the earlier estimate on d(A) that

This establishes JC(m^} > (3$, since E > 0 is arbitrary. If g = (0(1), 0(2), • • •) G /(*>, then lim \g(j)\ = 0 since /<*> C c0 so that

> 1 — lim < I—>00

sup

= 1. Similar to (3), we find r(A,l^} > 1 and JC(l^) > ^g, which completes the proof of (1). If $ ^ A 2 (0), then /3g = 1 by Theorem 1.1.7. Thus the last assertion follows from (1) since JC(X) < I for any Banach space X.O Theorem 2. For an N-function $ € A 2 (0), we

4.4 Bounds for

141

and JC(l*)

Proof. Since o;| = liminf ^_1i^l, for any n > 2 we can find a un satisfying 0 < un

1

1

and

(5)

Let fcn be the integer part of (2 n w n ) 1, then kn < (2nun}

-- Tun < knTun n We consider a set A — {fa : 1 < i < 2n}, where

1

< kn + 1 so that

< 1.

(6)

[A, /2, • • • , /2»] = an ei, e 2 , • • • e fcn(2 n_ (2 n -l)x2 n

with ,-1

(7)

For example, if n = 2, kn = 3 one has kn(2n — 1) = 9 and 1 1-1-1 1-1 1-1 1-1-1 1 1 1-1-1 1—1 1—1

[/i) /2,/3;/4] = 0"2 [ei,e 2 , • • • ,e 9 ]

1-1-1 1 1 1-1-1 1-1 1-1 1-1-1 1

or

/i = oa(

1,

1,

1;

/2 = a 2 (

1, -1,

-1;

/3 = aa( -1,

1, -1;

/4 = oa( -1, -1,

1,

1,

1;

1, -1,

-1;

-1,

1, -1;

1; -1, -1,

1,

1,

1, -1, -1,

1; 0,0, -1;

0, (),

1, -1;

0,0,

l; -1, -1,

1; 0, (),

IV Jung Constants of Orlicz Spaces

142

It is seen that \f{ |w = 1, 1 < i < T. Let f-

and a = — , '

1 < i < T. - -

Then \\gi ^ — 1 where ^ is complementary to $, and £ c^i = 0. Further, 1=1 n ) . If /(*)(A; n (2 - 1)) is the linear span of e i? 1 < « < /c n (2 n - 1), then by Theorem 1.5, r(^,^*)(jfc n (2 n - 1))) = 1 and

We shall simplify the right side of (8) and find an upper bound in terms of a^, using (5) and the fact that $~l is concave increasing. By (6) one has kn(T - IK > 1 - - - knun > i - I -n 1 n n 2 and hence, < <

kn(2"-l)Un On the other hand, we have 2

Hence from (8)-(10) it follows (cf. (5) also) that d A

( ^

< 1

_i2^9-n\i-lf

T < •

Consequently, *W ( * ) (M2 B -D))

1 - n-1 - 2

Since $ e A 2 (0), U lw(kn(2n - 1)) is dense in lW, the later is also a n>2

separable dual space, and -P/cn(2'i-i) : l^ —> l^(kn(1n — 1)) defined by Pfc B ( 2 n-i)/ = (/(I), • • • , /(U2 n - 1)), 0, 0, • • -)

4.4 Bounds for JC(l^} and JC(l*}

143

for / = (/(I),/(2), • • •) 6 /(*) is a norm 1 projection. Now we can apply Theorem 1.3 to obtain - sup JC(l^(kn(2n - 1))).

(13)

n>2

Thus, (12) and (13) imply (4).D Corollary 3. (i) /(*) is non-reflexive iff JC(l^) — 1; (ii) For reflexive l^, one has (14)

Proof. Similar to Corollary 2.3.Q Example 4. Let $>r(u} = e^ - 1, r > 1. Then JC l(^

< 1.

(15)

In fact, $~l(u] = [log(l + u)]r, u > 0 so that a°$r = ^Jp = 2~^ and $r 6 A 2 (0) n V 2 (0). Hence (15) follows from (14). Example 5. Let Then Cg = Hm

= | and by Corollary 1.1.9, 0

oO .0 $ = p$ = 7$ =

Hence from (14) we get

With the Orlicz norm we have the following analogs: Theorem 1'. Let ($, ^) 6e a pair of complementary N -function and ra$ and I® be the corresponding sequence spaces equipped with the Orlicz norm. Then we have mn , JC(l*)}J < 1 (16) l*} = 1 if $ £ A 2 (0). Theorem 2'. For a complementary pair (
144

IV Jung Constants of Orlicz Spaces

Corollary 3'. (i) /* is non-reflexive iff JC(l*) = 1. (ii) For reflexive /* one has -^ < max (f3l yZ \

-\) = max (^^] < JC(l*} < 1. £&$ J \£Oiy J

(18)

The other assertion, similar to Section 3, takes the following from. Theorem 6. Let $ be an N- function, $0(u) = u2 and &s be the intermediate N -function, 0 < s < 1. Then

, JC(i*-)} < 2-f .

max

(19)

Example 7 (Ivanov and Pichugov [1]). For 1 < p < oo, we have JC(lp] - max 2"i, 2p - 1 .

(20)

Indeed, here C£p = p, 7^ = 2"p where $ P (M) = |w p. Since lp = l^p\ it follows from (14) that m a x ( 2 ' p , 2 p - 1 ) < JC(lp),

(21)

and similarly with Corollary 3.2 we can conclude

Hence (20) follows from (21) and (22). [See Def. 3.1.1 about 7V(/ P ).] Example 8. Let <£ r (u) = e^ — 1, 1 < r < oo, as in Example 4. Then C$r — r, and with 3>s using this $(= 3>r] we have w ) ] u , 0<s
We thus obtain, by Theorem 6, that

4.4 Bounds for JC(l^} and JC(l*}

145

We have the following comprehensive statement: Theorem 9. For an N-function <£, let <&s be the intermediate N- function as in Theorem 6. If $ & A 2 (0) D V 2 (0), then

) = 2-*.

(23)

Proof. Similar to Theorem 3.5. In fact, if $ 0 A 2 (0) n V 2 (0), then

implying (23) by (19). D A final application is given by Example 10. Consider the TV-function Q whose inverse is , 0<M<e e-2
JC(l(Q]} = JC(/ g ) = 2-4. In fact, Kaminska ([2], p. 304) showed that if

(24)

r 0, u=0 3>-l(u} = I e ~W, 0 < |u| < \ ( 4e-V, < H < oo

then $ is an TV-function, Cg = lim^f^ = oo and $ 0 A 2 (0). But Q-1 = ($;1)s=i, so from(23) we get (24). Bibliographical notes. It took a long time, after Jung [1] introduced the constant, to get the values JC(LP] and JC(lp] for infinite dimensional spaces, (cf. Pichugov [1] and Ivanov and Pichugov [1]). The work included in Sections 2 and 3 is taken from Ren and Chen [1], while Section 4 is essentially from T.Zhang [1]. We note the following general result (cf. Rao [11], Proposition 6.10 on p. 98, the details of which are given as Proposition 7.2.4 later): On an arbitrary measure space (ft, E, fj,) if L^(/J,) is an Orlicz space of real functions and M. is a closed subspace, then it is the range of a positive contractive projection iff M. is a Banach lattice. Here Q — IN, S = power

146

IV Jung Constants of Orlicz Spaces

set, ^ — counting measure so that L^(^L) = l^ and take M. = m which is a Banach lattice, i.e., /, g € M^ =>• max(/, g) e m ($) and 0 < /„ /* f and / e l^ =>• f G m(*) which is a consequence of the dominated convergence theorem and the absolute continuity of norm of m^ . Hence there is a norm one projection P : l^ —>• m^ . Then by the first paragraph of the proof of Theorem 1.3, since l^ \= (m*)*l is a dual space, we get immediately that JC(m^} < JC(l^). This inequality is also true with the Orlicz norm, i.e., JC(m^} < JC(/*). Thus, Theorems 4.1 and 4.1' can be formulated as follows: For any TV-function $, we have Pi < JC(mw] < JC(l^) < 1 and

Chapter V Packing in Orlicz Spaces

The structure of a unit ball in Orlicz sequence and general space is revealed, in the sense of its geometry, when one attaches an invariant for each space (as in the preceding chapters), and finds exact bounds (or best inequalities) for them. After introducing the relevant concepts and immediate consequences in the next section, we study the problems for Orlicz sequence spaces in Section 2, and then consider the corresponding problems for the spaces on the Lebesgue measure triples in the following section. Finally, Section 4 refines some of these results when the spaces are reflexive.

5.1 Preliminaries In a Banach space X, a sequence of balls with centers xi,xz, • • • and a fixed radius r > 0 is said to be packed into the unit ball if the following two conditions hold: (\} \L) \\TIl x t

(ii) \\Xi -Xj\\ > 2r,i^j,i,j = 1,2, ••• With this concept, one can associate a constant (or an invariant) to X as: Definition 1.

A packing constant P(X] of a Banach space X is

P(X] = sup{ r > 0 : infinitely many balls of radius r are packed into the unit ball of X}. Thus if X is of finite dimension, then P(X] = 0. If X is an infinite dimensional Hilbert space, then Rankin [1] established

P(X] = —1-7=. l + v/2

147

(I)

V

'

148

V Packing in Orlicz Spaces

Slightly later, Burlack, Rankin and Robertson [1] generalized it as: P ( l p } = -V^> 1+ 2 ?

l
(2)

Much later, Wells and Williams [1] showed 1+2

P

P(Z7(0, !)) = <(

(3) 2 < p < oo.

and that for p — I and = oo, the value = 1/2. Both formulas (2) and (3) will be deduced from the Orlicz space results below in Sections 2 and 3 respectively. In 1970, Kottman [1] proved the inequality for any infinite dimensional Banach space X: (4)

Definition 2. For an infinite dimensional Banach space X , the Kottman constant of X is defined as K(X] =sup(inf||ii-:r,-|| : {x^ C S(X)\ , )

I i^j

where S(X] is the unit sphere of X. Clearly, 1 < K(X] < 2. The following relationship between P(X] and K(X] was noted by Kottman [1] (cf. also Ye [1]). Theorem 3. If X is an infinite dimensional Banach space, then one has

Proof. The minimal and maximal values of P ( X ) and K(X] clearly satisfy the equation (5). To see that the left side of (5) always dominates the right, note that, by definition of K(X), for given 0 < e < K ( X ) , a sequence {xi} C S ( X ) exists satisfying inf ||xj — Xj\\ > K(X) — E. Let K(X)-£ £ = n , rsfV\

r

'

2/» = (1 - r e ) X i ,

5.1 Preliminaries

149

and then, for i ^ j, we have ||y< - Vj\\ = (1 - re)||zi - zj > (1 - r e )[ff(*) -e] = 2re.

(6)

Hence, using the definition of P ( X ) , we get P(X] > r£. It therefore follows that (letting e \ 0)

For the opposite inequality, since (j)(x] = ^ defines a concave increasing function for 1 < x < 2, with 0(1) = |, it follows from (4) that P(X) < (j)(K(X}} for any K(X] when P(X) = |. Consider therefore the case that P(X) > |. By a result of Kottman ([!]. Thm. 1.3) for any e > 0, there exists a sequence {xi : i > 1} C S(X] such that for i ^ j

•'-

-'-

2P(X)

-e.

(8)

This and the definition of K(X] imply that

and since e > 0 is arbitrary, (7) and (9) establish (5). D Remark 4. (i) Elton and Odell [1] proved that for each infinite dimensional Banach space X there exist an e > 0 and a sequence {xi : i > 1} C S(X) such that \\Xi — Xj\\ > 1 +e for i ^ j, so that K(X] > 1 for all infinite dimensional spaces. This fact and (5) imply that for such a space, P(X) > 2+1(i'+e) > I since (f>(x] is increasing and > |. (ii) For an infinite dimensional Banach space X, we therefore have: 1 < a < K(X] < b < 2, iff -±-
l i , 2 '

Cleaver [2] found upper and lower bounds for P(l®} under certain conditions, which however were found to be restrictive, and in a review Zaanen observed that they are satisfied only for the lp spaces. Later Ye [1] and

150

V Packing in Orlicz Spaces

Wang [1] investigated P(l^) and P(/ $ ) respectively, under weaker conditions. Somewhat later Wang and Liu [1] generalized these results for certain more inclusive sequence spaces with simpler formulas for K ( X } . The following two results from their works will be used in the sequel. Theorem 5. (Ye [1]) (i) $ g A 2 (0) =» K(lW] = 2 (or P(/W) = 1/2). (ii) $ € A 2 (0) =>#(/<*>) =

sup { c > 0 : p * ( { ) = i}. ll/ll(*)=i A proof of (i) is in the lecture notes of Orlicz ([1], p. 180) and of (ii) is in Wang and Liu [1], as well as in Chen ([1], p. 149). Theorem 6. (Wang [1]) (i) $ £ A 2 (0) => K(l*} = 2 (or P(/ $ ) = 1/2). (ii) $ e A 2 (0) => K(l*) = sup

imi*=i

A proof of this theorem is again in Wang and Liu [1], as well as in Chen ([1], pp.149-151). Remark 7. (i) Ye and Li [1] (and Wang [1]) showed that /<*>[/*] is reflexive iffP(/(*)) < 1/2[P(/*) < 1/2]. The necessity of these assertions can be proved using the concept of p— convexity. Recall that a Banach space is p— convex if there exist £Q > 0 and HQ > 3 such that for all (xj, 1 < i < HQ} C S ( X ] , one has mm

* 7*^,1
(10)

A p— convex Banach space is known to be reflexive (cf. Kottman [1]), and if X is p— convex then K(X] < 2 so that P(X] < 1/2, by (5) [see also Chen ([1], Theorems 3.30, 3.45 and 3.46)]. From a result we show in the next section, the reflexivity of /(*>[/*] implies P(/W) < 1/2[P(/*) < 1/2] using the corresponding estimation for the Kottman constants ( see Corollary 2.4). (ii) Hudzik [2] showed that if X is a nonreflexive Banach lattice then K(X] — 2 and thus P(X] = 1/2, by (5). Since /(*) and I® are lattices, this implies the desired conclusion when $ 0 A 2 (0) fl V 2 (0). 5.2 Packing in Orlicz sequence spaces

We begin w r ith estimation of the constants K(l^} and K(l®). In view of

5.2 Packing in Orlicz sequence spaces

151

Remark 1.7, it suffices to consider reflexive spaces l^ and I®. We have the following: Theorem 1.

For an N-function $ 6 A 2 (0) D V 2 (0) one has max[^,^]<^W)<^

(1)

where

Proof.

To derive the left side of (1), we first show w 4r a$ < K(i )-

(4)

By definition of a$(cf. Sec. 1.1), there exist 1/2 > wn \ 0 satisfying lim

S-1^) o —- - a#.

n->oo $-l(2u n )

*

So for any 0 < £ < 1, there is no such that the last relation implies $[(aj + e)^1^)] > un,

n> HQ.

(5)

If kn = [3"], the integer part, then we can find cn > 0 satisfying A;n2wn + $(cn) = 1, n > n 0 . Since $(cn) < (/cn + l)2u n — kn2un = 2un —> 0 as n —» oo, we may also assume that <$(cn) < e for n > no- For convenience, let Vi = uno+i,mi — kno+i,di = cno+i and define the (m; + 1)— vectors, ^ = (0, 0, • • • , 0) and ht = ($"1(2^), • • • , $~ 1 (2v i ), di). Let /» = (0i» 02, • ' ' > 9i-i, hi, gi+i,gi+2, • • •)• Then /j E /^ $ ^ and p$(/i) = 1, i > 1. From (5) we get for i ^ j,

V Packing in Orlicz Spaces

152

1-e 1

2(11

{[i-

Hence ||/; - /J^) > ^^, which implies (4). Next we show

-V < tf (i(*}),

(6)

which is due to Ye [1]. For any integer /c > 1, let
i - 1, 2, • • •.

It is seen that p$(fi) = 1, i > 1 and for i 7^ j, (7)

Since k is arbitrary, by (2) and (7) we get (6): ^-^-:A; = l , 2 , . . . y = -^. For the right side of (1), due to Ye [1], let f(w) = ^r^,u > 0 and set = $~ 1 (2u) so that

For an / = (/(I), /(2), • • - , /(i), • • •) 6 / oo

y (3) i-l

<

A 2 (0)], we have

5.2 Packing in Orlicz sequence spaces

153

if f({) ^ 0 and u = |$(|/(z)|). Consequently by (8) = \P*(f) = £•

• _ -I

&

£

It follows from this and Theorem 1.5 that = sup

and hence (1) holds in its entirety, D We present a couple of consequences. Example 2. If $ p (u) = \u\p, I < p < oo, then lp = /<*"> and a?9p = a%p = a$p = 2~p. So that by the above theorem we have (2) of Sec. 1, namely, K(lp] = 2? and P(P) =

^-j.

(9)

Example 3. Let $ r (w) = e|u|r - 1,1 < r < oo. Then ) - K(lr) and P(/ ($r) ) = P(/ r )-

(10)

Indeed, $~ 1 (w) = [log(l + u)]r so that a$r = 2"^, and one finds that r 6 A 2 (0) n V 2 (0). Now one can compute a<s,r = mf<[H(u)]r

: 0 < u < -} ,

L

&)

where H(u) = log(l + w)/log(l + 2u). Since, as seen in Example 1.1.12, (•^H)(u) > 0 for u > 0, we get a^r = a%r and (10) follows from Theorem 1 and (9). Corollary 4. Let $ € A 2 (0) n V 2 (0). Then we have

2 or equivalently,

<2,

(11)

154

V Packing in Orlicz Spaces

where (<j) being the left derivative of <&)

(13) Proof. Since $ e A 2 (0),5£ < oo by Theorem 1.1.2. From (4) and Theorem 1.1.8, the first half of (11) follows. For the last half, by definition of (13), < t < < $ > - l l . But for 0 < h < tz < $-ll ,

,, ,

,

log5 —-—- = \ —7-rdi > A$ \ — = log 5 — It, $(i) ~ Jtl t

-

Taking t\ = 3>~l(u], t^ = <&~l('2u], 0 < u < |, one gets from the above —r^r > 2 ~ ^ ,* o r a $^^ — >2~i. ^R — 1 /r* \ —

(14) \ '

To complete the proof of (11), it suffices to verify that A$ > 1 in (14). But since $ e V 2 (0), there are numbers a > 1, t0 > 0 such that ^^ > a, 0 < t < t0 < $~l(l). Now for t 0 < t < $~l(l), one gets -

'

This shows that .4$ > 1 and (11) follows. The equivalent form (12) for is then immediate. D To find the values for K(l^} and P(l^), we need an auxiliary result on TV— functions $, due to Yan [1]. Lemma 5. For an N -function <3>(w) = /J' cj)(t)dt,

.(u) =

define :M

(15)

/or t > 0 anrf u > 0. If F$(t) decreases (increases) on (0, $~ 1 (?/o)] /or some WG > 0, ^/ien G$(u) a/so decreases ( increases ) on (0, ^]. Proof. This is a consequence of Lemma 3.3.4. D We now establish the following.

5.2 Packing in Orlicz sequence spaces

155

Theorem 6. For $ e A 2 (0) n V 2 (0), we have: (i) if F
(16)

C{= limF*(*); i->o+

(17)

where

(ii) if F$(i) is decreasing on (0, $ -1 (l)], then

Proo/. (i) If F$(i) is increasing on (0, ^^(l)], then C$ exists and AQ = B C$. Hence, (16) follows from Corollary 4. (ii) If F$(t) is decreasing on (0, $~1(1)], then G$(w) = $-1$) ^s a^so creasing on (0, |] by Lemma 5. Therefore,

and (18) follows from Theorem 1 and the above. D Example 7. Let ) = (1 + \v\) log(l + \v\) — \v\, then _#•(/(*)) —v/2) P(/( $ )) = (19) 1 + \/2 and K(lW) =

^ « 1.48699,

P(/ w ) w 0.42644.

(20)

Indeed, F(i) = ^^zf is increasing for t > 0. Since C$ = lim F$(t) = 2, (19) follows from Theorem 6(i). On the other hand, the function w v /

(l + t)\og(l+t)-t

is decreasing on (0, oo) and C^ = lim Fy(t) — 2, i.e., ^ e A 2 (0) n V 2 (0).

156

V Packing in Orlicz Spaces

Hence, (20) follows from Theorem 6(ii). Formula (20), due to Yan [1], is the solution of a problem raised by the authors [2] (Problem 2.7). Example 8. Let V

'

log(e +

then

X(/ w ) =

V w 1.46987 and P(/ ($) ) « 0.42361.

(21)

In fact, the function Tjl

O

/J-\

• t) log(e + t]

is decreasing on (0, $ -1 (l)] and Cj = 2, i.e., $ e A 2 (0) n V 2 (0). From Theorem 6(ii) one gets (21). Example 9. Let u\plogr(l + \u ),

1 < p < oo, 0 < r < oo.

Then ^p,r\2<

Indeed, we have for t > 0

and CQ P,r = t-+0+ lim F$ d ^

(t) — p + r > 1, i.e., <E> ,.

E1 /j.\ _;_, r*t>.r l1^

e A 2 (0) n V 2 (0). Since

r[log(l + t)/ — Jt] l "V /,

,

,\oi

5/-,

.

,\

^- r> ^ U)

fc

j. ^ n ^> U,

F$ p r (t) is decreasing on (0, $"^(1)]. Hence, (22) follows from Theorem 6(ii) In particular, for $1;1 (w) = |u log(l + M|) we have #/(*i.i)

=

i « 1.18806 and PZ(* 1 - 1 ) w 0.37266.

5.2 Packing in Orlicz sequence spaces

157

Example 10. Let $ p (u) = ^(1 + 1 log |u||),

P > i ( 3 + \/5).

Then ^~l(l) = 1 and $p(t) = tp(l - log*) for 0 < t < 1. Since

F<;,p(t} is decreasing on (0, 1] and C$ = lim F$p(£) = p, i.e., $p G A 2 (0) n V2(0). It follows from Theorem 6(ii) that

*('"•') = ^m^P V 2 / In particular, one has for p — 3 1.38378 and P/ ( * 3 ) w 0.40895.

To find the corresponding results for / $ , we need an auxiliary result. Lemma 11. For a complementary N—pair ($,\I>), define [1 + M*//)l »

(23)

* / > ! : H/ll* = -k [ l + M * / / ) l f

(24)

q*= inf : * / > ! : |/||* = ll*— i

= sup

A 2 (0) n V 2 (0), C = supjfg1 : 0 < u < 3>~l(l)} and

then 1

*

1 < 1 + — < g$ < Q* < , $

< oo.

(26)

158

V Packing in Orlicz Spaces

Proof. Since $ € A 2 (0), we have 2 < C < oo, $(2u) < C$(u) for 0 < u < $-l(l). Let / = (/(I), /(2), • • • ) € / * with ||/ * = 1, so that

*= and |/(i)| < S-^l),* > 1. Therefore, oo

Cp*(f)

> Z>( 2 /(0) = P*(2/) i=l

and l + i<

inf

l+ *

<

inf

*;

which establishes the first half of (26). Regarding the second half, since $ 6 V 2 (0), one has a$ > 1. Now let /||$ = 1, so that there is kf > 1 satisfying 1 = ||/||$ = ^-[1 + Then for any £ > 0, such that kf — £ >q$ - E > I , one has from (25):

E 1=1

or 1 1'

This implies the second half of (26) and hence the lemma. We now establish the analog of Theorem 1 for £*— spaces.

5.2 Packing in Orlicz sequence spaces

159

Theorem 12. For3> 6 A2(0)nV2(0), with \f as its complementary function, then max (2/3° , 20i) < #(/*) < -^, (27) v ' of® where. -i(-U -

o/ ^j,

(29) Proof. We follow the argument of the previous case with suitable modifications. Thus for 0 < £ < 1/2, there exist 1/2 > vn \ 0 for which ib~l(v \

F

>#-'

n> 1

--

(30)

Let kn be the integer part of ^-, and since —^- /* oo as v \ 0, one has

V Zi ( fofi ~\

\- ] J

Vfi

and

(32) Denote by

so that 6n —>• 0 and 0 < cn < 2\Er~1(2-|-) —>• 0, as n —>• CXD. Choose no such that 6ncn < £ if n > n0 > 1. For convenience, let t>0 = f n o , ^0 = kno,b0 = 6no, and CD = c no , and consider the A;0—component vectors ZQ = (0,0, • • • ,0),X 0 = (bo, b0, • • • , 6 0 )- Define /,- 6 /* by /i =

(ZQ, Z0j • • • , ZQ, XQ, ZQ, ZQ, • • •),

160

V Packing in Orlicz Spaces

where X0 is at the ith position. Then ||/;||$ = b0k0^^l(^-) = l,i > 1. From (31), (30) and (32), one has for i^j

>

(2/3° -

It follows that K(l®) > 20$. On the other hand, for any given integer k > 1, let Zk = (0,0, - . - , 0 ) , and Xk = [k^-l(^)]-l(l, 1, • • • , 1) with dim^ = dimXk = k, and define fi — (Zk-, Zk, • • • , Zk,Xk,Zk, Zk, • • •),

i > 1,

with Xk being at the ith position. Then /j||$ = 1, z > 1 and for i ^ j,

Since A; is arbitrary, one has >sup

(2x]/- 1 ('-M \ 2 ^ : f c = l,2,.

I ^ U/

Next we wish to establish the second half of (27). Let £(w) = ^^1, w > 0, so that $[^(w)$" 1 (2u)] = u. For any f e I* with ||/||$ = 1, there exists kf > I satisfying ||/| $ = ril + P*(%/)]• Since ^ e A 2(0) n V 2 (0), we can apply Lemma 11 with Ui — |$(/c/|/(z)|). Then by (24) and (26), one has $ (M/WI) < M*//) = */ - 1 < Q* - 1,

5.2 Packing in Orlicz sequence spaces

161

so that from (29), it results that

and

1=1

Hence from Theorem 1.6 we get K(l*) = sup linf [ c > 0 : p * ( — 1 -i U>i ll*=i This implies the second half of (27) and hence all the assertions. Example 13. Let $r be as in Example 3. Then r

) = 2r and P(l*r) = P(lr).

Indeed, by Theorem 1.1.15, for any pair (<£, \I/) we have that If $ 6 A 2 (0) n V 2 (0), from (27) one has

- < K(l*) <

I" log(l + u)

= 1.

(34)

-

Since

(33)

., i

is increasing on (0, oo), by (29) we find that a*^r = a$ = lim G$T(u) which implies (33). A useful consequence of Theorem 12 is given by:

162

V Packing in Orlicz Spaces

Corollary 14. Let $ e A 2 (0) n V 2 (0). Then we have 1< 2^ < K(l*) < 2^ < 2,

(35)

or equivalently,

1 _L O

^

where, with the notation of (24), (37)

Proof.

Similar to (14), we can show that 1

(38)

U£Q

On the other hand, Theorem 1.1.8 implies that

This and (38) with (34) imply (35). D To study a problem of the authors (cf. Rao and Ren [2], Problem 3.8), Yan [1] proved the following. Lemma 15. Let $(w) = /j $(t}dt, $(v) = /Q ip(s)ds be a pair of complementary N—functions and let

We have the following assertions. (i) F$,(t) increases on (0,?/>(so)] for some SQ > 0 if and only if F$/(s) decreases on (0, SQ]. (ii) Let a$ be defined by (25) and

(39)

5.2 Packing in Orlicz sequence spaces

163

then (40)

=1 ^ "<j> + F ^$

(iii) //$ 6 A 2 (0) n V 2 (0), and Q$ is in (24), then Q$ < b\. Proof, (i) For 0 < s < SQ and 0 < t = ip(s) < ip(so), we first assume that if} is continuous at s and 0 is continuous at t, so that t — ^[^(t)} and =

*1<WJ

F9(t)

=

*[(/>(£)]

t(f)(t) - $(t)

F$(t) -

or equivalently,

which implies the assertion. With no difficulty one obtains the general case. (ii) Letting s0 = ^^(l) in (i), we have by (41) and (39): ^ <,<«-'(!)

'

which proves (40). (iii) It follows from Lemma 11 and (40) that

Theorem 16. Le^ $ G A 2 (0) D V 2 (0) iwii/i \f 6em^ zfe complementary function. We have the following assertions: (i) If F$(t) = ^^ is increasing on (0, <£~ 1 (fr£, — 1)], where by is in (39), then *(t). (42)

164

V Packing in Orlicz Spaces

(ii) If F<j>(t) is decreasing on (0, '0[^ f ~ 1 (l)]] , then

Proof, (i) From Lemma 15 (iii), F$(t) is also increasing on (0, $~l(Q$ — 1)], so that Bl = Al = Cl and (42) follows from Corollary 14. (ii) By Lemma 15 (i), F*(s) = ~~ is increasing on (0, ^(l)] so that

or 6^ — 1 — ^(^[^"^l)]). It follows from Lemma 5 and the hypothesis that G*(u) = |^gy is decreasing on (0, ^[^(l)])] - (0, |(6^ - 1)] D (0, |(Q$ — 1)] and hence

On the other hand, again by Lemma 5, Gy(v) — ^i^l is increasing on (0, |], so that 2xl/~ 1 (i.) K(l*} > max (2/5° , 2ft) = 2ft = -^3^.

(45)

Finally, (43) follows from (44), (45) and Theorem 12. D Example 17. Let $(w) = e | u | -|w|-l and *(u) = (l + |u|)log(l + |i;|)- u|. Then we have AT(/*) - A (46)

and 1.496

< K(l*) < 1.498.

(47)

Indeed, F*(t) = ^ is increasing on (0, oo) and Cg = lim F$(t) = 2 (cf. Example 7). Hence (46) follows from Theorem 16(i). Since F\f,(t) = t-^ff- is decreasing on (0, oo) (cf. Example 7), from Theorem 16(ii) one has

$-1(1)

5.2 Packing in Orlicz sequence spaces

165

By a simple computation we get estimation (47). Remark 18. For the function ^ of Examples 7 and 17, we have that K(l^) < K(l*). It should also be noted that Problem 3.8 in the authors' paper[2] is still open. Theorem 19. Let be an N—function, $Q(U) = u2 and let $s be the inverse of 3>~l(u) = [$~ 1 (w)| [([^(w)] , 0 < s < l,w > 0. (48) Then we have ( 1 1\ 1 max -5-, — < #(*(*•>) < — < I1'*,

(49)

max (4-, 2^ + ] < K(l**) < 4~ < 21"1 • a a

(50)

and

\ ^s

/

$3

in particular, if $ 0 V2 (0), £/ien AT(/(*-)) = 2:-f = K(l**),

(51)

or equivalently,

Proof. Note that $s <E A 2 (0) n V 2 (0), 0 < s < 1 and that 2a^/3°+ = 1, where ^ is the complementary function of $s. By Theorems 1 and 12, to prove (49) and (50) we need only establish both the right sides of (49) and (50). Indeed, ^-~iffi) ^ 2 ^or a^ u > ® an(^ nence> 1—s r /— •} s \ . Ill \

(52)

which implies that min(a$ s ,Q;^) > 2~f ( - ) \£/

If $ ^ V 2 (0), then a% = \, so that als = limmf G*.(u) = (aj)1-^'! = 2f~ 1 .

(53)

166

V Packing in Orlicz Spaces

Thus, (51) follows from (53), (49) and (50). D Example 20.

Let M be the TV-function given in Example 3.3.18, so that 0,

0 < u < e2 u > e2.

Then K(l^) = 2f = K(1M}. Since there is a $ g V 2 (0) and C£ = 1, such that M -1 (w) = [$7 1 ( U )] S =J-! the above calculation holds. An equivalent statement for the packing constant is

1 1 + 2^ showing the utility of Theorem 19 for exact values. Theorem 21.For <3> e V 2 (0), let $~l be as in (48). The following assertions hold. is increasing on (0, ^(l)], then

(54) (ii) If F$(t) is decreasing on (0, $ *(!)], then 1-s

(55) Proof, (i) The hypothesis implies that G$(u) = $-1$) ^s a^so increasing on (0, |] (cf. Lemma 5). It follows from (52) that l-s

=2

(56)

Hence, (54) follows from (49) and (56). (ii) The hypothesis implies that G$(u) is decreasing on (0, |], so that l-s

(57)

5.3 Packing in Orlicz function spaces

167

Finally, (55) follows from (49) and (57). D We conclude this section with the following illustration: Example 22. since

Let $ r (u) - e |ui? " - 1 with 1 < r < oo. Then $r e V 2 (0)

Note that

^ ^ r MI

is increasing on (0, oo) D (0, |] U (0, ^(Q*^ — 1)], where $s is the inverse of

= ^f [iog(i + w)]1^ ,

o < 5 < i.

From (52), (49) and (50) we have

and

5.3 Packing in Orlicz function spaces Bounds for the packing constants of Orlicz spaces L $ [0,1], with Orlicz norm, were considered in Cleaver [2], and the corresponding results, with the gauge norm, have been obtained by Ren in 1985 and included in our companion volume ([1], pp.256-258). A more detailed discussion appeared in Ren [1]. We start here with lower bounds of A"(L<*)(fi)) and P(L ( *)(Q)) where £) = [0,1] or R+ with Lebesgue measure. Theorem 1.

For an N—function $, we have max (^, 2/9*) <#(£<*>[(>, 1]),

or max

(

J__,

/

] < p(i(*)[0,1]),

(i)

V Packing in Orlicz Spaces

168

and (2)

or max

Proof.

By definition of a$, there are 1 < vn /* oo, such that lim ^-i/^\ —

a$. So given E > 0, there is no > 1 satisfying ^-\^v ) ^ Q;* ~*~ £' n —n°- ^et w = i ^no+j so that Ui,

i> 1.

(3)

00

By taking a subsequence, if necessary, we may assume Z) ^T < 1. Let {G^ : z=l

z > 1} be a disjoint sequence of Borel sets of [0,1], with n(Gi) = ^- where /Lt(-) denotes Lebesgue measure. If /; = ^l(2ui}xGi,'i > 1? then ||/i||($) = 1 and for i ^ j we get P* [(a* + e)(/i - /,-)] = ^> [(a* + e)$-1(2

Thus ||/i - £||w > ^,t ^ j, so that A"(LW[0,1]) > ^ for (1). For the second one with /?$, by definition, there is a sequence 1 < un /* oo such that lim Q-\&\ — /^$- So, as above, for 0 < e < 1, there is n0 > 1 such that

Set MQ = uno, and construct a Rademacher sequence {Ri, i > 1} on [0, ^-], so that A;

Clearly,

k=\

5.3 Packing in Orlicz function spaces

169

and if fi = $~l(u0)Ri, i > 1, then ||/i||($) = 1 and for i ^ j we have

This and the previous inequality imply (1) for the Kottman constant and then the packing constant follows from their interlocking identity (cf. Thm. 1.3). For (2), by definition of a$, given e > 0, there is a VQ > 0 such that

00

.

Consider the decomposition of R+ as U EI where EI = [^, ^-) and n(Ei) = 2 . If fi = ^(Iv^XEi, then ||/i||($) = 1 and using (5), for i^ j we find

so that K(L^(R+)) > ±. For the second part of (2), let 0 < e < I be given, and by definition of ^$, there is 0 < UQ < oo satisfying

Again construct a Rademacher system {.R;,z > 1} on [0, — ) and let fi — ^~1(u0)Ri so that ||/j||($) = 1. As in the preceding case for i ^ j we find \\fi — /j||($) > 2/3$ — e. This and the previous result imply (2) itself. D We have the following consequence. Corollary 2. For an N— function $, and fi = [0, 1] or R+, with Lebesgue measure, we have or --= < P(LW(fi)) , 1 + v2 J

(7)

and if L^ (fi) is nonreflexive, then (8)

V Packing in Orlicz Spaces

170

Proof. For (7) it suffices to note, by (1), that \/2 < max(^-, 2/3$) since otherwise a$ > -4 > /3$, which is impossible because a$ < /5$ always. Similarly \/2 < max(^,2^ $ ), and (7) follows. For (8), $ £ A 2 (oo)n V 2 (oo) ( if ft = [0, 1] ) and by Theorem 1.1.7, p^ = 1, or a$ = | so max(^, 2/3$) = 2. Similarly if f2 = # + , we get the same bounds and (8) follows. D The corresponding results with Orlicz norm are as follows. Theorem 3. For an N~ function $ with ^ as its complementary one, we have max or max

2/3*,

(9)

, 1]),

<

<

and

(10)

or max

Proof. As above, by definition of /?#, there exists a sequence 1 < w n /^ oo, n such that n~->oo lim ,5_ 1 /" \ — pV and for given 0 < e < 1, we can find in this * (^Wnj sequence an no such that, letting vt = uno+l,

(11) oo

By going to a subsequence, if necessary, one can take ]C 2ii~ < 11=1 ' disjoint Borel sets of [0, 1], n(Gi) — ^- and define 2v-

Then

171

5.3 Packing in Orlicz function spaces and similarly if i ^ j, py(gij) — 1. Hence, with (11), for i ^ j, we get f fl ll/t - /J* = sup M (^ -

fj)gdfj.

v. J 0

>

/•i y (fi-fj)9ij

This implies /i:(L$[0,1]) > 2/5$. For the second part, similarly, by definition of a$, we can find a sequence 1 < vn /" oo, and for 5 > 0, an no such that

(12) Set i;o = vno, construct the Rademacher system {Ri, i > 1} on [0, ^-) C [0, 1], as in Theorem 1, and let /; = ^_i? \Ri,i > 1- Then one finds ||/j||$ = 1 and for i

by (12). This implies A"(L*[0, 1]) > ^, and (9) follows. By a similar argument, we establish (10). n Analogous to Corollary 2, we have the consequence: Corollary 4. For an N— function $, and Q — [0, 1] or R+, with Lebesgue measure p,, one has the following relations:

>V2,

or

v/2

(13)

and 2/Z/ $ (f2) is nonreflexive, then

(14) Remark 5. (i) The inequalities given in Corollaries 2 and 4, namely (7) and (13), need not hold for sequence spaces. In fact if 2 < p < oo, then

172

V Packing in Orlicz Spaces

K(V} = 2? < V2 (cf. Example 2.2). (ii) The equalities [i.e., the exact values ] given in (8) and (14) can also be deduced from Hudzik's results (cf. Remark 1.7 (iii) ). In the next section we show that every reflexive Orlicz function space has packing constant strictly less than 1/2. (iii) From Theorems 1 and 3, as well as Theorem 2.2.1 and 2.2.2, it is seen that the Kottman and (James')nonsquare constants for L^(£l) and Z/*(Q) have the same lower bounds. But this is not true for the sequence spaces l^ and /*. For the intermediate spaces, the following inequalities hold: Theorem 6. Let $ be an N— function, &Q(U) = u2 and $s be the intermediate function considered as 0<s0.

(15)

Then for Z/* S )(Q) and L*s(il) on a a— finite space (£7, £,//), max{K(L^(ty,K(L**(ty)}

< 21-",

(16)

or equivalently ~ .

(17)

Proof. Since by Lemma 1.4.7, $s e A 2 (0) n V 2 (0),0 < s < 1, it follows that Z/*')(n) is reflexive. By Theorem 1.4.14, for any ft € £(*•)(£}), l 0 with Y^ Ci = 1, one has 1=1

where c = max(l — Cj : 1 < j < n). Consider ||/;||($s) — 1 and for i ^ j, let ft ~ /,•!!(*.) > ^o > 0. Taking a = l,n > 2, in (18), we get

Letting n —>• oo in this expression, it results that RQ < 2 1 ~2, and since the sequence {fi : i > 1} with these properties is arbitrary, we conclude that

5.3 Packing in Orlicz function spaces

173

< 2 1- 2. By a similar argument, using Theorem 1.4.14 for the Orlicz norm, the other inequalities obtain. D It is remarked that, if also $ € A 2 , then s = 0 can be allowed in the above result, since (18) holds for $ s ,0 < s < 1 in that case (cf. Cleaver[2]). But the theorem as stated is more general. Taking £) = [0, 1] or R+ and fj, as the Lebesgue measure, we deduce the following: Corollary 7. For an N— function $ and the usual associated intermediate function <& s , 0 < s < 1, we have 1 max (—,2/0 < {K(LM[0,1]),K(L*'[Q,1])}<2 -* L J

21-*.

(19)

(20)

Proof. By Theorem 1.1.15, 2a$s/3^+ = 1 and 2a^s/3^,+ = 1, where ^ is the complementary A^— function of <3?s. Thus, (19) and (20) follow from Theorems 1, 3 and 6. D We can deduce the following exact bound, due to Wells and Williams ([1], p. 97), from our results presented above. Proposition 8. Consider the Lebesgue space L P (Q)) where D = [0, 1] or R+ with Lebesgue measure. If I < p < oo, then

(21) or equivalently, P(L"(n)) = max - 1-^, —-r . + 2 ~p 1 + 2 Proof.

(22)

Let $ p (u) = \U\P, then L p (fi) = L^")(n) and also

Q!$p = /5$p = a$p = ^$p = 2~? so by (1) and (2) we get (23)

174

V Packing in Orlicz Spaces

For the opposite inequality, consider 1 < a < p < 2, <3>(w) — \ua and <3?s as the intermediate TV— function where s — ^I") , so 0 < s < 1. Then s (u) = |w| p , so that by Corollary 7 (especially by the right side of (19) and (20)), K(Lp(ty] < 2 1 - f .

(24)

Since lim(l — |) = -, we get on taking the limit in (24) and noting that the left side is independent of a, we get the bound as 2?. This and (23) imply (21) for 1 < p < 2. If 2 < p < oo, let 2 < p < b < oo, and take $(u) = |w| b , s = ^Efj an<^ consider the associated intermediate function $S: which as in the previous case, becomes $s(u) = \u p. Since lim (1 — f ) = 1 — i, we find from the same 6->CXD P Corollary 7 that Z

K(Lp(ty} < 2^p,

2
(25)

From the previous inequality and (25) we again have (21). In other words, (21) is true for 1 < p < oo, and the bound on P(Lp(£l)) is then immediate from the relations between K(X] and P ( X } . D Theorem 9. Let $ be an N— function and <3>s be its associated intermediate function, 0 < s < 1. Then we have when $ ^ A 2 (oo) D V 2 (oo) ^(L(*^[0, 1]) = 2 x -f = ^(L*s[0, 1]),

(26)

K(L^(R+)) = 2 x -f = K(L**(R+}),

(27)

or

A

or

= P(L**(R **++)). Proof.

If $ ^ A 2 (oo) n V 2 (oo), then ^$ = 1 or a$ = \. Thus by (19), max — ,2/3$ a ] = 2 ^ 5 ,

5.4 Packing in reflexive Orlicz spaces

175

which gives (26) by (19). Similarly, using (20) one gets (27). D Let us conclude with the following illustration: Example 10.

Let (the ./V-function) M be given as the inverse of ,, , f eM, M~l(u) = < , (logu) i , [ U4[e J2,

0
Then or

where 17 = [0, 1] or R+ and /j = Lebesgue measure. Indeed, if we consider the N— function $ given by o

e


then M~l(u) = [$~l(u}}s=i, and C$ = oo so $ 0 A2(oo) in Theorem 9. Hence (28) follows from (26) and (27). It should be observed that, under the hypotheses of Theorem 9, we have

where J(X] is the nonsquare constant of X in the sense of James (cf.,e.g., Section 2.4). 5.4 Packing in reflexive Orlicz spaces The main purpose of this section is to establish that the packing constant of reflexive Orlicz space I/$(0)[L^)(J7)] is strictly less than half. We begin with the case of the Orlicz norm. Theorem 1. Let $ e A2(oo) f| V2(oo) so that L$[0, 1] is reflexive. Then for any fa G L$[0, 1], ||/i||$ = 1, i = 1, 2, 3, one can find an £0 > 0 such that /i - /J*, 1 < t, j < 3) < 2(1 - e 0 ).

(1)

176

V Packing in Orlicz Spaces

Consequently, K(L*[Q, 1]) < 2(1 - £0) < 2 and hence P(L*[0,1]) < \. Remark. A riormed vector space for which (1) holds is termed 'p(3)-convex'. Proof. The argument is divided into steps for convenience. I. We first derive some inequalities for $ e A 2 (oo) f) V 2 (oo) to be used in the proof of (1). Thus consider 1 < A < B < oo and define u0 > 0 such that (to use this work later)

«•-Ml ('-!)]•

2

<>

or equivalently, $(5u0) = 1 A _ 1 £( 5 2 V* A, We now take A and B as the particular numbers:

A = inf {k > I : ||/||$ = —(1 + and

B = sup {k > 1 : ||/||$ = —(1 + Then one has 1 < A < B < oo (cf.,e.g., Chen [1], Thm.1.35). With this choice let k e [^4,-B], satisfying

Thus by (3), we have with (j, again as the Lebesgue measure i /

K J[\f\>u0]

$(/fc/)^ = 1 - T - T f K

l

k J(\f\
- ~ (4)

which will be used later on in the proof. On the other hand $ G V^oo) and hence there is a 0 < 6 < 1 satisfying

B

\

(16)B , , V

(4)

5.4 Packing in reflexive Orlicz spaces

177

Now * /* and ^ /> as u /*, so that for A < a, 6 < 5 one has ^ < and so with (5):

\

/

-

B

\A

Hence aw] < (1 - 6} -r$(aw), \au\ > UQ. a+b ) a +b But $ € A2(oo) also. So there exists K > 2 such that , |w| > mm(u 0 ,

y4

).

(5)

(6)

This is because if the original A2(oo)-definition gives the inequality (7) for |u| > HI > 0 and a KI > 2, then for any 0 < u-2 < HI we have by continuity of the TV-function $, sup{^^ : u2 < u < HI} — KQ < oo and thus <&(2w) < K0$(u} for HI > u > u2, and taking K = max(A'o, ^i) yields (7) for |w| > u2. In the other words, by increasing K one can take a strictly positive lower value. Thus in (7), take u2 = min('Uo, ^a). Consequently, with this it is possible to find a C > 1 such that BK iSu

!K

u 0 .

(7)

Similarly one can find 0 < A < 1 (using (6)) such that

We now assert that for \v < |w|, where \u\ > min(uo, ^p-),

(u).

(9)

In fact, if v = 0, this is true and trivial, and if \v\ > 0, by (7) and the convexity of <£, one has

u

V Packing in Orlicz Spaces

178

giving (10) in all case. II. Set h = min(<52, £) > 0, and take 0 < e0 < 5, satisfying I2£0(K+$(u0)) < |(1 - J). For /i 6 L*[0,1], H/illa = 1, there are ^ € [A, B] for which

(10) Define a new function F by +

Ji ~~ Jii+l

.,

in which we set k4 — ki and /4 = f i . We assert that

(H) This key inequality will be established in III below. If it is given, then (1) can be deduced as follows. By (11) and (12), z+1

fi~

+ P*

1-eo

1 r = 2 y^ k— + J / Fd/j,, i^i i o

by definition },

of F,

6y(12),

Hence < 2(1 1=1

so that (1) is true, and it remains to prove (12). III. Inequality (12) is true. Since 0 < £Q < |> one 1-

+

, , and a =

l-£0

(3 =

5.4 Packing in reflexive Orlicz spaces

179

the last a + 0 — I is a convex combination. Then F of (12) may be bounded by the following: t+1

i -r

i -r i+i

i+i

^ t ^r^ f («» + w* fit^-w - /«)! + i=1

ACiACi+l

I,

L/Cj + Ki+l

J

by (7). For convenience let 3

Since 1 < A < ki and 0 < SQ < |, we get F <

-G + -

—G+—-—-—r$(uo) < G + 2K£QG + l2e03>(u0). (12)

Now let ki be the smallest of the A^s, and let QO = {t : max \fi(t)\ < UQ} C 0, = [0,1], and

Then

and by (4),

i)^>i(l-Tl-

(13)

We now estimate the size of F (or G using (13)) on fix, ^2, ^3- Consider Divide it into these disjoint (measurable) parts as follows: £i = { t e n i : ( / i / 2 ) ( t ) > 0 , or (/i/ 3 )(t) > 0},

V Packing in Orlicz Spaces

180

E2 = {t e OA^i : max(|/ 2 , /3 and

5_ ~K

E3 = {t e flA^i : max(|/2|, Now ifttEi

K

and (/i/ 2 )(t) > 0, then from (6): Aa&2 / ,

• (14)

u i / i j , since

But &4 = k\ > 1, /4 = /i and <3> is even and convex , so one has 3

~r '(/i-

z=2

_L -f-

,

0 +

"-i + "-i

(16)

It follows from (15) and (16) that for t G EI (with a similar argument to 3

f \

fa/1^

X

(17)

i)-

A;,-

Next let t e £^2 and suppose /2 (*) > ,/3|(t) so that |/2 (t) < -||/i (t), and by (10) and (6) one gets: ki + k-2

-^(/i-^))

<

1 + K-2

1 + ^ I/

4 j + A;2 / 1 + A . 6

'77 '

<

(18)

Similarly (18) holds if |/2|(t) < |/3|(t) in E2. Hence form (16) and (18) we find 3

*"""

^

i).

(19)

5.4 Packing in reflexive Orlicz spaces

181

If t e E3, and |/2|(t) > |/3|(t), then using (8) and (9), one has (/2 - /a)

<

I ,

3

$ ,

y

A;2

3

, *2/2

< ^r^-A^ l^fe/i

where we use (8) and the inequality

> Using the same method as in EI, we find for all t £ E3,

i=i

i

i)-

(20)

Thus on QI we have from the choice of h at the beginning: ,). i=1

Aj

Al

In case i G ^2? a similar computation shows that ^

t=l

"'i

K

1

22-

But |/2 1 (t) > |/i|(£) for i G £1% and fci is the smaller of the two /Cj. So,

(21)

V Packing in Orlicz Spaces

182

and hence (21) holds on (72 as well. The same argument can be repeated to conclude that (21) holds on £73 also so that that inequality is valid on Consequently by (13) one gets:

+2Ke

(22)

By the convexity of <$, one has on (23)

This and (13) give: / Fdfi <2 f Jfln

-

(24)

Jfli

Adding (22) and (24) and using (14), we get / ± CLLL ^ Zt I /o •//o 11

i/j£

y ' "^i

•

" lc -

»V^

CLLL [

, * V V\

^i

-+ 2Ke0 I 7n

A

SJL.

(25)

Since by (23)

j! and 0 < £Q < |, we have 2Ke0 f Jo,

(26)

Thus (25) and (26) imply the desired inequality (12), and with it the result follows. D The above proof implies also the following: Corollary 2. If $ € A 2 f| V 2 so that L^(R+) is reflexive, then it is p(3)convex. Consequently K(L*(R+)) < 2 and P(L*(R+)) < \.

5.4 Packing in reflexive Orlicz spaces

183

For the gauge norm, Ye, He and Pluciennik ([1], Th.2) and Kolwicz [1] established the following: Theorem 3. Let f2 be [0,1] or R+. I f l W ( f y is reflexive, then it is p(3)convex so that K(LW(ty) < 2 and P(Z,(*>(n)) < \. By Corollary 3.2 (ii) and 3.4(ii) along with the above results we can state the following: Theorem 4. Let $ be an N-function and X$, be any one of the spaces L$(0) or Z/*)(f2) ; where Q = [0,1] or R+ with Lebesgue measure. Then A$ is reflexive iff K(X^} < 2 or equivalently P(Xq>} < |. Bibliographical notes. Rankin [1] was the first to introduce the packing constant. Theorem 1.3 is essentially due to Kottman [1], but formula (7) there was given by Ye ([1], Lemma 1). The results of Section 2 are taken form the authors [2] and Yan [Ij. Sections 3 and 4 are adapted form Ren [1], and are a completion of Cleaver's [2] work for function spaces. We now proceed to other applications of Orlicz spaces to an analysis on Fourier series among others.

Chapter VI Fourier Analysis in Orlicz Spaces

This chapter presents an account of the convergence of Fourier series in Orlicz spaces on compact abelian groups, and especially contains an analysis of conjugate functions when the underlying space is a circle (torus) group G. We start with G = [0, 27r] and present Ryan's theorem that the conjugate function mapping on L*(G) is bounded iff the space is reflexive. Then we extend the result for certain subspaces (especially M$ determined by trigonometric polynomials) in L*(G) for a general class of TV-functions, $, and then give some extensions of these results for more general groups G. Only brief accounts of the latter can be included, since the necessary auxiliary results and applications are too numerous, demanding separate book length treatments. 6.1 Preliminaries on Fourier series Let 17 = [0, 2w] and p, be the Lebesgue measure, i.e., dp,(x) = dx, then for / G Ll (17) the Fourier series of / is given by 00 1 f f(x) ~ Y^ cketkx. ck = ck(f] = — / }(x]e" ,k= — oo ZTT Jfi J

\

/

/

j

"•

I

n>

*v \J

/

f\

I

*f

\

/

dx.

\

(I) /

The series is a formal expression, and called the Fourier series. It is necessary to discuss the sense (i.e., convergence properties) in which the series represents /. Now the partial sum Sn[f] is given by 2?r

Sn[f](x) = £ che** = - / f(t)Dn(x - t)dt, |fc|
(2)

^o

where r> 4- - J X J

184

n

— 0 1 2 •••

(3)

6.1 Preliminaries on Fourier series

185

is called the Dirichlet kernel of order n. Since /^(fi) is finite (p,(£l) = 2?r actually), we have L*(fi) C L1^) for any ./V-function $ and hence for / e L*(ft), 5n[/] is well defined. We start with the following two technical lemmas due to Lozinski [1] and [2], since they play a key role in this work. Lemma 1. //<3> is an N-function with (f) as its (left) derivative, then &

250

(4)

v(f>(v)

for v satisfying v<j)(v) > 1. Proof. Observe that ncos[(n+DO;] 2sinf

sinnrr 4(sinf) 2 '

implying (after an estimation) <

2n?r , 0 < x < TT, n > 1. x

We estimate on symmetric subintervals /^ = [xk — ^^^k + nj^] with xk = f^TT, A; = 1,2, • • • , n + 1 for fixed but arbitrary n. Thus x e I^} implies that 2n?r *tni — 4. —

[

2n+l

3n i

A>I\I

lOOn

10n2

A

2n+l

and

10n2 lOOn

2n (2Jfe-l)7r

n 5fe'

(5)

186

VI. Fourier Analysis in Orlicz Spaces

Consider v > 0 such that v(f>(v) > 1, and let /CQ be the integral part of For any 0 < E < (j)(v), define a function g on f2 = [0, 2yr] as f ((j)(v) - e}sgnDn(xk}, 0(2;) — < 10

x e f\ k = 1,2, • • • , k0, . fco / ^ x € fA U I •

Note that Dn(x)Dn(xk) > 0 if x 6 /£ , and consider the complementary H function ^ : v »—)• / ip(s}ds of $ so that: o

/Co

'50n :

4 < 1, £\n

(6)

since 5r is simple, (f> and ^ are inverse to each other, and the definition of kQ is used. Form (5) and (6) we get ko

n 5^^ ' '

250

j^[k ~

/

y

50n

')\ — £ . n 6 250 u0(u)'

establishing (4). D Another estimate of importance is given by: Lemma 2. For an N-function <J>; suppose that there is a constant A > 0 such that

\\Sn[f] * < A\ f \ \ t , /eL*[0,27r], n > l .

(7)

one has / n 4- (frf?;"! \

, 0 < u < oo.

(8)

187

6.1 Preliminaries on Fourier series Proof. have

With ^ denoting again the complementary function of <E>, by (2) we 27T

27T

- f

= sup

7T J 0 27T

1

f(t}Dn(t-x)dt g(x)dx

27T

r

- \ g(x)Dn(t - x)da f(t)dt

= sup

7T J

27T

=

f(t)Sn\g](t)dt

sup

(9)

But 1

\Sn[g](t)\ =

27T /

— / g(x)Dn(x — t)dx TT J

< —||^||(*)||-Dn||$> < -

— ^ i\\D ' n $, since

by Holder's inequality < 1.

(10)

Now Sn[g] is a trigonometric polynomial. By a classical theorem of Bernstein, one has ' d

dt Sn\g](t)

(11)

With the definition of Orlicz's norm, given e > 0 there is h£ satisfying : 1 such that 2?r

nll* -

<

Dn(x}h£(x]dx =

Dn(x - 7r)h£(x - 7r}dx

0

3?r

=

27T

/ Dn(x - 7r)g£(x)dx = I Dn(x - 7r)g£(x}dx

= 7rSB[&](7r), by (2),

(12)

188

VI. Fourier Analysis in Orlicz Spaces

where g£(x) = he(x — TT) and h£(x ± 27r) = h£(x),x G [0, 27r]. It follows from (11) and (12) that for t e [TT - ^, TT + £] „[&] (TT) - ~\\Dn\M}

(13) Let fu(i) = uxi(t),u > 0, so that p*(/u) - ^. By (9), (13) and also < 1) we have 27T

fu(t)Sn[g£}(t)dt 27T

>

(11 A,||* -

(14)

2n?r Hence from (7) and (14), one obtains

1+

2n?r (IIA.I Thus

IIAJI* < which, by the arbitrariness of e > 0, implies (8). D With these two useful estimates of the norm of Dirichlet's kernel, we have a key result as: Theorem 3. (Ryanflj)

Let 3>(u) — / (/)(t)dt be an N-function. If there o is an A > 0 such that (7) holds for all f G L*[0, 2vr] and all n > 1, then $ G A 2 (oo) n V 2 (oo); so -L$[0, 2?r] is reflexive.

6.1 Preliminaries on Fourier series

189

Proof. By Lemmas 1 and 2, for vcf>(v} > 1 and u > 0 we have 0(t;) log -^ < 250HAJI* < jf n + <^, v(p(v) u

(15)

where /C = 5007rA Let A > 1 be such that log (A - 1) > 3K and choose v0 > 0 such that

$(^0) > 1 and Y<£ (Y) > 1 for v > v0. A

V A/

(16)

But for v > ^o °ne has V

(17) Let n — 1 be the integral part [$(v)] of $(w) and by (17) we get

Also for ^ > ^o, = K[*(v)]

+ ! + *(„)

< 3K $W <

~ Now by (16), v may be replaced by | in (15) and then (18) and (19) give

0(T)< l o g ( A - l p " Letting / — -°s!^f

> 1, the above inequality gives : (f)(v),

V > V0.

This implies that $ 6 V2(cxD). In fact, if u > i;o, then Au

:

/ l(f) I — I dv XVQ

Au

<

/ ( \VQ

(20)

190

VI. Fourier Analysis in Orlicz Spaces

If a — |(/ + 1) > 1, (a < /) there is UQ > VQ such that , u > UQ.

(21)

, then tf(v) = Ci*(^-), and $i(u) < $ 2 (w) for w > u0 > 0 implies ^i(f) > ^^(v} for 02( w ) > w o- So (21) gives that there is a v\ > 0 such that Aa^(^) > ^(f ) for w>vi. Letting v - ^, we have ,

- . Aa

So ^ € A 2 (oo) or $ 6 V 2 (oo). On the other hand, by (7) and (9) one has for g € £*[0, 2?r] = sup< f

g(t)Sn(f}(t)dt

I 0

< sup{|M|w||Sn[/]||* : < A\g\\9. Hence H ^ ^ J H ^ < 2^4||p||^. We then can repeat the above procedure with ^ in place of $ to conclude that $ e A 2 (oo) so that $ E A 2 (oo) D V2(oo) and hence Z/*[0,27r] is reflexive. D The converse is also true, and it is established in the next section. We conclude this section with the following result on AMiinctions. Proposition 4. An N -function $ 6 V 2 (oo) iff for some A > 1, r mf • **('A")' > ^ i1. lim A proof of this can also be found in Chen ([l],p.9). Hence (21) implies that $ G V 2 (oo), also as a consequence of this result. 6.2 Conjugate functions and Orlicz spaces

Since L*(Q) is a subspace of L X (Q) and hence for each / G I/ $ (^), whose Fourier series is well-defined, it is natural to study the conjugates / of such /. We first recall the concepts of a conjugate function, and proceed to discuss some properties, and then to prove the converse of Theorem 1.3.

6.2 Conjugate functions and Orlicz spaces

191

Definition 1. For any / € L*(fi) cL1(ty,£l = [0, 2ir), the conjugate f of / is given by

'-***-

<»

It is a basic result in the subject that the conjugate function / of (1) exists a.e., and / is also given by

2 tan f I

2

£<|X|<7T

J

(For a proof, see Zygmund [1, Vol.1, p. 131].) The definition, as given, is unmotivated, but can be justified as follows. For each / and x € 0 consider its even and odd parts defined as:

(even) ex(t) = ^[f(x + t) + f(x - t)]; (odd) ox(t) = ~[f(x + t) - f(x - t)], u £ 00

so that f(x + t) = ex(t) + ox(t). If / is integrable, S[f](t) = £ cnemt and ~

S[f](t)

oo

.

mt

n=—co

~

= — i £ sgn(n)cne ; thus S[f] is the complex conjugate of S[f]. n=—co

The behaviors of S[f] and the conjugate series S[f] are crucially dependent 7T

on the properties of ex and ox at t = 0. In fact, if / |o x (£)|7cft < oo, then one jj

can show that the series S[f] converges at x to f ( x ) , the conjugate of f ( x ) (see Wheeden and Zygmund [1, Ch.12] for a quick view of these results). Thus if / G L1^), so that S[f] exists and the conjugate series S[f] is the Fourier series of some function g e Ll(tl), then g(= f) is the conjugate function of /. The following theorem, due to M. Riesz (cf., Zygmund [l,Vol.I, p.253]), which will be used in the following, explains the concept further. Lemma 2. (M. Rieszfl]) constant Cp > 0 such that

If I < p < oo, then there exists an absolute

< c II f II \I \ Jfli\ \ p S t-'pll/llp)

p

r (o\ /f (^ ^ Li (II),

so that the mapping T : f H-> / is a bounded linear operator on Lp(£l).

(?} (Z)

192

VI. Fourier Analysis in Orlicz Spaces

It is interesting to note that Pichorides [1] proved in 1972 that the best constant Cp in (2) is given by: Cp — tan^ 1 , if 1 < p < 2; = cot I1, if 2 < p < oo. Definition 3. For each measurable function / : 0 —> JR, let mf(X) = fi{x e fi : |/0r)|> A}. Then m/(-) is termed the measure function of / determined by fj, . If /.i(O) = 1 and Ff(X) — I — m/(A), then Ff(-) is called the distribution function of / in Probability Theory. Unfortunately some times in the literature, especially in trigonometric series, m/(-) is also termed a distribution, and we shall avoid this conflict by calling it just a measure function. Let T : Z/°(Q) —>• L°(Q) be a linear operator, where L°(fi) is the vector space of all real measurable functions. If p, q > 1 are any real numbers, suppose that for each / e £ p (ft), Tf <E Lq (17) and there is a constant C(= CPA > 0) such that \\Tf\\g < C \ \ f \ \ p , (3) then T is said to be of strong (p, q)-type, so that T is a bounded linear operator on LP(Q) into Lq(ty, and the least C is called the norm of T, denoted ||T||pig. On the other hand, let the linear operator satisfy for l < p < o o , l < ^ < o o and any A > 0 that mr/(A) = p{x € 0 : \Tf\(x) > A} < y l l / H p , / € /^(fi), \ A

/

(4)

for an absolute C(= Cp > 0), then T is called a weak type (p, q). Here T need not be bounded. The least 0 < C < oo in (4) is called weak (p, q)-norm of T denoted by iy||T||p)9. If no such C exists in (3) [or (4)], we set ||T||pi9 = +00 [or u>||T||p)g = +00]. It is seen that each strong type operator is also of weak type, but not conversely. We have the following. Lemma 4. (Zygmund [1, Vol.1, p. 134]) The mapping T : f H-> / of (1) is of weak (1,1) type, but not necessarily strong (1,1). In fact, ||/||i, A > 0 , for some absolute constant b > 0.

6.2 Conjugate functions and Orlicz spaces

193

This result will be useful for Theorem 7 below. First we need to introduce some more terminology. Definition 5. A mapping T, as above, is called quasi-linear if |T(a/)|= a||T(/)|, and \T(h + /2)| < CQTWl + |T(/2)|), for a constant C > I and it is termed sublinear when C = 1 here. If Tf — / in (1), then it is in fact a sublinear operator. The following result, due to Marcinkiewicz [1] (for detail, see Zygmund [1, Vol.11, p. 116]), will be used to prove the converse of the result in the last section, given as Theorem 7 below. A generalization of this, due to Riordan proved in (1956), is also available and presented in our earlier volume [1, p. 247]. Lemma 6. Suppose that a quasi-linear operator T is simultaneously of weak types (a, a] and (/3,/3) where 1 < a < (3 < oo,^j({7i) < oo,i = 1,2. // an N '-function $ € A2(oo); satisfies

7

J

and 1

for large u, then g = Tf, f e L®(/J,I}, is defined and satisfies: K,

(6)

for some K > 0 independent of f, where (fij, £)i, /z;),i = 1,2 are measure spaces on which the L®([j,i) are defined. We now can establish the previously announced: Theorem 7. (Ryan [1, Theorem 3]) Let $ € A2(oo) n V2(oo). If T is a bounded quasi-linear operator defined on I/p[0,27r) —>• Lp[0.27r) for each I < p < oo, then T is defined on Z/$[0, 2?r) —> L*[Q, 2?r) and is bounded. Proof. By Lemma 2, T : f —> /, / e Lp[0, 2?r), 1 < p < oo implies that it is of strong type (p,p), and by Lemma 4 it is of weak type (1,1). These imply that the first part of the hypothesis of Lemma 6 is satisfied. We now assert that there exists an TV-function $1 equivalent to $ (i.e., $ ~ $1 for large argument) such that <3>i satisfies (5) for some 1 < a < /3 < oo.

194

VI Fourier Analysis in Orlicz Spaces \u\

Let <3>(w) — / (f>(t)dt. o

For convenience we may take
l«l and strictly increasing on (0, oo), since
1< A* = liminf

< limsup^ = B* < oo.

-

(7)

Let 6 — \(A$f - 1) > 0 and put a = A$ - 6, b — B® +6. Then 1 < a < b < oo, so by (7) there exists MO > 0 such that 1 < a < - - - < b < oo, t>u0.

(8)

Hence for u > UQ we have u

u

. .

u

a /• 0m /" b -at < I —rrdt < / -at t ~ J $(t) " 7 t

UQ

MO

MO

and so (see also our volume [1], p. 26)

Now define [

3>(u),

\u\ > Mo-

Then $ ~ $1 and <j)i(t) = $\(t) satisfies K a < T < 6 < o o , t>0. $iW

(10)

We now verify that $1 establishes (5) for suitable constants a,/3. For a, b of (8), choose a, /3 such that l < a < a < f e < / 9 < o o . Then by (10)

6.2 Conjugate functions and Orlicz spaces

195

and 1 ^ — I[Wa *^n at > a IA* —r~r«t a+1 J t ~ J t

/fa*i(t J ta

(12)

Integrating by parts, gives (13) and

r d<&i(t) _ f <&i(t J ta J ta^ o o Combining (11)-(14) appropriately we get

(14)

fVl(t

J t?+l and a — a:

u

Consequently, $x satisfies (5). It follows then from Lemma 6 that there is a constant K > | such that 27T

+ 1 -

If ||/||($l) = 1, then p ^ ( f ) = 1 ($ e A2(oo)!) so that (15) gives

o

N

'

o

and hence ||T/||($l) < IK for H / H ^ ) = 1. This implies

Since $ ~ $1, there exist 0 < di < d2 < oo such that di||/||$ < d2||/||* so that IIT/II* < 4X||/||$ or T is bounded. D

(15)

196

VI Fourier Analysis in Orlicz Spaces

The announced main assertion can be given by: Theorem 8. (Ryan [1]) Let $ be an N-function. Then the following are equivalent where f denotes the conjugate of f : (i) L*[0,27r) is reflexive. (ii) There is a constant C > 0 such that for all f e L®[Q, 2?r) 11/11* 0 such that for all n > 1 and f € L®[0, 2?r) < 41/11*-

Proof. (i)=>(ii). Let Tf = /, then by Lemma 2 and Theorem 7 this implication follows. That (iii)=>(i) is already proved as Theorem 1.3. Thus it is to be shown that (ii)=>-(iii). For this consider the averaged sums

with 5J[/] = So[/], as in Zygmund ([l],Vol.I,p.50). Then

= gn(x) sin(na:) — hn(x) cos(nx),

(16)

where gn and /in are conjugate functions of gn and hn given by ^n(a?) = f(x) cos(nx), /in(a;) = f(x) sin(nx).

(17)

(Cf. Zygmund [1], Vol.1, p.266.) Clearly gn,hn G L*[0,27r) and by (ii) Il£ n ||« < C\\gn\\* < C7II/H*, |fc||* < CII/1,,11* < Cll/ll*.

(18)

It follows from (16) and (18) that

(19)

6.2 Conjugate functions and Orlicz spaces

197

However, Sn[f](x)

=

S^[f](x)

0

and hence 27T

0

But by the Jensen inequality we have

dx 0

|_

n

\ "* "v*)

x

27T

,. 27T

Hence noting that / \f(t)\dt = \\f\\iX[o,2v) o

an

d (21)

from (19)-(21) we obtain HS»[/]||# < ll^[/]||# + 2||/||* < 2(C + 1)11/11*, which is (iii) with A = 2(C+1). D Corollary 9. For on N-function $, we /iaye Um||S B [/]-/||*=0

(22)

/or a// / e L $ [0,27r) z^<E> e A2(oo) n V 2 (oo). Proo/. If $ e A2(oo) n V2(oo), by Theorem 8, ||5B[/]||* < ^||/||*,/ € L®[0, 27r),A > 0. Since trigonometric polynomials are dense in L$[0, 2?r) = M*[0, 27r), for any e > 0 there is a polynomial n

n

Pn(x) = — + X] (a*cos ^x + bk sin kx) 2

*=-n

198

VI. Fourier Analysis in Orlicz Spaces

such that | / — Pn||* < s/(A+l). This implies since Sm\pn} = pn for m > n, \\Sm[f - pn] \*->• 5n[/]. Then {-£„[/], ra > 1} is norm bounded by (22) for each / € L*[0,2yr). Since the latter is a Banach space, by the uniform boundedness theorem we have sup ||5n|| = A < oo so that (iii) of Theorem 8 holds, whence <$ 6 A 2 (oo) D n>l

V2(oo) as desired.

D

Corollary 10. For a complementary pair of N-function ($, \&), if $ e A 2 (oo)n V 2 (oo), i/ien (7>?/ i/ie Parseval's result) we get for f G L $ [0,27r) anc? ^ G L*[0, 2yr) that f(x}g(x}dx=

^ ck(f)c_k(g}

is convergent, where c^(f} and Ck(g) are the Fourier

(23) coefficients.

Proof. Since by hypothesis L*[0, 2yr) is reflexive, so that the trigonometric polynomials are dense in both L*[0, 2yr) and L*[0, 27r), the result follows. D Example 11. Let $ p (w) = w| p , 1 < p < oo, then by Corollary 9, ||5n[/] /||p —> 0 as n —>• oo for every / e I/p[0, 2?r). This is the second part of Riesz's theorem while Lemma 2 is its first part. Example 12. If £ + J = 1,1 < p < oo, and / 6 L p [0,2?r) and g e L 9 [0,2?r), then (23) holds for these /, p. (Cf. Zygmund [1], Vol.I.p.267.) Remark 13. (a) Theorem 7 can also be proved by using an interpolation theorem due to Gustavsson and Peetre [1]. (See Maligranda [2], p. 149.) (b) For a result on Banach function spaces, yielding the proof of Corollary 9, see Bennett and Sharpley [1], p. 158. (c) Let ($,\I>) be a Young complementary pair, such that $ e A 2 (oo) and vp G A 2 (oo) so that L*[0,27r) is reflexive. Then ($, #) is equivalent to a complementary pair of ./V-functions by Proposition 1.3.4 and Corollary 1.3.10, so that reference to the Young pair is not necessary, as in Ryan [1].

6.3 Conjugate series and convergence in subsets of Orlicz spaces

199

We now conclude this section by the following observation on pointwise convergence, which is a more delicate problem. Indeed, in 1876 du BoisReymond constructed a continuous function with a divergent Fourier series at a point, and extension of his work shows that the same is true for a dense set of points in [0, 27r), which however has Lebesgue measure zero. Then in 1915 Luzin conjuctured that the Fourier series converges a.e. for each / E L 2 [0,27r). In 1926 Kolmogorov showed that there exist / E Ll[Q,2n) whose Fourier series deverge a.e.. Then in 1966 Carleson [1] finally proved Luzin's conjuture, namely that the Fourier series of every / E L2[0, 2?r) converges a.e. His proof was extended by Hunt [1] to cover all functions / E Lp[0, 27r), 1 < p < oo. Thus we have the final result as: Theorem 14. The Fourier series of f E Lp[0, 27r),l < p < oo, converges a.e. (and this is known to be false both for p = 1 (Kolmogorov) and p = oo). Corollary 15. If <& E V2(oo) is an N-function, then the Fourier series of f E L$[0, 2;r) converges a.e., and also in mean if further <E> E A 2 (oo). Proof. By definition, <£ E V 2 (oo) implies the existence of a > 1 and UQ > 0 such that u \a *&(u) — 1 < VT—r, u > UQ, UQ/

®(UQ)

(cf.(9)). Then it follows that L*[0, 27r) C La[0, 27r), the inclusion being continuous, i.e., ||/||a < C||/||$ for some C > 0. Hence Theorem 14 implies the a.e. convergence assertion. The last statement in mean convergence follows from Corollary 9. d

6.3 Conjugate series and convergence in subsets of Orlicz spaces Recall that for pairs of TV-functions $1? 2, one can introduce a partial ordering "-<" defined as: $1 -< <&2 [ or $2 >- $! ] iff &(u) < 3>2(Ku),

U > UQ,

(1)

for some K > 0 and UQ > 0. With this, the set of TV—functions becomes directed. Again we consider the circle group fi = [0, 27r), and p, as Lebesgue measure on it. For each / E LI(^L) and 0 < r < 1, define the Poisson integral

200

VI. Fourier Analysis in Orlicz Spaces

of / by the convolution equation 00

/(re") = (P(r, •) * /)(*) = £ r^f(n)eint,

(2)

n=— oo

where as usual n Then it is well-known that P(r, •) * / is harmonic in the open unit disc of the complex plane, and lim/(re lt ) = f ( e l t ) , a.e. The harmonic conjugate f of f r—>1 is defined as

/(re*) - ~i E sgn(n)rH/(n)e int

(3)

and it is also known that lim/(re lt ) = g(t), a.e. for each / £ r—>1

/(e«) = o or <3>0 -< $ -< $!. // M® is the closed subspace generated by the simple functions (i.e., the MorseTransue subspace) 0/L*(/u), then for each f £ Ai$, its conjugate f exists and f £ .M*. Moreover, there is a constant K$ depending only on $ such that ll/ll* < ^«ll/ll*. (4) We use this result only when $ £ A2(oo) fl V2(oo) so that .M* = L^(^JL) and it is reflexive. In this event (4) and the proposition itself is a special case of Theorems 1.3 and 2.7, due to Ryanfl]. The case when $ does not satisfy this growth condition, which is possible (cf. Krasnoselskii and Rutickii[l], p. 28, the example given there shows that there exist such $ ), then the argument is long and tedious, and we shall omit it here. A possible method is to use Green's formula and extending the argument of M. Riesz's. An outline of it has been included in Rao [12], which is already long and not revealing. The result is stated here with the hope that some simple proof

6.3 Conjugate series and convergence in subsets ofOrlicz spaces

201

may be found in future. Note that M* ^ L*(IJL) in general and Ryan's results (given in the preceding sections) show that the inequality (4) may not hold for / G L*(/z)\jM*, as a warning note to future analysis on this topic. We now consider some results on the strong convergence of the partial sums of Fourier series of /, which is related to conjugation / of /. Here we should like to unify and include the n— torus case, i.e., Q = [0, 27r)n. Since / 6 L*(p,) C LI(IJL), f has a Fourier expansion, namely, if f ( n ) =

sn[f](t) = £

j=-n

We then have: Proposition 2. Let $ 6 A2(oo) n V2(oo) and f G £*(//). Then Sn[f] ->• / strongly (i.e., in norm) as n —t oo. Proof. We recall that a Banach space (X, \\ • ||) contained, as a set, in admits conjugation by definition if each / 6 X has a conjugate / again in X. By Ryan's theorem (or Prop. 1 above) X = L?(IJL} admits conjugation. Moreover, the norm || • ||(— || • ||$) has the following additional properties since it is also absolutely continuous, i.e., ||<7X>inlk ~* 0 f°r anY Ai 4- 0 : (i) ||/en||$ = ||/||$ for any en : 1 1-> emt,t G [0, 2?r) = Q, called the character of fl; (iij (rhf)(t) = f(t -h), h £ SI (addition mod(27r)), then ||rh/||# = ||/||#; and (iii) ||(T)U — r/i2)/||$ —> 0 as h\ —>• h^. Any Banach space (X, \\ • ||) contained in Ll(i 0 as n -> oo. [see below in Prop. 3 for details of this result.] Thus the assertion for reflexive L^(n) spaces is a consequence of these facts. D This result in the case of L $ (^/), 1 < p < oo, has been known when 0, is replaced by a compact abelian group, using methods of complex analysis (cf. Rudin [1]). We now present a few extensions to Orlicz spaces in the manner of Proposition 2 above. Let us set up the problem, in general terms, to be specialized later by adding further restrictions. Thus let O be a compact group with // as its normalized Haar measure. [If Q = [0, 27r)n,

202

VI. Fourier Analysis in Orlicz Spaces

group operation being addition (mod(27r)), then d/j, = j^^duj] When $1 is not necessarily abelian there is no dual group, and we preceed differently. Let G be a compact group and p, be its normalized Haar measure, i.e., p,(G) = 1. Since there are no nontrivial characters of G, we define the Fourier transform on G by using its unitary representations on a Hilbert space as follows. Let U(a) = {u<*, i,j = 1,2, ••• ,da}, a 6 A, be a complete set of unitary irreducible inequivalent representations of G intoL 2 (G,yLi), where A is an index set and 1 < da < oo are integers (da = I iff G is abelian ). Also u = u and ,.

= 1,2, • • • , d a , x e G.

/

This is a consequence of the classical Peter- Weyl theorem. For details of unitary representations and the famous Peter- Weyl theorem, we refer the reader to Hewitt and Ross ([l],Sec. 27). Then C L2(G» is a complete orthonormal set. For / G L 2 ( G , f i ) , one defines the Fourier transform / as ,

(5)

here the factor \fd^ is put in for convenience, to normalize wg's, and will be omitted later on. Let J be the class of all finite subsets of A, directed by inclusion (/i , 72 € J, then 7\ -< 72 iff /i C /2 defining a partial order, and for any pair I,JeJ there is a K € J7 such that / -< K, J -< K, namely K — I U J, so J is directed) and consider the Fourier partial sums defined by Sr(f) = E V^ E /(*^»«5' a&I

7e J

-

^

i,j=l

Then the Peter- Weyl theorem implies that ||S/[/] - /||2 —>• 0 as / ->• oo in the above partial order. I f p / 2 , l < p < o o and L P (G,^) or I/*(G,/^) is the corresponding function (Lebesgue or Orlicz) space, and if G is also connected and abelian,

6.3 Conjugate series and convergence in subsets of Orlicz spaces

203

we can define the conjugate function / of / 6 Lp(G,/j,) or L$(G, /^), using the fact that G has the dual group F which is discrete and, what is more decisive, F admits a linear ordering compatible with its topology. Namely, there is a closed semi-group P C F such that (i) P(JP~l = F, (ii) PnP" 1 = {0}, P"1 = {p~l : p G P}, so that for x, y e F one can define x < y iff yx~1(= y - x) G P, (cf., e.g., Rudin ([1], p.197, 8.1.8). Using "+" for group operation and "— " for inversion (set ufj = ua since da = 1 for a G A = F here) , one defines a trigonometric polynomial as h = ]P a7w7 (finite sum)

(7)

and its conjugate function (also a similar one ) as h = — i ^ a7u7 + i ^ a7u7. 7>0

(8)

7<0

Let (Tyf)(x) = f ( x y ~ l ) , so ry is the translation operator. The desired extension of the result used in Proposition 2 above, on homogeneous Banach spaces, can be presented as: Proposition 3. Let (X, \\ • ||) c Ll(G,(j,) be a homogeneous Banach space, i.e., (i) || • ||i < || • ||, (ii) 1 1 /it 1 1 = ||/w||, u G F, the dual of G (the compact abelian group), f G X, and (iii)||ry/ — ryof\\ —> 0 as y —>• y0, / G X. Then X admits conjugation iff it admits convergence in norm. Proof. For convenience, the argument is given in steps. 1. For each / G X,j € F, let /7 = f f(t)j(t)dfji(t), where 7 ^ 7(i) = G

(t, 7) is a duality mapping of G —>• C, so that (0, 7} = 1 = (i, 0), {— i, 7) = (t,j}~1 = (t^>, and fata, 7) = fa,7>fa,7), <*,7i + 72> = fa7i)fa72) for ti, t-2, t € G and 71, 72, 7 6 F. Here the same operations "+" and "— " in both G and F are used. Let / be a trigonometric polynomial and / be its conjugate given by (7) and (8), so a7 = fa in their definitions. Set F/ = ^2 / 7 (-,7) in 7>0

the above chosen ordering, so that Ff = \(fo + f + if) • We claim that the mapping T : / H-> Ff is linear and bounded iff / G X. To see this, note that if / e X, then ||/|| < C||/|| by definition, whence \\Ff\\ <

(\\f\\ + \ \ f \ \ + C \ \ f \ \ ) =

204

VI. Fourier Analysis in Orlicz Spaces

and T is bounded (and clearly linear) on the set of trigonometric polynomials. These are dense in X by condition (iii) of the definition of a homogeneous space, so that T is bounded on X. Conversely, if T is bounded, since / = -i(2F / -/-/o), by definition, ||/|| < 2(||r|| + l)||/||, with (1) of the definition of the space X. Consequently, / G X. The general case follows again by the density of trigonometric polynomials in X using (iii). 2. We next assert that X admits norm convergence iff the set of mappings S/ : X -* XJ e J, is uniformly bounded, i.e., ||5/[/]|| < K0\\f\\Je X,I e J, for an absolute constant K0 > 0. In fact, by the density of trigonometric polynomials in X, for each e > 0 and each / € X there is such a polynomial h£ € X for which ||/ — h£\\ < £/(K0 + 1). Now 5/[/e] = h£ for large enough / G J. Then for this / € X, St[f} —> / in norm as / —> oo, since \\Si[f] - f\\ < \\Sjlf] - he\\ + \\h£ - f\\ < (K0 + 1)11/1, - f\\ < e. Conversely, if Sj[f] -» / in norm as / -> oo, then the convergence set being (pointwise) bounded, we have H/S/f/JH < Kf for each / G X and I e J. Since X is a Banach space, it follows by the uniform boundedness theorem that Kf < K0\\f\\ for some absolute constant K0 > 0. Thus our statement is established. 3. It is now asserted that for each / e X, the function Fj e X iff X admits norm convergence and this will establish the proposition in one direction. Indeed, let / e X and fn e X be a trigonometric polynomial such that fn —>• / in norm. Since Ffn e X, it suffices to prove that {F/n,n > 1} is Cauchy. Let Jn = (71, • • • , 7n) G J ', with 7; > 0. If u G F is arbitrary, let Hu : F —>• F be a shift defined as Hu(-y) = j+u. Then 77U is a homeomorphism of F onto F, and if In = Hu(Jn) G J, consider (9)

To find the connection between Fn and Sjn , note that (;u)SIn[fu}=

E(/")7<-.«>= E /«+7 (10)

6.3 Conjugate series and convergence in subsets of Orlicz spaces

205

Now using condition (ii) of a homogeneous space and 2. above, we deduce \\fn(f)\\

= ||5/n(/ii)|| < #o||/t*|| =

tfoll/ll

(11)

for an absolute constant KQ > 0. Thus the mapping / (-)• Fn(f), being evidently linear, takes / into a trigonometric polynomial. For any / 6 X and e > 0 there is a trigonometric polynomial h£ E X such that ||/ — h£\\ < e and Fn(he] = h£ for large enough n. By the argument of 2., {Fn(f), Jn € J} is Cauchy, so that Fn(f) ->• Ff e X, where Ff = E /7{-,7> e #. Thus the 7>0

assertion of this step follows, i.e., F/ €. X iff X admits norm convergence. 4. For the converse implication, suppose X admits convergence. Then by the preceding step, it suffices to show that the "partial sum operations" {£/[•], / € J} is a bounded set. By the present hypothesis, / e X =$• Ff € X(Ff = /). Again consider the mapping Hu : F —>• F, defined in the preceding step. By (9) and (10) for each / e X, 5/n and Fn are defined and replacing / by fu in (11) it follows that the boundedness of [Fn — FJn, Jn e J} is equivalent to the same property of {Si, I G J}. Since X admits conjugation, g i->. Fg is a closed linear operator on #, hence by the closed graph theorem it is bounded. Now if In e J is arbitrary (finite) index, there exists a 0 < u < T such that Jn = Hu(In] is finite and has positive entries, since if In = (71, • • • , 7n), then we can find u € F, u > 0 such that 7, + w > 0 for all z [since 7^ € P or P"1 by definition, and if 7^. e P, let v^ = 7"1 G P and take u = v^ H ----- h ^n > 0, similarly other possibilities are handled]. Thus 7>0

= E/7(-.7 + «> = E(/«) 7 <-.'y> 7>0

7>0

= Ffa-Fn(fu), where Fn(fu)

=

E

0<7<w

(12)

(/«)-{-, 7> and Jn = ff u (/ n ) = {7 : 0 < 7 < u}, a '

finite set. Since \\g\\ = \\gu\\ = \\gu\\ and g •->• Fg is bounded by 3., we get \\Fn(fu)\\ < \\Ffn\\ + \\FfU\\ < K0\\fu\\ + K0\\f\\ = 2K0\\f\\. Thus ||5/B[/]|| = \\Fn(fu}\\ < 2K0\\f\\ for an absolute constant K0 > 0. D We record the following consequence.

206

VI. Fourier Analysis in Orlicz Spaces

Theorem 4. Let G be a compact group and f € L2(G, yu), where p, is the normalized Haar measure on G. Then the partial Fourier sums S/[/] of f converge in L2(G] as I /** under the inclusion order of the directed set J of all finite subsets of A, the index set introduced above. If moreover G is connected and abelian so that the dual F has an order (relative to a certain closed maximal subgroup, used above and proved in Rudin [1], p. 195) and if Sj[f] is the (finite) partial sum of the Fourier series relative to this order, then 5;[/] ->• / for each f e LP(G}(= LP(G, //)), 1< p < oo. Proof. The L2(G, /^) case is simply a consequence of the classical Peter-Weyl theorem, already noted above. In the general LP(G], I < p < oo case, while G is now also a connected compact abelian group, it is known under this hypothesis that X = LP(G) admits conjugation (cf. Rudin [1], Thm. 8.7.2). It is trivial to see that X is a homogeneous Banach space contained in Ll(G). Since the same ordering is employed in Proposition 3, the result follows from it. D Remark. An obvious question is to extend this to reflexive Orlicz spaces L®(G). This needs an extension of Ryan's theorem to L?(G}. This is the same as extending Theorem 8.7.2 from Rudin [1] to this case. However, the proof there does not easily generalize to Orlicz spaces, and a different attack is desirable. We now include a result on the existence of an unconditional basis in an Orlicz space, due to Gaposkin ([1],[2]). Recall that a sequence {xn,n > 1} of elements of a separable Banach space A" is a (Schauder) basis if every oo

x G X can be expanded as x — £] a,i(x)xi for a unique coefficient sequence 1=1 ai(x), and the series converges (i.e., the partial sums converge) to x in the norm of X. If the above series is unconditionally convergent to a;(i.e., if the indices can be permuted), then it is called an unconditional basis of X. If X has a basis, then the space is necessarily separable! For a long time it was an open problem about the existence of a (Schauder)basis for each separable Banach space, but P. Enflo in 1973 has constructed a separable X which does not admit a basis. It is known before that the space C[0,1] of continuous functions of the closed interval [0,1] with the uniform norm does not have an unconditional basis (by a result of S. Karlin's, cf., Day [1], p.77). Consequently the following result is of interest in applications. Theorem 5.(Gaposkin [1],[2]) The Haar system of functions in I/*[0,1], with

6.3 Conjugate series and convergence in subsets of Orlicz spaces

207

Lebesgue measure, forms an unconditional basis iff $ £ A2(oo) n V2(oo), so that L*[0, 1] must be reflexive. We present a short proof of this result utilizing the indices of TV— functions and a general result on symmetric function spaces due to Semenov [1]. The argument is from Ren [1], and illustrates a nice application of the properties of the quantitative indices of Chapter I. First we need to recall the definition of a symmetric space, which is related to the concept of decreasing rearrangement of functions. For a real or complex measurable function / on (17, E,//), if A/(y) = IJL{UJ G 17 : |/(w)| > y}, which is nonincreasing, its inverse /* defined by /*(t)=inf{y>0:A/(y)<«}>

*>0

(13)

with inf{0} = +00, is called the decreasing rearrangement of / (cf. e.g., the companion volume, Rao and Ren [1], p. 406 for some of its properties). Then /* < /| if |/i < |/2|, and /i,/2 are equi-measurable when ^/i (?/) = ^h(y}iV > 0- If (X, II ' II) i§ a Banach space of measurable functions on (17, E,^), then it is symmetric whenever: (i) /2 € X, \h\ < |/2|, =* /! e X and \\h\\ < ||/2||; (ii) | /i | and |/2| are equi-measurable, f i e X => /2 e X and ||/i|| = ||/2||. Also X is called rearrangement invariant function space if / G X, g is measurable and /* = g*, then g e X and ||/|| — ||^||. It is verified that X is a symmetric space iff it is rearrangement invariant. Thus one sees that an Orlicz space L^(^JL) is a symmetric space. The general result that we would use is: Theorem 6. (Semenov [1]) The Haar system is an unconditional basis in a separable symmetric function space (X, \\ • ||) on (17, E, /x) — ([0, 1], B, a} with a. as the Lebesgue measure iff

1 < lim inf t-*>

< lim sup

F(t) ~

t-+o

F(t)

< 2, '

(14) v '

where F(i) = ||x[o,t]||, 0 < t < 1, is the fundamental function of (X, \\ • ||). For convenience, we recall the definition of Haar functions on [0, 1], as follows. Let #<)(•) = 1 and H2k (•) = 2* (x[o,2-*-2) - X[2-*-*,2-*)) ,

A; = 0, 1, 2, • • •

208

VI. Fourier Analysis in Orlicz Spaces

and if 1 < j < 2k,k > 1, set H2k+j(-) = H2k(- -

J2~k)x[j2-k,(j+i)2-k)-

Then {Hn,n > 1} forms a complete orthonormal system in L2[0,1] and is a (Schauder) basis in Lp[0, l], 1 < p < oo as well as in L*[0,1] if $ e A2(oo). We turn now to the Proof of Theorem 5. Let Z/*[0,1] be reflexive, which is equivalent to $ e A2(oo) n V 2 (oo). Now the norm F(t) is given by

Letting u — (2t)

, we have r mf • F(t]

lim sup

and similarly lim sup

But by the relations between these indices in Chapter I, we find that 1 < P$l < ctj1 < 2 so that (14) holds. Consequently the sufficiency part of this result follows from that of Semenov's) Theorem 6. Conversely, suppose that the Haar system is an unconditional basis of L*[Q, 1]. Hence this space is separable, which implies that <3> G A2(oo) (cf. Rao and Ren [1], Prop.III.5.2 on p.91), since the Lebesgue closed interval [0,1] is a separable measure space. To see that <3> e V2(oo) as well, in the contrary case we have a$ = | so that the second half of (14) fails, since Z/$[0,1] is a separable symmetric space. Thus by the corresponding part of Theorem 6, the Haar system fails to be unconditional in L*[0,1], contradicting the hypothesis. Hence the result holds as stated. D Remark. We observe that the same result holds with Orlicz norm, since it is equivalent to the gauge norm and the unconditionality is a topological invariant. This also can be proved by a similar computation, on noting that (L*[0,1], || • ||$) is a symmetric space. It is natural to ask whether the basis problem can be considered for

6.4 A Hausdorff-Young

theorem for Orlicz spaces

209

subspaces of L$[0, 1], such as M* which is not an Orlicz space in general (it is just the Morse-Transue space). The question can be answered for certain spaces if $ is restricted as in Proposition 1, but the method of proof is different. It depends on the HausdorfT- Young inequality. The latter has independent interest for many other applications. Hence we consider this in the context of a general compact (or locally compact abelian) group as in Proposition 3 above. The elegant interpolation method, first noted by A. Weil [1], in the L*(G) context does not give anything new unless one finds an independent proof for one L?(G] with $ not a power. The classical method of Hardy and Littlewood fortunately extends, and we present a result in this way which may be of interest in other contexts. 6.4 A Hausdorff- Young theorem for Orlicz spaces To motivate the subject, we recall that if ^(IR) denotes the classical Lebesgue space on M and / e LP(JR), and its Fourier transform / exists so that / = T/, then the Hausdorff- Young theorem asserts that, for 1 < p < 2, / exists and satisfies the inequality ,

It is known that this result is false if p > 2, i.e., / does not exist for some / G Lp(£i) if p > 2. A quick proof of this result is obtained from the fact that ll/lloo < ll/lli and (by the Plancherel theorem) ||/||2 = \\f\\2 so that, using the Riesz convexity theorem (cf., e.g., the companion volume, Rao and Ren [1], Thm VI.3.5, suitably specialized), \\f\\q. < ||/||p. where i = ^£ + A, 0 < s < 1 and here p0 — 1, pi = 2 and "7 = ^ + "7 w^h QQ = +00, q\ — 2. But this procedure does not bring in any Orlicz space that is not a Lebesgue space. So it is necessary to go back to one of the methods, not depending on interpolation theory. We extend a result of Hardy and Littlewood [1] and obtain an inequality for a genuine Orlicz space, and then present some applications. We use the partial ordering -<' between the Young functions $1, $2 i.e., $i-<'$2 iff $1(0;) < $2(kx)), x > x0 > 0 and some k > 0, discussed in Section 3 above, and they also demand that $2(cix) < C2$i(x) for \x\ < x\ with positive constants c\ , c2 and x\ . Lemma 1. Let $1, $2 be continuous Young functions vanishing only at the origin, and i^'2- Then for their complementary functions (^1,^2) °ne

210

VI. Fourier Analysis in Orlicz Spaces

has ^'2^,'^i(and conversely). Proof. It should be noted immediately that -<' is the same as -< for large values (cf. Rao and Ren [1], Thm.II.2.2 and the same for small values, so that for finite measures L*1 (//) D L$2 (fj.) but for counting measure I®1 C /*2 hold. This combination is needed in applications to Fourier analysis [e.g., (L*(G)) A C /* for such groups as G — [a, b]]. Thus the lemma implies for large values $1 -< <£2 =>• ^2 -< ^i as in the above noted reference. [This fact is a consequence of the inequality a < $~l(a)ty^l(a) < 2a, a > 0.] The relation for small values is also simple and established as follows. By definition $i(ax) < 6$ 2 (^), 0 < x < x0 and consider the Young function $(x) = ^i(ax] < $2(2), and ^ be its complementary function. If p, pi are the (left) derivatives of <&, $1, then we have $(z) = F p(t}dt < F P2(t}dt = $ 2 (a;), JO

x > 0.

(1)

JQ

If q, q-2 are the inverse functions of p, pi, then the point (x, q?.(x)} gives equality in Young's inequality for ($2) ^2) but inequality for ($, ^) so that one has < $[fc(a:)] + *(x),

x > 0.

(2)

From (1) and (2) we deduce that V 2 ( x ) < V(x) for 0 < x < y0 =
relative to the unitary, inequivalent, irreducible (finite) d(a) dimensional representation of G on L2(G, n) relative to some basis, so that

1 < «, j < d(d),

a eA>

6.4 A Hausdorff-Young

theorem for Orlicz spaces

211

is an orthogonal set. We used a different normalization from that in Section 3 above. Define F(a, f) as *—. /

,

«\ O

-^

V1""^

i * / .

a a

( ) i,j=i

.

«/'

\ |O

\

f A\

/1 '

V /

For the work below it is convenient to normalize the Young complementary pair ($, #) to satisfy $(1) + ^(1) = 1. This can always be done, and it entails no loss of generality. [See Remark 1.2.23 on this point, and related discussion.] Thus we define the (gauge) norm of / and F as: ./*(/) = inf and

J*(F) = inf \k > 0 : £ a€A

I $

$

Then {L (G, /A), «/$(•)} and {/ (^1), «/$(•)} are Orlicz spaces and they have equivalent norms if the normalization is omitted, as seen from the discussion of Remark 1.2.23. We can now present the desired Hausdorff-Young inequality for a class of real Orlicz spaces as follows: Theorem 2. Let ($, ^) be a continuous complementary pair of normalized Young functions (hence $ is an N -function) such that (i) $^'$0, $0(aO =c|z| 2 ;

(ii) V'(x) < c0xr,x > 0,

(6)

for some r > 1, c, CQ being positive constants. If (G, fj,) is a compact group with normalized Haar measure, as above, andM®(G,iJL) is the closed sub space ofL^(G,^i), then for f E M?(G, //) and F/ = F ( a , f ] we have (a)

J»(F/) < AoJ*(/); and

(b) J9(f) <

fciJ«(f»,

(7)

where ko, k\ are some constants depending only on $; -<' and CQ, c\ > 0. The proof depends on the following extension of a crucial inequality due to Hardy and Littlewood [1] in the ZAcase:

212

VI. Fourier Analysis in Orlicz Spaces

Proposition 3. Let ($, ^) be as in the preceding theorem and AQ C A be a finite set. Define a simple function fAo as

fAo(x] = £ d(°^ E c(i, j^X/x),

x G G,

(8)

where c(i,j, a) are complex constants. Then f CLJ

u\j/ (1? AQ ) _ *^0 *J<& \j Aft )'f

Q'f^O'

( uj

*Jty \J A.Q ) _ "•'!*'$ \-^ An / 5

v /

where FAo is a function defined as in (4) with fAo, i.e., is given by ri2 /

\

i -1

d(a) V"^ I / •

•

\|2

( \

k0, ki are some positive constants depending on $, -<' as in (7) but not on AQ. Proof. We present the argument in six steps for convenience. Step I. Consider fAo given by (8). Then f A o ( i , j , a ] = c(i,j,a)xA0(a)Hence F(xJAo)xA0 = FAo(a), and set J^[F(-JAo)] = S*(fAo). If / € L?(G, ft), let c(i,j,a) = f ( i , j , a ) and let fA be given by (8) with c for c there. Now define the numbers M and M', following Hardy and Littlewood [1], as M = M*(A0) = sup

I M/J

:/ / 0

(10)

and similarly of 8 with a11 c

'

We claim that M < oo [and then show later that M' — M}. Since the quantities are ratios of norms, we can and do assume that Sy(fAo) = I in computing this bound. Then for the simple function fAo, using the continuity of ^ and the definition of the gauge norm (kQ = 1 there) we get a&A

6.4 A Hausdorff-Young

theorem for Orlicz spaces

213

Since d(ot) > 1 and F(a, JAO} = 0 for a e A — A0, it follows from the equation above (since 0 < \I>(1) < 1), for at least one term on the left side, it is > | where a = #(A0], the cardinality of A0. Let a0 G A0 be such a term. Then

On the other hand, from (4) one gets

d(a)

F(aJAo)

<

c(i,j, a)I, since a2 + b2 < (|a| + |6|)2,

d(a) *5± /.

1

d(a) «w

= / l / W IiJ=l E W>

'di(«) ("/ G G

d(Q)

1

d(a}dn(x), by the Cauchy-

Schwarz inequality, = d(a) I \f(x)\dn(x},

since (u^(x)} is a d(a) x d(a)

G

orthogonal matrix for each x G G, < d(a} J*(/), by Holder's inequality and J*(l) = 1.

(13)

Hence for a = a0, one has from (12) and (13) that

./,(/)

-

(14)

The right side is independent of /, and hence taking the supremum on / such that Sy(fAQ) = 1, we conclude that M < oo. Step II. We now claim that M' < M(< oo). For this consider a step function g as

(15) where /^0 is given by (8). Then by definition J^(g) = 1 and moreover, there is equality in Holder's inequality between fAo and g (cf. Remark 1.2.23 and

214

VI. Fourier Analysis in Orlicz Spaces

Zygmund [1], Vol. I, p. 175). Hence «M/Ao) = j 9(x)f^0(x)djj.(x),

fAo is the complex conjugate of fAo,

G d(a)

=

E d ( a ) E £(M,")/A O (»,.?, <*)*, Parseval's formula, a&A0

<

Y

i,j=l

d2(a}F(a,~9Ao}F(aJ~Ao}

a&A0

< S^(fAo)S^(gAo) < S*(fAo)MJ*(g), by Step I, =

MS*(fAo).

(16)

Dividing by S(fAo) and taking the supremum over all constants c in fAo, one gets that M' < M. Since M < oo by Step I, we conclude that M' < oo. It is to be shown next that M' — M. Step III. We assert that M < M' so that M = M' will follow. For this purpose, define another simple function hAo from fAo for / € L $ (G, //) with 5*(/40) = 1as: Ao)]^,^^*.^),

(17)

where V(x) - J*' (f), x > 0, with *(0) = 0 and k = J*(F(-,/ Ao )) > 0. Then a calculation similar to the one above, using Parseval's formula, gives J f(x)hAo(x)dn(x)

= J

fAo(x)h*Ao(x)dn(x)

6.4 A Hausdorff-Young

theorem for Orlicz spaces

215

by the equality condition in Holder's inequality and the fact that Jy[F(-, fAo}} = S#(/4 0 ). Thus we have from (16) *(hAo).

(18)

G

On the other hand, by construction we see that

It then follows that S*(/u0) - J [ F ( - , h A o } } = 1. So by (18) [S*(/40)/(fA0)] holds. As a consequence, for such fAo, there is equality through out in (16) so that for the corresponding simple function g of (15), we have S*(~gAo) = M- and J*(fAo) = MS*(fAo}.

(19)

We fix this pair fAo and g in the following computation and observe that M > 1. Indeed, taking / = 1 and noting J$(l) = 1 (by the fact that = 1 and $(1) + *(1) = 1), we find that

=

> 1 for some a0 6

so that M > [5^(/4 0 )/J r $(/)j > k0 > 1. Next we have by Bessel's and Holder's inequalities

Sl(~9A0] = E

d2

M E \9(i,M\* < I \9(*)\*dn(x) < J9(
(20)

VI. Fourier Analysis in Orlicz Spaces

216

since J$(l) = 1. Let *i(x) - #(x 2 ) so that UJVtfi (v&i is an W-function) and since g is bounded, if a2 = Jy(g2)(< oo), then one has

(21) x

G

'

G

whence J^>l(g) = a. So from (20) and (21) we conclude that 82(9AO) < «7*i(0) - a-

(22)

r

We now use the condition that fy'(x) < c0\x\ , r > 1 to obtain the absolute bound independent of AQ. Since by (21), Jy^g) = a > 0, there exists a /?o > 0 such that 1

r

f_g_\

~ / 'Uv dfj, l», by (15), \f.Ao\

t, by the inequality on ^',

*(/^0

(23)

dp,

where ^2(2:) = ^i(x2) and ^ = (cQ/a(30)r > 0. Thus ^2 is a Young function satisfying ^V^-x'^- Hence (23) implies the following key inequality. Namely, there exists a /?2 > 0 depending only on ^2 such that > 0.

(24)

Since «/* 2 (-) is a norm, one has on using the definition of /3\ above: 2

.

°

> /?2 I — I Jvi(g) — fizJy 1(9)1 (say).

We thus have

'*2! so that ^^'^i^'^^'^o and c /

$1

c Z* c /*° = / 2 c /* c i*1 c J*2

(25)

6.4 A Hausdorff-Young

theorem for Orlicz spaces

217

by Lemma 1. One obtains from the preceding, the following chain of inequalities: 1 < M = M* = S9(gAo) < k2S2(gAo)

3

3

k2 r

o

-

*

4

Af,

(

'

}

(26)

where *3(*) = -^T|*r(r+1), SO

r+I $3 is complementary to ^3 and S^a(fAo) < P±S<s>(fA0} for some /34 > 0 depending only on $3 and $. Thus (26) can be finally written as

1 < M£+1 < A;3rM;3,

(27)

for some &3 > 0 depending only on $, $1, $3 and the ordering constants. If q = 2r(r + 1) > 2, then L*3(G, /z) = L9(G, /i) and the L9(G, /z) bound (from classical Hardy-Littlewood account for G = [0, 27r) and the general compact group case, by Hirschman [1], a short proof of which we include in the next step for completeness) implies that M$3 < k± < oo, where k± depends only on c0 and r. Thus (27) yields 1 < M$ < k5 < oo,

(28)

where k5 = (ksk4)r^r+l^ which is independent of AQ. Step V. For each / e L P (G,//), 1 < p < 2 (so g = ^ > 2), and F(a, /), o; e ^4, we have

(a)

• (29)

In fact, let lg(d2) be the Lebesgue sequence space with weight {d?(a), a G A}, d(a) > 1, and T : If(G,fj,) -)• / 9 (d 2 ) be an operator defined as T/ = F(-,/) so that |T(a/)| = |a||T(/)| and |T(A + /2)| < \Tf,\ + |T/2|, i.e., T is sublinear. By Parseval's relation =EF(ajrd2(a) a&A

= \\f\\ljl2(d2},

(30)

218

VI Fourier Analysis in Orlicz Spaces

and by definition

\Tf\\q= supF(a,/)/d(a),

q - oo.

a£,4

However, by (13), we have (31)

Hence by the interpolation theorem (for sublinear operators, this is an extended version by Calderon and Zygmund [1] of the classical Riesz-Thorin theorem, and is further extended for Orlicz spaces by Kraynek [1], and is detailed in our companion volume, Rao and Ren ([1], p. 374), we get from (30) and (31), ||T/||9 < ||/||p, 1 < p < 2. This gives (29) (a). Then (29) (b) can be obtained by a similar argument or can be deduced from (a) by a duality trick, given in ZygmundQl], Vol.11, p. 103), and we omit the repetition. This result establishes (28) fully. Step VI. Let ko — k$ = k\ in (28) to obtain (9). In the Lebesgue case both are unity, but here they are > 1. Note that (28) and the work of Step III implies that \I/2> ^3 are determined by ^, the complementary function to <1>, and all the constants are determined by $-<'$0 and perhaps on c0 and r. This establishes the proposition. D We can now complete the Proof of Theorem 2. Let / 6 M$,(G, p,} and AQ C A, a finite set and fAo be the corresponding function of (8) where we take c(i,j,a) = f(i,j,a). From the structure theory of Orlicz spaces, it is known that the set {/,i0, A0 C A} is a dense subspace of A1*(G, n). If {F(a, /A O ),O: G A0} is the resulting function of /*(rf 2 ), then we conclude that Jim J9(f - f~Ao) = 0, Jim J9[F(; fAo}} = J»(F).

A0CA

A0CA

From this and (9) (a), we deduce (7) (a). Also by Fatou's lemma and (9)(b), one has J*(/) < liminf J (f ) A CA 9 Ao 0

< ki Alim J*(/A O ) = k i J * ( f ) , CA 0

6.4 A Hausdorff-Young

theorem for Orlicz spaces

219

giving (7) (b). Hence the limits as AQ varies in A are taken using the inclusion partial ordering as in Section 3 above. Thus (7) holds as stated. D Remark. As noted before, between any two Young functions (even those corresponding to the Lebesgue spaces) one can construct a Young function which does not satisfy any growth conditions. We shall show below examples of L®(G, //) spaces satisfying the hypothesis of Theorem 2, that are not Lebesgue spaces. Thus the preceding result includes new spaces that are not Lp-spaces for any 1 < p < oo . We now present an extension of Theorem 2 if G is a locally compact abelian (LCA) group. The corresponding result for nonabelian G involves difficulties even for a definition of the Fourier transform. We include some discussion on this point latter. The extension of Theorem 2 for LCA groups is as follows. Theorem 4. Let G be an LCA group with a normalized Haar measure n and $ be an N-function satisfying the order conditions of Theorem 2 (i.e., $^'$0 and V'(x) < c0xr, r>l,x>Q). Then for each f e M*(G, /z) C L*(G,//), Tf = f exists and MTf) < W*(/), (32) where k0 > 0 is a number depending only on $ and the ordering constants in <£-<'o and c0 > 0 above. Proof. We use the structure theorem for LCA groups, namely G = G\ x Mp where JRP is the p-dimensional Euclidean space and G\ contains a compact subgroup H such that G\/H is discrete and hence isomorphic to Zq for some q > 1 (cf. Rudin [1], p. 40). Here and below, all equations between groups are isomorphisms, as usual. Hence we can invoke the classical Weil formula connecting integration on a group and the quotient by its compact subgroup, and the subgroup itself (cf., e.g., Reiter [1], pp. 59-60). Thus for any compactly based continuous / we have

J

J f(xy}dy

i/H Lff Iff

dx = J f(x]dx J

Gi

and

J J f(yz)dz GI URP

dy = j f ( y ) d y ,

220

VI. Fourier Analysis in Orlicz Spaces

where the Haar measures dy, dx, dx, dz can be normalized to have the above equalities. Combining these we get the relation: f(x]dx — G

j f(yz}dz GI

Jfis

dy= J

f

f f f(xyz}dzdydx.

(33)

Gl / H ** Zft"

Replacing / by $ (£) for a suitable k > 0, we can find a bound for the right side integral in each of the three cases and combine the result on the product space with the measure dx — dzdydx through (33). Since we already completed the work for the compact group H in Theorem 2, we only need to consider bounds on the other two groups^ 9 = Gi/H and 1RP, and combine the results. This is done in steps for convenience. Step 1. Let G = JRP. We claim that (32) holds for this case. For simplicity, we take p = 1, since p > 1 is an easy consequence. The method used is an extension of that of Titchmarsh's [1]. Thus let / 6 M®(]R} be of compact support. We use the fact that M is cr-compact, and utilize the result of the preceding case for the compact group G = (—TT, TT]. Thus if o; > 1 and A > 1 are fixed numbers, let n = [a\] — 1 where [x] is the integral part of the real number x. Let .i±L

a,-= a,- A = = J/L

A

ft((xx))ddxx,,

and

(x} = > fluel]X gqnn(x) 0

The reason why such a gn is considered will appear only later. For G the measure p, is given as dfj, — dx/1-n and later we consider for G\ — (—TrA, ?rA], dn\(x] — dx/ZnX. For the special case of G — (—7r,7r], we can apply the result of Theorem 2, by which we get from (7)(b) M9n) < W*(Fn),

(34)

where Fn — {flu, — n < j < n} is identified as a simple function in /*, i.e., the "function" Fn is a sequence with zeros for \j\ > n, and having values a, for \j\ < n. Note that k\ depends in (34) only on $ (and TT). On the other hand, by Jensen's inequality (since A > 1) we have , \

j+i

,

6.4 A Hausdorff-Young

theorem for Orlicz spaces

221

Adding on j, one gets (a,) < f *(f(x))dx. J-ot

3=-n

(36)

Replacing / by f/J$(fxA) in (36), it follows from the definition of the gauge norm that with A = (—a, a}

(37) Hence (34) and (37) imply for any A > 1

.

(38)

Now let n —> oo [so that A —>• oo since a is fixed] we see that on remembering a,j = cij(X) defined above, gn(x] -» JAeltxf(t)dt uniformly for x in compact intervals of JR. Then using the Fatou property of the norm J$ and the continuity of \& [$ being continuous as an iV-function], one finds that from (38) A

If we define

then for any b > a > 0, (39) implies - F(0, •)] < *i

Since / e Ai^(JK), one can take limits as b —> oo, and then a —> oo to conclude that the right side tends to zero, so that {F(a, -),a > 0} is Cauchy and converges in norm to a function F. Then one can conclude (since we get convergence of norms immediately) J*(F) < *!,/*(/),

(40)

and F = f = Tf here. This implies (32) for / of compact supports in A^*(jR), which is a dense set in the latter. Hence T has a unique extension for all / € M*(1R), and (32) holds in this case.

222

VI. Fourier Analysis in Orlicz Spaces

Step 2. Let G =$q, the discrete group, q > 1, an integer. We again take q = 1 for simplicity and establish (32), which suffices. Then the dual group G is compact (the torus) and is identified as (— TT, TT]. Consider the closed subspace M®($} of / $ , and let / G M*($} be a simple function (i.e., is nonzero for only a finite number of coordinates). Again set f(x] = E /(n)e' M , and so E *(/(n)) < oo. Let gn(x) = E /(j)e^. Then gn is nS£

j=-n

ne£

defined on G and is bounded. Consequently, applying the inequality (8) we find for a fixed constant k2 > 0 depending only on $ (and G) such that M9n) < k2J*(Fn}(< /c 2 J*(/)),

(41)

where Fn = {0, /(j), — n < j < n}, as in the preceding step, is the restriction of / to a finite set. But gn(x) —>• /(re) as n —>• oo, uniformly in x, and (41) implies J*(/) < liminf J*(<7n) < A 2 J*(/). (42) This shows that (32) holds in this case also. Step 3. The general case follows from the preceding work. As noted before, the structure theory of LCA groups implies that such a group G is topologically isomorphic to the product of JRP, $q and a compact abelian group GI. Let / e M^(G] with compact support, and if AI, A 2 , A3 are the character groups of iRp, $q and GI, then the structure theory of LCA groups implies the following assertion (cf. Rudin [1], p. 55). If L C G is a closed subgroup and F(x) = I f(xy}dnL(y], J Jtv

where x is the coset containing x, and IJ,L is the Haar measure satisfying the Weil's quotient integration formula recalled for (33), then we get FI = / Ai , the restriction of / to A;, and Fi, f satisfy (32) for each i with an absolute constant ki depending only on <$. Let kG = max(A;i, A;2, k3) and (for nontriviality) suppose that k0J^(f} > 0. Then by using (33) for the product measure dp, on GI and letting Gi/H for^ 9 , H or GI and G\ = G/MP, we get tJi(x)

(43)

6.4 A Hausdorff-Young

theorem for Orlicz spaces

223

by our previous analysis. This implies, since / —>• F is well-defined, F = Tf for / 6 M®(G] with compact supports and by (43), that MTf)

< J*(F) < V*(/).

(44)

Since T is evidently linear, (44) implies it has a bound preserving extension to the closure of such / e M*(G), which is M*(G) itself. This gives (32) in complete generality, d Remarks, (a) The Hausdorff-Young inequality of Theorem 2 plays a key role in all that follows and the conditions on ($, ^} are fully utilized in the LCA case also. Since an application of the Riesz-Thorin convexity theorem applied to Ll and L2 gives only the Z/p, 1 < p < 2, via Plancherel's theorem and since between any two Lpi-spaces, 1 < pi < 2, there exist general Orlicz spaces L^i (\U\PI -< $»(w) -< |w| p2 ), the result of Theorem 2 demands a different technique. The method of Hardy and Littlewood for Theorem 2 and a special trick of Titchmarsh's for its extension to LCA groups (via the structure theory) seem necessary. (b) The normalization of TV-functions in the preceding result is done for convenience only. Because of the topological equivalence for the normalized and general definitions, the same results (of Theorems 2 and 4) hold with different finite absolute constants depending only on the $ and <3>o(c) In the case of the Lebesgue spaces LP(G) with 1 < p < 2, conditions for equality in (7) have been investigated by Hirschman [1]. The corresponding study for M®(G] will be interesting but seems to be quite involved. Since in the previous chapters we have used interpolation results to extend the work from a pair of Orlicz spaces, we can similarly present a corresponding statement on Fourier transforms for LCA groups. We devote the next result for this purpose. Theorem 5. (a) Let ( M*(G}), then T is a densely defined, one to one and closed-linear-operator. (b) If ($i, ^i), 2 = 1,2 are two pairs of complementary and normalized N-functions and T : Ai^(G') -> M^^G], i = 1,2, with domain all of M^^G], then for the intermediate N-function $s which is the inverse of S;1 = (3>il)l-s(3>zly,

0 < s < 1,

224

VI Fourier Analysis in Orlicz Spaces

and similarly V~l is defined, we have T : M*'(G) -> M*'(G), 0 < s < 1, as a bounded linear operator and moreover

J9.(Tf)
feM*'(G),

(45)

where kSt^s is a constant depending only on s and $s, and «/$(•) is the gauge norm defined earlier. [If (<&i, \&J, i = 1,2, satisfy the conditions of Theorem 4 (or 2), then the hypothesis of this theorem are satisfied, but the assertion of (a) is true more generally.] Proof, (a) Let P c L°°(G) be the set of positive definite functions and / 6 P fl M?(G], be of compact support. Then it follows from the classical inversion theorem that / is bounded and integrable so that / € M. *((•?), (cf., e.g., Loomis [1], p. 143), and moreover that every step function can be approximated by elements of PC\M*(G). Indeed, Llr\L°° is dense in M*(G) and if / e L1 nL°° and u is an approximate identity chosen from the same set (i.e., given V, a neighborhood of identity, there is a u > 0, supp(w) C V and / udfj, = 1, and moreover the convolution f * u approximates / arbitrarily G closely), then \\f *u — /||$ is arbitrarily small. This is also a classical result (cf. Loomis [1], p. 142) again if M®(G] is a Lebesgue space). If V is the linear span of such /, then from the structure theory of M*(G), it follows that Z> is dense in the former, and T(D] C Ai*(G). It remains to show that T is closed, since it is clearly linear. Let {/„, n > 1} C T> and gn = Tfn be such that «/$(/„ — fm) —» 0 and Jy(9n — 9m} —»• 0 as n, m —> oo. Then there exist / and g as limits of these sequences, / e M*(G] and g G M*(G). Let h e M*(G)r\P, with compact support, where P is the set of positive definite functions on G. Then, as noted above, such h form a dense set in M*(G}. We show that f £ T> and Tf = g, by the following computation. Consider / g(x)h(x)djj,(x)

— lim / / < x,x >* fn(x)h(x)d(j,(x)dfj,(x),

J

^

^^ J

< -, >

J

is the conjugate of the character < •, x > of G, =

lim / h(x)fn(x)diJ,(x),

n-*oo 7 G

h = Th is the transform of

of h and if GO = G, then GQ — G,

6.4 A Hausdorff-Young =

=

theorem for Orlicz spaces

225

/ h(x) f (x)diJ,(x) , since {fnh, n > 1} is Cauchy G with limit fh in Ll(G), 11 < x, x >* h(x}f(x}dfj,(x)d^(x],

by the inversion

G G

theorem, =

/ h(x)f(x}dp,(x], G

by the Fubini theorem.

This implies j(g-Tf}(x}h(x}dfi(x}

= Q.

G

Since such h are dense in J(4®(G), g = Tf a.e., so that / e £>, and T is a closed operator. (b) If the domain of T is M*(G] itself, then, the latter being a Banach space, it follows from the closed graph theorem that T is bounded. By hypothesis, T : Al$i(G) —>• M.9i(G), i = 1,2 is bounded on both spaces so that If ($s, tys) are defined as in the statement, then it follows from the interpolation theorem in Orlicz spaces (cf. Rao and Ren [1], p. 226) that T is defined on M*'(G) into A4*'(G) and

Setting ks$3 = k\~sk%, 0 < s < 1, we get (45). D The above result has the following version for general compact groups which we include for comparison. Proposition 6. If G is a compact group and (j, ^j), i = 1,2 are complementary N -functions, and T : jM $i (G) —> A4*'(G} is defined everywhere, where Tf = F ( - , f ) of (4) and G is the dual object of G (of all unitary, inequivalent, irreducible [finite dimensional] representations), then ^.m-,/)]<4A;.,*.J» J (/) J / € M * ' ( G ) , 0 < 5 < 1 ,

(46)

226

VI. Fourier Analysis in Orlicz Spaces

where k$j$s > Q is a constant depending only on s and $s, the couple ($s, ^ s ) are given in Theorem 5(b). Proof. The argument is similar to the preceding result and (46) is just the counterpart of (45). The only crucial difference is that T is a sublinear operator, and so one has to use the corresponding interpolation theorem for such operators. Fortunately the necessary result is available. It is due to Kraynek ([1], Theorem 2.1.1) and it is also included in the companion volume (cf. Rao and Ren [1], p.367). We omit the repetition of the detail.D It is of interest to observe that examples of N- functions <3> of the type studied in Theorem 2, include those considered by Riordan in his thesis. He treated <£(x) = ^-L(x], I < p < 2, where L(-) is a (finite number of) product of iterated logarithms and fy(x) = ^-Li(x), x > 0, where L\(x) = [Z/(a;)]~p for large x and q — -^-. Although ^ is not necessarily complementary to <£, it is equivalent to the complementary function. Actually, these are the principal parts of a pair of complementary TV-functions (cf., e.g., Krasnoselskii and Rutickii [1], Theorems 7.1 and 7.2 on pp.58-59). The particular form allows a more detailed analysis involving complicated calculations. Some of J. Riordan's unpublished results, related to an extension of Marcinkiewicz's interpolation theorem, have been included in our companion volume (cf. Rao and Ren [1], p.242). Applications of his results to Fourier analysis are possible, but we shall not consider this further here. A basic motivation of the above work is to study the conjugate function / o f / e L*(G), i f p = 1. Then / <£Ll(G),G = [0, 2yr), but will be in £?(
6.5 Fourier analysis on generalized Orlicz spaces

227

omit the abstract study of this subject, and turn briefly to the generalized Orlicz spaces with F-norms (cf. Rao and Ren [1], Chapter X) and some results on conjugate functions based on [0, 2?r) spaces. 6.5 Fourier analysis on generalized Orlicz spaces The spaces to be considered here include L p ,0 < p < oo, and we recall these definitions for convenience and immediate use. A function vanishing only at the origin, increasing and continuous is called a ^-function. In other words, a continuous function 0 : 1R —> 1R+, such that (j>(—u) = 0(w), (•) /* oo and (j)(u) = 0 iff u = 0 is a 0-function. We then have a generalized Orlicz space L^^T) on (Q, £,/x) as before, i.e., 1/^(0) = {/ : ]R —> M measurable, p^(oif] < oo for some a(= a/) > 0} where Jn 0(/)c^u- The metric or F-norm || • \\^ is then given by

We defined M*(fl) = {/ e L*(fl) : p(af) < oo for all a > 0}, so one finds M*(n) = L*(ft) iff 0 € A2(oo) when /z(ft) < oo. A few of the basic facts for such spaces may be found in the companion volume (Rao and Ren [1], Chapter X). We restate the following pair of results for use below. Proposition 1. (a) (L^(fi), || • ||0) is a Frechet space, i.e., a complete metric space under the metric d(f,g] = \\f — g\\^; (c)\f\<\g\,a.e. =» ||/||, < \\g\\+; (d) \\fn\\(j> —> 0 as n —> oo =^ fn —> 0 in /ji-measure; (e) lim ||/||0 = 0 iff (j) e A 2 (oo),w;/ien^(Q) < oo. The following order relations hold between pairs of 0-functions. Proposition 2. If <j>i,i = 1,2 are (^-functions and /z(fi) < oo, then the following statement are mutually equivalent. (a) There are constants c, k, UQ > 0 such that ^(u) < c4>\(ku} for u> UQ (in symbol i >- fa); (b) L+^ty cL^ 2 (fi); (c) M^Cfy CM^ 2 (17); (d) H/nlU, — > 0 =» ||/n||02 -^ 0, for fn € L^(fi).

VI. Fourier Analysis in Orlicz Spaces

228

We present an example and use it later in computations. Example 3. Let 0P = |w|p, 0 < p < I and Q, = [0,2yr),// =Lebesgue measure, then

{

2?r

, , p

/2?r

1

\ P+T

k>0: [ ^ dt
If p = 1, then equivalent to ||/n

jj/IT so that L*(ti) = L1^) and 0. Moreover, for 0 < p < l,ZA(

ll/nlk — > 0 as n —> oo iff ||/n||, = /|/ B (t)|'dt

.

n||^ —> 0 is = LP(S7) and

> 0.

0

Hereafter we consider / : [0, 2?r) —> JR as one defined on JR by periodicity, i.e., /(x + 2fc7r) = /(z),0 < ar < 27r,fc = 0,±1,±2, • • - . For each / € L[0, 27r), let / denote its conjugate (cf. Def.2.1). As noted at the end of the last section, our aim here is to develop the generalized L^(0)-spaces to establish that the mapping T : f i-> / of functions into their conjugates, satisfies T(L1(fi)) C L^(fi) for a 0-function and that T is continuous when (and only when) 0 satisfies an integral growth condition. The treatment mainly follows the work of Lesniewicz [1]. Recall that if / 6 L1^), then its conjugate / is given by

where /z is the Lebesgue measure on ^ = [0,27r). This is equivalent to the representation (seen after a computation of each term of (1) with f ( x — 2?r) = 2?r

/(*) = -- lim

Z7T e->0+

f ( t ) cot

2

+ f J

f ( t ) cot

(2) 2

which is obtained by a change of variables, and the fact that the limit exists as e \ 0 (see Remark 10(3), below for details). In this section we essentially characterize the continuity of the linear conjugate function mapping from Z/^O, 2?r) into a generalized Orlicz space. The next result plays a key role in this effort and its converse is given later in Theorem 6.

6.5 Fourier analysis on generalized Orlicz spaces

229

Theorem 4. // 0 is a (^-function such that —du < oo,

(3)

then for each f 6 I^fO, 27r), the conjugate function f 6 Z/^[0,27r) and mapping T : f i—> f is a continuous linear operator. Proof. Define an equivalent ^-function 0i as

)w 2 ,

0 < u < 1,

Then (/>i ~ 0 (i.e., 0i >- 0 and 0 >- (/>i) and so by Proposition 2 above L^[0,27r) = L^[0,27r). Also /^ 0i(u)w~ 2 du < oo. This implies

(2v)2

v

4v '

Hence = 0.

(4)

For v > 1 and / € L^O, 2?r), consider the set E(v) = {x e [0,2?r) : |/(x)| > v} and ^(T;) = fj,(E(v)}, where p is the Lebesgue measure. Then g satisfies the key inequality, by an important result in the subject (cf. Zygmund [1], Vol.I.p.134; and Lemma 2.4), the so-called weak-L1 inequality 9(v] < £||/||i

(5)

for some C > 0. We then have (x)\)dx ^ 'I'

=

" f y^ /l~,(m— , ! >, « , , L—J ,^1

( — ^ I"/! f —

Vn / r V n

V

-I

d>i(\f(x)\}dx /I/

_,„,,.- T T - i - M r f V

(f>i(u)dg(u)

v}I - JJq V(—v

)

\n

VI. Fourier Analysis in Orlicz Spaces

230

as n —> oo by definition of the Stieltjes integral. Integrating this by parts and using (5) we get
r

<

-

A

+

Letting v —> oo, the left side tends to p
Let a = gives

> 0 and k = a [C fR+ ^f]

5

, so that replacing / by f , (6)

j = ,k. ^ r r i i n If — 9i du P^\T\
Since <^i ~ 0, the F-norms are equivalent by Proposition 2 one has ||/n||i 0 =$. ||/n||0 —>. 0. So T : / H4 / is continuous. D Regarding the cotangent function, (cf. Lesniewicz [1])

6.5 Fourier analysis on generalized Orlicz spaces

231

Lemma 5. Let fo(x) = cot f , then /0 6 Z^[0, 2?r) iff(f> satisfies (3). Proof.

If 0 satisfies (3) , then 2?r 2?r

"" /•

/

7r

x\

/ pf)O — \

/•

/ TT

-J } dx < 2 <j> (- dx / 0(|/o(z)|)dz = 2J/ 0 \ — <-> x/ *^ \jC v Sill 2

1 u*

< oo,

so that /0 e L^[0,27r). Conversely, let /0 € L 0 [0,27r), so that p^(A/ 0 ) < oo for some A > 0. But then

f

by change of variables. Hence (3) holds for 0.

d

We can now establish the following: Theorem 6. I f T : f i - > f i s 0 satisfies (3).

continuous from Ll[Q, 27r) info Z^[0, 27r),

Proof. Since T is continuous, given £ = 1 there is a 6 > 0 such that if II /||i < <5 one has ||/||0 < 1. Consider a sequence fn defined by /„ = n6x[oti], on [0, 2?r), n = 1, 2, • • • . Then ||/n||i = 0" so that ||/n||^ < 1. Hence by Proposition 1 (b), 27T

l. But by (2)

i

2-n- J

0

t —x , 2

v

.1 n'

(8)

232

VI. Fourier Analysis in Orlicz Spaces

Hence by Lebesgue's differentiation theorem, fn(x] —> ^ c o t f f°r x ^ (0, 2?r). We then find from (8), using Fatou's Lemma, that p(^fo) < 1, so that /0 € I^[0, 27r) and Lemma 5 then shows that (3) is true for this . D Example 7. The condition (3) shows that (f) can not be a convex function! Here are some illustrations of such (u): 0 1 (w) = [log(H-|w|)]r, l < r < o o ;

and vov / — i / i i i \ > log(e + |w|)

^ ~- f ^ ••••

Let f G L^O, 27r)(c Z^[0,2?r)) and / be its conjugate function, where 0 satisfies (3). Denote by Sn[f] and Sn[f] the nih partial sums of the Fourier series of / and its conjugate Fourier series respectively, so that n

^ and

k=i n

x — bkcoskx), k=l

where 1

27T /

ak — — I f (x) cos kxdx; 7T J 0

1

27T /

bk — — I

f(x)sinkxdx.

7T J 0

Then we have the following analogs of the case in which is convex. Theorem 8. 7/0 satisfies (3) , then for f € Ll[Q, 2?r) one has n lim||5 B [/]-/|U

= 0 and Jim ||5B[/] - f\\ = 0-

(9)

The proof is an extension of the classical argument, using Theorems 4 and 6 appropriately. We shall omit the detail, which may be found in Lesniewicz [1] (pp.632-633). As a consequence with 0(w) = \u\p, 0 < p < 1, in Theorems 4, 8 and Example 3, we deduce the following classical result.

6.5 Fourier analysis on generalized Orlicz spaces

233

Corollary 9. (Kolmogorov [I]) The mapping T : Ll[Q, 27r) ->• 1^(0, 2yr), 0 < p < 1, defined by T : f (->• / is continuous and linear. Moreover, Urn ||5B[/] - f \ \ P = 0 = Um ||5B[/] - f \ \ p . Remarks 10. 1. Condition (3), called Lesniewicz condition by Zhizhiashvili [1] (p. 71), is related to certain ordering relations and growths of 0-functions. Lesniewicz [1] showed that if 0 satisfies (3) , then 0 -< 00 where (/)Q(U) = \u\, and by Proposition 2 one has Z/^O, 27r) C L^[0, 27r). Using the order relations given in our companion volume ([l],Ch X), we get actually -<^ Q. 2. We recall that 0i is essentially stronger than , in symbol 0i >->- 0, if for each e > 0

= +00,

and 0i is completely stronger than 0, in symbol 0i >j 0, if for each £ > 0 there are u0 = u0(s) > 0 and K = K(e) > 0 such that

Thus 0 >j 0 iff 0 G A2(oo) and 0i >->- 0 =>• 0i >{ 0 => 0i >- 0. Let 0 satisfy (3) and 0i = 00 above, then by (10) and (4), 00 >->- 0. So L x (fi) C M^(Q) with //(fi) < oo (cf. our volume [1], p.403) and by Proposition 7 on p.404 there, which holds for 00 and 0, so that if fn e Ll(Q,), \\fn\\i < 1 and /„ —>• 0 in measure, then ||/n||0 —> 0, (whence the converse always holds, 3. The verification of (2) is as follows. Definition (1) means that 2 tan \

(11)

For any e > 0, we have

where, by setting u = x +1, II =

IJe

tor. tan,t

dt =

du IJx+e f^cot ^T~ 2

(13)

234

VI. Fourier Analysis in Orlicz Spaces

and, by setting v = x — t,

JE

tan |

JX-E

2

/Z-TT '^'C0t 2 U /"° ., x w - x . rx~e ,, . i; - a; , = / /(v) cot —-— dv + / f(v)cot——dv 2

JX--K

2

JO

= / 3 + / 4 , (say)-

(14) M

y

Further, since f ( u — 2?r) = /(w) and cot( ^ — TT) = cot ~, by letting u = v + 27T, one has in (14) /*27T

^3 = /

J X+TT

Y/

/(u-2?r)cot(—

'T* 2

/*27r

Tt)du=

JX+TT

T/ — 07

f(u)cot

2

du.

(15)

From (12)~(15) we have

27T

=

27T

/

J f(u)cot v

'

u —x

du

Letting e —)• 0 in the above, we obtain (2) from (11). Bibliographical notes. This chapter concentrates on certain results from Fourier series, in the context of Orlicz spaces. The basic result contained in Sections 1 and 2 is an extension due to R. Ryan [1] of the classical result, first given by M. Riesz [1], on conjugate functions and Fourier series. This is an important characterization and an extension to Orlicz spaces based on the torus group identified as [0, ITC}. Some of the results (independently worked out of Ryan's work) for certain subspaces of L®-spaces based on compact groups, and locally compact abelian groups are in Rao ([6] and [12]). Ryan's result shows that $ € A2(oo) n V2(oo) if we consider the M. Riesz theorem for I/*-spaces. If these are not assumed, then $ and ^ (the complementary TV-functions) must satisfy a different set of conditions. This

6.5 Fourier analysis on generalized Orlicz spaces

235

work, extending the classical Hausdorff-Young theorem, is given in Sections 3 and 4. Also conjugate series are considered there. It is interesting that an extension of Hardy and Littlewood method is needed in obtaining a proper Orlicz (non Lebesgue Lp) space result. In the final section we have presented the conjugate function result for Z^fO, 27r) into generalized Orlicz spaces due to Lesniewicz [1], who discovered a key condition for the 0-function to admit conjugate mapping from L1 into lA Thus we have included some hard analysis in the context of Orlicz spaces related to the classical harmonic analysis. Many other results can be extended with similar methods. More on generalized Orlicz spaces will be found in Chapter X when 0 is slightly specialized. Some other applications are considered in the following chapters.

Chapter VII Applications to Prediction Analysis

The classical least squares prediction, originated by Gauss, finds a natural home in Orlicz spaces when the optimality criterion is considered as a nonnegative symmetric convex function vanishing at the origin, the quadratic function being a special case. This is also closely related to aspects of approximation theory. Here we present a brief account of the existence, uniqueness of best predictors for convex (loss) functions, and consider a method of construction of them for a subclass, using the available Orlicz space theory. Then we discuss conditions under which the prediction operators become linear, since they are typically nonlinear in the non least squares problems. This chapter attempts to show how Orlicz space theory forms a crucial part of this analysis which is important in probability theory as well as abstract approximation and other areas.

7.1 Best predictions with convex loss Let XT = {Xt,t € T C M} be a random process on a probability space (0,E,P) such that Xt e L*(P),t € T, where $ : JR —> JR+ is a Young function. The family XTl, T\ C T is given (i.e., is observed in an experiment). If to G T — TI and Xt0 is the unobserved random variable, it is desired to predict or approximate Xto by using a function g of X^, such that g is closest to Xt0 in the sense of the norm of L*(P) [or the modular defined by <&]. It is called the prediction problem relative to the convex loss function <&. More precisely, this is stated as follows. Definition 1. Let M c L$(S) be a closed subspace determined by XTl — {Xt,t G TI e T, TI ^ T}. This means either (i) M is a closed linear span of the Xt,t G T! or (ii) M = L*(BTl) where BTl = a(Xt,t G TI), the aalgebra generated by the Xt,t G TI. [So the first space is a proper subspace

236

7.1 Best prediction with convex loss

237

of ^(B^), if at least one Xt is nonconstant.j If Xto G !,*(£) — A4, then a unique X G Ai, if it exists, is termed the best predictor of Xto relative to the convex (loss) function $, if \\Xto - X\\9 = inf{||Xto - F||$ : Y G A4}. [Alternately, we can also understand it as: / $(Xto - X)dP = inf { / (Xto - Y)dP : Y G M} , 7n Un J

(1)

assuming that these integrals are finite. In particular this holds if $ G A2. Here problem (i) is called linear prediction, and (ii) nonlinear prediction. We consider only (ii) hereafter since (i) uses different tools.] It must be noted that the best predictor, using the distance (or norm) functional and the modular in the alternative definition, are not necessarily the same in contrast to the Lebesgue spaces. This is not unexpected since in the more inclusive Orlicz context, the norm and modular convergence need not be the same. One can better utilize the geometry of L$(E)-spaces with the norm than with the modular concept in (1). We present first an auxiliary result, using the norm topology, to understand the problem better. Proposition 2. Let L $ (E) be a separable Orlicz space on (Q, E,P) and M. = L®(B), called a measurable subspace. If X G L $ (E) and YQ G M. such that \\X — IQ||($) — mf{||* — ^||($) : Y G M}, then there exists a closed linear operator P§ : X \—> YQ such that (i) i f f e M , then f G P(PJ), the domain of Pj, Pjf = /, a.e., (ii) P§ is closed and linear, i.e., fn G D(P§}, fn^fin norm, gn = P§fn and gn —> g in norm, then f G T>(PBx) as well as g = P£f, a.e., (in) if BI C #2 C E are a-subalgebras, then P^P^X = P^X, a.e., (iv) if f is B-measurable and g, fg G T)(P£), then P§(fg) = f(P§g), a.e., and (v) ifX>0, then PgX > 0, a.e. Proof. Since L $ (E) is separable, the subspace M. is also separable. Then there exists (non uniquely) a quasi- complement M' of M. (i.e., M. D N1 — {0}, M. 0 J\P is dense in L $ (E) and the linear subspace M can be enlarged [or contracted] using finitely many elements so as to remain a quasi-complement, cf. e.g., Murray [1], p.93, and Mackey [1], p.322). Given X G L $ (E) and YQ G M such that \\X - y0||(*) < II* - ^ll(*),^ e M, let Z0 = X - Y0 so that ZQ £ M. when X ^ M.. If ZQ G A/"', then X = Y0 + Z0 is a unique decomposition, but if ZQ £ A/"', let M be the enlarged quasi-complement of M. (i.e., J\f = J\T Q){ZQ}}. The theory of these complements shows that this is possible (in any separable Banach space). Then there exists a closed (linear) projection on T> = M ® A/" onto Ai which we denote by P§ (since it depends

238

VII. Applications to Prediction Analysis

on X). We assert that (i)-(v) hold for this P§. Since by construction it is a closed linear operator on the dense subspace £>, (i) and (ii) are immediate. Regarding (v), if X > 0 a.e., so that PjX — F0, let Y — YoX[y0>o] so that Y G M and \\X - Y\\(^} < \\X - Yb||(*) with strict inequality if Y0 < 0 on a set of positive measure, (i.e., Y improves upon YQ, for X > 0) which is impossible by definition of Y0. Thus Y0 > 0 a.e. as well. Regarding (iii) and (iv), let Yi G Z/*(Bj),^ = 1,2 be the closest elements to X, and X = YI + Zi = Y2 + Z2 defining Zi G A/;, the quasi-complements of Mi = £*(#*), so that P§}X = Yt,i = 1,2. Thus Yl G Ml = L*(Bi) C £*(B2) = M2 and Y2 G M. But the decompositions X = YI + Zi = Y2 + Z2 are unique so that Z\ G A/i D A/2- Consequently, PB2(Pi1X} = PSaY1 = P§1Xt

(2)

and also PB^PB^)

= P^P^Y. + Z,}} = P^YI, since PJ2\M2 = id., = yl = p^X, since Pj^Mi = id.

(3)

From (2) and (3), property (iii) follows. Finally for (iv), if / e M and g 6 T>(P§) such that fg 6 X>(P|), note that g — g\ + hi for a unique gi £ M and /ii G M. Then /g = /^i + fh\ =>• /51! G Ai since it is ^-measurable, and gives a decomposition for f g . Note that if fhi G M then with / ^ 0, /ii must be 5-measurable so that hi G .M => /ii = 0 since ^ G A/". Also P£(M) — M, and it results that

Thus (iv) holds, and all properties of Pj are established. D Remarks. 1. As noted in the proof, if {X\, X2, • • • , ^n} is a finite collection of elements of £*(£), and {Fi, Y2, • • • , Yn} are the corresponding closest elements of M = L®(B), then nonuniqueness of quasi-complements allows one to choose the PQ that works for X\, • • • ,Xn, such that P^Xi = Yi,i = 1, 2, • • • , n. This may even be done for a countable collection, but not for all the elements of L*(E). The dependence of the closed (usually unbounded ) PB on such a collection of X'ns may be exploited in some problems. 2. The existence of quasi-complements is a nontrivial result in Banach space geometry. It was shown by Lindenstrauss [1] that in every reflexive Banach space (not necessary separable), each closed subspace admits a

7.1 Best prediction with convex loss

239

quasi-complement, and that for nonreflexive nonseparable spaces (in contrast to Mackey's result) this is no longer true, and constructed spaces X that contain m(F), the space of bounded functions on an uncountable set F, having subspaces that include isometrically co(F), the space of bounded functions that vanish at "infinity" , which do not admit quasi-complements. [This was shown by him in [2].] Thus the separability used as hypothesis is needed. 3. The operator PB has a dense domain in L*(E) so its adjoint (PB)* exits as a closed projection, whose domain is weakly*-dense in (Z/ $ (E))*. If I/$(E) is reflexive, and either the range or the null space of PB [or of its adjoint] is finite dimensional, then PB is bounded. However, even then PB is not necessarily a contraction and if Z/ $ (E) is not a Hilbert space, the above operator, in Proposition 2, need not coincide with the conditional expectation, although the properties noted as (i)-(v) are characteristic of a conditional expectation operator. For a given X, the unique minimal element Y0 assumed in the above proposition will now be established in the following result. The existence question for the modular functionals was solved in De Groot and Rao ([1], [2]), and for the Z/1^) case by Shintani and Ando [1]. Uniformly convex Orlicz spaces will be treated later as the techniques are different in all these cases. We start with convex loss functions satisfying some smoothness (i.e., differentiability) conditions. The concept of best predictor relative to a convex loss function, as introduced in Definition 1, will be used. Also, define the mapping U : f \—> /n$(/o - f)dp,, f e Z/*(/z), where (S7,E,/z) is the basic probability space. Theorem 3. Let (f2,E,/z) be a probability space and Z/*(/z) an Orlicz space based on it, where $ : ]R —> 1R+ is an N-function. Suppose $ G A 2 and the derivative $' exists and satisfies the following condition: for each a > 1 there is a Ka > I such that $'(ax) > Ka$'(x),x > 0. Then for each asubalgebra BofS and fo G L®(p], there is a unique f (depending on fo) in L®(B) = {/ € Z/$(/z), / is B — measurable} such that (4)

Moreover, the function f is the solution of the integral equation: f $'(l/o - f\)8gn(f0 - f)dfjL = 0, Jfi or equivalently the solution of: I $'(|/o - fDdfj, — I $'(|/o - f\)dfj,. [/o>7] [/0<7]

(5)

(6)

240

VII. Applications to Prediction Analysis

[Here, as usual, sgn (x) is the signum function.] Proof. First observe that U(f) < oo because $ 6 A 2 . Let a0 = " g € L®(B}}. Since for nontriviality we can assume that /o ^ £*(#), it follows that 0 < Q!O < oo. The conditions on $' imply that $ is uniformly convex, and (with $ 6 A 2 ) that the space L^(^L) is uniformly convex. (Cf. the companion volume, Rao and Ren [1], p.288.) By definition of infimum, there exist fn € L*(B) such that U(fn) —> CXQ. Hence the sequence {/n,n > 1} is norm bounded, i.e., is contained in a ball. Since L^(IJL) is also reflexive, its subspace L®(B) is reflexive. Then {/ n ,n > 1} is relatively weakly sequentially compact; whence there is a weakly convergent subsequence fnk —> f1 with f1 € L®(B}. Since e A 2 , the norm convergence and the modular (or mean) convergence in £*(//) are well-known to be equivalent. Hence fnk —> f1 in norm by a classical result (due to F. Riesz) in any uniformly convex Banach space. [See Theorem 4.2 below on a generalized version.] It follows now that U(fn) —> U(fl) = Q>Q. Since the space is strictly convex and the functional U(-) : L?(B] —> M+ is convex, it follows that the minimum is attained at a unique point. So f1 — f of (4) and the first part of the theorem is proved. To derive (5), we now differentiate the functional U(-). Let h ( t , g ) = ~ + tg},g 6 L*([j,). Since U(J] < U(f + tg) for all* e R and g € with the unique minimum at /, it follows from elementary calculus considerations that /i(0, g) = 0 for all g e L $ (/z). To evaluate h(t, g} observe that , on setting / = /o — /,

—$(|/ + tg\) = $'(|/ -f tg\) — (\f + tg\). at

(7)

at

From known results (cf. Dunford and Schwartz [1], V.9.1) that |/ -I- tg\/t is nondecreasing in t as t \ 0, and ^(|/ + tg\) < \f\ + \g\. Hence the known results in Orlicz space theory imply that the right side of (7) is integrable. Consequently, by the dominated convergence theorem the integral and derivative can be interchanged in (7). This gives

d

~

-7i(\f + tg\) at =

f Jn

dp, t=o

*'(f)(8gnj)gdn.

Since g G L*(n] is arbitrary, setting it to be a nonzero constant g = a e L*(/z), and cancelling, we get (5), and then (6) is an immediate consequence of the definition of the signum function. D

7.1 Best prediction with convex loss

241

With a slight restriction of the problem another more usable form of the integral equation (6) can be obtained. But this depends on the disintegration property of the probability measure/u and a restriction of the range of /o, the element to be predicted. We now outline a procedure to show how the conditional measures (and their distributions) are essential for this specialization, and the problem is of independent interest. Recall that, if g € Ll(n} and B C E is a a-algebra, then the set function vg : A i—> fA gd[i, A 6 B is a-additive and if HB = H\B, one has vg « HBSo by the Radon-Nikodym theorem g = ^p- exists a.e., and g € Ll(B). It satisfies vg(A) = I gdfi = I gd^B, A G B. (8) The mapping EB : g \—> g is a positive linear contraction, and taking A = fi in (8) one gets E(g) = i/ fl (fi) = E(g) = E(EB(g}). This identity, which is valid for all g € Ll(^), and all a-subalgebras B of E, is needed to derive an alternative form of (5). The minimization of U(f) of (4) is simply that of

h(t, g) = -rU(J+ tg] = —E(*(\f + tg\)) = -E(EB(^(\f + tg\)}),

(9)

where g e I/ $ (E),/ € L*(B) and f = f0 — f . Under the same integrability condition as before, we get Jy.~

V"VI^

• "y | / /

i •

V-*-"/

Writing /j,B(A) = EB(XA},A 6 E, one verifies that p,B : E —>• L1^) is a a-additive (vector) measure and its variation is ^, i.e, |// B |(-) = //(•)» where \VB\(A) = sup { E ||/zs(^)||i : Ai C A\ . Now nB(A)(uj) = Q(A,u), defining u=i J Q : E x £1 —> M+, such that Q(A, •) is ^-measurable, but Q(-,u) need not be a (scalar) measure. This Q(-,-) is called a regular conditional measure iff (a) Q(-,(*}) is a probability measure for each u e Q and (b) Q(A, •) is Bmeasurable for each A 6 E. When such a regular conditional measure exists, then (10) is simplified for almost all (a.a.) uj, [//], as:

= J^'(\J\}sgn(J}g(u}dQ(-^},

J = f0 - fj € L*(B),

242

VII Applications to Prediction Analysis

- I/ *'(/M-/o)0MdQ(-,w), • [/O
(11)

for a. a. (a;). Thus by (11), under this regularity condition (and as before the differential and the integral commute), taking g(uj} = 1, a.e.,

)

(12)

is obtained for a. a. uj (under the /^e-measure) . Thus (6) becomes (12) when these conditions hold. It should be noted that under the regularity assumption on Q(-, •), (12) gives a solution f(uj) for each uj € f2, and one has then to verify that /(•) is measurable [or equal to a measurable function for a. a. (a;)] to call it a predictor. This needs an additional argument, but it is not too difficult (cf., e.g., the proof of Theorem 3.2 below). The convexity of the modular [/(•) can be used to show the measurability, by approximating the integral with a sequence of monotone summands each of which is measurable. With this remark, we now summarize the above alternative form, for reference, in the following: Theorem 4. Let (£7, E, p,) be a probability space, B C S a a-algebra such that the conditional measure p
J[fo>f(u)]

*'(/o - f(u))dQ(-,u) = I

J(fo
#(/M - / 0 ))dQ(-,a;),

(13)

for a. a. uj G fi (under fi^)- In particular, if <&(x) — x1 , corresponding to the quadratic loss function, the solution f(co) of (13) reduces to f(u)= [ fQdQ(-,u}), Jo. so that f = EB(fo)

a.a. u e fi, [p,B],

(14)

is the conditional mean of /o relative to B.

Discussion. It is useful to note the relation between the existence of regular conditional measures related to a given probability space (Q, E,/z) and another concept called disintegration of measures in mathematics.

7.1 Best prediction with convex loss

243

Consider again the identity E(g) = E(EB(g]} for all g 6 Ll(p,), given in (8). This is written explicitly, if a regular conditional measure Q(••>•) of ^('}('} exists, as: = I I g(u'}Q(du',uj)nB(duj}, Jn. Jo.

where /J,B = H\B and |Q(-,-)l XA, A £ E, this becomes

=

(15)

variation of Q(-, •) = //(•). Taking g =

(16) or symbolically //(dw) = Q(du}',uj)[j,B(du>}, and thus /j, is "broken up or disintegrated" relative to fiB into a family {Q(-,ui),uj 6 fi} of probability measures. Such a disintegration for a general measure is not always possible. In fact, there is a close relation between regularity of conditional probability measures, disintegration of a probability measure and the existence of a "(strong) lifting operation" on L°°(B). A detailed discussion of this may be found in the book (Rao [21], Ch.5). If, for instance, B is generated by a vector X = (/i,/2, • • •) of random variables (B C E) on fJ, and if the range X(Q) C JRW, is a Borel set, then the regular conditioning [j,B(•)(•) exists. This is always possible if the fa are "coordinate functions" fi = TTj : 17 = jRw —> JR. These questions are discussed in the above reference, and we shall omit them here. Finally, it may be shown that the best predictor given by (13), when regular conditional measures exist, is valid for a convex "loss function" $ which need not satisfy a A2-condition and hence L*(ii) need not be reflexive. This is established in De Groot and Rao [1]. We shall discuss this and a more specialized case of $(x) = \x\ later to appreciate the nonreflexive (and non strictly convex) cases clearly. If {fn,n > 1} is a sequence of observations, let Bn = v ( f i , i < n] and BOO = !)• Then by the preceding theorem, for each n > 1, there is a (unique) best predictor /„ based on^fi, • • • , fn in the sense that U(fn) = mm{U(g} : g e L*(Bn)}. Similarly, /c» is the best predictor of /0 based on all the observations so that /oo e L^(Boo) and is the closest to /0. The relation between {fn,Bn,n > 1} and /0 is described by the following: Theorem 5. Let $ : M —> M+ be the N-function and L*(IJL), L*(Bn) be as in Theorem 3. If U : L^(^} —> M+ is the modular functional and {fn, Bn, n > 1} is the best predictor sequence obtained above, then fn —> /^ as n —>• oo in U-(modular) mean, so that £/(/<» — fn) —> 0 as n —> oo. [It can also be shown, under the same additional hypothesis that fn —> /<00; a.e.J

244

VII. Applications to Prediction Analysis

Proof. Here we establish the mean convergence, but omit the pointwise part which is better considered by using a detailed probabilistic analysis. Now under the present hypothesis, L®(p,} is uniformly convex (hence reflexive)^ By Theorem 3, for each n, there is a unique /„ G L*(Bn) such that U(Jn] = min{[/(/) : / e L*(Bn)}. Since Bn C Bn+l implies L*(Bn] C L*(# n+ i), it follows that U(fn) > C/(/n+1), so that the set {/n,n > 1} C (J L?(Bn] C £^($00), is norm bounded, and hence is weakly sequentially relatively compact. The argument of the proof of Theorem 3 shows that there is a weakly convergent subsequence fnk —> /0 (say) , and that the convergence (by Riesz's theorem used there) is in fact strong. The strict convexity of the functional £/(•) also implies that every convergent subsequence of { f n i n •> 1} has the same limit so that the whole sequence /„ —> /o in norm and /0 € I/* (#00), where B,x, = a( U Bn) and that I/^^oo) = closed linear span( \J L^(Bn}}. Note that the equivalence of the norm and modular n>l

convergences is utilized here, and so the assertion holds, d Remark. The pointwise convergence, using a probabilistic argument is established even for a more general class of "Riesz-Kothe" function spaces whose norm is absolutely continuous, strictly convex and has the "Kadec property" as well as obeying a uniform monotonicity condition. These are all automatic for the Orlicz spaces considered above. This general result, due to Bru and Heinich [1], is detailed in a recent book (cf. Rao [24], p. 475), and will be omitted here. Instead, we shall present a characterization of linearity of the prediction operator -Pg(-), for all re, (cf. Proposition 2 above) in the next section, extending a basic result due to Ando [3]. 7.2 When is a prediction operator linear ? It was seen in the last section that in a separable L*(E), if M. — L®(B] C L*(£), then for each X € L $ (S), the existence of Y0 e M such that \\X y0||(*) = inf{||X - r||($) : Y <E M}, implies Pj : X H-> F0 defines a linear operator depending on X. Here we wish to consider a subclass of spaces L*(£), not assumed separable, for which unique minimal elements exist in M = L*(B), so that PB : X ^ PBX is defined for all X e L*(E) but is also linear. If L $ (E) = L P (E), 1 < p ^ 2 < oo, such a characterization was obtained by Ando [3], and its extension to Orlicz spaces was considered by Rao [11]. We shall detail the salient part of the latter for the problem here, as it is interesting and nontrivial, revealing a new aspect of Orlicz spaces. One says that a subspace M. C £*(//) is a Chebyshev set iff for each / e

7.2 When is a prediction operator linear ?

245

there is a unique #/ G Ai such that ||/ — #/||() = inf{||/ — /i||($) : /i £ i.e., Qf is the best predictor of /. A pair of subspaces {M, jV} C £$(AI) is complementary if Ai fl A/" = {0} and the direct sum ./M 0 A/" = L* (//) (not merely dense!). The following simple but crucial observation is basic for the characterization problem of linearity of the prediction operator on L$ (/^) into M, denoted by PM. Proposition 1. Let L?(IJL} be an Orlicz space on (Q, E,/^) and M. C be a Chebyshev subspace so that PM : L? (n} —> M is the prediction operator which exists. Then PM is linear iff there is a complementary subspace M of M. which is the range of a contractive projection on Proof. Since M. is a Chebyshev subspace the mapping PM '• f '—> 9f G -M. exists, but generally it is nonlinear. If it is linear, then Q — / — PM is a (linear) projection with range JV which is complementary to M. Moreover, for / € L*(n) and h e M (1)

Taking h = 0, it follows that ||Q/||($) < ||/||($) so that Q is also a contraction. Conversely, let M be a complementary subspace of Ad, that is the range of a contractive (linear) projection Q. Then PM = I — Q : L®(/J,} —>• M. is a (linear) projection with range M. and null space A/". Also, for h £ M. \\f ~ PM/\\W = \\Qf\\W = \\Q(f ~ *OII(*) < ll/ - ^ll(*)-

(2)

This follows from the fact that M is the null space of Q and Q is a linear contraction. Since PM/ 6 A4, it is clear from (2) that PM/ *s tne Dest predictor of / in M. which is a Chebyshev subspace, so that PM is a linear predictor on L*(ti). D Note that for this statement, we did not use any property of the Orlicz space, and thus it is valid for any Banach space X in place of L*(//). However, the structure of L^(^) will be used for a characterization of the pair M. and A/" in order that PM is linear. To solve our problem, we restate the above result in a different but equivalent form which uses the Orlicz space structure in a crucial manner. We assume hereafter that $ is normalized, i.e., $(1) + \P(1) = 1, [cf. the companion volume, Rao and Ren [1], p. 35]. It is classical that in a Banach space X with M as a closed subspace and M.L = {x* € X* : x*(M] = 0}, the annihilator, then the quotient space X IM. with the quotient norm, i.e., if x = x + M. 6 X /M, \\x\\ = \id{\\x + y\\ : y € M}, is isometrically isomorphic to M.L whose norm is that

246

VII. Applications to Prediction Analysis

of X*, the adjoint space (cf. e.g., Dunford and Schwartz [1], p. 72). Moreover, if P : X —>• X is a bounded projection with range At, and null space A/", so that X = A40A/", then X* = A / f- L 0A/ r - L and the adjoint operator P* : X* —> X* is also a projection with range A/""1 and null space A4-1. In our case, X = L $ (//),A1 is a Chebyshev set, and PM : X —> JV[ is linear (hence also a contractive projection) iff / — PM = Q : L®(/J,) —> A/" is a contractive projection by the preceding proposition. Then the adjoint operator Q* is a contractive projection on (L*(/i))* with range JviL(= L*(ju)/A1). Assuming £*(//) to be a real space, so that it is actually a Banach lattice under the pointwise a.e. ordering, the quotient space L*(//)/Af inherits a partial ordering, or ML C (L* (//))* is also a Banach lattice. Since in general this adjoint space is not a real function space (it has set functions), we now restrict
ll/ - P/||(*) = ||Q/||(*) = HW - $)||(*) < / - 0||(*).

(3)

Thus Pf is a minimal element and then it is unique so that P : f i->- P/ is a (linear) prediction operator. But now ML C I>*(^) is a Banach lattice. However M^ and A/"* (as well as A'f* and A/"1) can be isometrically identified, preserving lattice operations. Hence A^ is a Banach lattice as well, and is a subspace of L*(n). Thus the preceding result can be restated as: Proposition 2. Let At C L*(n) be a Chebyshev subspace with PM : L® (fj,) —> M. as its prediction operator and $ e A 2 fi V 2 . Then PM is linear iff L*(yLj)/Af (identified as a complementary subspace of M.) is a Banach lattice (because PM/ > 0 f°r / > 0 always). This result demands that we characterize Banach sublattices of the (Banach) lattice !/*(//). In general, this is an involved problem. A complete solution was given by Ando [3] when $(x) = \xp,p > 1, and it was extended by Rao [11] when $ 6 A 2 . Since these details are numerous, we shall present the general form completing the last proposition and then discuss some consequences. A set 5 G E is called the support of a subspace M C L*((JL) on a afinite measure space (17, E,/i) if S is the supremum of {S/ : f € A'f} where Sf = {uj : f(u}} / 0} is the support of /. It can be verified that such

7.2 When is a prediction operator linear ?

247

a supremum exists in any complete cr-finite measure space (fi, E,//). [It is even true for all "localizable" measures, but here we restrict to finite IJL for simplicity.] Then one has the following with $ 6 A2, an A^-function: Proposition 3. Let L®(p,} be an Orlicz space with (j, complete and finite. If M. C L®(/J,} is a Banach lattice, then there exist (i) a function 0 < g$ € -M. such that g$ has the support a. e. of that of M., and (ii) a a -algebra B C E such that M. is representable as M. — go • L*(fi, B, Ate) = [gof • f e L*(n,B,Ate)} and ||#o/||($) = ||/||(*),/ e L*(Ate) where HB is fj, restricted to B. Here gQ need not be ^-measurable, but S = supp(g0) G B can be chosen and then EB(g0} = xs a.e. With this structural result, A4 can be related to a projection operator on the space L®(p,) as follows (it was already noted at the end of Chapter V). Proposition 4. A subspace A/" C L$ (//) is the range of a contractive projection iff it is a Banach lattice. Proof. If A/" = P(L $ (//)) where P is a positive contractive projection, it is a complete subspace. Also if / G A/", then / = Pf since P|A/"=identity, and I/I = |P/I < P(|/|) a.e. But ||/||w < ||P(|/|)||(*) < ||/||(») so that by the monotonicity of the norm, P(|/|) = |/| a.e. and hence |/| G A/". Thus A/" is a Banach lattice. Conversely, if A/* is a Banach lattice, then by the preceding result, A/" = go • L $ (/^B), where 0 < g0 G A/" is an element with support S £ B which is also the support of A/". But the conditional expectation EB on L*(fj,) has range L®(p,B] and is a positive contractive projection. Let P : L?(IJL) —> Af be denned as P(g) = goEB(g),g G L*(/z),^ 0 > 0. Then it is asserted that P is a positive contractive projection on L^(p] with range A/". Indeed, by the isometry of the mapping (cf. Prop. 3) and using the fact that EB is a positive contraction , we have

It remains to show that P2 = P. By construction S = supp(g0) G B , hence also EB(go) = xs- From this one has with the "averaging property" of EB, the following relation: if g G L*(//), then P2g = P(g0EB(g)) = 9oEB(9oEB(g)) =

B

9oE

(g)EB(g0)

248

VII. Applications to Prediction Analysis

Hence P is a contractive positive projection with range A/". d It should be observed that when L*(fj,) is not a Hilbert space, a bounded projection with range a given subspace Ai C L®(p,) need not exist. The above result shows that such a projection (even a contractive one) exists if M. is a Banach lattice. When it exists, the projection need not be unique. However, if A^ is a Chebyshev subspace in addition, or if L®(p.) is strictly convex and some further conditions hold, then again the uniqueness property obtains. Based on the preceding propositions, we can present the following general statement on the linearity problem. Theorem 5. Let L*(/u), $ € A 2 Pl V? be a (reflexive) Orlicz space and M C L*(/J) be a Chebyshev subspace. Then the prediction PM '• L*((J>) —* M is linear iff the quotient space L$(/i)/.M is isometrically isomorphic to an L $ (fi, B, jT) where $ is as before and (fj, B, Jl) is some (finite) measure space. Moreover, the isomorphism preserves the maximal supports of spaces. Sketch of proof. If PM is linear, then L $ (/i) = M. ® A/", where M is a complementary subspace of M which is isometrically isomorphic to L®(n}fM. and is the range of a positive contractive projection. It is then equivalent (=isometrically isomorphic) to an L®(B) — L®(Cl,B,jl} by the preceding proposition. Conversely, if L®(p,)/A4 can be isometrically identified with an L*(#), then (L*(ii)lMY = ML, the annihilator ofM. This implies, since $ € A 2 n V 2 , ML = L9(B). But ML can be identified with a closed subspace of !/*(//) and the above equivalence implies that there is a unitary operator U such that U(A4±) is a Banach lattice so that it is the range of a positive contractive projection T on £,*(//). Consequently, T* : (L* (//))* = L*(n) —>• L*(/J) is a contractive projection with a similar property (after some work). Thus P = T* is a projection on L*(IJ,) with range A/" and null space M, whence A^©Ai — L*(/^). Since A4 is a Chebyshev space, it follows that PM = / — P : L®(n] —;>• M. is a linear prediction operator, as in (3), completing the sketch. D Some remarks on the solution are in order. From the point of view of applications, it is hard to verify the conditions of the characterization given in the theorem. If M. itself is a subspace I/ $ (// e ), based on a a-subalgebra B C S, [i.e., M C L* (//)], then B has to be "small" so that L*(//s) is a small space, and the direct form of such a space on which PM is a linear projection is really uninteresting for applications. The next best things are the isomorphism results. This can be appreciated if we recall that any reflexive Orlicz

7.3 Nonlinear prediction for nonreflexive spaces

249

space L^(fj,), which itself is not smooth, is isometrically isomorphic to a uniformly convex and uniformly smooth Orlicz space Z/$1 (//) where <3> ~ $1 (cf. e.g., Rao and Ren [l],p.297). This fact exemplifies the reasons behind the preceding considerations. If one does not require the linearity of prediction operators, Theorems 1.3 and 1.4 present integral equations for finding the best predictors for any given element of L®(fi), under reasonable conditions. We consider some complements to the results of the preceding sections. 7.3 Nonlinear prediction for nonreflexive spaces Most of the geometric properties of spaces used in the preceding sections are not available for general nonreflexive spaces. So here we indicate some methods for such spaces by considering the special case of real Ll (fi) and a class of real L?(IJL) both on a probability triple (fi, S, fj,). For this analysis, we need to consider regular conditional probability measures as recalled in Section 1 above. A somewhat more general problem will be presented, and the following preliminary result will be useful. It follows De Groot and Rao [1]Let W : JR —> 1R+ be a symmetric measurable convex function that is not a constant. Hence W(-) is actually absolutely continuous with right and left derivatives existing at each point of JR. Define U : 1R —>• 1R+ as: (1)

where n : M —>• M+ is a probability measure. Assume that U is finite, so that, since it is clearly convex and continuous, it has finite right and left derivatives at each point. Since a convex W has its (right and left) derivatives to be nondecreasing, and in fact the difference quotients ^[W(i — e) — W(t)] are known to be nondecreasing, by the monotone convergence theorem, the finiteness of (1) implies that /

J[t>8]

WR(t - 5}dn(t] < oo; and t

J[t<6]

WR(t - 8} dp(t) < oo, (2)

and a similar statement if WR is replaced by WL, where WR(WL) is the right (left) derivative of W. The fact that W(-} is not a constant implies that lim W(t) = lim U(6) = oo. Suppose, for simplicity, that WR(t) = WL(t] t—>±oo J—>±oo for alH 7^ 0, we then have: Proposition 1. Let (M,B,/ji) be a probability space and W(-) a symmetric convex function with U(-) defined by (1). If U(1R} C JR+ , W exists at all

250

VII. Applications to Prediction Analysis

points except the origin (and W ^ Q), then the set of values I C 1R at which [/(•) has a minimum value is a nonempty compact interval, and 6 & I iff

t

W(6 - t)dn(t) > I

W'(t-6)dM,

(3)

I

W'(6-t)dfjL(t)<

W'(t-$}dv(t).

(4)

J\t<5]

J\t>5]

and J[t<S]

f ~ J[t>5]

In case either W'(Q) — 0 or /^(-) is nonatomic, then 6 G / satisfies the integral equation

f

J[t<S]

W'(6-9)dfj,(t)=

f

J[t>5]

W'(e-6}dfj,(t},

(5)

with I having exactly one point when W(-) is strictly convex in addition. Proof. Since lim U(t) — +00 and U(M) C M+, U being convex, the set |t|-»00

of minimum values constitute a compact nonempty interval. To verify the validity of (3), let cn \ 0 be a sequence of numbers, and note that for any 6e R, gn(t) = -[W(t-6-en)-W(t-6)] &n

is nondecreasing as n —> oo, and n—>oo lim gn(t) = —WL(t — 6). Hence UR(6) — lim —[U(6 + en) — U(6)] exists and by the monotone convergence theorem UR(6) = - f WL(t-S)dfi(t}.

(6)

JIR

But by the symmetry hypothesis, WL(t) = WR(t) = -WL(-t),t ^ 0, and WR(ti) = -WL(0). Hence (6) becomes UR(S) = f

J[t<5]

W'(6 - t}dn(t} - I

J[8>t]

W'(t - 6}dn(t).

(7)

Since UR(6) > 0 for S € /, (7) implies (3). Similarly (4) is established, and (5) is obvious. D With the above result, we can present a solution of the (nonlinear) prediction problem. If {Xi,---,Xn} or {Xt,t < tQ} is an observation set of random variables and XT,r > to, is to be predicted, or approximated by a "best" element 60(= 6(Xi, • • • , Xn) or 6(Xt, t < t 0 )) that is closest to XT in the sense that U(S0) = inf ( / W(Xr - 6(X))dP : all such 6\ < oo. Un

J

(8)

7.3 Nonlinear prediction for nonrefiexive spaces

251

This will now be made precise, and a solution obtained using the preceding result. Let B be the cr-algebra generated by the observed set, i.e., t
where 9? is the Borel a-algebra of JR, and let 6(-) be a B-measurable function. It is the best predictor, or estimator or approximant of XT relative to the convex "loss function" W, if E(W(6)) = Jn W(6)dP < oo, and if there is a S0 (B-measurable) such that:

U(XT - S0) = mm{E(W(XT - 6)) :6 is B- measurable, E(W(6)) < oo}.

(9)

B

Using the identity E(W(X}} = E(E (W(X}}} for any random variable X for which E(W(X)) < oo, we may translate (9) into the problem of Proposition 1, for conditional probability measures under some restrictions involving regularity, as follows. It was already seen, just prior to Theorem 1.4, that the identity E(X) — E(EB(X)) for any integrable X : fi —> 1R and
(10) Expressing this in differential form P(du}') = PB(duj')PB(du)), the inequalities (3) and (4) as well as the equation (5) can be written with the classical Fubini theorem, if PB(duj'}(uj} can be taken as a measure for each LJ £ $7, as follows. The best (nonlinear) predictor is a function SQ(-) based on the observed variables {Xt,t < t0} and should be such that (i) it is Bmeasurable, (ii) E(W(5}} < oo, and (iii) E(W(Xr - 5Q}} = mm{E(W(XT 6)) : 6 satisfying (i) and (ii)}. If T> is the set of all 6 that satisfies (i) and (ii), then for (iii) we should minimize the functional U(6) on V, i.e.,

U(6)

= I W(XT - 5}dP Jo, , by (10),

=

f]R[f]RW(t-6(x))(PBoX-l)(dt)(x)^dG(x),

(11)

252

VII. Applications to Prediction Analysis

by the image law of probability, (cf. e.g., Rao [18], p.19) when PB is regular. It follows, under the regularity hypothesis that (PB o X ~ l } ( - ) ( x ] = Q ( - , x ) is a probability measure so that the integrals in (11) are in the Lebesgue sense, that the pointwise minimization of the inner integral yields the desired solution SQ(-). Thus it is required to minimize the integral fKW(t~8(x))Q(dt,x] for each x and show that solution 8Q(x) can be chosen to be a measurable function 6(-) which will thus be in T>. This is solved below. Theorem 2. Under the conditions of the existence of regular conditional probability function PB and letting Q(-, •) be the image measure PBX~l, there exists a (minimal or) best [nonunique generally] predictor SQ G T> which is a pointwise solution of the integral inequalities when W'(-) exists (except at 0): I J[t>8o(x)]

W'(t - d0(x))Q(dt, x)> f

W'(60(x) - t)Q(dt, x)

(12)

W'(t - S0(x))Q(dt, x)< f

W'(60(x) - t)Q(dt, x).

(13)

J[t<S0(x)]

and [ J[t>So(x)]

J[t<5Q(x)}

Further if W'(0) = 0 and W(-} is strictly convex, then (12) and (13) reduce to a single integral equation with an a.e. [PJ unique solution SQ: f

J[t>S0(x)]

W'(t - 60(x))Q(dt, x)= f

J[t<S0(x)}

W'(60(x) - t)Q(dt, x).

(14)

Proof. The result follows immediately from Proposition 1, in view of the regularity of the conditional distribution [or measure] Q(-,x),x € 1R, if we show that the pointwise solution SQ(-) is a ($)-measurable function of x, [B — a( U Xj~l(?R.))}. To prove this fact consider for each arbitrarily fixed

xeJR,

t
U(S\x) = U(6(x)) = ! W(t- 6(x))Q(dt, x). (15) Jm, By the regularity of Q(-,x), [i.e.,of PB(-}(UJ)] it follows that U(6\-) can be chosen to be a measurable function on 1R (relative to 5R, the Borel cr-algebra). Also for each x € ]R,U(-\x) is convex, and let I ( x ) be the closed bounded interval that contains the values 8 for which U(8\x) is a minimum, as guaranteed by Proposition 1. Let SQ(X) be the left end point of I(x] for definiteness, x € M. Observe that for any a e 1R, 5Q(x) < a iff UR(a\x) > 0, where UR denotes the right derivative of the convex function U(-\x}. But for any en \ 0 one has rrRf i \ r U(a + en\x)-U(a\x) U (a\x) = hm — — , x 6 JR.

7.3 Nonlinear prediction for nonrefiexive spaces

253

Now the right side element is nondecreasing and measurable in x, relative to 3ft, for each n. Hence x \—> UR(a\x] is a Borel measurable function, so that UR(a\-) is Borel. Since by choice of S0, we have {x : 60(x} 0} € 3ft. Thus 80(-) is a Borel function, and hence S0 6 T>. As already noted, the other statements now follow from Proposition 1. d Remark. If W(-} is not strictly convex, then the solution of the inequalities (12) and (13) is not unique. In particular if W(x] = \x\, then WR, WL exist every where and WR(0) = — WL(—0) so that there are many solutions, and each is termed a (generalized or a B—) median which will now be denned, and an explicit form of Theorem 2 will be presented in this case. If W(x] = \x\, then (12) becomes on substitution of W'(x) = 1 and adding the right side integral to both sides gives, 2 I

Q(dt, x) < j Q(dt, x} =

or

Q[txJ 0 (x),rr]<

(16)

and similarly from (13) one has Q[t<5Q(x},x}<1-.

(17)

Such a 6o(x} satisfying (16) and (17) is called a median of the conditional distribution Q ( - , x } . Thus Theorem 2 implies in case W(x) = x\, that conditional median <50(-) is a best predictor of XT and there are many such. To analyze this further, consider the functionals [/* and U* defined by

U*(x) = sup {y : Q[(-oo, ar), y] < i} .

(18)

Both t/*,C7* are measurable and £/*(rc) > 60(x) > U*(x). If Q(-,x) is independent of x, i.e., the case of unconditional (or classical) situation, then the medians defined above are the classical concepts used in elementary statistical theory. Since x \—> Q[(—oo,x),y] is left continuous (as a distribution function for each y, by regularity), the functions £/*(•) and £/*(•) are measurable, we have the following with additional properties:

254

VII. Applications to Prediction Analysis

Theorem 3. Let [hn € Ll(B),n > 1} be a sequence such that irn \Xr-hn\ i = so that g is a best predictor of XT. Then {hn,n > 1} is relatively weakly sequentially compact, and if h is a weak limit of hn , then h E T>. Proof. In Ll(P] on a finite measure space, (relative) weak sequential compactness of a set is equivalent to its uniform integrability (cf. e.g., Dunford and Schwartz [1], IV. 8. 11). To establish the latter property, consider the identity for any An C £1, \XT ~ hn\XAn + \Xr - hnXAll

= \Xr\XAn

+ \XT - hn\.

(19)

Since \XT — hn XAn > \hn XAH ~ \XT\XAn, substituting this in (19) and integrating one gets dP+ [ \Xr - hnXAcn\dP < I \XT - hn\dP + 2 I \XT\dP. (20) Jn. Jn. JAn Taking limits as n —> oo, and noting that fAn \XT\dP —> 0 for An \ have limsup / n->oo

JAn

0, we

\hn\dP + liminf / \XT — hnXAc \dP n^oo

JQ

< lni m XT - hnll, = \\XT - g l l , .

"

(21)

Since if An e B the middle term is never smaller than \\Xr — g\\i, then (21) holds iff the first term vanishes. This means {hn,n > 1} is uniformly integrable as asserted. Thus the first observation that {hn,n > 1} is relatively weakly compact is established. Then there is a convergent subsequence hnk —)• h weakly, and since Ll(B) is weakly complete, h 6 Ll(B). But P(A^) -> 1 in (21), and this implies that h G T>, and in fact every weak limit of the sequence {hn, n > 1} must belong to T>. n From (18), it follows that for regular conditional distributions Q ( - , x ) [or measures PB: (-)(-)], U* and U* are (conditional) medians. By Theorem 2, every conditional median is a best predictor of XT for the "loss function" W(x) — \x . Since V is the set of all minimal elements (or best predictors) for XT, we can conclude that U*,U* G T>, and that every g e T> satisfies U* < g < U* so that V is a closed "band" in Ll (B) following the terminology of Banach lattices. [There is an independent proof of this fact in Shintani and Ando [1]]. Using these results, we can state:

7.3 Nonlinear prediction for nonrefiexive spaces

255

Proposition 4. If {hn 6 Ll(B},n > 1} approximates XT as in Theorem 3, (without hn being in T> ), then hn V [7* —>• U* [and hn /\ U* —>• U*J in L1(B} — norm (V = max, A = min). Proof. It was already noted that U*,U* e Ll(B], and let ^4n = [hn > [/*]. Then ^n e #. Since hnVU* = hnXAn + U*XA'n and hnf\U* = XAnU* + XAf,hn, we have a = \\Xr-U,\\l

= j \XT-U*\dP+ j \Xr-U*\dP Acn

An

\XT - hn MJ*\dP - j\XT-U*\dP+

f\XT-hn\dP, Acn

An

so that \Xr - U*\dP <

\XT - hn\dP.

(22)

Thus using the definition of the minimum value, one has a < I \XT - hn V U*\dP < j\XT- hn\dP -> a n n as n —>• oo, so that by Theorem 3, {hn\/U*,n > 1} is a relatively weakly compact sequence in Ll(B}. Hence it has a convergent subsequence with limit h(£ Z>), and so \\XT — h\\\ = a. But, as noted already, U* is the upper bound of T>( the closed band) and so h < [7*. Consequently, 0 < limsup||/in Vt/» - t/»||i = limsup n—>oo

n—>oo

n

= lim sup I hn V C/*cLP - / C/*dP n—KX> 7 7 n n

= [ hdP - f U*dP < 0, by the fact that the sequence has a weak limit h(£ T>}. Thus hn\/U* —> [7* in L1^). Similarly one shows that hn /\U* -> E7* in L1^). D

256

VII Applications to Prediction Analysis

We finally obtain an analog of Theorem 1.5 for the I/1-case. Because of the nommiqueness of best predictors, the argument there does not apply here. If the observations are taken for t < tk,tk < £ fc+1 , and Bk = a(Xt,t < £&), then Bk C Bk+i and if gk e Ll(Bk) is a best predictor of XT where T > tk for all A;, and lim t^ = t^ < T, then the problem is to discuss the convergence k—too of the sequence [gk, k > 1}, and its relation to g^ if B^ = a( IJ Bk). This is k>l

outlined here. A sequence of (not necessarily monotone) a—algebras Bn C E is termed strongly (or a.e.) convergent to a a—algebra #00 if the sequence EBn(f] —» EB°°(f) in norm (or a.e) as n —> oo for each / G Ll(E) where (£7, E,P) is a probability space. If Bn C Bn+i, then this is an immediate consequence of a classical martingale convergence theorem. Here we are assuming just its conclusion to gain some needed generality. The following important result on strong convergence of best predictors of a given XT E L1^), relative to Bn, is due to Shintani and Ando [1], and the proof uses the structure of Ll(E). Theorem 5. Let XT € Ll(Y,), Bn C S and hn be a best predictor of XT in the norm ofLl(E) for Bn. IfBn -» B^ strongly and U? and C7* are the maximum and minimum conditional medians of PBn, which is assumed regular, then the sequence {hn, n > 1} is uniformly integrable, hn V t/" —» U^° and hn A U* —>• U^ strongly in I/1 (#00), where U™ and U^ are the corresponding maximum and minimum conditional medians of XT, i.e., of PB°° oXT. Proof. For clarity, the argument is given in four steps. I. For convenience let us write £/" of Xr as Usn(XT), and [/* as VBn(XT). Then their definitions imply UBH(XT) — — V g n ( — X T ) , and if XT > 0, then UBU(XT) > 0, a.e. We assert that \UBn(XT)\,\VBn(XT)\<2EB-(\XT\),n>lt

a.e.

(23)

Indeed, by definition of UBn(XT), we have the inclusions: [A < U^Xr)] C fi < PB"(XT > A)| , a.e., L^

J

using the regularity of PBn. Taking expectations for their (set) characteristic functions one has P[UBn(XT > A)] <

j B

[l<2P n(XT>\)]

=

B

l-dP< 2 IPBn(XT > X)dP fi

2E[E »(X[xT>x}')]=2P[XT>X].

(24)

7.3 Nonlinear prediction for nonreflexive spaces

257

On the other hand for any 0 < / < Ll(£], UBn(f) > 0 which implies OO

E (uBn (/))

= / p [Usn (/) > A] dX, by the image law o [cf. e.g., Rao [18], p. 19, Thm. l(iii)], oo

< 2 1 P[f > X}dX, by (24), with XT = /, o = IE ( X[/>A] ) = 2P(f > A). (25) Replacing / by f\A,A € Bn, using UBn(xAf) = XA %„(/), one gets from (25) E (XAUBn(f)} < IE (XAf) = IE (xAEB»(f)} , so that the Bn— measurable functions satisfy UBn(f)<2EBn(f)-

(26)

To extend this for all integrable /, (i.e., for / = XT), let XT = X+ — X~ , and then (26) gives (UB*(XT)F < UBn(X+] < 2EB"(X+] < 2EB»(\XT\), a.e. Similarly (UBn(XT)r = (VBn(-XT))+ < (UBn(-XT)}+ < 2EB»(\XT\), a.e. Combining these two we get \UBn(XT)\ < 2EBn(\XT\), a.e., as asserted. Because VBn(XT) = —UBn(—XT), the inequality for VBn(XT) also follows. II. Since we have already noted that UBn(XT) > hn > VBn(XT] one has, from this and the preceding step, \hn\ < UBn(XT) V (-VBn(XT}} = UBn(XT) V UBn(-XT) < 2EB»(\XT\). But {EBn(\XT\), n > 1} is a uniformly integrable family and hence the subset {|/in|,n > 1} has the same property. This implies a similar conclusion for {hn V UBn(XT),n > 1} since it is dominated by the uniformly integrable collection {\hn\ V UBn(XT),n > 1}. Then the former set has a convergent subsequence. We need to show that each such sequence has the limit UBn (XT) which implies that the whole sequence converges to the same limit, using the assumption on the Bn,n > 1, sequence at this point.

258

VII. Applications to Prediction Analysis

III. We observe that the UBn(XT} (and Vsn(XT)) converge in to UBoo(XT] (and VBoo(XT)). Since VBn(XT] < hn < UBn(XT), and the hnsequence was shown to be uniformly integrable, it follows in particular that the end point sequences, i.e. {UBri(XT),n > 1} and {Vsn(XT},n > I } , have the same property. Then the desired L1 — convergence will follow from Vitali's theorem if the concerned sequence is shown to converge in measure which will now be established. For this, by standard results in Real Analysis, it is sufficient to show that each subsequence has a further subsequence that converges a.e. to UBoo(XT). Now by hypothesis Bn -> B^ strongly. [The same argument below works for a.e. convergence also.] By the regularity of PBn, consider their conditional distributions, denoted Q(-,x)(= PBn o X ~ l ( - ) ( x ) ) . The assumption on BnS implies that Q ( - , x ) —>• Qoo(-,x) as n —> oo, for any arbitrarily fixed x. Then using a diagonal procedure (as in the classical Helly selection principle) one can find a subsequence Qnk(-,x) —> Qoo(',x) at all rational (and then at all) points x G JR. Now the Borel a— algebra 3? and hence UBnk(XT), may be chosen to satisfy \imsupUBnk (XT) < UBoo(XT). On the k—>oo

other hand UBn(XT] G ^(B^.n > 1 so that by definition of best predictor, U B ( X r ) being such a one, liminf UB (XT) > t/R (J£T) always holds. Hence n—>oo UB (XT) -> UBOO(XT)J in norm. Thus from the earlier reduction it follows that UBn(XT) -> UBoo(XT) in P-measure. So hn V UBn(XT) -> UBoo(XT) in measure, and then in norm. Similarly hn A VBn(XT) —> VBoo(Xr} in norm (or the result can also be obtainable from the above). IV. Since {hn,n > 1} is uniformly integrable (hence relatively weakly compact), there is a convergent subsequence hnk —> h^ weakly, and h^ € Ll(£,). However, for each n, VBn(XT) < hn < UBn(XT], a,e. Consequently VBOO(XT) < hoo < UBoo(XT] a.e. must also hold. If it is shown that hoo is ^oo— measurable, then h^ € Ll(Boo) and then h^ G V so that hnk —>• h^ in norm. Every subsequence will have the same property, and this proves the desired statement. Thus to establish the #00 — measurability, let g G L°°(S) and consider EBn(g}. By the strong convergence of the Bn, EBn(g) —> EB°°(g), in L 1 — norm. Also by the boundedness of g,hnkg -» h^g in Ll(E), as well as a.e., and the sequence is uniformly integrable. So using Vitali's theorem,

/ n

gh^dP

-

/•

lim / ghnkdP /c->oo J n EBnk (ghn )dP, by a property of EB,

7.3 Nonlinear prediction for nonreflexive spaces =

259

\imoJ'hnkEBnk(g)dP, since hnk G Ll(Bnk), n

=

/ lim (hn,EBni*(g}}dP, by uniform integrability, ' n

=

/ h00EB°°(g]dP^ by the preceding analysis, n

J fc-KX> \

= j E^fr^gdP, n

because EBeo is "self-adjoint".

(27)

Since # G L1^) is arbitrary, this implies E^fooo) = /i^ a.e. But £|B°°(/IOO) G Ll(Boo) so that ^oo is B^— measurable, and by the earlier reduction /i^ is a best predictor, and hn V UBn(XT] -> hx V UBoo(XT) = UBoo(XT), a.e., the convergence now being in norm. Similarly one gets hn A VBn (XT) —> hoo A^BOO(-^T) = ^BOO(^T)> in mean or norm. D Remark. A similar analysis works in the above if Bn —> B^ a.e., except that in (27), one has to use Fatou's lemma. In this case one gets VBoo(XT) < limmfVBn(XT) n-+oo

< lim sup UBn (XT) < UBoo(XT), a.e. n->oo

(28)

Note that every limit point of the [hn,n > 1} sequence will give the same (VB^UBJ(XT}. We now discuss briefly on alternative approaches to the existence of best predictors in Ll(B), for a dense set of elements of L X (E), that may "simplify" the analysis of the preceding work. Thus for X G L p (£),l < p < oo, let Y G LP(B), B C S a a— algebra, be the unique closest element (Y = YX,\\X-YX\\P = i n f { p r - Z | | p : Z e LP(B)}}, and UP(X\B) = YX,VP(X\B] = \\X - Yx\\p. It is seen that IIP(-|B) : L^(E) -> U>(8) is continuous (but not necessary linear), so that VP(-\B) : LP(E) —>• 1R+ is also continuous. One can verify that for X G LP(E), and 1 < p0 < pn < p, lim Upn(X\B) = HPO(X\B) a.e., and Vpn(X\B] ->• VPO(X\B). It was shown Pn\PO

by J.A.Cuesta and C. Matran [1] that the above limit operation implies for any sequence 1 < pn < p,pn ->• 1, lim n pn (X|B) = ni(X|B), a.e.(P) and IIi(X|jS) is a best predictor of such an X in Ll(B}. However, it is known that U L P (E) ^ L 1 (E), and for an equality it is necessary that Orlicz spaces

260

VII. Applications to Prediction Analysis

be allowed in the union (cf. e.g., Rao and Ren [1], p. 6 after Cor. 3). Thus the above argument does not cover all of Ll(Y,}, and moreover, there are many other elements in Ll(B] that are best predictors of X. Hence the problem has to be approached using different methods that do not depend on the L p (E),p > 1, spaces. This is an important reason for our special study, and a complete analysis involves lattice properties which were crucially utilized by Shintani and Ando [1]. In the next section we indicate how the prediction analysis proceeds for vector valued functions, which has some novelty and complements the preceding work. It is also useful for other applications.

7.4. Some extensions to vector valued functions In this section, we present some extensions of the preceding results if the functions (i.e., processes) take values in (infinite dimensional) Banach spaces. When once the proper formulation is set forth, the results generalize, and such statements will be useful for applications. Since conditional measures are important in this work, they depend on the Radon-Nikodym theorem which is not always valid in a Banach space. So we first state the concept. Definition 1. Let (ft, E,/z) be a finite measure space, X a Banach space, and v : E —> X a a— additive function. Then X is said to have the RadonNikodym (or RN-)property if for each such v which vanishes on fj,— null sets, there is a strongly measurable /„ : ft —>• X (i.e., /j,(ft) C X is separable and f~l(A) 6 E for each Borel set A C X] and /||/I/||G^ < oo satisfying n v(A) — f fvdp,, this being a Bochner integral. [Thus v has finite variation.] A The Orlicz space £*(//, X], of X— valued strongly measurable / : ft —> X, is defined as those elements / for which the gauge norm is finite: = inf k > 0 :

$

dp < i

< oo.

(1)

It can be verified that {L®(/j,, X),\\ • ||($)} is a Banach space and several properties of the scalar valued case extend. Also, it is known that Banach spaces X with separable duals X* have the RN-property. In particular all reflexive spaces have the RN-property. A set S C Ll (p, X) is uniformly integrable if for each e > 0 there exists a 6£ > 0 such that (J,(A) < 8e implies / ||/||efy/ < e for all / e 5, which is a direct extension of the scalar A concept. Analogous to the scalar case, M®(p, X] C L®(fj,, X] denotes the closed subspace determined by the simple functions of L®(/J,, X] in the norm

7.4. Some extensions to vector valued functions

261

|| • ||($). There is also an equivalent Orlicz norm || • ||$ given by = sup^

(f,h}dn • \\h\\m < 1 ,

(2)

where # is the complementary JV—function of $,{•,•) is the duality pairing of X, X*, so {/, h) : ft —> ]R (or C] is measurable and the integral in (2) is the usual Lebesgue integral. Recall that a Banach space X is said to have the Frechet differentiable norm \\ • \\ if for each x € S ( X ) , the unit sphere, ^,_

x

-V~,*/

,. -t~Q

\\x + ty\\-l t

exists uniformly in y e S(X}. This is denoted as property (F). A Banach space X is said to have property (H) if X is strictly convex and {xn,n > 1} C X converges weakly to x e X, and ||a;n|| —» \\x\\ as n —>• oo, then ||a:n — a;|| —>• 0. We shall later present simple conditions on L*(/z, X] to have these properties. (The notations (F), (H) and others below follow Day [1].) If M C X* is a subspace that is total on X [or determining, i.e., x*(x] = 0, W G M => £ = 0], then the M-topology of X, denoted as a(X,M] is defined by the system of neighborhoods of the point x 6 X :

N(x,x\, • • •,<,£) = {y G X : \x*(x] - x*(y}\ < e,z* e M, 1 < i < n} , (3) for each e > 0. Then o(X, M) is a locally convex Hausdorff topology for X and if M = X*, it is the weak topology. If in the definition of (H) above weak convergence is replaced by a(X, M}— convergence and the conclusion still holds, then it will be termed the (#M) condition. The following abstract version for best predictors xn of a given point XQ e X in monotone convex sets Cn C Cn+i, —oo < n < oo, will be established and specialized thereafter for vector valued elements in Orlicz spaces such as Theorem 2. Let (X,\\ • ||) be a Banach space, M C X* be a norm determining (or a total) subspace for X, and Cn C Cn+\ be (norm) closed convex sets with COQ = W( \J Cn}, the closed convex hull, and C-QQ = f| Cn. Supn>l

n
pose that: (i) every bounded sequence in C^ is relatively a(X,M) —compact and (ii) X has the property (HM)- Then, for each XQ G X, (a) there exists a unique xn e Cn which is the best predictor of XQ, i.e., \\x0 — xn\\ = inf {||;TO -y\\-y£ Cn}, -oo < n < oo, and (b) xn -» x^ € C^ (or X-^ e

262

VII. Applications to Prediction Analysis

C-oo) as n —> oo (or n —> —<x>), as well as \\xn — XQ\\ —>• ||:r±oo — #o|| as n —»• ±CXD. Moreover, conditions (i) and (ii) above are satisfied when (1}X* has property (F), or (II) X* is separable and X is weakly sequentially complete, having property (H). Proof. For convenience, we present the proof in steps. 1. Let C C X be a closed convex set satisfying (i) and (ii) of the hypothesis. Given XQ G X we assert that there is a unique yQ G C which is the best predictor of XQ. Indeed, let F(x) — \\x — XQ\\. If XQ £ C, then y^ = XQ is obviously the desired solution. So let XQ G X — C and d — inf{F(y) : y G C} > 0. Then there exists a sequence {yn,n > 1} C C such that lim F(yn) = d, and clearly {y n ,n > 1} is norm bounded which by (i) is a(X, M)— relatively compact. Since C is also &(X,M)— closed and the closure of {y n ,n > 1} in this topology is complete, there exists, a subsequence yni converging in this topology to an element yQ G C so that x*(yni) -» x*(y0) for all x* G M. It follows that x*(yHi — XQ} -» x*(y0 — x0},x* G M, and hence \X*(XQ - y0) < liminf ||a;*||||zo - y n j| = ||z*|| lim F(yHi) = d\\x*\\. I—>OO

(4)

But M is determining for #, so the supremum over x* e S(M), the unit sphere, on the left of (4) gives 0
(5)

Hence yo £ C is minimal and by the strict convexity of X, it is also unique. This establishes the assertion. 2. We first consider the case that the Cn are increasing. Let xn G Cn be the minimal element of XQ as guaranteed by 1., 1 < n < oo. We assert that l l ^ n l l ~* l^ooll) where x^ G C^. [The decreasing case is in 5. below.] In fact, since Cn /* C^, it follows that Cn verifies (i) and (ii) for each n (since C^ does), and hence by 1., the desired xn G Cn exists. Let dn = F(xn] = \\XQ — xn so that dn > dn+i —> d' > d = F(x00). But for each e > 0, since x^ G C^, the closed convex hull of (J Cn, there is ne such thai n>l

n > n£ =$• xn G Cn with \\xn — Xoo < E. Consequently d
(6)

Hence d = d', since £ > 0 is arbitrary. 3. We assert that {xn, n > 1} of the above step converges in the cr(X, M) -topology.

7.4. Some extensions to vector valued functions

263

For, since the sequence is bounded, it is relatively &(X, M) compact. Hence by 1., there is a convergent subsequence {xni,i > 1} of {x n ,n > 1} such that x*(xni] —> x*(y00}, for all x* 6 M and some y^ G COQ. Then d < \\XQ — yoo II < liminf ||x0 — xni\\ = liminfd ni = d.

(7)

Z->00

By the strict convexity of Co^yoc = x^. Repeating this argument for each subsequence, which necessarily has a convergent subsequence, and has the same limit o^, so that the whole sequence must converge to XQQ in the a(X, M)-topology. 4. We now observe that {xn, n > 1} is Cauchy. This is so since xn -» XQO in a(X, M)—topology, and ||a:n|| —>• H£OO|| so that by (ii) and the hypothesis that X has (Hyi) property. Thus \\xn — XOQ\\ —>• 0 as n —>• oo. 5. Finally consider the decreasing part, which is simpler. Since CL^ = fl Cn, the set {x_n, —l>n> —00} C C\ is bounded, because for n < — 1,

n>l

||a;_n|| < \\x-n - x0|| + Ikoll < Ik-oo - ^o|| + ll^oll < oo.

(8)

But then there is a subsequence {x_ni,i > 1} of the a(X,M}— compact sequence {x-n, n > 1} having a limit y_oo- Since for each n > ra, C-m 3 C*-n> it follows that y_oo G C-m for every ra, and hence y-oo & C_oo- By the strict convexity of X, we conclude that x_oo = y_oo as above, and then ||o;_n — aj-ooll —* 0? by the same argument. This proves the main part. 6. Let us now verify that the special conditions (I) or (II) imply the main hypothesis. (I) Indeed, if X* has property (F), then by using several geometric properties of Banach spaces (cf., e.g., Day [1], pp. 111-114), one can deduce that X is reflexive and has property (H) with M = X* here. This is a consequence of some classical results of V. L. Smulian [1]. It follows that X is both strictly convex and each of its closed bounded sets is weakly sequentially compact. Thus the main hypothesis is satisfied in this case. (II) The fact that X* is separable implies the same for X, and the closed unit sphere S(X] in its weak topology is then a metric topology (see Dunford and Schwartz [1], Sec. V.5 here and for the following consequences). Also X is weakly sequentially complete so that S(X] is a complete metric space. By the same argument, since X* is separable, the closed unit sphere S(X**} is a complete metric space in its weak-star topology, and S(X**} is compact in the latter topology (Alaoglu-Birkhoff-Kakutani or ABK-theorem). Also S(X) C S(X**} is weak-star dense. But since both S(X] and S(X**} are complete metric spaces in the same topology, it follows that S(X] = S(X**),

264

VII. Applications to Prediction Analysis

where X C X** under a natural embedding is used. Hence X is reflexive (by the ABK-theorem again). Since X has property (H), the main hypothesis is implied, and the result follows as before. D Remark. It is possible to prove 6., of the above theorem using a diagonal method and the Eberlein-Smulian theorem, in a more elementary fashion, to show the reflexivity of X. The above short argument is due to C. Foia§, and both are given together with the theorem and some complements in Rao [9]. However, it should be observed that the weak sequential completeness is essential for this result. We now translate the conditions of the above abstract result for Orlicz spaces, which is our main concern here. Let us recall that if X is a Banach space and I/*) (//, X} is the Orlicz space of strongly measurable functions / for which ||/||($) < oo, given by (1), then L^(p,, X] is reflexive iff L^(n) and X are both reflexive. Moreover, by Theorems 7.2.3 and 7.5.1 of our companion volume (cf. Rao and Ren [1], p.280 and p.304) L^(n,X) has a weakly (or Gateaux) differentiate norm if $ G A 2 fl V2, $' is continuous and X,X* have also weakly differentiate norms. If X* has property (F), and ty' is also continuous [it is already true that the complementary function \l> G A2 n V2 is also], then since every countable set of elements of !/*(//, X*) has a separable range in X*, it follows that I/*(yu, X*} has property (F). Hence these facts and Theorem 2 imply the following result. Theorem 3. Let G A2 n V2 with its derivative $' continuous and strictly increasing. If X is a reflexive Banach space with property (F), then L^(n,X) [on a probability space (£2, £,//)/ is reflexive. Suppose Bn C Bn+i C E, a—algebras, —oo < n < oo and Cn = L^(Bn, n,X}. Then the Cn satisfy conditions (i),(ii) of Theorem 2 [actually L^(^X) has property (F)], and if fQ 6 L^(p,, X) then there exist (uniquely) best predictors fn € Cn of /o such that fn —» /oo(/-oo) in norm as n —> oo(—oo). Somewhat strengthened conditions on <$ can be presented to obtain pointwise convergence of the best predictors in the above theorem. However, we omit that discussion. [For related results and further discussion on this problem, see the recent book, Rao [24], pp.470-478.] We discuss briefly the smoothness [or F—differentiability] properties of L®((j,)— spaces to explain how "close" one is from a Hilbertian structure. Recall that (X, \\ • |) has twice F—differentiate norm iff a2 \\x0 + ay\\ = \\x0\\ + aG(x0, y) + —TXo(y, y) + o(a 2 ),

(9)

7.4. Some extensions to vector valued functions

265

for all x0, y € S(X], the unit sphere, where G(XQ, •) and T IO (-, •) are the first and second F-derivatives at XQ of the norm, as a —> 0, the limit existing uniformly in y e S(X}. Also G(x0,y) is linear and T x o (-,-) is a positive definite quadratic form. The following simple characterization of a Hilbert space can be stated for comparison. Proposition 4. Let X be a Banach space with X* as its dual space. If either X or X* has a twice F—differentiate norm, then X is a Hilbert space iff one of the following conditions holds: (i)sup{-±i(\\x + ay\\- \\y + ax\\) : a > 0} < oo, x,y £ S ( X ) ,

(ii) sup £(||z* + oy*|| - ||y* + az'||) : a > o < oo, z*,y' 6 S(X*). Proof. Suppose (i) holds. Then by (9) for x,y € S(X) a2 \\x + ay\\ = l + aG(x, y) + —Tx(y, y) + o(a2) and

a2 \\y + ax\\ = I + aG(y, x) + —Ty(x, x) + 0(0?}.

Subtracting and dividing by or one has

lim

Q-H)

\x + ay\\ - \\y + ax\\

G(x,y) - G(y,x)

a*

a

=

2

-[Tx(y,y)-Ty(x,x)}.

Since the first term is bounded, by (i), and the limit exists, the second term in | • | must also be bounded as a —>• 0. This holds only if G(x, y) = G(y, x). But G(x, x) = \\x\\2 and G(x,~) € X*. Hence G(x,y) = [x,y] is an inner product and the norm of X is given by it. So X is a Hilbert space. The case that (ii) holds is similar. The converse is trivial. D For the Orlicz spaces, we give some results of interest in applications. The study on the RN—property and property (H) of the Orlicz spaces seems to have first appeared in Rao [8], Propositions 6.3 and 3.7), which is now completed in Chen [1], Theorems 3.32 and 3.14). The Gateaux and Frechet differentiabilities of the gauge and Orlicz norms are also detailed in Chen ([1], Theorems 2.50, 2.52 and 2.57). See also our volume ([1], pp. 278-283). Proposition 5. Let (<&, \I>) be a pair of complementary N—functions, (f2, E, fj.) be a a—finite measure space with fj, being nonatomic. We have the following assertions. (RN): LW(n) (hence L*(p)) has RN-property iff$eA2(3>e A2(oo) for (J,(£l) < oo).

266

VII Applications to Prediction Analysis

(H): L,W(n) (or I/ $ (//)) has H— property iff $ is strictly convex and $ G A 2 ($ G A 2 (oo) for fj,($l) < oo). (G): L^(n) (or I/*(^)) has G — differentiable norm iff 3>' is continuous and $ € A 2 ($ G A2(oo) /or //(ft) < oo). (F): L^(n) (or Z/ $ (^)) ^fl5 F — differentiable norm iff & is continuous and $ G A 2 n V 2 ($ € A 2 (oo) n V 2 (oo) /or /z(fi) < oo). From the above results, we assert that if $ G A 2 n V 2 and $' is continuous and strictly increasing (hence \I> and ^' satisfy the same conditions), then Z/ $ )(//),Z/*)(^) (with the gauge norm) and L $ (/Lt), Z/*(/x) (with the Orlicz norm) have properties (RN), (H), (G) and (F). We note that the nonatomicity is needed only for the necessity proofs. Proposition 6. If $ is an N— function, <£' is strictly increasing and B C £ is a a—subalgebra with n(£l} < oo, then L®(B) C !/*(£, /^) is a Chebyshev set, (although not every closed convex subset of L®(/J.) has this property). Proof. Let A = L®(B). Then A is strictly convex, closed, and its sequential compactness in cr(I/*,M^*^) can be shown as follows. It is well known that !/*(//) is sequentially complete in the o~(L $ , M^*^)-topology (cf., e.g., Rao and Ren [1]), p. 135, Theorem 4.4.1). So if {fn,n > 1} C A is a Cauchy sequence in a(L*, Af(*))-topology, then by the preceding statement of completeness, there is an /o 6 L^(^JL) such that vn(E) = j fndv -)• v(E] = / /Od/z, i/

£

E e S.

(10)

J

£

In particular, vn(B] —> ^(J3), B E B. Since # is a a— algebra, by the classical Vitali-Hahn-Saks theorem, v(-) is countably additive on B and the countable additivity is uniform in n in that m— lim v (B } = 0 for Bm \, 0, Bm € B, uni>-oo n m formly in n. Hence z^ is a— additive on B and since /i(S) < oo, by the RadonNikodym theorem /0 = ^ , a.e. so that /0 is B— measurable. Hence /0 € A, and so A is weakly sequentially complete. It has property (HM)- Thus it will be a Chebyshev set if each closed bounded set is sequentially compact in this (weak) topology. This is shown more generally in the next result which thus completes the proof. Regarding the parenthetical statement, it should be observed that in a strictly convex Banach space it is known that every closed convex set is a Chebyshev set iff the space is reflexive. Since our L*(/^) need not be reflexive, the statement follows, n We now state a general result on the weak compactness and uniform integrability for vector valued functions, for convenience.

7.4. Some extensions to vector valued functions

267

Proposition 7. Let X be a reflexive Banach space and L l ( [ i , X ] be the Lebesgue space of X—valued Bochner (or strongly] integrable functions on a probability (or just a finite measure] space (£7, £,//). Then the following statements are mutually equivalent for a subset H. C L1^, X] : 1. H is uniformly integrable. 2. 'H C L®(n, X] is a bounded set for some N—function $ with ' continuous. 3. *H C L1 ()U, X] is relatively weakly sequentially compact. 4- *H C L?(IJL, X] is relatively sequentially compact in the &(L®(n,X), M*(/z, X*]] topology, where as usual M*(//, X*] is the subspace of X* -valued functions in L*(n,,X*] determined by its simple functions, ^ being complementary to the N—function <$. 5. 'He Ll(/j,,X] is weakly relatively compact. 6. 'He. L®(II,X] is relatively compact in the cr(L$(/z, X ) , M*(/z, X*]] -topology, where (3>, \I/) is a complementary pair of N—functions with continuous <£>', ^'. Proof. Most of these assertions are classical, and we merely outline the arguments here. Now 1. <=>• 2. is a consequence of de la Vallee Poussin's theorem (c.f., e.g., Rao and Ren [1], p.3) and the definition of Bochner's integral. That 1. •<=> 3. is obtained from a simple extension of the classical Dunford-Pettis theorem (cf. Dunford and Schwartz [1], pp.293-295). Also 3. «=> 5. and 4. <=> 6. are consequences of the classical Eberlein-Smulian theorem. The details are omitted here. Since 6. =$• 2. is evident, it suffices to show that 3. =£> 4.. We now sketch this detail which is somewhat similar to the proof of Proposition 6. Thus let 3. hold so that 'H C Ll(p,,X] is weakly sequentially compact. If {/n,n > 1} C H is a weakly convergent sequence, then vn(E] = / fndfj, E

converges for each E e E to some VQ(E). Then v0(-) is a—additive and //—continuous. Since X is reflexive, it is known to have the RN—property. So there is an /o € Ll(p,, X] such that 7^ = /o a.e., and this implies lim /{/„ — /o, ti}d[i = 0 for each h G L°°(/J,, X*]. Since by 2. <=£> 3., noted above, there is an A7"—function $ satisfying the conditions of 4. such that H C L®(/J,, X] is bounded. To see that /0 6 L*(p,, X], for any h e L°°(^, X*], consider (fo,h)d[j,

< liminf / n < K\\h\\*«x>,

(11)

268

VII. Applications to Prediction Analysis

by Holder's inequality for Orlicz spaces and the fact that L*(/z, #*) D I/°°(^, /%"*), since p,(Q} < °° and 'H is bounded in L $ (//, A"). Taking supremum over \\h\\y < I in (11), it follows that (by the inverse Holder inequality in this space) | /o||($) < oo. Since I/°°(/z, X*} C M*(/Lf, X*} and the former is dense in the latter, it follows that nlim

j ( f n - f0,h)dn = 0,

h€

n

This implies that /n -> /0 in <j(Z/ $ )(//, X ] , M*(/J, A""))— topology, which is the assertion 4. D Remark. If X is not reflexive, but has the RN— property, the preceding work gives the implications: 1. <^ 2. •<= 3. =4> 4. •£> 6. => 2. and 3. <^> 5.. Further discussion will be omitted, since this generality is not needed here. The following example clarifies the relations between differentiability of norms and of the TV— functions. The F— differentiability of norms is much more stringent, as examples show where $, \I/ are infinitely differentiable, but do not have F— differentiable norms. Example 8. Let $(2;) = (l+a;)log(l+a:)-a;,a; > 0, so \P(j/) = ey-y-l,y > 0. Then $, \l> are infinitely differentiable and $ € A 2 . Let L^(/J,) (with the gauge norm) and !/*(//) (with the Orlicz norm) be (scalar) Orlicz spaces on a finite measure space (ft, E, /u). Then M<*> = L^(M) but M* ^ L*(/z) and the set M* is nowhere dense in Z/*(/u). By Proposition 5, L^(^) has properties (^?JV), (H) and (G). But L ($) (A«) and L*(t*) do not have F- differentiable norms. Note that M* has F— differentiable norm, since (M*)* = L^(/J.) and L^(/J,} is locally uniformly convex (see Chen[l], Theorem 2.28(ii)). Both L^\p] and L^(/J,} are strictly convex. [Recall that M* is not an adjoint space if p, is nontrivial, we analyze this space further in Example 11 below.] Consider $(z) - a(f + ^) so *(j/) = _« + §[(! + £)i - l] , x , y > 0, and 0 < a < 1. These spaces L^(n) and L^(n] are uniformly convex and uniformly smooth Banach spaces whose norms are seen to be twice F-differentiable. Since $(z) ^ ^,L( $ )(//) [= (L*(/z))*] is not a Hilbert space. It can be given an equivalent norm in terms of which it becomes a Hilbert space. Thus L^(/LA) is isomorphic and topologically equivalent to a Hilbert space but itself is not one. Since the Milnes-Akimovic theorem on the isomorphism of the uniformly convex Orlicz spaces with reflexive Orlicz spaces is invoked in this chapter, more then once, we shall present a short proof of it here. In our earlier volume (cf. Rao and Ren [1], pp. 296-298), a proof was given, but a shorter

7.4. Some extensions to vector valued functions

269

argument (due to Ren) based on the indexes A^,B^ of Chapter I, will be included here for completeness and variety. We recall the following. Definition 9. An N— function $(u) is uniformly convex for large u (for small u ) if for each 0 < a < 1, there are a 6(= 6(0) > 0) and a u0(= u0(a) > 0) such that (12) for all u > u0(Q < u < u0). If (12) holds for all u > 0, (i.e., u0 = 0) then <£ is termed uniformly convex on [0,oo). Using the notations A 2 (oo), A 2 (0) and A 2 etc., from earlier chapters, we have: |u|

Theorem 10. Let $(w) = / (j)(t)dt be an N— function which satisfies $ G o A2(oo) n V2(oo) (or $ e A2(0) H V2(0), $ € A 2 n V 2 ). //

and \I>o is its complementary N— function, then we have the following: (i) $o ~ $> (equivalence), (ii) $0) ^o are strictly convex, (Hi) $0 e A2(oo) n V2(oo) (or $0 e A 2 (0) n V 2 (0), $0 e A 2 n V 2 in the corresponding cases), (iv) $0^0 are both uniformly convex for large arguments (or for small arguments, for all arguments). Proof. (i) and (iii) are well-known properties (cf., e.g., Rao and Ren [1], p. 20). Moreover, $Q exists, strictly increasing and continuous on (0,oo). Also 3>()(0) = lim$' (t) = lim *I*1 = 0. Then *' uv u is the inverse of ${, u which is ' t^o 0u v ' t->o i continuous and strictly increasing. It then follows that (ii) is true. We establish (iv) when $ 6 A2(oo) n V2(oo). Observe that 1
f- < limsup

*

-

= B$ < oo. *

(13)

V

^

Let a = j(l + A$), 6 = 1 + 5$ so that 1 < a 0, such that

b,

t> UQ.

(14)

270

VII Applications to Prediction Analysis

Hence, given e > 0 and u > WQ, we have (l+e)w

f

J

(1+e

a , f -dt< t J

implying log(l + e)a < log ' L ^ ( u p J < log(l + £) b .

(15)

Taking exponentials and rearranging, we have from (15) for u > UQ

The first half of (16) implies that for u > u0

Hence <&o(w) is uniformly convex for large u by a well known result (cf., e.g., Rao and Ren[l], p. 284). Also the second half of (16) gives, for u > u0, that

Let f = $o(u) an(l ^o = ^o( w o), so that u = ^o(v] for f > VQ. From (18) one obtains (1 + e)V'Q(v) < ^'0[(1 + E}b-lv], v > v0. (19) Given £l > 0, let £ = (1 + eji^ - 1 > 0 or (1 + e}b~l = 1 + £i. Hence (19) implies, for v > VQ, the following inequality:

Since (1 +£1)^ > 1, ty0(v) is uniformly convex for large v. Thus (iv) follows in this case. The others are similarly verified. D As in our treatment of the result (cf. Rao and Ren [1]. p. 297) the general theorem of Milnes and Akimovic can be stated in other forms, which are immediate consequences of the above theorem. Example 11. Let the complementary JV-functions (, ^) be as in Example 8. Then, as noted there [<&' and \&' are strictly increasing and continuous] $, \l> are strictly convex on IR+. We verify that \& is uniformly convex on M+. Given £ > 0, let K£ = ee and UQ — I , so that for u > UQ tf'[(l + £)u] = eu£(eu - e~U£) > e£(eu - 1) = Ke^'(u).

7.4. Some extensions to vector valued functions

271

Hence ty is uniformly convex on [1, oo). Also ,.

hm

U-+Q

V'[(l+£) U] J (

\. /

fy'(u)

..

= hm

«->-o

e(1+£)u-l u

n

— = 1 + e.

e —1

Let A'g = 1 + |. Then there is ue > 0 such that for 0 < u < ue, one has \£'[(l-l-e)w] > K'£ty'(u). So ^ is uniformly convex on [0, ue]. Since ^ is strictly convex, it is uniformly convex on [u£, 1}. Hence fy is uniformly convex on JR+. But (i>), although is uniformly convex for small v, does not have the same property for large v, since
272

VII. Applications to Prediction Analysis

hope that our treatment of Section 3 focuses better on the real mathematical problems lying underneath the analysis of the subject. To make it better appreciated, we have devoted Section 4 for vector valued functions. Here Theorem 4.2 is the abstract version of results in a general Banach space and the others are useful specializations of it in the context of Orlicz spaces. These are based on papers by Rao [9], [10], and other references given in the text. In the Lp(fj,, X] context, the result of Proposition 4.6 was treated by Cuesta and Matran [1]. It is interesting to observe that the whole (nonlinear) prediction problem depends on important aspects of the geometry and other structure theory of the L*(fj,)— spaces. In the next chapters we consider other applications, related to stochastic analysis and PDE, which also use the Orlicz spaces as essential components in obtaining new perspectives.

Chapter VIII Applications to Stochastic Analysis

This chapter is devoted to certain classes of stochastic processes and fields that take values in Orlicz spaces. First we show how Orlicz spaces appear quite naturally in important parts of stochastic analysis, by discussing an aspect of large deviations for independent random variables and then for other processes such as Brownian motion. These lead to vector Orlicz spaces and they are also considered. Then we include, in some detail, the regularity properties of stochastic processes and fields that take values in Orlicz spaces where the TV-functions appearing there are typically of exponential type. Here the sample path behavior of processes that are so important in many applications is analyzed. Proceeding further, we study a subclass of processes, namely martingales. Also presented include some vector valued extensions of these processes. There are elaborate theories on each of these subjects. We restrict to those parts where Orlicz space theory contributes for a deeper understanding of the subject. The work shows the extent that Orlicz spaces with Young functions growing exponentially fast have special roles to play in this analysis, as distinct from those based on growth restrictions (such as A2 type conditions) in applications of the preceding two chapters as elsewhere. This exemplifies the fact that all Orlicz spaces are useful in applications. 8.1 Large deviations and Young functions Let us motivate the subject by a simple result in probability theory. Thus consider a sequence of independent (Bernoulli) random variables {Xn,n = 1,2, • • •}, where Xn = I if the n-th toss of a coin is head and 0 if it is tail each with the same distribution, P[Xn = 1] = p and P[Xn = 0] = l-p = g , 0 < p < l , n > l . Let Sn = £ Xt and Yn = Sn/n. So Yn 1=1 denotes the number of heads in n independent tosses. Then Yn —> p in measure, i.e., for e > 0, P[\Yn — p\ > e] —> 0 as n —> oo, by the weak law of large numbers. In practical applications, it is desirable to find the rate at which these probabilities tend to zero, i.e., one wants to know the

273

274

VIII. Applications to Stochastic Analysis

probabilities of events such as An = [\Yn — p\ > e] = [(Yn —p)2 > £2] for large n, or probabilities of the large deviations from p. Note that the convergence statement is simple, since P(An}= I dP<\ I JAn

E2

(Yn-p)2dP

JAn

HE2

where we used the Chebyshev inequality. However the rate of decay of P(An), as seen below, is exponentially fast, and this needs a more detailed analysis, utilizing the further information that the Yn are actually bounded random variables. For illustration we consider p — \ — q. Other results of intrinsic interest will follow later. Proposition 1. Let {Xn,n > 1} be independent Bernoulli random variables with P[Xn = !] = ! = P[Xn = 0], and Yn - \ £ X,. Let Qn = P o Y~l be the image measure of Yn on M. Then for each 0 < e < | and A = {x E IR : \x — | > E}, one has for large n: O (A) = e~ n -^2+ £ ) -f- o(n}

(2)

where (1-a;) log 2(1-a;), x € [0,1]

Proof. The above formula depends on the Stirling approximation of n! for large n. Thus for the Borel set A C M of (2), if An = {k : |£ - || > E}, then the probability of Yn 6 y4(^=4> A; e A n ) is given, using an elementary combinatorial argument, by the expression:

(3)

Q,(A) = E (I) Yn' But (3) can be bounded as max I IfC- A \

I — < Qn(A) < (n + 1) max ( t-c 4 \

Z" // Ol A/c:^in \ A/ ^

I —,

K*// OTl /vC/T-n \ Ay ^J

(4)

since the sum in (3) has just (n + 1) terms. This may be simplified by taking logarithms, since the "log" is an increasing function. So one gets the bounds max — log n

j — j

8.1 Large deviations and Young functions

275

(„

By the Stirling approximation, n! ~ %/2~7rne~nnn, which on substitution for \k)~ (n-k}\k\ giyes after a simplification:

r / n \ 1]

1

1

k

k

= l°g --- l°g -6 2 n 6n

- log , — n &[\kJ2n\

(

k\

(

k

1 -- log 1 n-n 8

(6)

Taking A = |£j, a singleton, in (5) using (6), one finds

Substituting this in (5), since {x e [0, 1] : x = £ for some k e An} C .An [0,1], one has 1

lim — logO nn(^4) = lim max

f

(k

—/ I —

ik\ = — lim min / [ — ] = — min/(x), n-K»A;€An \nj xeA v '

n

v(7) ;

since I(x] = +00 for x £ [0,1], and /(•) is continuous on [0,1] so that lim min I(x] = min I (V x ) . Now /(•) is also a convex function with a unique V w M n-+ooxeAn

'

xeA

'

minimum at x = | and is symmetric about the point |, the minimum value on A is attained a t x = | ± £ , 0 < £ < |. This gives (2). n The result implies that one may have to consider in such applications, exponential Orlicz spaces with e®^ as the principal part of a Young function and that the convex function can be symmetric about a point "a" where a 7^ 0. In the above, the convex function /(•) is termed a rate function of the large deviations problem. We generalize this simple but basic result. It can be considered for the multinomial case, the multidimensional Xn, empirical measures for Yn, and also more general sequences or processes (or multidimensional parameters, termed random fields). All these are of interest, and one can refer e.g., to Ellis [1] for some of these extensions. Here we discuss the role of Orlicz spaces which appear in such applications.

276

VIII. Applications to Stochastic Analysis

To see the ideas clearly, we formulate the large deviations principle. Let Xi,X2,--- be a sequence of independent integrable random variables on a probability space (Q, E, P) with a common distribution. Then

A C R,

n

(8)

is called an empiric distribution or measure, if A is a Borel set. Thus Q n (-)(u>) is the proportion of the random variables X\ , • • • , Xn that take values in A, and it is again a consequence of the (weak) law of large numbers that Qn(A] —> 8P(A}, where p = f^XidP and 6P(A) = 0 if p £ A, = I if p e A, for all open sets A C JR. The problem of large deviations is to find the speed of convergence of Qn to 6P, the point mass, i.e., want to determine limsup — logQn(A), n-4oo n

and

liminf — logO n (B), n -*°° n

(9)

for any closed A and open £?, and show that these bounds are equal for large classes of sets A and B. This implies the exponential convergence of Qn when the limits exist, and the latter turn out to be related to TV-functions. Consider the Laplace transform of X\ , also called the moment generating function. Thus

f(t) = I etXldP, t£M Jn is the transform. Since / > 0, one can define A(£) = log/(£), giving the cumulant (or semi-invariant] generating function of X\. It is interesting that A(-) is a convex function and clearly A(0) = 0. Indeed, if 0 < a = 1 — ft < I , then by Holder's inequality = log (7 \./n

Hence one can define the Young complementary function A(-) of A(-) by A(s) = sup{st - A(t) : t <E M}.

(10)

The function A : s \—> A(s) is convex but not symmetric in general, and is also termed the Legendre transform of X (in honor of A. M. Legendre who

8.1 Large deviations and Young functions

277

lived in the 18-th century and briefly considered such transforms earlier than W. H. Young). Thus (10) implies the Young inequality st< A(*) + A(s), s,t>0,

(11)

and since A(0) = 0, A(s) > 0 for all s e JR. Actually A(m) = 0, m = fnXidP = E(Xi), the mean. To see this, since x = loge x , x € JR, one has tm = I tXdP = f logetxdP < log / etxdP = A(i), Jn Jn Jo, where we used Jensen's inequality for concave functions. Thus from (10) this gives, on using the positivity of A, that 0 < A(m) = sup{tm - A(t) : t e JR} < 0, implying A(m) = 0. So A has a minimum at ra, and A(-) is a shifted complementary Young (or Legendre) function at m. In the case of Proposition 1, one can verify that A(-) = /(•), the rate function of the problem with m = p = |. We now show that this holds more generally. Note that

— = - sup{ts - A(s) : s e 1R} > s0 e M, X

6

so that limsup^ = co, and since A > 0,^ < —s,s > 0 and thus t—too

lim -4Q = oo, or A(-) is the principal part of an ./V-function. f—^"itOO I I The significance of the functions A(-) and A(-) will be better appreciated if we discuss, in detail, the following classical result due to Cramer [1] which became a basis for the study of large deviations. Theorem 2. Suppose that A\, X%, • • • is a sequence of independent random variables on (fi, E, P) having a common distribution for which the cumulant generating function A(-) exists. If A is the Legendre transform and Pn = n PoY~l is the image measure ofYn = ^^,Xi on the Borelian space (1R, 5R), i—l then for each A £ 3?.

- inf A(x) < liminf-logPjyl) x€A°

n-^oo

n

< lim sup log Pn (,4) < - inlA(rc),

(12)

278

VIII. Applications to Stochastic Analysis

where A° is the interior and A is the closure of A. [As usual inf {0} = +00 andsup{0} = 0. Thus equality holds if the boundary of A has measure zero.] Remark. This result shows the importance of the convex function A(-) and its conjugate A(-), the latter qualifies, by (12), to be the rate function. These two functions are also crucial to establish links with Orlicz spaces. After demonstrating this result we comment on extensions leading to several interesting applications as well as to new aspects of this analysis. Proof. For convenience we present the details in steps. 1.^ Recall that if A : 1 1—> logE(etXl] is the cumulant generating function and A is its complementary function, m — E(X\), then A(m) = 0, and A > 0 is convex increasing to the right of m and decreasing to the left of m. Consequently for q > m, one has A( 0} since qx — A.(x] > qy — A.(y] for y < 0. Similarly K(q) \

(13)

for q < m, so that

A(g) = sup{qy - A(y) : y e 1R} = sup{qy - A(y) : y < 0}.

(14)

Hence for q > m P[Xl >q} = P[exXl > exg], for all x > 0, < e~xqE(exXl], by Markov's inequality, =

e -[^-A(i)] )

x

> 0.

Since the left side does not involve x, it is bounded by the infimum of the right side, so that

Similarly, if q < m, by (14) and the same procedure, one gets P[A"i e~xq] < e^.

(16)

tXl

Now considering Yn, E(Yn) = m, and if M(t) = E(e ) is the moment generating function, one has Mn(t] = E(eiY»]

since the X\,-- • ,Xn are independent and have the same distribution. Thus An(t) = logM n (t) = nlogM (-) - nA (-) . \ / i /

\ Iv /

(17)

8.1 Large deviations and Young functions

279

The complementary function An of An is then, using (17), given by A.n(x) = sup{tx - A n (t) :t€ fft}

( tx f t\ 1 ~ = nsup^ -- n A ( - } : t € 1R\ = nA(x). In Vn/ J

(18)

F[F n >m] < e - n A ( m ) ,

(19)

Hence by (15) and (16) applied to Yn. Writing Pn = P o Y~\ it follows from (19) that, for any closed set F C [m, oo), - logPn(F) < -A(m) < - inf A(z), n x£F

(20)

since A is increasing on [m, oo). Similarly for F C (— oo, m], using (19) and (16), one gets the (same) inequality ilogPn(F)<-mfA(z),

(21)

Taking limits as n —> oo, (20) implies the upper bound of (12). 2. The lower bound of (12) needs a new method, namely a change of measures. [This dichotomy (of using two procedures for upper and lower bounds) is a standard procedure in these problems.] Here we make a reduction of the result for a simpler case. Since the left side of (12) follows if it is shown to be true for any open set O C M, and since it is well-known that an open set of JR is a countable disjoint union of open intervals, it is enough to establish it for (x — 6, x + 6} for any x G 0, 6 > 0. This follows from (—5, 5), by translation. In fact, let Y = Xi — x and observe that Ay(A) = \QgE(eXY] = log[e-^E(eXXl)] = -Xx + A(A), and hence A r (y) = s u p { A y - A y ( A ) : X e JR} so Ay(-) = A(- + x), and therefore liminf — \ogPn((x — 6, x + 6)} > - inf A y (A) - - inf A(x + A) - -A(x). Afc/K

(22)

280

VIII. Applications to Stochastic Analysis

We thus only have to establish this inequality with x = 0. 3. Let fj, = P o Xi1 be the image measure. Suppose the support of // is a bounded subset of JR, and that "0" is an interior point of the support, so that /z((— oo, 0)) > 0 and /^((O, oo)) > 0. Then A.(x) /* oo as \x —> oo, and since supp(^) is bounded, A(x) < oo, x € JR. The convexity of A implies, under these conditions, that A(-) has a minimum, say at XQ G 1R. Construct a new measure jl from (JL by dji(x) = exox-*(lo)dij,(x}.

(23)

The existence of moments of X\ implies that A(-) is differentiable (in fact is real analytic!) and A'(XQ) — 0. Note that // is a probability measure, since J1(JR\ = e ~ A(a;o) / exoxdp(x] = e (-A(*o)+A(* 0 )) = 1; JK and Jl and fj, are equivalent. Also Xi, ^2, • • • are independent and identically distributed relative to /L Moreover

=

/ Jm.

If Pn(Pn} is the induced measure on JR under n(p,) of Yn, then for any e > 0 P ((—f p}} *n\\ £i£))

— P(\3 s n?} — I — * v P n <- n£) —

r > e~~ne\x°\ I

e,,-ne\xo\+n£\XQ\ arjp

n

*^o / ^ -**-i e «=i o?Pn) since the sum

J[\Sn\
is at most ne on this set, dP, by (23).

From this, since the last integral is Pn((—e,e)) for any 0 < E < 6, one gets \immflog Pn((-6, 6)) n—>oo j^

> liminf-logP n ((-e,e)) 7i— >oo YI > A.(XQ) — £\xQ\, by the above computation,

since ^ -> 0 in Jl probability by the weak law of large numbers; so P n ((— e, e)} —> I. From this (22) follows since e > 0 is arbitrary and x0 is fixed. We now remove the boundedness restriction on the support of p,.

8.1 Large deviations and Young functions

281

Suppose that supp(^} C M is unbounded and that ^(JR+) > 0, > 0. Thus let k > 0 be chosen large such that //((-&, 0)) > 0, , k)} > 0, and consider Vk : X (->• f*keXxd/j,(x), and a new probability measure ^ defined by 4.

fc.^eA)

(24)

'

Then A\, A^, • • • are independent having distribution i/^ with values in (—k, k}. Thus the previous case applies. For 6 > 0 if we define

,---,n}) >

'

/!((-*,*))"

*

}

then we get

so that Hence (12) holds for Qn, in that for 0 < e < 5 Una infi log
(-e:,5)) > Afc(a;o) -e|xo|.

(26)

If Ajfc(-) is the corresponding cumulant function of ^, so that Ajt is its complementary, and Afc — log V^ — log/^((— k, k)}, then one has

(27) AfcXK

But the left side is independent of A; > 0, so that if /* = limsup/jt, then (27) becomes

A;—>-oo

(28)

Now Afc(-) is convex and increasing, so — /* < A^(0) < 0. By definition, Ik < oo for all A; and —/£/*. By the preceding step, /* < oo, or — /* > — oo, and

282

VIII. Applications to Stochastic Analysis

the sets {x : A fc (x) < — /*} are nonempty, decreasing, closed and bounded, i.e., compact. So they must have a nonempty intersection. Let XQ be a point of this intersection. Thus A(2r 0 ) = lim A fc (zo) < -I*.

(29)

fc->oo

Hence liminf -\ogPn((-6,6)) > -I* > inf A(z) n->-oo

~

~

This shows that (12) holds if supp(n) is unbounded and has nontrivial intersection with M+ and 1R~ . However, if p,(JR+) = 0 or //(.R~) = 0, then A(-) being monotone, inf A(x) = log/z({0}). Thus by independence,

> o, so that liminf-logP n ((-<J,<J)) n-»oo n

->

log/X{0}) O A - V I j / = inf Av(a;) / = -A(0) v / > -oo.

Thus the conclusion holds in all cases, and (12) follows. D The result contains that of Proposition 1, where A(x] = I ( x ) given there. We included all the tedious details for the result, since it is the basic proposition of large deviations theory, and the key functions for it are the cumulant generating function A(-) which is convex increasing, and its complementary Young function A(-) which is always convex and nonnegative. Neither of these is symmetric. Thus Orlicz space theory is an ingredient but simple adjustments (such as symmetrization) are necessary. The maximization (or the variational) problems involved for (12) above also have a key relation with a classical method due to Laplace. We indicate it here. If h : [0, llJ —> IR is a continuous function with a — min h(x) = 1 V '

0<x
'

h(xQ),xQ G [0,1], expanding h(x) by a Taylor series when it is differentiable, one finds lim - log f1 e-nh(x]dx =-a = min h(x). (30) n-»00 n

JQ

0<X<1

This is also valid for any continuous /i, by approximating it with a smooth function uniformly (e.g., by Weierstrass's theorem). The usefulness of (30) is seen from the following consequence establishing the Stirling approximation. Thus for an integer n > 1, consider /•CO

n\ = T(n + 1) = / Jo

r

xne~xdx = \ Jo

8.1 Large deviations and Young functions

= nn+1

283

rOO

e-n( (31) Jo if we set x = nt. Here h(t] = t — \ogt, so at t0 — l,h(tQ) — mmh(t). Hence the dominant contribution of the integral comes just from the interval (1 — e, 1 + e) for E > 0. Expanding i , (*-!) 2

h(t) =t- \(t-l) one gets

(32) l-£

l-£

The right side for large n, is nearly the same as m.

the standard Gaussian integral. Thus (31) becomes 27J-

n\ ~nn+1— e~n, as n -» oo,

(33)

which is the Stirling formula. So n! = n n e~ n \/27m(l + £n)^n — ° ( n ) < For those who like to see a probability argument, the following elementary application of the central limit theorem (for comparison) may be instructive. Let A"i, Xi, • • • be independent Poisson random variables given by a common distribution P[Xt = k] = ^, k = 0, 1, 2, • • • and so E(Xi] = 1, E(Xi I) 2 = 1. Let Sn = Xi + • • • + Xn. Then the central limit theorem says that for any Borel set A C JR, =

lim P

Since E (^^E) = 1 for all n, {^p, n > l} is uniformly integrable and so

(34) V y However, one has an explicit evaluation of the left side as: Sn-

n- k

284

VIII. Applications to Stochastic Analysis e

v/n

n ^k+l -^ '^_ -*

l.\

lk=Q

""

n nk~\ v-* "'

p~nIU nn+\ ^

2-^* L.\ \

n\

fc=0

"" J

f^} '

'*•

Hence by (34) and (35), lim

which is (33). Returning to Cramer's theorem and Laplace's method, one can abstract the following facts. Let Xn,n > 1, be a sequence of random variables on a probability space (£1, E, P) with values in a topological space X, usually taken as a separable complete metric (=Polish) space, with its Borel a— algebra B. If Xn —> XQ 6 X, in probability, then the sequence is said to obey the large deviations principle (LDP) if the convergence is exponentially fast. More precisely, if there is a mapping / : X —> [0, oo), continuous (or only lower semi-continuous), called the rate function, with the property that {x : I ( x ) < A;} is compact for each 0 < k < oo, and for each A € B - inf I(x] < liminf -logP[Xn e A] ~ ™->°° n < limsupilogP[X n e A] < - inf /(:r), n—>oo

n

(36)

x&A

where A° — int(A) and A = d(A). Thus Cramer's theorem shows that the large deviations principle is valid if the Xn are independent, identically distributed jvith the cumulant generating function A of Xi existing and X — M, I = A, the complementary Young function of the convex A, as the rate function. An immediate question is to formulate and establish the same type of result if X = JRk,k > 1. In this case one defines for each x* e X*(= X]

where (x*,Xi) on X* x X, is the duality pairing. Then A can be defined as A(z) = sup{(x*, x) - A(z*) : x* e ^*},

(37)

which is a complementary Young function in X =• Mk. To compute the minimum value of A, which will be the mean vector of Xi, one has to invoke the minimax theorem, so that in the computation for a minimum of A one can commute the inf and sup on a compact convex set C C X. This means: inf A(x) — inf sup {{x*, x) - A(x*)}

8.1 Large deviations and Young functions = sup inf{(x*,a:)-A(x*)}. x£C x*ex*

285 (38)

New problems arise in the multidimensional case. To proceed further, it is necessary to generalize (i) the Laplace's principle to the topological set X [usually for X a separable Banach space], and then (ii) establish, for the large deviations principle, the inequalities in (36). For the latter some aspect of independence (or conditional one at least) is needed, in addition to the variational calculus for (38). The desired result was first obtained by Varadhan [1], on which the following developments are based. Then the second part for a number of different types of Markov processes was also established by Donsker and Varadhan in a series of papers (one can refer to the survey by Varadhan [2] for references). We state the basic results on both aspects without details and then will indicate the underlying Orlicz (or more exactly Fenchel-Orlicz) space analysis. A generalized Laplace method, i.e., of (30), can be obtained as a consequence of the large deviations principle. This is the content of the following theorem, due to Varadhan [1]. Theorem 3. (a) Let {Xn,n > 1} be a sequence of random variables on (J7,E,P) with values in a separable complete metric (=Polisti) measurable space (X, B), whose distributions [or image probability measures] Pn = P o X~l obey the large deviations principle with a rate function I : X —> [0, oo), so that /(•) is lower semicontinuous, convex and {x : I(x] < k} C X is compact for all k > 0. Then for any continuous function f : X —>• ]R which is bounded below , one has iin log

e~nf(Xn)dP = -inf{I(x) + f ( x ) : x <E X}.

(39)

Moreover, the rate function I of the sequence is unique. Conversely, if I : X —>• [0, oo) is a rate function satisfying (39) for all bounded continuous f : X —> JR, then the sequence {Xn,n > 1} obeys the large deviations principle on X with rate function I. (b) If y is another Polish space, h : X —> y is a continuous mapping and I : X —> [0, oo) is a rate function, then J : y —> [0, oo), given by J(y) — inf{/(x) : x € h~l(y)},y 6 f(X], and = oo otherwise, is a rate function on y. Further if {Xn,n > 1} obeys the large deviations or the Laplace principle on X with rate I, then { f ( X n ) , n > 1} obeys the large deviations principle on y with rate function J. Since generally J takes fewer values then /, the result of (b) above is termed the contraction principle. We omit the proof of this result and refer

286

VIII Applications to Stochastic Analysis

to Varadhan [2] or to Dupuis and Ellis [1] for details. In general the key rate function /(•) is difficult to compute, as seen in Theorem 2. If A' is a vector space, i.e., is a Banach space here, then Dinwoodie [1] shows that the following method can be used for its construction. Let $(/) be the left side limit of (39). If (y,x) is the duality pairing of X x X*, define for each number A;, (y,x) A k, the minimum, and set #(z) = lim $({y, x)hk), /c->oo

x £ X*.

(40)

This limit exists by (39). Then Dinwoodie shows that, if /(•) is convex and lower semi-continuous, it can be obtained as I(y) = sup{(i/, x) - V(x) : x 6 A"}.

(41)

In the nonconvex case, difficulties arise, and the 'lim inf and 'lim sup' for (36) can be different sandwiching /(•), properly containing it in the interval. Taking X = JRd, d > 1 finite, we get from the above result the multidimensional version of Cramer's theorem, which we state for reference as well as for discussing a generalization in the next section. Theorem 4. // {Xn,n > I } , Xn taking values in IRd satisfies the large deviations principle, then (39) holds with rate function /(•) given by I(y) = sup {{x, y) - log I e{x>Xl}dP : x e Md\ . I Jn J

(42)

Moreover /(•) is convex, continuous and nonnegative with minimum at a = E(Xi), where the Xn are independent, identically distributed and having a well-defined cumulant generating function. Since for all this work /(•) plays a key role and / : X —> [0, oo) is defined, we discuss its properties for special vector spaces X. Here again convex function analysis plays a significant role. That X may be infinite dimensional is seen from the next result on the Brownian motion, which extends the preceding theorem. It is due to Schilder [1], and thus motivates the general Fenchel-Orlicz space analysis for us. Let {Xt, 0 < t < b} be a Gaussian process with mean E(Xt] — 0, and continuous covariance given by r ( s , t} — E(XsXt) — u ( s / \ t } v ( s V t } , u ( - ) , v(-) > 0 and C%)(t) strictly increasing. If v(t] = 1 and u(t) = t, 0 < t < oo, the resulting process is called the (standard) Brownian motion (or BM, or Wiener) process. It has independent increments, since with Gaussian hypothesis this is equivalent to showing for 0 < s < t < a < b, E[(Xt — Xs}(Xa — Xb}} = 0

8.1 Large deviations and Young functions

287

which is clearly true. In case u(s) = cr2e^s and v(s) = e~^s, so that r(s,i) - a2e~^a~^,ff2 > Q,0 > 0, it is called the Ornstein-Uhlenbeck (or 0. U.} process. It may be shown that both have continuous sample paths, and the BM Xt takes values in X — Co[0,6], the space of continuous functions vanishing at t — 0 and with the uniform norm. If / G X, then Xt+f(t) is also a BM (or 0. U. process) with mean function / and covariance r(s, t). Let P0 and Pf be the image measures of, or induced by, the processes [Po — PoX~l and Pj = P o (X + f)"1] which are Gaussian on the cr-algebra B of X, with means 0 and /, but having the same covariance r(•,•). Now by the fundamental Gaussian dichotomy [special case of the classical Hajek-Feldman] theorem, Pf and PQ are either mutually absolutely continuous or singular. If Pf « PQ [hence P/ ~ PQ] then / is called an admissible mean of PQ. It is known that the set of such admissible means is a vector space H C £1 which can be endowed with a Hilbert space metric [or norm], and then the RN-derivative -^- can be calculated. It is given, even for general u, v above, in the following. Theorem 5. Let X = {Xt,t E [a, b]} be a Gaussian process with mean zero and continuous covariance r(s,t) = u(s A t)v(s V i),s,t e [a, b], where the process is canonically represented by (fi, E, P) in that fi — JR^'^,^ is the cylinder a-algebra on which P is a Gaussian probability measure. Also, u, v > 0 are a pair of functions such that ^(t) is strictly increasing, the derivatives u'', v' exist on [a, 6] and are of bounded variation. Let f be an admissible mean so that the processes X and X + f — {Xt + f ( t ) , t G [a, b}} have their induced measures PQ and Pf to be equivalent. For simplicity let /(a) = 0. Then the RN-derivative j^- (which exists) is given by

jp v - /

~"f i r> /

fu\i\"/~~ I

../.A

/ I'

v^°v

where [j is taken as 0, and the integral is a well-defined simple stochastic integral [and can be understood as a Bochner integral after a formal integration by parts, also called the Paley-Wiener-Zygmund integral ]. This is a specialized version of a general theorem for all Gaussian process, due to Pitcher [1], and sharpened when the covariance function is factorized, as here, by Varberg [1]. All these results and a unified treatment of the work may be found in the resent monograph (Rao [24], Section V.I and the above result is given as Theorem V.I.9 there). So the details will not be reproduced here. For the O. U. process, fi = C[0, b] and (43) may be simplified, since

288

VIII. Applications to Stochastic Analysis

u(t) = a V, v(t) = e~0t, $ > 0, a2 > 0,

f\f

+~

+ /}/)2(t)e-<"d( ,

(44)

and for the BM, where v — 1, w(£) = t, b = 1, (43) simply becomes = exp { f1 f(t)dXt l^o

- \ f\f(t)]2dt} }

2 Jo

.

(45)

It should be observed that with a1 — I in the O.U. process, one can verify that Y = {Yt,t > 0} given by (Xt is written as X ( t ) ) ^-logi], t > 0 ,

(46)

is a BM. Thus they are closely related and both have continuous trajectories. For a process with mean zero and a covariance function r(-, •) , if e > 0 and X£(t) = y/eX(t), then |X e (£)| —> 0 in probability, as £ \ 0, since for any 6 > 0 one has P [ \ X e ( t ) \ >6}<

E(\X(t)f) =

_^ 0 ) a S e^ 0;

(47)

by Chebyshev's inequality. Thus the problem is to analyze a class of processes with range space A', of infinite dimension, for which the above convergence is exponentially fast. Here we discuss the BM process whose range is Co[0, 1], the space of continuous functions on [0, 1] vanishing at 0. Let ve denote the probability measure determined by {X£(t),t > 0} on C*o[0, 1]. We now follow the method of Varadhan's ([2], Sec. 5) and establish the following result due originally to Schilder [1], which is an infinite dimensional extension of Theorem 4, with X = C0[0, 1]. Theorem 6. Let {f £ ,£ > 0} be the probability measures induced by the BM {X£(t},t > 0}. Then the large deviations principle holds for it with X = C*o[0, 1] and the rate function I : X —> [0, oo), given by 2 Jo

*,

(48)

for f G C*o[0, 1] which is absolutely continuous and derivative f G I/2[0, 1]; /(/) = +00 otherwise. So doml = {/ e C0[0, 1] : ||/'||2 = /
8.1 Large deviations and Young functions

289

Proof. First we derive the lower bound. Given e > 0 and / 6 SI — Co[0, 1], let U be a neighborhood of /. Then we can find a g € U and a neighborhood V of g, V C U, such that I(g) < /(/)+£, and g is twice continuously differentiable. This is possible since polynomials are dense in X by the Weierstrass approximation. Thus ve(U} > ^ e (V), and we assert that liminf elogi/ e (V) > -I(g) > -/(/) - e, e\0

(49)

which gives the desired lower bound. Consider the translated process Xe(t] — g(t) so that V — V — g is a neighborhood of 0', and let v£ be the translated measure of ve for X£(t)—g(t). Thus ve(y] = ^e(V). By the choice of g it has a square integrable derivative, g(0) = 0, and is an admissible mean of the ^"-process. [The set of admissible means forms a Hilbert space in £), and g is close to / in this topology so g is admissible.] Hence ve « ve, and by (45) we have the RN-derivative ^(X) = exp (1 I' g'(t)dXt - 1 dve L e Jo 2,e Jo = exp

j X t /(t)dt -

^(l)X! -

fl(g'(t}fdt} ) /(^) ,

(50)

on integrating the first integral by parts and using (48). Now choose 6 > 0 , small, such that Ve contains a ball -6,5(0) of radius 6/2 at "0". Then one has

However, we also have the estimate tl X^'^dt-g'^X, Jo

<

<

Joo

fl\Xt\\g"(t)-g'(l}\dt

= \\g"\\[/Q1\Xt\dt+\X1\},

(52)

where ||0"ir= sup (\g'(t)\ + \g"(t)\] < oo, the Sobolev norm of g. But Xe -x 0 te[o,i] in ^-probability as e —>• 0. It follows that, since /Q \Xt\dt < | and |Xi| < |, inf

X ~ > exp i--I(g) - -||/|fl Ny *\

€ ^'

5

"J

290

VIII. Applications to Stochastic Analysis

Thusfirstletting 6 \ 0, and then E \ 0, we get from (50) and (51) liminf£logi' e (V) > —I(g) > —/(/), e\0 since i/e(J3j(0)) —> I as s ->• 0 by (47), for any 6 > 0. So (49) is proved. For the upper bound, we need to use the symmetry of BM about the origin, and also invoke the reflection principle of D. Andre for this purpose, as well as certain other properties. We sketch the essential details [leaving a few (nontrivial) computations to the reader]. It is to be shown that, for each closed set C C 17 — Co[0,1] arid £ > 0 one has lira sup ve (C} < - i n f / ( / ) . (53) e-*Q

f£C

For this we find upper estimates if C C A U B, so v£(C) < v£(A} + ve(B] for a proper choice of A,B. In fact, given a 6 > 0 let C6 = (j S(f,S), /ec where S(f, 6) is a sphere of radius 6 and center /, so that C5 C A U B since C C C5. To use the independent increments property of BM, consider a mapping Y[n : 17 —>• 17 which defines a polygonal approximation of the BM, with step size -, i.e., n n / = ^/ i x [ kii i i ) , fi = Hence letting A = {/ <E ft : H n / e C5} , B = {/ e 17 : ||Hn/- /|| > <5}, one gets C"5 C A U B for each fixed n. Thus vE(C] < vE(A}+ve(B).

(54)

Next we estimate ve(A) and i/e(B) in (54). Let ls = inf{/(/) : / e and note that (Il n /) is identifiable as the sum of n independent normal AT(0, ^), random variables under P and 7V(0, ^^f) under VE. Then f (H n /) 2 is distributed as a chi-square random variable with n-degree of freedom. [This is the same as saying that ve o II"1 has the stated distribution.] Hence we have the probability of the cylinder set {/ : I(YLnf) > /,?}, given as v£(A)

<

ve({f:I(nnf)>ls}) r

(f) (55)

8.1 Large deviations and Young functions

291

by using the form of the distribution of chi-square random variable and Laplance's method of finding the maximum by Theorem 3 (cf. (39)). From (55) one finds that limsuplogz/ e ({/ : /(!!„/) > 6}) < -ls. £->0

To estimate the second term v£(B] of (54), since the increments of BM on equal intervals are identically distributed and the distribution of a BM is symmetric, by using the key reflection principle noted above (cf., e.g., Doob [1], p. 392) and the Laplace's method again, we get: i/e(B) < v£({f : \\Unf -f\\> 5}} < nvAlf:

sup |/(*)-/(0)|>

< 2 r o / 1 / : sup \f(t)-

f(Q)\>

= 4m/! H / :/(I) -/(O) > ^ H > by reflection, 4n

e ** dy

-^— + o ( - j L as £->Q. OS

Hence

(56)

\£/ J

r?^2 limsupelogi/ e ({/ : ||Hn/ - /|| > 6}) < - — . o

e-)-0

Substituting (55) and (56) in (54) and taking logarithms, one gets

Is e l, o go2 - e m m -, X2

- min ls, — ) , as e \ 0.

(57)

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VIII Applications to Stochastic Analysis

Now let n —>• oo in (57) to obtain, for fixed 6 > 0, ) < —lg-

(58)

E->0

But from definition of CS(D C), one has

(59) by the continuity of /(•). Hence (58) and (59) yield (53). Thus from (49) and (53) we deduce the assertion of the theorem. D Remark. The computations for the lower bound use only the sample path continuity of the process { X £ ( t ) , t > 0} and the fact that the covariance function is "triangular", i.e., of the form u(-,-)v(-). The work appears to extend to the O. U. process. But the upper bound uses the symmetry and reflection principle for the BM, and hence the computations do not easily go over to the O. U. process. So these properties are not available for the latter. It is known that a centered Gaussian process with a triangular covariance such as r — uv above (including the O. U. process), is Markovian. Using other methods a great many of the large deviations results are extended by Donsker and Varadhan in a series of papers of fundamental significance. Another important method due to Freidlin [1] will now be described briefly. It is appropriate here to indicate how Schilder's theorem admits a extension to a large class of (not necessarily Markovian) Gaussian processes, following Freidlin (cf. Freidlin and Wentzell [1], Sec. 3.4), based on the crucial observation that centered Gaussian processes with continuous covariances are linear integral transforms of BM processes, so that the LDP theory of the latter may be extended. Here is a more precise description. Let X = {Xt,t 6 T = [a, &]} be a centered second order process with a continuous covariance function r : T x T —> JR. Then the associated integral operator R : f \—> (Rf)(t) = ]~Tr(s,t)f(s)ds, is compact (actually Hilbert-Schmidt) on the Hilbert space L2(T,dt), nonnegative definite , and hence has a unique (positive) square root R* . [This is true if only r is square integrable.] The subspace Ri (L2(T, dt)) has the following remarkable property, noted by T. S. Pitcher [1]: Take (fJ, E, P) as a canonical space (i.e. f2 = MT, E = cylinder a— algebra, P : E —> [0, 1] a probability) as described prior to Theorem 5 above, let / : JR —> JR be an admissible mean of P so that Pf « P where P/ is the measure of X(t, •} + f ( t } . If Mp is the set of all admissible means, then Mp C Ri(L2(T, dt)) and there is equality if moreover X is Gaussian (and not otherwise, cf., Rao [24], Sec. V.I on this

8.2 Infinite dimensional extensions and vector Orlicz spaces

293

topic and for details). Further Mp is a Hilbert space under a norm given as follows: fi e Mp => fi = R*hi for some hi € L2(T,dt),i = 1,2 and the inner product of Mp is obtained by (/i, /2) = (R*hi, R^h2) = (Rhi, /i2), the latter is the scalar product of L2(T, dt). Using the spectral theorem (cf., also Dunford-Schwartz [1], VI.9.59) one finds that R* is given by a kernel G as: (R*f)(t)

= f G(s,t)f(s)dsje

L2(T,dt)

J J.

and one has r(s^t) = fTG(s,u)G*(u,t)du (G*(u,t) = G(t,u)). If we define a new process X(t) = fTG(s,t)dBs, where {Bs, s e T} is the standard BM, then { X ( t ) , £_E T} is a Gaussian process with mean zero and covariance r, so that X and X can be identified. This means that X = X(= fTG(s, -)dBs) is a linearly transformed process of B, observed earlier. Remembering that R~l and R~2 exist uniquely on the range of R. so that R~*f = h iff / = R* h, define the rate function / : L2(T, dt) —>• M as /(/) = / f H^/ll 2 , if f € range(R), | +00, otherwise.

(6Q)

Thus /(/) = ^ ( R ~ l f , /), in the above notation of inner product of the space Rz (L2(T,dt)). Then Freidlin's extension is given by: Theorem 7. Let Xe(t) = eX(i),£ > 0, be a centered Gaussian process for t G T = [a, b], with a continuous covariance function. Then X£ obeys the LDP with the rate functional I: L2(T, dt) —> [0, oo), defined in (60). The proof is given in the above monograph, even for a process with values in ]Rd without much difficulty, and we omit it here. It has alternative procedures, excluding the representation through the BM, and many other developments are possible. We leave this discussion for specialized treatments, and observe that /(•) is again a convex functional on L2(T,dt). So we need consider vector Young functions and the corresponding Orlicz spaces on X, possibly of infinite dimension, which is the topic of the next section. 8.2 Infinite dimensional extensions and vector Orlicz spaces The finite dimensional Cramer theorem, i.e. Theorem 1.4, admits an extension to infinite dimensional spaces containing all the finite dimensional sets as subspaces with a suitable (and natural) compatibility condition on overlaps. The collection of spaces (of different dimensions) are then termed a "projective system" for which we can extend the above result. Let us make

294

VIII. Applications to Stochastic Analysis

these terms precise and establish an abstract version which also shows some new features leading to general Fenchel-Orlicz spaces. We recall the concept. A family of probability spaces {(0Q, £ Q , PQ), a G J}, where J is an index set, is called a projective system if (i) J is directed (i.e., there is an order relation < such that for a, /? G J, a < /3 is defined for at least some pairs, and for any a, (3 G J, there is a 7 G J such that CM < 7 and /3 < 7), and (ii) there are connecting mappings and gap : ftp —>• QQ for each o; < /3 in J, satisfying gaa = identity, and for a < (3 < 7, Qapogpf — 9a-yAn example of such a system, which actually will be of interest below, is: ftQ = JRQ and gap : JR? —>• JRQ for a < /3 with a = (1, 2, • • • , m) and £ = (1, 2 , . . . , m, m + 1, • • • , n), "<" is inclusion (in other cases a lexicographic) ordering, so IRa and IRP can be identified as the m and n-dimensional spaces (m < n), with gap as coordinate projection mappings, g a p ( x i , - • • ,xn) = (xi, • • • ,xm). We may denote such gap as \[mn : Mn —> IRm,m < n, to specify this case. Consider a set f2 C x ae yQ a such that the "thread", uj = (uja, a G J) is in f2, iff for each a < f3, gap(u}p) = uja. In the above example gap — l\ap, Qa — M01, then £7 can be identified with xa£j]Ra, the cartesian product. In the general case Q can be empty (e.g., let £)n = (0, ^ ) , n > 1 and for n > m let gmn : Qn -4 Qm where gmn is an inclusion, then {(^ n ,#mn), m < n} is a projective system , but Q = 0). We need to restrict the spaces so that Q is nonempty. A good sufficient condition for this is sequential maximality (s.m.) introduced by Bochner in his basic formulation of this subject. Thus the family is said to obey the s.m. condition if for each sequence Oi\ < a-2 < ••• from J and any ujQi G f2Qi satisfying <7aia i+ i( w ai+i) = ^Qi,i > 1, there exists an element uj — (u> Q ,o: G J) G 17 such that ga(u] = ua where ^QOO : XQGJ^Q —> ^Q- When this holds ga = gaoo is onto, i.e, ga(&) — ^a for all a £ J. The s.m. condition is automatic if gap(= T\a/3}i9a = Tla(: MJ —> lRa) are coordinate projections. Moreover we can also show it to be automatic if each Qa is a compact space. The measure spaces (f2 Q ,£ a ,.P Q ) are said to form a projective system, if gap : $lp —>• fiQ is measurable, i.e., ^(E"1) C Ep for a < /3, and Pa = Pp o g~p so that Pa is the image probability measure of Pp. Thus {(fia, E Q , Pa, ^a/g) : a < /? in J} is termed a projective system of probability spaces, which generalizes the Fubini-Jessen concept of infinite product measures. The relation that Pa = Pp o g~p is just the consistency requirement for these measures and if £7Q = JRa,£lp — M^, then this simply means that Pa is the marginal probability of Pp. But the general case is also of interest since the random variables can take values in different spaces. Thus if Q = \im(£la,gap), called the projective limit of the

8.2 Infinite dimensional extensions and vector Orlicz spaces system, with ga : ft —> ftQ, one defines EQ = U 9al(^a)

295

an

d observes that

a£J

EQ is an algebra and let E = cr(Eo), called the cylinder cr-algebra. Then a set function P0 can be defined as Po(A) = P0(g~l(Aa)) = Pa(Aa), A e E 0 , where A = g~1(Aa) for some a 6 J by definition. It is not hard to see that P0 is uniquely defined (does not depend on a) and that PQ is finitely additive on EQ. In general one can not assert more. It is here we need to assume that each ftQ is a topological space, gap : ft,g —> ftQ is continuous (it is clearly onto and ga : ft —> ftQ is required to be continuous), each EQ is Borel, and what is decisive, each Pa is a regular probability measure. Under these conditions P0 is cr-additive (but still need not be regular!), and if P is its (unique) extension to E, then we set P = limPQ and (ft,E,P) is called the projective limit measure space. It satisfies immediately P o g~l = Pa for all a € J. [For a proof of these statements, see e.g., Rao [20], p. 354, and we shall not reproduce the proof here.] In order to conclude that P is regular, one has to demand the Prokhorov (e, K£)- condition, namely for each e > 0, there exists a compact set K£ such that Pa(toa-9a(Ke))<£,

« € J,

(1)

so (ft, E, P) will be a regular projective limit space. [For a detailed proof, see the above reference pp. 359-364.] This condition is automatic if each ftQ is compact, and then ft = lim(ft Q , gap) is also compact as well as P regular. With the above setup we can present an abstract large deviations result from (f2 Q , E Q ,P Q ) to their limit. It is essentially due to Dawson and Gartner [1]. The result depends on the contraction principle discussed after the statement of Theorem 1.3. Theorem 1. Let {(£7 Q ,£ Q ,P^,# Q/ g),a: < ft in J} be a regular projective system of probability spaces depending on a given positive parameter e, and suppose that it verifies the s.m. condition so that there is a unique projective limit (0,S,P £ ) = lim(f2 Q , E Q , P^gap), depending on E. Then as e —> 0, the system obeys the large deviations principle (LDP), iff the limit triple does. When this happens, if Ia and I are the corresponding rate functions, then I(u)=supla(ga(u)),

weft,

(2)

a£J

where Ia(x) = inf{/(w) : u G g^dx})},

x € fia, a e J.

(3)

296

VIII. Applications to Stochastic Analysis

Proof. By hypothesis ga : 17 —> 17Q is onto (a consequence of the s.m. condition) and ga and gap are continuous. Suppose that the limit measure Pe of (fi, E, P£) = lim(fi a , £ Q , P*, gap)a 0 is compact and by the compatibility condition, P^ = P£p ° 9a$ and J^(-) is a rate function, so that for a < (3,Ayp which is compact satisfies gap(Ayp) = Aya. Thus {(A^, gap) : a < ft in J} is a projective system of compact spaces for each y > 0 and so Ay = lim(^, gap) exists, nonempty and is compact by the earlier result. To conclude that /(•) is a rate function, one has to verify that it satisfies the appropriate upper and lower bounds of the LDP as e \ 0. Thus let O C 17 be a nonempty open set and u € O. From the fact that 17 = lim(17Q, ga/3) and the definition of the projective limit topology, there is an open set Ua C 17Q, g"1^) C O and ga(u) € Ua. So from P£(0) > P£(g-l(UQ}} and / - sup/ Q o9a, we have liminflo g P £ (0) > E\0

e\0

Taking sup over a £ J and since u e O is arbitrary, the lower bound for / is obtained. Regarding the upper bound, let F C O be any closed set, consider a compact level set Aya,y > 0 for Ia. On Ava,Ia(x) < y, and the upper bound for / is obtained immediately to be y on F D Ay, where Ay = \\m(Aya, gap), a compact nonempty set, if this is nonempty. So let F n Av — 0 and suppose inf{/(x) : x e F} > y0 for some y0 > 0. We obtain the upper bound for the LDP in this case also to conclude the argument, since the result follows if there is no such yo- Now the projective limit theory implies that if Pa = 9a(F}, then P = lim(P a ,^ Q ^) where PQ is the closure of PQ, although Fa may not be closed. To see this, for a < ft,ga/3(Fp) C Fa since the gap are continuous. Thus P C \im(Fa,gap). On the other hand if w 6 Pc, the complement of P, there is an open set Up C ftp, ft € J such that (jj 6 g~p(Mp) C Fc. Hence gp(u) e Up C Fjj implying that gp(u) £ Fp and hence uj ^ lim(P Q , gap). This implies that the above asserted equality

8.2 Infinite dimensional extensions and vector Orlicz spaces

297

holds. Now Fa n A%> is compact for each a, since Ia is aerate function, and F n Ayo = lim(F0) n lim Aya° = lim(FQ r\Aya°). But each FQ n Ava° is compact and F n Ayo = 0 so that FQ n A^° = 0 for some a = ao essentially because F n Ayo is an intersection of compact sets. But (f2 QO ,E Qo ,P^ 0 ) obeys the LDP and so limsuplogP^ 0 (F) < -yQ. This shows the upper bound holds in e\0

this case also and (17, E, P£) satisfies the LDP. D To apply this result, it is useful to get some classes of spaces (QQ, Ea, P*) for which an LDP holds. If each S7Q is a finite dimensional vector space, then Theorem 1.4 applies to each of them. However in a locally convex topological space X, the projective limit of all finite dimensional subspaces, whose collection J is ordered by inclusion, is known to be larger than X. More precisely, let J be the collection of all subspaces, of a topological vector space X, of finite codimension, so that F e J iff X/F, the quotient space, is finite dimensional. Then J becomes directed if for a, ft E J, a < ft iff a D ft. Let FIa/8 : «^9 —*• %a where Xa = X/a and Ha : % —* %a is the coordinate projection, a 6 J. With this understanding, {('Va, FIo^) : ex < ft in J} becomes a projective system of vector spaces, and each is finite dimensional. If X is assumed to be a locally convex metric space (e.g., a Banach space), then Q — lim(^'a, Ylap) exists, but usually larger than X. If X* is the adjoint space of X and (X*)'(D X**} is the algebraic dual of X*, the projective limit topology of fi is weaker than the topology of X, and if it is endowed with <j((X*)', A"")-topology, then il and (X*)' can be identified in the sense that there is a linear bijection between these two spaces. [See, for instance, (Rao [22], Prop.l, on p. 34) on this problem, and references therein.] It can also be shown that, with EQ denoting the Borel cr-algebra of Xa, the limit of the projective system {(Xa,T,a,Pa,Hap) : a < ft in J} exists and equals (X,Y,,P] iff it satisfies the (e, Ke) -condition (1) of Prokhorov's, as discussed in the above reference. Since Theorem 1.4 implies that, for e > 0, the (Xa,Ea,Pa) satisfies the LDP where Xa is finite dimensional, the following is a consequence of the above theorem and discussion which we record for a convenient reference. Corollary 2. Let X be a locally convex topological vector space and {(Xa, OQ^) : OL < ft in J} be a projective system of finite dimensional subspaces of X. Consider the projective system of probability spaces {(X*, E a ,P^,n a ^) : a < ft in J} such that (X* is the adjoint space of Xa): Ia(x) = \ims\ogE£(e^'x^£),x€ Xa, £

(4)

5rfU

exists and Ia(x) < oo , where the expectation E£ is computed under PJ.

298

VIII. Applications to Stochastic Analysis

Suppose that /«(•) is a "generalized N-function", i.e., Ia(-} is convex and weakly (or in JRd) Gateaux differentiate in the sense that t H-> Ia(x + ty) is differentiable at x in the direction of y in Xa. If Ia(x*} = sup{(x*, x) - Ia(x) : x G Xa},

x* G X*,

(5)

then /Q(-) is the conjugate of the Young function I a ( - } , which is the rate function of the system {(X*,Ea,P^,l\ap) : a < /3 in J ] , and if I(x*} = s\ipIa(Y[a(x*)),x G X' , the algebraic dual of X with a ((X*}' ^X] -topology, Q

then ((X*}', E, P£] satisfies the LDP with the above rate function I where The result is really a restatement of Theorem 1, where one uses the standard results of convex analysis in Md (cf. Rockafellar [1]). This is useful for applications, and to show how Theorem 1.4 is extended to infinite dimensions. Now if A" is a separable Banach space and X : ft —> X is a mapping from a probability space (ft, £,P), then X is termed Gaussian if for each x G X*, the scalar variable ( X , x ) : ft —> IR is distributed as a normal random variable with mean mx and variance vx. For x, y G X* , (X, x} and (X, y) are jointly normal with means mx, my and covariance Cx,y- Thus in order for such a P to exist one must have the characteristic functional (or Fourier transform) of X, given by (with mx = 0, and Qp goes with the measure P] = e-*Q*(x'x\

x G X\

(6)

where Qp(x, x) = vx and Qp(x, y) = Cx^y so that Q p ( - , •) is a positive definite bilinear form defined on X* x X*. The existence of such a P (or X or Q) is not obvious. Let us illustrate it for the Wiener or BM process, where X = Co[0, 1] so that X* = M[0, 1], the space of regular (signed) Borel measures on [0, 1] with the total variation norm (by the Riesz representation theorem). Then (Xt,X)(u) = f£ Xt(u)dX(t), where {Xt,t G [0,1]} is the BM process, A G M[0, 1], and (6) becomes with Xt(u) = u(t},u) G ft, =

/ exp{z(o;> «/ 1 2

- exp{-A p (A)} (7)

with r(s, t} — s At. To verify (7) in a simple case, suppose A is a discrete mean

sure having jumps A 1; • • • , A n at i l 5 i 2 , • • • , tn so that

(X, A) = J] X^. (co>)Aj

8.2 Infinite dimensional extensions and vector Orlicz spaces n

Mj*

an

299

d one finds by the image measure theorem of Probability

.7 = 1

Theory: exp{i(X,X)(u)}dP(u>)

r

=

J

r

I ••• I

( n

J R

n

\

exp < i Y^ XjXj > dFtl,...,tn (xi, x2) • • • , xn) ^ j=l

J

which is (7) for this particular case of a discrete measure A e X*. The existence of P satisfying (7) can be proved^by Kolmogorov's fundamental theorem, or using Wiener's original construction on C0[0, 1], which gives many special properties of this measure at the same time. Thus for a centered Gaussian probability measure P on a separable Banach space X(=£i), one has = I

(x*,x)2P(dx).

JX

By the CBS-inequality, one can verify that AP(z*) < \\x*\f f

JX

\\x\\2dP(x}.

Then Ap(x) = sup{(:r*, x} - Ap(a;*) : x* 6 X*}, xeX,

(8)

is convex and nonnegative. [If X — JRn, then A p (-) = A p (-), which need not be true for general X, c.f. Rockafellar [1], p. 104. However, this will be true if A" is a reflexive Banach space.] It is a rate function. Freidlin and Wentzell [1], and in all their publications, call it a normalized action functional. We shall use the shorter name, as before. Let P£ be defined as P£(A) = P(V/L4), where a vector space multiplication a A = {ax : x G A} is used. We then have the following result due to Donsker and Varadhan (for different proofs, see Freidlin and Wentzell [1], Sec. 3. 4, and Deuschel and Stroock [l],p.86 ff).

300

VIII. Applications to Stochastic Analysis

Theorem 3. The family of Gaussian measures {P£,£ > 0} defined above satisfies the LDP as e \ 0, with rate function Ap given by (8). Moreover, if a — inf{Ap(:r) : |x|| = 1} and b = sup{Ap(x*) : \\x*\\ = 1}, then ab — 1 and lim — logPfx : l l x l l > r) = —a.

r-»0 |-2

~~

(9)

Further, fx exp{a\\x\\2}dP(x) < oo /or 0 < a < a, and = CXD in the case that a < a < oo. One of the important points raised by this as well as the previous result is to find ways of evaluating the rate function A p (-), which is the (Young) complementary function of A p (-), the "cumulant" generating function of the process (or the measure P). The actual calculation is not simple, as already observed (cf . (40) of Section 1 above) . A class of processes were considered by Pinsky [1] for an evaluation of A p (-) which the reader may consult for an appreciation of the problem. Here to explain this further, we include some functional analysis properties, by discussing Orlicz spaces based on the pair Ap(-) and A p (-) and present a few results in that direction. If Ap(-) (and A p (-)), considered as a convex function on lRn (and on n (M )* = Mn), is desired, then one may turn to Rockafellar [1] for an incisive analysis, and we refer to this work when needed. Since the rate function for Brownian motion, and that related to general Gaussian processes, demands that we consider infinite dimensional spaces X (rather that X — Mn only), they are discussed below to explain some details underlying our (large deviations) problems where our chief interest however is a calculation of Ap. Although A and A (written for Ap and A p ) are convex, they are not necessarily Young functions in general in that, A(x) = A(— x) > 0, A(0) = 0, A(-) convex does not imply A(to) —>• oo as t —> oo for x ^ 0. For instance, even for X = M2,$(x,y) = ^(ar^Cy), where $! : M —•> M+ is a Young function and $2(y) = (X[y=o] + °°X[z//o])> is convex and its complementary function $(x, y) = ^i(:r), where ^i is complementary to $1. Then $ is convex, M be a Young function, namely A.(tx) —> oo as t —> oo for any 0 ^ x G X. The above example shows that this itself does not make A a Young function on X* . It may be verified that A is also a Young function iff the closed linear span of {x : A(z) < 00} is X, a Banach space. For more details on these functions and certain other related questions we refer to Turett [1]. In our applications A(-) is the cumulant generating function. It is convex, A(0) = 0, A(fcc) —>• oo

8.2 Infinite dimensional extensions and vector Orlicz spaces

301

as t —)• oo if x 7^ 0 , but A(—x) = A(x) is not satisfied. Also we assume that {x : A.(tx] < oo for some t > 0} = X, so that A is also a Young function, although A(—x*) = A(x*) is not satisfied usually, since the random variables X need not be symmetrically distributed. However, one can often symmetrize the variables and apply the results by using a standard procedure in Probability Theory. So we assume hereafter that A(-) is a Young function and that {x : A.(tx} < oo for some t > 0} = X. Definition 4. Let X be a Banach space, A : X —> JR a Young function with {x : A.(tx) < oo for some t > 0} = X, A(—x) = A(x) and (fi, E, /u) a measure space. Then the Fenchel- Orlicz space L (/i, X) is the set of (strongly) measurable / : Q —>• X such that the (Bochner) integral /n A.(kf)d/j, < oo for some k > 0. The gauge functional (essentially norm) || • 11(A) : LA(fj,, X) —> 1R+ is given as II/II(A) = inf | k > 0 : ^ A (- J d/z < 11. Using the standard arguments, extending the classical Orlicz spaces LA(II) as in Rao and Ren [1], one can establish the following: Proposition 5. The space (LA(n,X},\\ • ||(A)) is a normed linear space, when equivalent classes under the norm functional \\ • ||(A) are identified. If either X is finite dimensional or A(-) is such that A(x) > o;||x|| + 0 for fixed a,(3 > Q,(x ^ 0) then LA(n,X] is also complete, i.e., it is a Banach space. The last condition is equivalent to the continuity of A(-) at 0, or alternatively equivalent to the assertion that \\fn — /||(A) —> 0 implies convergence in measure, i.e., that ^JL{UJ € £t : \\fn — f\\ > e} —>• 0 for any e > 0. The condition A(x) > a||x|| + 0 is of "support line" type, and it is equivalent to saying that lim inf{A(x) : ||x|| = n} = +00, termed uniformly large at infinity. This is always satisfied if X is finite dimensional. Without such a growth condition LA(fj,, X] need not be complete; but it is always satisfied for the cumulant generating functions. This is also equivalent to saying that the complementary function A(-) is continuous at 0. With these observations the completeness proof is a modification of that given usually (cf., e.g., Rao and Ren [1], p.67). It will be omitted , and a detailed account can be found in Turett [1] who extended the standard Orlicz space analysis to LA(/j,,X) and characterized (LA(n, X))* when A satisfies a A2-condition, defined below. Indeed much of the theory of the corresponding scalar Orlicz theory extends (nontrivially). We present a few analogs of the classical analysis (cf., Rao and Ren [1], Sections 4.1 and 4.2) for our applications.

(10)

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A Young function A : X —> 1R is said to satisfy a A2— condition for large values (for all values) if A(-) is bounded on balls of X and there exist Ki>Q and K2 > 0 such that A(2x) < K2A(x) for \\x\\ > KI (for Kl = 0). Also let M A (/^, X] be the closed subspace of LA(/J,, X] determined by simple functions. We recall that a simple function / : 1) —>• X is given by n

/ = ZX**' ^ e ^,A Z e E,A,n^ = 0, i ^ j ,

(n)

and p,( 0 Ai) < oo. Similarly an additive G : £ —>• X is of finite A— variation 1=1 (relative to p,) if | G||(A) < oo, where (12)

/

and the A— variation I\(G] is defined by

IA(G)

= sup

disjoint, 0 < /^(A) < 00} •

(13)

The space (V^(^, X ) , \ \ • | (A)) of additive set functions G, with oo, is a Banach space, and it has been analyzed by Uhl [1]. The subspace S\(/JL, X] C VA(/^, X.\ of countably additive elements, is of interest here. We can present, extending the classical scalar results, the following: Theorem 6. Let A be Young function as in Definition 4, A be its complementary function, and M A (/^, X] be (as defined above) the subspace of LA(/J,, X] determined by simple functions. Then each continuous linear functional I : M A (/i, X] -—>• IR is uniquely representable as

where the integral is the classical bilinear vector integral (in the sense of Bartle [1]), and = sup{|J(/)| : / e MV X\ ||/||{A) = 1} = = sup { / fdG : / e MA(/., AT), ||/||(A) = l} . L

Ji t

y

(15)

8.3 Regularity of stochastic functions and Orlicz spaces

303

If further, X* is separable, or more generally X* has the RN-property, then & = g exists and g G L A (//, X * ) , \\G\\% = \\g\\^, where

|M|x = sup{jf^ :||/|| (A ) = l}

(16)

is the Orlicz norm, equivalent to the gauge norm: \\9\\n\ < \\9\\^ < In particular, i/A(-) satisfies a ^-condition (recalled preceding the theorem), then MA(^,X) above can be replaced by LA(p,,X) everywhere. The proof needs many details. Although they are extensions of the scalar case, we shall not include them here. Essentially these may be obtained from the computations in Turett [1] with easy modification. It is of interest to remark that the full dual space (LA(p,, X}}*, itself is not available since this leads to "singular" functionals, and a complete analysis of the space VA(A*, X) is needed involving purely finitely additive vector measures, which are the analogs of Sections 4.2 and 4.3 of our book [1]. A detailed treatment of the vector space case is not available, and may be considered as a problem for a future investigation. Reflexivity, uniform convexity and other geometric properties of the space £ A (M, X] utilize Theorem 6, and we omit further discussion of such a study, but just include the following statement to indicate the type of expected results. Theorem 7. Let A : X —> IR be a Young function as in Definition 4 and A : X* —>• JR be its complementary function. The space LA(/j,,X) on (ft, £,//) of strongly measurable and "A.-integrable" functions is defined as before. Then L A (/^, X) is reflexive if both A, A satisfy ^-conditions for all values and X is reflexive. On the other hand, LA(/J,, X] is strictly convex (in that the sphere has no line segments) if A is strictly convex and satisfies the ^-condition. We omit the details (cf., Turett [1]) and turn to the classical spaces in which the Young function A(-) has exponential growth (as in, e.g., Rao and Ren [1], Chapter II). We then can immediately apply the analysis to stochastic functions.

8.3 Regularity of stochastic functions and Orlicz spaces Recall that a Brownian motion, or BM, {Xt,t 6 [a, b]} has continuous sample paths. This property is useful in the work on stochastic integration and numerous other applications. The same property may be expressed also as X : Q —> C[a, 6], a continuous (coordinate) function on £7; so that

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Xt(uj] — u}(i), and hence X(.)(u)) = w(-) e C[a, b]. If [a, b] is replaced by a set D C Md, then X(.) is called a random field. Continuity properties of stochastic functions are important, and Kolmogorov obtained early in the subject, the following result for verifying this property. Kolmogorov's Criterion. A process {Xt,t 6 [a, b]} with values in a complete separable metric space X is continuous at t if there exist N-functions $i,$2 such that P[u : d(Xt+h(u),Xt(uj)}

> $;(/>)] < $ 2 (/i),i e [ a , b ] , h > 0,

(1)

where d(-, •) is the metric on X and the $; satisfy: fh $'i(t) I dt < oo, Jo t

fh $2(1) / dt < oo. J o t

This result is discussed in Neveu ([1], p.95), and in many other books. For random fields, i.e., t — (ti, • • • , td) € D C JRd, and (X, || • ||) a Banach space, a simple extension of (1) is as follows: there exist £ > 0, r > 0, C > 0 and d on > 0 with E Qt ^~ < 1> such that i=l

A proof of (2) [for random fields] may be found in, for instance, Kunita [1]. We consider these results as a beginning for studying stochastic regularity. For any A^-function $, we set A.(x) — e®^ — 1. Then A is an AMunction of exponential type. It satisfies the A 2 —condition for large x, i.e., there exist constants K > 1 and XQ > 0 such that A 2 (z) < A(Kx),

x > x0,

(3)

by definition. This follows from the inequality A 2 (x) = e2*^ + 1 — g2*(x) _ ]_ < e*(2i) _ i = A(2x), x > XQ = 0. In general, A^-functions of class A 2 grow exponentially in that for A 6 A 2 , there exist a > 0 and x\ > 0 such that ex° < A(x], x > xi. [For a proof, see, e.g., Rao and Ren [1], p.40.] A slightly weaker condition is V, which is "opposite" to A', and given by A(;r)A(y) < A(kry), x , y > y 0 , b > 0,

8.3 Regularity of stochastic functions and Orlicz spaces

305

so that A 2 =>• V. A discussion of these conditions is in the above reference. It will be of interest in the following application to consider the Orlicz space £A(A*) where A(x) = e®^ — I . If p is finite and A(y) is the complementary JV-function to A.(x), then L~(Ai) C LA(/,) C 0 Lp(»] C (J Ufa) C Ln) C L1^). P>I p>i

(4)

Indeed, for any 1 < p < oo, - + - = 1, we have ^ -< A(x) and A(y) -< ^ for large values. Thus LA(/^) C Lp(^) and L9(//) C Z/V) (cf., e.g., Rao and Ren [1], p. 155, p.50), proving (4). To have some feeling for the space L A (/z), we present an easy characterization, and some related results. Part (a) is from Buldygin and Kozachenko [1] and Part (b) follows Fernique [1]. Theorem 1. Let A.(x) — e*W — I , where $(•) is an N-function. Then (a) f € £A(AO on a probability space (Ti, S,//^, iff there exist constants C = Cf > 0 and D = Df > 0 such that V[u> € a : I/Ml > x] < Cexp[-*(|L)],

x

> 0,

(5)

and when this condition holds, one has

(6) = |2;|p,p > 1 and N(f) = sup 1%^, ^en JV(-) and || • ||(A) are equivalent norms, in fact

n>no

aN(f) < ||/||(A) < bN(f),

f € L A (/,),

(7)

for some absolute constants 0 < a < 6 < oo. Hence L^(fi) is a Banach space under either norm. Proof, (a) If 0 ^ / G £ A (ju), we have

since 1> / A

VIII. Applications to Stochastic Analysis

306

and

= 1- Hence by the Markov inequality AH6^ • I/ 1 — x\

M ] w • exP ^ 1

pr

/ x

' I/I V I/I

11 / - [ III/ J\J x 2ep - [ *u ^

JH

I

> exp

(A

^

exp

\

'I(A)/.

which is (5) with C — 2 and D = ||/||(A). For the converse, suppose (5) holds with some C > 0 and D > 0. By the image law of probability (cf., e.g., Rao [18], Theorem l(iii) on p.19), if F\f\ = /j, o I/I" 1 , the distribution of [/ , we have for a > 0

A \J-

j iL* \V a // Ji , a / j dx

L

(9)

It is thus enough to show that the integral in (9) is finite if (5) holds. Note that in the above reference, the integral is given for g(x) = x\p and here g(x) — exp[<$ f |J], but the same proof applies. Let G(x) — I — F\f\(x) and note that G(x) — p,u> e /| > x] to use (5). Thus for a > D, by (5) and the convexity of $, we get

/o < C

6X

P

d\*(a

exp <

C I /o

exp

CD < GO, a-D

(10)

in which inequalities $(;§) - <E>(f) > $ (§ - f) and $(|rc| + |y|) > $(a;) + , are used. From (9) and (10), we conclude that / E LA(n). Then the

8.3 Regularity of stochastic functions and Orlicz spaces

307

right side of (9) is bounded by 1 if ^j — l = l o r a = ( l + y).D and hence by definition of the gauge norm, ||/||(A) < a = (1 + f ).D, i.e., (6) holds. (b) If $(z) = \xp,p > 1, then A(z) = e™p - 1 = £ ^-. Let n=l

Af (/) =sup

|np

(11)

where \\f\\np is the usual Lebesgue norm. If an = (n^~np\\f\\np, then np n=l Ml-/ H ( A ) .

so that an < ||/||(A) for all n > 1, i.e., M(/) < ||/||(A) • On the other hand, A

implying ||/||(A) < 2pM(/). Now || • ||(A) and TV(-) are clearly norms and

(12)

M(f) <

or M(/) ~ ||/||(A)- To see that (7) follows from (12), we first note that ll/lln /* ll/ll oo and hence the supremum is unaltered if n > n0. Next we use the Stirling approximation to n! (see (33) of Section 1) to get l\nf

sup( — 711

sup

e"n~" X/27T7T,

k. IV* *

= sup

,

sup-^. n>l V n

Substituting this in (11) gives M(/) ~ A^(/). Thus, N(f) ~ ||/||(A) by (12), and this is alternatively written as (7). D Remark. The above result and proof are valid for any continuous Young function <$, and p > I. An analogous result to (7) is given by Ciesielski, Kerkyacharian and Roynette [1], p. 190. It is not always easy to obtain exact values of the constants a and b of (7). We illustrate this for an embedding constant in C Lp(n] where A(z) = e^^ — 1 and p > 1 , and then present an

VIII. Applications to Stochastic Analysis

308

example of interest in signal extraction. The fact that LA(n) is continuously embeddable in Lp([i) for any p > 1, implies that

(13)

IP -^

We now show how C(— Cpi$, > 0) can be given a good approximation following Buldygin and Kozachenko [1], both for this and the next example. Example 2 (Embedding constant). To compute the coefficient "C" in (13), we need to find an inequality between \x\p and A(x). Let ^ be the complementary to the TV-function $. By Young's inequality one has p
(14)

since

p • exp

p

But (14) implies the numerical inequality

P

xp <

IP

(15)

Put x = ]/M|/||/||(A) if 0 / / e L A (/i), and integrate (15) to get (A)

Thus, the above and (8) give

fll <-^ /HP ^ ,T,_I, So C — pip 1^ l(p) is a "good" estimate of the constant in (13). The following example describes the sample paths of a process with values in the Orlicz space defined by an exponential TV-function, and can be modeled to discuss a type of telephone signal traffic. This uses Theorem 1. Example 3. Suppose that, for simplicity, there are two telephones receiving cells which are connected so that one or the other is handled with equal probability, and the incoming calls are described by a continuous parameter process {Xt,t > 0}, each taking only integer values. The system is then

8.3 Regularity of stochastic functions and Orlicz spaces

309

assumed to satisfy X0 = 0, and, for s < t,Xt — Xs are independent of Xa. They are distributed as Poisson variables (a practical and standard assumption) with stationary independent increments and integer values. This means for any 0 < t\ < £2 < £3, Xt3 — Xt2 and Xt2 — Xtl are independent and P(Xt2 - Xtl =k} = e-*to-V[c(t2-*lW,

* = 0,1,2, • • • ,

(16)

where c > 0 is called the intensity parameter. Let Y be a symmetric Bernoulli random variable, so that P[Y = — 1] = | = P[Y = +1], describing a pair of telephones connected together. The incoming calls can then be described by Z(t] = Y(-l)Xt,

t>Q,

(17)

and it is reasonable to suppose that Y and the Xt,t > 0, are mutually independent. The problem then is to describe the process, by computing the moments, and obtaining the regularity properties of its sample paths. In fact, we show that Z(t) € LA(P) for all t,A(x) = e*(l) - l,x e M, and discuss the boundedness and other path properties of t \—> Z ( t ] , for almost all points of fi. First observe that E(Z(t)} = E(Y}E((-l}Xt] = 0, by the independence of Y and Xt using the fact that E(Y) = 0. Similarly, for s < t, E[Z(s)Z(t)\

E[Y2(-I)x*+Xt] E(Y2}E[(-I)Xt-x°]E[(-l)2X°]

= =

= I - £(-l)2*e- ' ^ E(-l)V^) - M^ _

c

k=0 -2c(t-s)

K

-

k=0

K

-

>

since the increments of the Xrprocess are independent. Thus for any 0 < t, s < oo, E[Z(s)Z(t)] = e-2c\t-s\. (18) So {Z(t),t > 0} has a covariance which is translation invariant, and such a process is called weakly stationary. On the other hand, the increments Z(t) — Z(s), s \7(+\ 7(o\ — nl r \Zj\i,) — ^\°) "J

— _

Y — oit- — nU, i1, • • •]1 •*P\Y L t — •'"•a ZK) K 00 (rc\f l _ goh2* -c\t-s\ \^ \ \ l/

h (2A;)!

VIII. Applications to Stochastic Analysis

310

= e~ c|f ~ s| cosh(c\t - s|) = a(s, t), (say]

(19)

and then P[Z(t) - Z(s) = b} = -[I - a(s, t)], b = 1, -1.

(20)

Since each Z(i) is bounded (the increments taking only 0, +1, —1), it follows that Z(t) € L°°(P) C L A (P) C U>(P) where A(z) = e*W - l and p > 1. Actually ||Z(t)||( A ) can be calculated as (let £ = Z'(t) — Z(s) for simplicity)

= i n f[L > o : [LAV( — ) + A V( -a / J a. J

2

+ A (0)a(s, t) < 1

= inf [a >0 : A (-) ( l - a ( s , < ) ) < l) I \aJ ~ } A -i

But in any interval (0,t), the sample paths are not continuous with positive probability since P[\Z(t) - Z(Q)\ > 0] = 1 - a(0,

,-ct

(22)

— e}\ > 0] —>• 0 because as although at every point to, P[|Z(to + z) — £ \ 0, a(0, e) -> 1, from (22). We now consider more general process than {Z(t),t > 0} to be in L A (P), using Theorem 1. It is interesting to note that, when the exponential A^function defined by Ap(u) = exp{|w|p} — l,p > 1, the corresponding Orlicz spaces L Ap (P) are closely related to the classical Lebesgue spaces. This fact allows us to discuss the regularity of stochastic functions with values in such spaces using some sharper tools through Besov-Orlicz spaces which are extensions of Orlicz-Sobolev spaces. We recall these classes and then present the desired results on stochastic processes. To analyze the smoothness of functions in L Ap (P), where A p (-) is as above, one considers the modulus of continuity of a function and adds it with the norm condition to isolate subsets of Orlicz spaces having these additional properties. [For the Sobolev norms, derivatives are included.] Thus for / £ L A "(P), where 0 = [0,1], E = Borel a-algebra and P : E -> [0,1], is a

8.3 Regularity of stochastic functions and Orlicz spaces

311

probability, the modulus of continuity function UAP(-, •) for / is derived from (A/JXz) = f(x + h}- f ( x ) , if 0 < x < 1 - h on [Q,t] C [0,1] as: "A, (/,*) = sup ||Afc/||(Ap).

(23)

0
This is modified by a scale factor Ua,p(f} — ta (log]-) ,t <E [0,1], to get a suitable norm of /; and we define the norm functional as: (24) W

<*,0V)

oo

Let

w Q ^,oo) = {/ e LA"(P) : ||/||(Ap),^i00 < oo}. Then the vector subspace jB(Ap,o;Q)^, oo) of L Ap (P) is seen to be a Banach space under the (stronger) norm (24), and is called the Besov- Orlicz space with values in 1R or Mn, n > 1. Unfortunately this space, just as the L°°(P), need not be separable for the norm (24). It may be noted that in (24) one can also use the equivalent Orlicz norm || • ||Ap, which may be given using only Ap without invoking the complementary TV-function of Ap (cf. Rao and Ren [1], p. 69, and p. 61). The above class of function spaces is of interest here, since the Brownian motion and, more generally, a large class of diffusion processes (i.e., those that are solutions of Ito-type stochastic differential equations) , take their values in these spaces as seen below. Actually their values will be in separable subspaces of Besov-Orlicz spaces. The following conditions "nearly characterize" the latter subspaces. These are obtained in terms of series expansions of / in B(A.p,u>a>p, oo) using a Schauder basis of the space. They are the indefinite integrals of Haar functions which are orthonormal, but the Schauder functions, although not orthogonal, are continuous and linearly independent which are convenient in this context. The idea is to expand each element / G P>(Ap,u;Q)/3, oo) in a (Fourier-Haar) series and consider the linear span of those whose nth term (or the "tail" part) tends to zero at a suitable rate, so that this space will be separable. The details follow. Recall that the Haar functions on [0, 1] are given as: hi = 1 and for j > 1 and 1 < k < 2j where Akj = [^r, £) are dyadic subintervals of [0, 1] (see Sec. VI. 3). The corresponding Schauder functions are the indefinite integrals of h^j. Thus 9k j '• t i—> ^hkj(u)du are obtained for 1 < k < 2 J , j > l,<7o = l,<7i(t) = t.

312

VIII. Applications to Stochastic Analysis

The orthonormalized set (by the Gram-Schmidt process) from g's is called "Franklin functions" which are often used in the "wavelet analysis" , but will not be needed here. With these g's one has the expansion of / as:

f(t] = fo9o(t) + fi9i(t) + j>0k=l

where /0 = /(0),/i = /(I) — /(O), and fjk are the coefficients, given recursively as: fjk

-

-

/ -

. . -

(25)

Now one defines J3°(AP, u;Q)/g, oo) as the subspace of those / G B(AP, w Qi/ g, oo) for which the fjk of (25) satisfy the condition (26)

and then one can prove that J3 0 (A p ,a; Q)j g,oo) is a separable Besov-Orlicz subspace. These properties are not obvious, but are discussed by Ciesielski et al [1], already referred to. We now present a result showing that not only the BM process, but several diffusions take values in 5°(A 2 ,u;2,2, oo) emphasizing the importance of Besov-Orlicz class. Here we follow Eddahbi and Ouknine [1]. There is also a two parameter extension of this result (see Boufoussi, Eddahbi and N'zi [1]). We remind the reader that a diffusion process is a solution of the ltd stochastic differential equation:

dXt = e(T(Xt)dBt + be(Xt)dtt

*€[0,1],

(27)

with an initial condition X0 = x e Md, where a : Md ->• Rd ® Ml and b : lRd —> ]Rd are matrix and vector functions which are bounded and satisfy a standed Lipschitz condition, so that (27) has a unique solution. Here b£ : Md -> Md is a sequence such that b£ -» b uniformly as e \ 0. Proposition 4. The solution process {X*(t),t € [0,1]} o/ (27), which exists with the initial condition X*(Q) — x, belongs to the Besov-Orlicz space B 0 (A2,cj2,2)Oo), with probability l,i.e., X* : [0, 1] x 17 —»• M and X*(-,(jj) e J3°(A 2 ,c<;2,2,oo) for almost all uj G D (relative to the P-measure).

8.4 Martingales and Orlicz spaces

313

The second paper above extends this result if [0, 1] is replaced by the square [0, 1] x [0, 1]. Here one has to give analogous conditions for the existence of a unique solution of the stochastic PDE replacing (27) and also the Besov-Orlicz space for which the modulus of continuity of functions of two variables is needed. The extension is not automatic, and the necessary detail will be found in the above cited paper. Actually in both these papers, the main aim is to present conditions for {X*(t),t > 0} to satisfy the LDP. For space reasons, we do not include these results here. The theory may be extended if the BM process {Bt,t > 0} in (27) is replaced by a suitable martingale process, and it will be an interesting research problem. We also include a result on LAp(/^)-boundedness of a general process or a field {X(t),t G T} taking values in this space. The following is from Weber [I]Theorem 5. Let (T,d) be a compact metric space and (T,T,fJ,) a Borel probability space. Let X : T x ft —> ]R be a mapping where (fi, E, P) is another probability triple, such that X is jointly n®P -measurable for J '<8>£. If d(-, •) is a metric on T, define do = diam(T). Suppose also that for some 1 < Q < q' < oo rd° / Jo

log 1 + -=fj,{s : d(s, t) < u}

du, t € T ,

(28)

where KQ is an absolute constant, and rrd0

-{v®v[(s,t}:Q
We refer the reader to the original paper for this interesting result. It has been further extended by Talagrand [1] for a large class of processes and TV-functions A. The reason for our inclusion of these results here is to exhibit deep connections between certain classes of (exponential) Orlicz spaces and important aspects of stochastic analysis.

8.4 Martingales and Orlicz spaces Let us introduce the martingale concept for vector and operator functions. Thus if X, y are Banach spaces, then B(X,y) denotes the Banach space of

314

VIII. Applications to Stochastic Analysis

bounded linear mappings from X into y under the operator (i.e., uniform) norm. If (Q, E, P) is a probability space and / : f2 —> B(X, y} is a mapping, we wish to define the conditional expectation of / suitably, then introduce the martingale concept, and present some (convergence) results explaining the role of Orlicz space techniques in the analysis. To discuss the convergence of vector valued martingales, it is necessary to recall conditional expectations of these general objects. Thus, if / : £1 —> B(X, y} is a function, it is termed strongly measurable if the vector function fx : fi —> y has the (strong) measurability property for each x G X, i.e., n

there exists a sequence gn(— g%] of simple functions gn = Y^aiXAi,Q 0 as n —>• oo. Then /n fxdP can be defined as a Bochner integral if Jn f x dP < oo. For simplicity this integral of a strongly measurable function will be called a strong integral here. Thus vx : A i—> fA fxdP defines a cr-additive function vx : E •—-> y, of finite variation given by |^ z |(fi) = f^ \\fx\\dP < oo. [The same norm symbol is used in both X, y, and if necessary, || • \\y will be written for the norm in y etc.] Since x is arbitrarily fixed, it suffices to consider X = M, [so B(X, y) — y] and let g : Q —> y be a Bochner integrable function. Then the existence and immediate properties of conditional expectation of g, relative to a a-subalgebra B C E are deduced from the following result. Theorem 1. Let (17, S, P) be a probability space, B C E a cr-algebra, y a Banach space and g : Q —> y a Bochner (or strongly) integrable function. Then there exists PB(— P B)-measurable unique strongly integrable g such that the mapping £B : g \—> g is a constant preserving contractive linear projection on L1^, E, P, y} into Ll(ft,B,PB,y)Moreover, if BQ C B C E are a-algebras and £B°, £B are the corresponding operators, then they commute, i.e., £B°£B=SB'£B° =£B°. n

Proof. Let / = ]C &iXAi, ai G 3;, Aj 6 E, disjoint, and B E B. Then 1=1

vf(B}

n

=

If fdP=y^(ii If XA-dP, by definition JB ~^ JB of the vector integral, n

— ^2 alvAi (B},

vAt'— v(Ai n •) : B—>• JR+ is a measure,

AtdPg, since \iAi « PB and g^ — RN — theorem, PB being finite,

, by the

8.4 Martingales and Orlicz spaces = /'(Ea*9At)dPB= JB

~

i=l

315

I fdPB,

(1)

JB

n

where / = £ cagAi '• ^ —* JR is ^-measurable and P& is uniquely defined. z=i The mapping EB : x/ij '—> fl'A; is unambiguous and we set g^i = EB(xAi}- It is clear that EB : L^h^P,^) —>• Ll(tt,B,PB,]R) is linear and positive. Let £B : / —>• /be defined as £B(f] = Y^ aiEB(xA-}- It is easily seen that 1=1 £B is linear on the simple functions of Ll(£l, £, P, y} into L1^, #, P#, y), and £ B (a • 1) = aEB(l) = a, a.e. [PB]. By the PB uniqueness of the RNderivative, it follows that EB(EB(h}} = EB(h] so that £B(£B(f}} = SB(f] for / e I/1(E,P, 3^) also, and SB is a projection. Moreover, with the basic properties of the Bochner (and Lebesgue) integrals, we get

Thus £B is a contraction on the simple functions of Z/ 1 (E, P, 3^) and since such functions are norm dense in this space, it follows that £B extends uniquely to all of L^S, P, y) with the same properties, i.e, is a linear contractive projection, and preserves the constants. Only the last property remains to be verified. To establish commutativity, let B0 C B C S be a cr-algebra. Then for any B € B0 C B, one has, if / G L X (E, P, y) ! £B°(f)dPBo

./B

= I fdP = I £B(f)dPB, 7fi

./B

since B e B,

= / f*(5 B (/))dP flb , since 0 for / > 0, and hence EB is

316

VIII. Applications to Stochastic Analysis

order preserving. Several properties of conditional operators EB, written for both the scalar and vector cases unless the distinction needs an emphasis, can be deduced, e.g., it is a linear contractive projection on Ll(E,P,y). A discussion of the vector conditional expectation is given in Diestel and Uhl [1] and also in the recent book by Dinculeanu [2] with applications. An easy extension of the above theorem for operator functions can be stated as follows. Corollary 2. Let f : Q -> B(X, y) be a strongly measurable and (strongly) integrable function on (Q, E, P] and Bi C S be o-algebras, i = 1,2. Then there exists a mapping £Bi : f i—> £ B i ( f ) , an essentially strongly measurable (and integrable) function for Bi, given by f fxdP = / £B>(f}xdPBi JA JA

(= f £B*(fx}dPBi] ,AeBi, xeX. \ JA /

(3)

The operator £Bi is linear (independent of x) and satisfies \\£*(f)x\\y<E*(\fx\\y),a.e.

(4)

Moreover, if BI C B2, then £Bl£B* = £B2£Bl = £Bl, i.e., is commutative, and £Bi is a projection operator. If X is separable, then \\f\ is measurable and (4) may be strengthened to a.e.,

(5)

where \ • \\ is the uniform norm of B(X,y}. The only new property is (5). This follows from the classical fact that, when X is separable, there is a denumerable norm-determining set of elements F so that sup{||/a:||y : x E F, \\x\ < 1} = \\f\\ and the left side being countable and each \fx\\ measurable, ||/|| is a measurable (E) function (even though / itself need not be measurable for E). Then \fx\\y < \\f\\\\x\\xwhich gives (5) on taking supremum on F. With these observations, the concept of an operator or vector martingale can be introduced as follows. Definition 3. Let Bn C Bn+\ C E be a sequence of <j-algebras from (0, E,P), a probability space, and /„ : ^ —> B(X,y] be a strongly Bnmeasurable strongly integrable operator sequence, denoted by {fmBn,n > 1}(C B(X, y)). It is called a strong operator martingale if £Bm(fn) — fm, a.e., in that for each x € X , £Bn (fn+ix) — fnx, a.e., setting x — I if X — JR. When X = H, it is a (strong) vector martingale sequence in y. If X = y = M, and EBm(fn] > ( < ) f m , it is a (scalar) sub (super)martingale.

8.4 Martingales and Orlicz spaces

317

We recall that a Banach space X is said to have the Radon-Nikodym property, if each vector measure v : E —> X, with a finite variation measure //, has the RN-derivative ^ which is Bochner integrable on (£),E,/x). It is known that all reflexive Banach spaces and those that are separable adjoint spaces have the RN-property. On the other hand, Ll[Q, 1] and CQ, or any Banach space which contains an isomorphic copy of either of these spaces do not have the RN-property. The following result can now be presented. Theorem 4. Let {Bn C Bn+\ C Ejf 3 be a-algebras of a probability space (17,S,P) and {fn,Bn,n > 1} be an X-valued martingale, fn : J7 —> X, where X has the RN-property. 7/sup£^(||/ n ||) < oo, then fn —> f^ a.e., as n -> oo,£(||/oo||) < lim.mfEfll/j'). In fact, if vn : A .—> fAfndPBn,A j

OO

<=

c

Bn,n > 1 and v(A] = n-Kx> lim vn(A), A e U Bn , then /oo = -jp a.e., where n=1

VQ is the P-continuous part of the a-additive component VQ of the additive v, exist by the (vector) Yosida-Hewitt decomposition and the RN-property of X. There are several proofs of this result. For instance, Diestel and Uhl [1], p. 130, or Rao [16], p. 190, and references to other sources given there. We therefore, omit the proof here. Using a Stone representation theorem of (fi, E, P) onto a nice topological space, the proof can be reduced to the case where v is a-additive. To assert that it is also mean (or norm) convergent, the hypothesis has to be strengthened which then brings in Orlicz spaces, via a classical result due to de la Vallee Poussin. We now present this and indicate its extension to operator martingales. Proposition 5. Let {/„, Bn, n > 1} be a martingale on (f2, E, P) with values in a Banach space X having the RN-property. Then the following statements are equivalent. oo (i) There exists an /^ which is measurable relative to B^ = a( (J Bn], n=l

such that { f n , Bn, I < n < 00} is an X-valued martingale. (ii) The set {fn,Bn,n > 1} is uniformly integrable, i.e., it is Ll(P, X}bounded and

uniformly in n, A 6 B^. (Hi) There is an N-function $ such that supE[$(\\fn\\)] < oo. n>l

(iv) There is an /^ : fi —> X such that lim ||/n — /oolli = 0.

318

VIII. Applications to Stochastic Analysis

(v) For the <$ of (Hi) and f^ : f] -> X, measurable for B^, and some (XQ > 1, one has lim E

= 0.

Proof. (!)•<=> (ii). We shall outline the argument which is standard, by reducing it to the classical scalar case. In this work, we note that by (4) ll/m|| = \\£Bm(fn)

<EB-(\\fn\\),a.e.,m
so that the sequence {||/ n ||,# n ,n > 1} is a positive submartingale. If there is an /oo as given, then it follows from definition that ^"(H/odD > ||/n|| and hence { \fn |,n > 1} is uniformly integrable. This implies that (i)=>(ii), easily. In the direct part no RN-property of X is used, but the latter is essential for the converse. Namely, if {/ n ,^ > 1} is uniformly integrable, then vn : A h-» JAfndP,A e Bn, is a-additive and has finite total variation. oo

Also v(A) = nlim vn(A),A G U &m shows that z/(-) is well-defined since ~>°°

n—l

vm = vn\Bm by the martingale property, and is additive. But the uniform integrability implies that v(-) is a-additive and then it has a unique extension oo

to BOO = a( U Bn). Moreover, v « P and by the assumed RN-property of 71=1

X5 /oo — ^p exists as a Bochner integrable function. It can then be verified that £ Bn (/oo) = fn, a.e., showing (ii)=>(i). (ii)^=>(iii). Since {fn,n > 1} is uniformly integrable, {||/n||,^ > 1} has the same property. By the classical de la Vallee Poussion theorem (cf., e.g., Rao and Ren [1], p.3), there exists an TV-function $ satisfying (iii), and by the same theorem if such a <E> satisfying (iii) exists, then the sequence {fn,n > 1} is uniformly integrable, which is (ii). (For this equivalence also the RN-property of X is not required, but for finding /^ it is needed.) (iii)=>(iv). This follows from its equivalence with (ii) and the RNproperty of X together with the Vitali convergence theorem. Similarly, that (iv)^=>(ii) is classical and has nothing to do with martingale theory. We shall sketch the detail that (iv)=^(v) and (v)=>(iii) to complete the proof. (iv)=^(v). If {fn,Bn,l < n < 00} is uniformly integrable and hence {||/n||,#n, 1 < n < °°} is seen to be a positive submartingale, whence uniformly integrable. So there is an TV-function $ such that sup£ l ($(||/ n | )) < n>l

oo. Moreover, {<£(||/n ),Bn,n > 1)} is also a submartingale because of the conditional Jensen inequality

8.4 Martingales and Orlicz spaces

319

since <&(•) is increasing. Hence 71—^OO lim £"($(||/n||)) = ^(^dl/oojl)) by the classical scalar submartingale convergence theory. It follows that for some aQ > 1, Wll/n - /ooll/ao), n > 1} is uniformly integrable and $(||/n - /oo||A*o) -> 0 a.e. So E($(\\fn - /oo||A*o)) -> 0, by Vitali's theorem. So (v) holds. (The a0 > I is used to ensure integrability since may not be in A 2 , as otherwise we can set a0 = I . Using the argument of [16], pp. 20-21, we can actually choose <$ e A 2 and take a0 = 1.) (v)=4>(iii). From the convexity of <$, one has

2a0 J ~ 2

V

a0

and so V

Letting $(x] = $ (2^) , £ > 0, it qualifies to be the TV-function of (iii), so that (v)=>(iii). D The above argument also establishes the following: Corollary 6. Let {/n, Bn, n > 1} be given by fn = £Bn(f] for some Bochner oo integrable f : £3 —> X, an arbitrary Banach space. If B^ = a( \J Bn) and n=l

/oo = £B°°(f), then fn —>• /oo a.e., and in Ll(P,X] on (ft,£,P).

The preceding result for vector martingales extends to strong operator martingales, which are useful for applications. As usual, the second adjoint of y is denoted y*\ Theorem 7. Let X, y be Banach spaces of which y has the RN-property. If (fi, E,P) is a probability space, fn : £1 —> B(X,y) is strongly measurable for Bn where Bn C Bn+i and {fn,Bn,n > 1} is a strong martingale, suppose that sup/ n ||/n:r||dP = Kx < oo for each x G X. Then n>\

there is a strongly measurable and strongly integrable f^ : £1 —> B(X, y**} such that lim \\fnx — fooX\\y — 0; a.e., for all x £ X. If moreover X is separable and s\ip{Kx : \\x\\ < 1} < oo, then \\fn — /oo|| is uniformly measurable [i.e, strongly measurable in the Banach space Z = B(X,y**}], lim \\fn — foo\\z — 0 a.e., even though / n ,/oo ar^ n°t necessary uniformly measurable. If X is separable, for the strong martingale {fn,Bn,n > 1} in B(X,y) the seven statements below are equivalent.

320

VIII. Applications to Stochastic Analysis n j ^ n j l < n < 00} fc's fl strong operator martingale for some (ii) The positive sequence {||/n||z,n > 1} is uniformly integrable. (Hi) There is an N -junction <£ such that sup E ( Q ( \ \ f n \ \ z ) ) < oo. n>l

(iv) There is an N -function $ and an a$ > I such that nlim

£?($(||/n -/ 00 ||/a 0 )) = 0.

(v) There is a strongly measurable /^ : fi —> B(X,y**} such that

(vi) There is a strongly measurable /^ : £7 —> B(X, y**} such that

(vii) There is a real integrable g : f] —> IR such that \\fn\\z < EBn(g), a.e., for n > I. Proof. By the martingale property of the /n sequence, the functions A

fnxdP, A£Bn, x^X

(6)

are of finite variation and z^+1|#n = ^n) Iknll(^) = /n ll/n^lly^-P — -^x < ooBy the RN-property of y and Theorem 4, there exists a strongly measurable fx such that fnx —> fx, a.e., and Jn ||/x||ydP < Kx < oo. The mapping x i—> fx(ui) is linear and bounded. So there is a strongly measurable /^ : Q —> B(X,y**} such that fx(u) — foo(u)x f°r almost all o> 6 f2. Taking /oo(^) = 0 on the null set, one gets ||/no; — /oo^lb(^) —^ 0 as n —)• oo for a.a. (w). It is to be shown that ||/n — /oo||y(') is measurable if X is separable. By the classical theory, if X is separable, there exists a countable dense set F C X so that fnXi —>• fooXi^a.e.^Xi G P. If NXi is the exceptional null set, NQ = U 7V X ., then P(NQ) = 0, and on (f2 — A'o), /n^ —>• /oo^ for each i>i x £ X. This implies, by separability (cf., e.g., Dinculeanu [1], p. 102), that the uniform norms ||/n||, ||/oo|| and ||/n — /ooll are measurable o;-functions. To see that |/n — /ooll —> 0, let e > 0 and for each n we can find x£n e F of unit norm, such that

8.4 Martingales and Orlicz spaces

321

Also for each u; G ^, one has lim ||/n(u;):r|| = 0 uniformly in n (cf. Dunford ||z||->0

and Schwartz [1], Theorem II.1.17) and that ||/ooM|| < liminf ||/n(o;)|| < Cw < oo. If T£ = {<} C T, then /nzf — »• /^f , i = 1, 2, "- ~n (fi - N0). Taking a diagonal sequence one concludes that ||(/n — /oo)^!!^ —> 0 on (ft - NQ). This and (7) imply on (fi - JV0) Jim ll/.- /coll <e.

(8)

Thus ||/n - /oo|| —>• 0, a.e., as n -> oo, although /„ and /^ need not be uniformly measurable. Assume X separable, so that ||/n||,n > 1 are measurable, one can use Vitali convergence theorem and extend the argument of Proposition 5 to establish properties (i)-(vii) of the theorem. Details will be omitted, n A consequence, corresponding to Corollary 6, can be stated as follows, which is recorded for some applications. Corollary 8. Let X be a separable and y be any Banach space. If Bn C Bn+i C S, of a probability space (fi, E, P), and f : fi —> B(X,y] is a strongly measurable and strongly integrable function, let fn = £Bn(f) be a strong operator, defined in Corollary 2. Then there is a strongly measurable /oo : 17 —> B(X,y**} such that fn —> /«> strongly a.e. Also \\fn - /oo|| is (uniformly) measurable and tends to zero a.e., as n —> oo. Moreover, there is an N -function $ and OLQ > 1 such that

Here one does not need to assume the RN-property of y, since the strong martingale {fn,Bn,n > 1} is uniformly integrable and /oo = £B°°(f] is the desired limit as in Corollary 6 with values in the larger space B(X,y**). For decreasing martingales, i.e., Bn D Bn+i and {/n = £Bn(fi),n > 1}, the uniform integrability is built in. So the preceding work slightly simplifies. oo However, the limit if (A) — n->oo lim L^ fndP, A 6 fl &n is a-additive, but its n=1

density is not guaranteed unless 3^ has the RN-property. Otherwise the procedure is the same, and we can make the following assertion. Corollary 9. Let {fmBn,n > 1} be a strong operator martingale on Q —> B(X,y], where X,y are Banach spaces with y having the RN-property. Then there is an /^ : Q —> B(X,y**) such that fn —> /oo strongly a.e., the limit f^ being strongly measurable and {/n, Bn, n > 1} uniformly strongly integrable.

322

VIII. Applications to Stochastic Analysis

We now give two applications of these results. The first one characterizes the strong convergence of a sequence of partial sums of independent stochastic processes, and the second establishes the structure of a sequence which is a martingale in both increasing and decreasing directions at the same time. The first result is related to the work by Walsh [1], and is as follows. Proposition 10. Let (Jl, S, P] and (T, J , //) be probability spaces with (T x fi, J ® S, p, <8> P) as their product. Suppose Xn : T x fi —>• IR, n > 1 are measurable functions such that {Xn(t,-),n > 1} are mutually independent, t 6 T, Xn(-,uj) € L^(IJL) for a Young function $ € A 2 n V 2 , /or almost all uj £ Q, and E(Xn(t, •)) = 0. Le£ XQ 6e in the unit ball of //*(// 0 P) and n

Sn(t) — 52 Xk(t) . Then the following statements are equivalent: fc=i (i) nlimJ|Sn(-,c*;) - X0(-,a;)||w = 0,a.a.(o;). (M) For each g e F C L^(^), where F zs norm determining for I/*(^), ^ 6em^ £/ie complementary function of $7 one /ms Jim^^, (X 0 -5 n )(-,a;)) =0,

a.a.(w),

where {-, •) denotes the duality pairing of (the reflexive) £*(//) and I/*(yu). When either of these equivalent conditions holds, then there is an Nfunction <J>i such that Sn(-,u)) —> XQ(-,UJ) in $1 -mean for a. a. (uj). Proof. Since by the Holder inequality \(g, (X0 - Sn)(-,uj))\ < 2\\g\\w • \\(X0 - 5 n )(-,w)|| ( *) — )• Q,a.a.(u), by (i), we obtain (ii) for all g G Z/*(/z). Conversely, let (ii) hold. By hypothesis E(Sn(t, •)) = 0 and hence by Fubini's theorem (and L*(n) C Ll(iJi)},E((g, Sn(-, •))) - (^, ^(5 n (-, •))) = 0. But by (ii), Sn —> XQ in L*(/i), and all quantities are integrable. Since L^(n} C LI(IJL) , again by Fubini's theorem Sn(t) —> X0(t) a.e., and since Sn(t) is a partial sum of independent centered random variables (so is a martingale) as well as the hypothesis that Xo(t, •) e I/*(P) C Ll(P), it follows from a classical scalar martingale convergence theorem (cf., e.g., Doob [l],p.337), that E(XQ(t}} = 0 - E(Sn(t)). Next let Bn = a{Xk(t),t 6 T, 1 < k < n}. Then {(g, Sn), Bn,n > 1} is a martingale, since
= (g,E*»( E Xk)) k-n+l

8.4 Martingales and Orlicz spaces

323

Xk)}

= (g,E( £ k=n+l

= 0.

Now EBn commutes with the //-integral, Bn is independent of Xk, k > n and so E(Xk) = 0 for all k. It follows that EBn(X0) = Sn, a.e. Thus {Sn, Bn, n > 1} is a uniformly integrable martingale with values in Ll(P,L* (//)). Then by Corollary 8, in which X = M, Sn —> 5^ a.e., and hence SQO = X0 a.e., from the preceding work. Moreover, under either of the conditions (i) or (ii), {Sn,Bn,n > 1} is a uniformly integrable martingale with values in the Orlicz space L^(fi). So by Corollary 8 again, there is an TV-function $1 and an a0 > 1 such that lim E($i(\\Sn - X0\\^)/Oi0)) = 0. This is the last statement. As already observed, here <3>i(the de la Vallee Poussin's function) can be chosen to satisfy a A2-condition (cf., e.g., Rao [16], p. 21) so that Q.Q = 1 is possible. Thus all the statements are established, D In the above result, if one can weaken the hypothesis to allow that <£ jumps to infinity at a finite point, so that L*(/J,) may be identified with L°°(n) and (i) of the proposition becomes uniform convergence of Sn(t) —> Xo(t) outside of a P-null set. Since L°°(p,) is isometrically isomorphic to the continuous function space C(T) on a compact (Stone) space, replacing T by T one may eliminate the P-null set. This stronger result with T as a compact metric space was established by Walsh [1] using direct methods. An interesting consequence of the latter is the following. If {£n,n > 1} is a sequence of independent standard normal random variables and {F n (-),n > 1} are Schauder functions in L2[0, 1] , then taking Sn(t) = ^nFn(t), one 00

has Xo(t) — 53 Sn(i), converging with probability 1, defining the Brownian n=l

motion, the convergence is uniform as well as in L 1 (P)-norm where $1 is an TV-function satisfying the A2-condition. This implies, on taking for the derivatives, F^(t) — sin27rnt and cos2?m£, the uniform convergence of the series 1 - cos 2?rnt

1 ^ sin 2,-imt

& + ^E-^

where {^ n >^n> n ^ 1} are mutually independent standard normal random variables. It includes a classical theorem of Paley and Wiener [1] who established it for a subsequence of the partial sums. Our proposition above does not apply to this situation, but may be extended to cover it. We now turn to the second application which is somewhat curious at first sight. Note that there is no limit operation here and no RN-theorem is

324

VIII. Applications to Stochastic Analysis

needed. It is as follows. Proposition 11. Let {Xn, Bn,l < n < n0} be a strongly measurable operator martingale on (J7, E, F), Xn : 0 —>• B(X, y) where X is a separable and y a reflexive Banach spaces. If the reversed sequence {Xno-i+i, Bi,l < i < HQ} is also a martingale, where Bn = v{Xno-i+i, 1 < i < n}, then X\ = X2 = • • • — Xno a.e., so that the sequence reduces to a single element. Proof. Since X is separable, {||Xn||, 13n, 1 < n < no} is a uniformly integrable submartingale. Hence by Theorem 7, there is an TV-function $1? which can be taken strictly convex, such that sup£'($i(||X n ||)) = £(i(||Jf no ||)) < oo. n

This implies for 1 < m < n < n 0 , by conditional Jensen's inequality )},a.e.,

(10)

with a strict inequality on a set of positive F-measure, unless ||/n|| = ||/m||, a.e. Since by hypothesis, the reverse sequence is also of the same type, the result of (10) with m and n interchanged also holds. So integrating and combining both statements, one gets, on using the identity E(EB(Y}} — E(Y] for any a-algebra B C £, and Y real integrable,

This is impossible (since ! is strictly convex) unless \\Xm\\ — \\Xn\\, a.e. for all 1 < m < n < n 0 , so that \\Xi\\ = \\X-2\\ = • • • = \\Xno\\, a.e. Let g = \\Xn\\ be the common value. If g = 0, there is nothing to prove. So eliminating the set where g = 0 (i.e., by restricting the measure space to its complement), we may assume g > 0 a.e. for the rest of this proof. Now define an = g~1Xn, so an: 0 -» B(X,y) and ||o;n|| = 1 a.e. for all n. Since g is BI-measurable, for m < n one has gam = Xm = £Bm(Xn] = g£Bm(an], a.e. But g > 0 a.e., so that {an, Bn, 1 < n < n0} is an operator martingale, and by a similar argument it is also a reversed martingale. We now assert that an does not depend on n. Suppose the contrary and am ^ an on a set of positive measure. Then there exist x G X and y* € y* such that y*(an — am)x / 0 on a set of positive measure. Also {y*anx, I < n < n0] is a uniformly bounded set of random variables. Thus for m < n 0 < / \y*(an - am)x\2dP

8.4 Martingales and Orlicz spaces -

y*(anx)EB»(y*(amx}}dP +

325 (y*(amx)]2dP:

(12)

using the fact (established above) that the an sequence is both a martingale and a reversed martingale. But then the right side of (12) is zero and we have a contradiction. This proves that y*(am - an}x = 0 a.e. for all y* e y* and x G X. Since X is also separable, this shows that the measurable function \\&m — &n\\ — 0 a.e. for all m, n. Hence oti = a<2 = • = ano, and then Xi = X-i = • • • = Xno a.e. as asserted. D If X = y — scalars, the above result was established differently in Doob ([l],p.314). If the martingale of the proposition is square integrable, at least when X = y =scalars, then (12) shows that the assertion of the proposition is immediate, but the general case is clearly more involved. Bibliographical notes. Applications in many parts of mathematics, some times not only use important results from Orlicz space analysis, they also show the essential need for extensions of the latter theory. This is well exemplified by the applications in analysis of large deviations treated in Section 1. They show the necessity of the use of Young (and N-) functions on multidimensional spaces. The complementary TV-function is called a rate (or an action) function in this theory. One appreciates the subject better, if the work on large deviations is seen from the underlying Orlicz space structure as well. A detailed account of large deviations is included to explain the importance of this application, in our context. We have already given references in the text. Here one has to reiterate that the important work of Cramer [1] and its deep generalization and expansion in the hands of Donsker and Varadhan in a series of fundamental papers have shaped this subject. Several monographs, after Varadhan's [2], have detailed the work. These include Dembo and Zeitouni [1] , Deuschel and Stroock [1], Dupuis and Ellis [1], Ellis [1] , and equally for the continuous parameter processes Freidlin and Wentzell [1]. The convex analysis of Rockafeller [1] is an indispensable companion in the study of Young functions on JRn, n > 1. Since we need to consider this multidimensional case as a natural problem, and thus the section has grown longer. An application of large deviations to processes, as in the work of Schilder [1] included already, shows that one needs Young functions defined actually on infinite dimensional vector spaces. Some of it can be reduced to finite dimensions, using the projective limit theory of measure spaces on spaces such as JRn. This and the direct analysis of Orlicz, or what are properly called Fenchel-Orlicz, spaces is discussed in Section 2. The work of Turett's [1], and the earlier work of Uhl's [1] are relevant in this situation. But much

326

VIII. Applications to Stochastic Analysis

remains to be done from a functional analysis point of view. We thus see several problems that are natural in this context, motivating further research. From another angle, notice that the less frequently used classical Orlicz space L^(fj,) when A grows exponentially has a deep interest in analyzing the regularity of stochastic functions. We showed some of this in Section 3. A detailed account of many more such regularity problems for processes was included in Buldygin and Kozachenko [1]. An interesting aspect of these Orlicz spaces, defined by exponential 7V-functions, is that they have a close relation to Besov-Orlicz spaces, and we indicated it in Section 3. The work gives a good motivation why one should study these, somewhat special spaces. It was noted there that the boundedness of sample paths and the large deviation of random processes and fields as solution of diffusion equations actually need Besov-Orlicz spaces. They are related to Orlicz-Sobolev spaces (cf., also Adams [1]), p.218 ff.) of interest in PDE analysis, to be discussed in the next chapter. From another view the results of Orlicz spaces related to uniformly integrable sets, or equivalently bounded sets in L®(fj,) when $ is an /^-function, are of interest in martingale analysis. We presented it for a general operator valued case, since it reveals the structure of these processes better without invoking too many special results. The basic vector measure theory, as found in Diestel and Uhl [1], and Dinculeanu ([1], [2]), is used at some places. The introduction of conditional expectation of operators is not difficult, and we included the discussion in its natural generality. The treatment in Section 4 follows a streamlined account from Rao [8]. Our aim in this chapter has been to show how substantive problems in various parts of analysis (here stochastic processes ) lead to a better appreciation of Orlicz space theory. We continue this point of view with PDE and related nonlinear problems in considerable detail in the following chapter.

Chapter IX Nonlinear PDEs and Orlicz Spaces

This chapter contains an outline of applications to PDEs with the ideas and flexibility afforded by Orlicz spaces as well as their differential structure. Several classical results admit satisfactory extensions, often giving optimal results, in the context of Orlicz spaces. These include removable singularities, comparison principles, embedding theorems utilizing the structure of Orlicz-Sobolev spaces as well as some inequalities of the Hardy-Littlewood type useful in those spaces. We also consider continuity of certain nonlinear operators on Orlicz spaces that are of interest in composition and Nemitsky type operations. Although explicit solutions of PDEs are difficult, we indicate such a possibility through stochastic differential equations. The areas opened up for this treatment are many, and we show how these function spaces expand possibilities in applications.

9.1 Orlicz-Sobolev spaces for PDEs One of the important problems in the classical theory of partial differential equations (PDEs) is solving the Dirichlet's equation which is written as !/(/) = g, where L is an elliptic (partial) differential operator given by

L(f) = E (-l)lalDa(aa\Daf\?«-2Daf),

(I)

M>o

where aa > 0, pa > 1 and (the other symbols will be defined presently) a boundary condition is Daf \dG — 0 for 0 < |a| < ra < oo. The various symbols and G are specified as follows. Now the domain G C Mn is an open connected set, and /, g are functions belonging to certain classes B of measurable or smooth functions on G. Also if a = (a\, • • • ,o:n), on > 0, /5 = (/?!,••• ,/? n ), fa > 0, are integers, then |a| = a\ + • • • + an, and if x = ( x i , - - - , X n ) , then D{ = /-, £>? =identity, so Da = D^ • • • D%n,

327

328

IX. Nonlinear PDEs and Orlicz Spaces

(a + /?) = (ai + p\, • • • , cnn + /3n), a! = a.i\ • • • anl, a < /3 iff cti < /% and /} i — i /, i

P /

V Pi /

i o

, the product of binomial coefficients. Here Daf is

V Pn

an a:*'1 weaA; (or generalized) partial derivative defined as

I fD*hdn=(-l}\a\ I gahd^ JG JG

(2)

for all infinitely differentiable functions h on G with compact supports (/i G CZ°(G}), if a <7Q satisfying (2) exists. Indeed if / is actually continuously differentiable a times, then (2) holds by an integration by parts formula; and is taken as a definition in the general case (dG is the boundary of G). With this notation, (1) is a (nonlinear) PDE, and it is desired to solve for a weakly differentiable /. Thus a natural and simple class to consider (as motivation) is the set of all /: G -> M, having generalized partial derivatives of order m, such that ||/||m,p < oo where

E|a|<ml|£°7IIJ]', I < P < O O

a e I II r^n sup Q < m \\D f U,

p = oo.

,.v

v /

The desired space then is denoted Wm>p(G) = {/ : G ->• jR, ||/||m,p < oo}. One can verify that {I4/m>p(G), || • ||m,p} is a Banach space and search for functions / 6 Wm'p(G) that solve the corresponding equation (1). Such spaces, incorporating the smoothness (or differentiability) properties for norm (3), were constructed and analyzed by S. L. Sobolev in [1], now called Sobolev spaces, and they play a key role in this analysis. For a finer analysis, it is necessary to embed these spaces (continuously) into LP(G) with Lebesgue measure, denoted Wm'p(G) c->- LP(G}. The following comprehensive result was established early by Sobolev for a suitable class of domains G C JRn. Theorem 1. Let G C JRn have the following: the cone property, i.e., at each x G G there is a finite cone Cx C G congruent to a fixed right spherical cone C so that each Cx is obtained by a rigid motion ofC. If Rk is a k-dimensional hyperplane (I < k < n), let Gk = G D Rk- Then for all integers 0 < m < n, and real numbers 1 < p < oo, one has (i) mp < n => Wm'p(G) ^ Lq(G], p < q < np(n - mp)'1, and m p W > (G) ^ Lq(Gk), n - mp < k < n, and p < q < kp(n - mp)"1 with k = n — m when p = I ; (ii) mp = n=> Wm'p(G) «->• Lq(Gk], I < k < n, p < q < oo and if p = 1, then q = oo is also possible. This result needs a detailed analysis for various cases, breaking up the argument into different steps and to consider all possibilities. It is the central

9.1 Orlicz-Sobolev spaces for PDEs

329

idea running through the whole methodology. A proof of the result may be found in Adams ([1], p. 99). It is possible to consider an extended result for Orlicz spaces, and taking the above as a basis and a motivation we introduce the desired function spaces, naturally called the Orlicz-Sobolev spaces, and establish results for this general (and flexible) class. [We observe that a ball in lRn with center deleted does not have the cone property, but a finite union of star shaped regions has that property.] First we present a classical result to show how the Sobolev spaces help in solving PDEs. Consider a special case of (1), where the operator is L=

D*(Caf>Dfi)

£

(4)

|a|,|/3|<m

and where the coefficient functions Cap are defined in a bounded domain G C Mn and are infinitely differentiate. If g G Lfoc(G,n), p, being the Lebesgue measure (also sometimes written dn(x] as dx), then / e Lf oc (G, //) is termed a weak solution of (*) L(f) — g, if for each h e C%°(G), one has (/, L*(h}} = (g, h) holds where {•, •) is the inner product in L2(G,n) and L* is the formal adjoint of L is given by

L*=

£ (-l)+WDfi(Ca()Da).

(5)

\a\,\P\<m

The existence of such a solution in the Sobolev space is the content of the following classical theorem due to Friedrichs. Theorem 2. A solution of the equation, L(f) = g, has square integrable distribution (or weak) derivatives of order 2n + p in a relatively compact open subset G\ of G (written G\ CC G) if g has similar weak derivatives of order p. This means f e Wp+2n(Gi). If p = +00, then there exists an /o e C°°(Gi} such that f = fQ a.e. on GI. Taking Cap as constants, one sees the importance of Sobolev spaces for the PDE analysis from the start. Again in (5) if m < oo, one has the equation of finite order, and if m = oo, it is of infinite order. A proof of Theorem 2 is not simple, and may be found in Yosida ([1], p. 178). This result also explains why Sobolev spaces are so important in PDE studies. Before proceeding further, let us illustrate this for a very special case, within the Hilbert space setting so that the generality in the Orlicz-Sobolev space result included later is better appreciated. We first consider a linear PDE of second order, also called a linear Cauchy equation:

L(f] = £ aij(x)DXiXj(f)

+ Y/bi(x)DXi(f)

+ c(x)f = g,

(6)

330

IX. Nonlinear PDEs and Orlicz Spaces

and the coefficient matrix A(x) = [aij(x)] is assumed to be real symmetric, DXi — -j^r. and DXiXj(f) = DXj(DXi(f}} = dx.gx. = fXiXj, (by the continuity of mixed partial derivatives), for the equation to be solved in a region G C Mn. Suppose that at a point XQ E G, the eigenvalues of A(XQ) are \I(XQ), • • • , Xn(xQ) which are such that n+(xQ) = the number of \i(x0) > 0, n_(x 0 ) = the number of Xl(x0) < 0, and n0(x0) = the number of \(XQ) = 0, so that n = n+ + n_ + HQ. Then (6) is termed elliptic at XQ if n — HQ or n — n_; it is hyperbolic if n0 = n — 1, n_ = 1 or n+ = 1, n_ = n — 1; (and ultra-hyperbolic if no = 0 and 1 < n+ < n — 1); and parabolic in case no > 0, (n± > 0). If these conditions hold at all points of G, then they are so termed on the region G itself. If 50 C G is a "smooth" surface of (n — 1)dimensions (e.g., SQ could be the boundary of G} and x0 E 50, then one is given the initial value f\s0 = /o,

d!_ dl where V/ is the gradient of /, i.e., V/ = (D Z l (/), • • • , DXn(f}}, and the vector l(x] = (/i, • • • , l n ) ( x ) , x E 5o, ||/|| > 0, that is not tangent to SQ. Under these conditions one has the classical Kovalevskaya theorem asserting that if the initial values in SQ are analytic then there exists a unique analytic solution of (6) in a neighborhood of XQ. To make this more concrete, we analyze a particular case of (6), writing C*(G) = {Dlf e C(G) : / E C(G) = Proposition 3. Let G C JRn be an open connected subset and a G C(G), b e Cl(G], a > 0, 6 > b0 > 0 onG. Consider the (elliptic) PDE L(f] = b^fXlXl 1=1

+f^bxJXi 1=1

-af = g, f\9G = 0, fx. - Dx.f,

(7)

where g e L2(G}. Then there exists a unique generalized solution f satisfying (7) and for an absolute constant C depending on G (but not on g) one has the bound: (8)

where the norm on the left is given by (3) with m = 1 and p — 2. Proof. The argument here (and in all such cases) depends on an application of the integration by parts formula, and the Riesz representation theorem of continuous linear functionals on a Hilbert space (or a function space

9.1 Orlicz-Sobolev spaces for PDEs

331

where such a theorem is available). Thus to use the concept of a generalized solution, let h be a G£°-function (i.e., with compact support in G), and multiplying both sides of (7) by h and integrating by parts on the left, one rrpf'c gt;t&

/ (aV/ • Vh + bfh)(x)dp,(x) JG

+ I (gh}(x}d^(x] = 0, JG

(9)

where we used Vf • Vh for the dot product of the gradients of / and h, and the fact that f\QG = 0. Since a > 0 and b > bQ > 0, we have a scalar product in Wl*(G), given by
(6V/ • Vh + afh)(x)dp(x\

(10)

JG

and using (g, h)^^) for the usual inner product of L 2 (G), (9) becomes (/, h)w^(G) = -(g, h)L>(G), h E CC°°(G).

(11)

By the CBS-inequality in L 2 (G), (11) implies {/, h)wi,2(G) < \\g\\L*(G)\\h\\L*(G) < \\9\\L*(G) • C\\h\\wi,,(G),

(12)

by (3) with m = 1 and p = 2 (D°f — f here) for some constant C > 0, independent of /, h. Since (f,-)w1<2(G) ls linear and C%°(G} is dense in W1'2(G), it follows from (12) that (/, •)w1<2(G) 1S continuous and has a unique extension to all of W1'2(G). Then by the Riesz representation theorem, there is a unique A £ (W 1>2(G}}* = W1'2(G) such that

This is the desired result. D Remark. The argument and result, which follow Mikhailov ([1], Chapter IV), both extend to Wm>p(G) if we consider C£°(G) C Lq(G] as norm determining for the latter and {•, •) is the duality pairing. It will also extend to Orlicz-Sobolev and Orlicz spaces if Wm'p(G) and Lq(G] are replaced by Wm'®(G) and L*(G) where ^ is the complementary Young function to (e A 2 ), recalling the definition of Wm^(G}. This can be done more generally as follows (cf. Rao and Ren [1], p.379). Definition 4. If (#, $) are Young functions and G C MH is an open set, then an Orlicz-Sobolev space W™p (G), with weights /3(a) > 0, is given by W%*(G) = {f € L $ (G, M) : N?f(f)

< oo},

(13)

332

IX. Nonlinear PDEs and Orlicz Spaces

where the gauge norm in this space is defined as

J V T C / ) - mf k > 0 : £ 0 |a|<m

^

£(«) < 1 V

/

(14)

J

Write Wm*(G) if 9(u) = \u\ and /3 = 1. When $(u) = up, p > 1, 0(u) = \u\ and £(a) - 1, (14) reduces to (3) and (13) gives Wm>p(G). The following result (Theorem 7 below) on the structure of this space will be of interest here, as it completes and complements the Sobolev embedding theorem (i.e., Theorem 1). Relations (8) indicates the importance of embedding operations in the analysis of PDEs. On the other hand in Sobolev's embedding Wm'p(G) c-> Lq(G), p < q < oo when mp — n, and p > 1 but q = oo is not admitted, so that the target space (on the right side) in the Lebesgue classes is not the best or smallest one. However the situation changes if we consider the more extensive Orlicz classes, and then a best target space can be found. We now present a result that extends Theorem 1, and brings us to our class of spaces. Recall that a Banach space X is embedded in another Banach space y, written X <-» y, if there is an absolute constant C > 0 such that \\x\\y < C \\x\\x, x e X, where x is the image of x in y. This is evidently not possible for all Banach spaces. To see this clearly, let us specialize if X — Wl'p(G) and y = L"(G), where G C JRn. Identifying / and / with / and Df(= V/), we must have ||/| q < C\\Df\\p for all / G Wl'p(G) and some absolute constant C > 0. For simplicity let G = JRn, and compute the norms, if fa(x] = f ( a x ] , x € JRn,a > 0 with / 6 C%°(G) where ax = (axi, • • -axn}. Since p, is Lebesgue measure in G, dp,(ax) = andfj,(x), we have

and similarly \Dfa\\p= \ap-n I L

^G

Hence we must have for any a > 0 \f\\9
(15)

and this can not hold unless - + 1 — - = 0 o r g = ^-. Thus a necessary condition for a continuous embedding of W1:P(MH) into Lq(JRH) is that the

9.1 Orlicz-Sobolev spaces for PDEs

333

relations 1 < p < n and q = p* = ^- hold. It is interesting that these relations are also sufficient for the embedding. More precisely the following result known as the Gagliardo-Nirenberg-Sobolev inequality holds. Namely for each 1 < p < n there is an absolute constant C(= Cp;n) > 0 such that for all / e C™(Mn) one has H/llp. < C||Z?/||P. (16) This p* — ^- is called the Sobolev conjugate of p and p* > p. The proof is a careful use of the Holder inequality for l/)^ 1 and the components of the gradient D f , first for p = I and then extending the result for 1 < p < n. For details, one may consult Evans ([1], p.263), where extensions and applications of (16) are also found. Here we discuss how this may be extended to OrliczSobolev spaces. It is noted that p* is not the conjugate exponent p' of p (i.e., p* = -^- ^ p' = -^j), and depends on n also. Since p* > 1, we can generalize the relation between the convex functions u i-> \u\p and x \-> \x\p* or their inverses as

1 rx _ L _ I ,

j.

l rx UP

XP* — — / UP* du = — / p* Jo p* Jo

n

du

1 rx UP I fx 3>~l(u) = /M* — // j-du = — / T—du. 1 -I- — in* / 1 -I- —

(17)

Let <E>*(:r) = (p*)~ p *|z| p *- Then (17) can also be expressed as -i

fx ^ -1 (w)

Jo

u1+"

We now take this as a definition of the general concept if $ is a continuous Young function with <3>(w) = 0 iff u = 0, so that $-1 is a continuous concave increasing function on [0, oo). Then $~l is also a continuous concave function since for any 0 < a < 1 we have $„ (ax) = aJQ —^dv>a

- ^ —^-dv>a^ (x}

(19)

for x > 0. However if $ is an N-function we can sharpen this statement as follows. Proposition 5. Let $ be an N-function such that <£* given by (18) satisfies $~1(1) < oo and $~1(oo) = oo. Then $* is also an N-function that is strictly convex in addition.

334

IX. Nonlinear PDEs and Orlicz Spaces

Proof. Note that t is strictly decreasing from +00 to 0 as t increases from 0 to +00, because is an N-function so that ~ and £" are strictly decreasing from +00 to 0. Thus for 0 < v\ < v2 one has from (18) vi

o /o

2 Jvi

1 / fvi =

- I /

2 Uo

2 J^4p

rv2\

$-l(t)

+ /

T^rf*

7o / t 1+ n

implying that <£~ 1 is strictly concave, or <£* is strictly convex. It remains to verify that <£* is an N-function, i.e., <3> (n\

<J> (n\

(a) lim ^± = 0, and (b) lim ^^ - oo. u->0

u

w->oo

u

(20) '

By definition of <&* in (18), we have

,.

lim

^(v)v *

,.

Irv3>-l(t)t

,

($-l(v)\fl\

' = lim - / -y-dt = lim -^

— = +00,

where we can invoke the mean-value theorem for integrals and the fact that for an N-function <£, lim v ^ 0 (^ -1 (f)/^) = +00. Thus (a) of (20) is true. Regarding (b) of (20), since we saw already that $ -1 (t)/t 1+ n is strictly decreasing to zero, it follows that (with the mean-value theorem again) lim

v—foo

Thus (b) and hence (20) follow. By the fact that ^(l) < oo and ^(oo) = oo, it is immediate that <£*(0) = 0 and *(oo) = oo. This shows that $* is a strictly convex N-function. Q Following the case that $(t) = \t\p, p > 1, $*(x) = (p*}~p*\x\p* where -^r = - — ^ > 0, we call <£* of (18) the (generalized) Sobolev conjugate of t> in the context of Orlicz-Sobolev spaces. We can now define higher order

9.1 Orlicz-Sobolev spaces for PDEs

335

conjugates if 3> is replaced by <3>* in (18). Indeed, given an N-function $, we let $0 = $, $1 = $*, and for A: > 1, define inductively the sequence $ fc , k > 1 by letting $k+i = ($fc)*> i-e., $fc+i is the inverse of (assuming $fc(*) < oo for |t| < oo)

'0

t^n

Since all the $k depend on n, there exists a smallest J > 1 such that

r

7o

< oo

(22)

and J(= J$) < n (so $j+i is not an N-function any longer!). Just as the importance of simple functions in the structural analysis of the L?([i) spaces, the set of functions Cm(G] in the Orlicz-Sobolev spaces (see below), play a special role. Thus using the notation of Definition 4, o

let W™f (G) be the closed linear span of CC°°(G) in the norm (14) and let H™j^(G} be the similar span of Cm(G] in the same norm, where Cm(G] is the space of / : G —> JR with Daf continuous for |o;| < m. Also let L$(G} be the subspace of L®(G) determined by C(G)(= C°(G)). The following structural result for Orlicz-Sobolev spaces was established in our earlier volume and will be stated for use in proofs below (cf. Rao and Ren [1], Theorem 3, p.382) and it extends an earlier result of N.G. Meyers and J. Serrin for the classical Sobolev spaces. Theorem 6. The set (W™/(G], N^/(-}) is a Banach space and H^/(G] is a closed subspace of W™p (G). Moreover if $ € A 2 (globally if G is o

noncompact, and at infinity if G is compact), then H™j^(G] =W™if' (G} = W$*(G). This result will be needed in establishing a generalization of Theorem 1 to the Orlicz-Sobolev spaces to be presented below. It is due to Donaldson and Trudinger [1] for bounded domains, and an extension for the unbounded case is due to Adams [1]. We consider 0 = 1 and 6(x] = \x\ in what follows. Theorem 7. Let G C JRn be a domain with the cone property and G^ = G fl Rk where R^ is a k-dimensional hyperplane, 1 < k < n. Let $ be an N-function, if $0 = $, $1 = $*; and for k > 1, $k+i = ($k)* with a J > 1 satisfying (21), (22) and ff° ^l(t)r(l+^dt = oo, k < J, but /i°° $^l(t)t~(l+^dt < oo. Then we have the embeddings:

IX. Nonlinear PDEs and Orlicz Spaces

336

(a) Wm>*(G)

and in

fact> tt

is true that

J. (b) If, moreover, JQ ^^_l(t)t^^l+^dt < oo and there is a 1 < p < n such that 3>(t) = <£ m _i(£p) defines an N-function, 1 < m < J, and n — p < r < n or_p = 1 andn-l < r LJ(Gr) and Wm^(G] ^ L®(Gr), where § = ($m)" is an N-function. Here we include details for the main result Part (a) of the theorem and explain how the rest is obtained by extending the argument. Moreover, we concentrate on the case m = 1 since the general result then follows by induction. First we derive some useful properties of the N-function $ or $* (the Sobolev conjugate of <£), to utilize it later in the proof. Lemma 8. Let $ be an N-function with <$* as its (generalized) conjugate, and suppose that (a) Q ~ l ( I ) < oo, ^^(oo) = oo, and (b) for some I < p < oo, $ : t H-> $(t?) is again an N-function. If p* — ^-, the classical Sobolev conjugate of p, then we have >£

(i) for any A > 1, $** is again an N-function (so in this special class, taking A = ^£- > 1, we get <£* to be an N-function, but proved generally in Proposition 5 for all N-functions); (ii) [$*(t)]$ < (p*)-r$(t) for allO^(1); (in) for any e > 0, there is a constant K£ > 0 such that

Proof, (i) By definition, <$(t) = $(£?) = ?/ so that ^ > ~ 1 (?/) = t = [
—•

1

and

dy

£_ yXp

where

Xnp

\p\p

n)

^ (23)

9.1 Orlicz-Sobolev spaces for PDEs

337

since \p > 1. By hypothesis
= oo 1

— 0. Further if 0 < s < t, the concavity of I*" implies and li l that $(s) = $>~ (\t} > -t$~l(t), and hence (23) gives

L

(Q-i)'(t)

Consequently (Q"1)'^) is positive and decreases monotonically from oo to 0 as t increases from 0 to oo. Thus Q is an N-function and (i) is proved. It may be noted that Q~l(y] can be represented in the form-

where g(y) denotes the right side of (23). Then, similar to the proof of Proposition 5, we can show that for 0 < v\ < v-2

and

,. lim

w->0

u

= n0, rlim

— oo.

i.e., Q is a strictly convex N-function. (ii) Let 0 < t < 1. Since p* = ^ > 1 for 1 < p < n, one has tn < 1 < (p*)P or by J = 1 - £, this becomes tl~£ < (p*)p so that t<

Hence by definition of $*, and $ being an N-function, one has

•iw' V

=

r ^M ^=

Jo ul+^

o

* ul

or

u

>

—

du

338

IX. Nonlinear PDEs and Orlicz Spaces

Let s — $~ 1 (i), then t — 4>*(s) and (ii) follows from the above. (iii) This is a consequence of (i) and (ii). In fact, let

so that g ( t ) / h ( t ) = [$*(*)]" -> oo as i -» oo by (i), and -> 0 as t ->• 0. Hence given e > 0 there is a tQ > 0 such that [g(t)/h(t)] > ^ for t > tQ. Thus £$*(t) > [$*(*)]**" for t > tQ. On the other hand, by (ii) h(t) is bounded on [0, to], say by a constant Ke, so that [$*(£)] p * < £$*(t) + K£$(t) for all t > 0, giving (iii). D We now proceed to the Proof of Theorem 7. The argument is presented in steps mainly detailing Part (a), and outlining the second part. We again denote d/j,(x) by dx. (I) First let 0 ^ f € Wl'*(G) be bounded so that / e^1'* (G) and also /Q &~l(t)t~(l+n"ldt < oo. Plence by definition of the gauge norm, one has Jo ' ll/Ik*., We need to show that there is an absolute constant K > 0 such that

!/!(*.)< Will,* So let u0 = max(||/ | w , ||/

(<M ),

p = 1 so that p* = ^_, V(«) = [$.(«)]£

and ^(x) = F(^^). Then ^ is bounded and since / eW1*1 (G), it has bounded support (or supp(/) is compact), with V(-) satisfying a Lipschitz condition on the (bounded) range of f\, and DjV( $\}(x) = (V'(\f\)Djf)(x) by the chain rule for (weak) derivatives, since D j V ( \ f \ ) — V (|/|)sgn(/) Djf. But G has the cone property, so by the classical Sobolev embedding (Theorem 1), there is a KI > 0 (not depending on /) such that \\g |p. < K\ \g\ i^. Using this and the fact that, for bounded /, JG $»(^^-)rfx is a continuous function of A;, we have a uniquely defined

*.=[/

[Jo

such that

Ko = \\9\\r

U0

>0

9.1 Orlicz-Sobolev spaces for PDEs

339

'(—)II(*)I|JVII(*) + Ki[v (^} dx, UQ JG \ UQ I

(24)

by Holder's inequality for Orlicz spaces, \& being the complementary function of <E>. To simplify each term of (24), let s = $*(t) and using the definition of (^(s), and differentiating t = $*l(s] implicitly, one has, from s1+« = $~l(s}%; and 5 < $~ 1 (s)*~ 1 (s), that % < sn^~l(s) or

<

dt

This gives

since -r = 1 — -. Note that the gauge norm is monotonic (i.e., |/i| < \fa\ a.e =^ ll/i||(*) < 11/21 !(*))> and then (25) implies Ifl 1 , Ifl ||^'(—)!!(*) < —11^ [^*(—)]||(*)UQ

p*

UQ

(26)

On the other hand, UQ > ||/||($.) implies that (using the modular py] I/IM1 I

I f

F

> = /

/

J

J\)

3>

*

G

(\f(x)\\

I -—-—— I /IT
l

/

\

UQ J

f /

— /

^

JG

( \f(x)\

I —•—-———

* I II f\\

Vll/ll(*.)

showing ||^ f " 1 [^*(^)]ll(*) ^ 1- (The above also shows that K0 < 1 in (24)). Thus by (26) we have i/i i ;•

(27)

For the last integral in (24), taking e = ^- and p = 1 in Lemma 8(iii), one can find a K£ > 0 such that

G

\ UQ

<

G

\ u0

G \ uQ

* n/(^)h . (28)

340

IX. Nonlinear PDEs and Orlicz Spaces

since ||/||($) < u 0 , KQ < 1 and p* > 1. From (24), (27) and (28), it follows that 2nKl KO KIK£ \f\\ 0 < <1 /111,*++Z~7T+^ UQ -- \\f\\l l / ll(*)» UQ

K K

and hence itfouo < 2nKl\\f\\lt9 + KiKe\\f

w

< (2nKl + tf)||/||i,*.

Letting K3 = 2(2nKi 4- K)/K0 which depends only on n, <£>, and the cone property for the region G and its measure, we get ||/||($») < u0 < ^3||/||i,$, o

giving (a) in this (bounded) case. But such functions / are dense in Wl>® (G) and KS does not depend on /. So the result follows by continuity on all of o

Wl>* (G}. It remains to extend the result to Wl>*(G). Let 0 < / G W 1)$ (G). Then there exists a sequence of functions /; which are bounded, nonnegative with bounded continuous derivatives such that fi eW1'* (G) and fa /* / a.e., since Clc is norm determining for Wl>*(G) from the standard structure theory of L**(G). By the Fatou property of norms on W 1>$ (G) and on L**(G), together with the fact that the Sobolev inequality | /i||($») < ^3||/i||i,$ holds, for an absolute constant K% > 0, the result is valid for / on letting i —> oo. So Part (a) holds when m — 1, since the general case is easy. (II) We now consider the condition that /g1 <£~ 1 (t)t~( 1+ ")d£ = oo. In this case $* is denned slightly differently, so that it again satisfies the hypothesis of Step (I). Then we choose l < r < n ( o r p < r < n i n the general case), and observe that there exists a tQ < $~l[^$(l)] such that 3>(t0) < KtrQ, where K is chosen to satisfy the equation X{$~1[|$(l)]}r = 5^(1). This is because in the opposite case, $(t) > Ktr for all t near zero, so that < £~ 1 (£) < ( ^ ) ™ for 0 < t < 6 for a 6 > 0. But then /Q $~ 1 (t)t~ (1+ » ) dt < oo holds and we are back in Step (I). Thus let ti > t0 be chosen such that $(ti) = Kt\. Then ^o < *i < & - l [ % $ ( l ) ] , and we take ^(t) = Ktr if 0 < t < h; = $(*) if t > t\. Then $! satisfies the hypothesis, i.e., $ is the principal part of <&i and $1* is now defined using <3>i. Thus the same proof of Step(I) applies and the result follows. The new $1* serves as the principal part of
9.1 Orlicz-Sobolev spaces for PDEs

341

are quite similar. So we omit the details which can be found in Adams' [2] paper. This and the iteration to the case m = J are discussed in the same article to which we direct the reader. D A sharp form of the embedding theorem for Orlicz-Sobolev spaces has recently been obtained by Cianchi [1]. Using a simplification of the Addendum of the paper, we shall present a new form here for comparison. Theorem 9. Let n > 2 and n' = ^-. Let $ be an N-function and define another N-function $n(u) = $[Hn(u)], where n'-l

dt

v>0.

(29)

IfGd ]Rn is an open subset of finite Lebesgue measure (hence G is bounded), then there is an absolute constant K > 0 depending only on $, the measure of G and n, such that II/||(* B ) < K\\Df\\(*)> f e^1'* (G}-

(30)

Further, (30) holds when G is not necessarily bounded, provided that J0"\I/(i) £-(!+«') dt < oo for all 0 < u < oo, where ^ is the complementary function of , and then K is independent of the measure of G (but depends on $ and n). In both cases, L®n(G) is the smallest space for which the inequality (30), o

of Poincare-type, (i.e., the embedding W1'® (G) <—)• L®n(G)) holds true. A proof of this result, as in the case of Theorem 7, is fairly long and we refer the reader to the original paper. This sharper version of the OrliczSobolev embedding is employed in several applications as well as theory, and we are happy to include it here. Other related work will be discussed in Section 3 below. For now we point out that Hn, defined by (29), is an Nfunction so that $n = $ o Hn is also an A^-function. In fact, (29) shows that n'-l

is strictly decreasing from oo to 0 as v increases from 0 to oo, since $ is an JVfunction. Similar to the proof of Proposition 5, we may show that [H~1(v)]n> and H~l(v) are concave functions, so that Hn(u) is actually a strictly convex A^-function. So far we discussed the case of Dirichlet's equation (1) of finite order. But its analog of infinite order has some new features. We briefly indicate

342

IX. Nonlinear PDEs and Orlicz Spaces

this case, which is not yet thoroughly investigated, to complete the picture. Recall that W™p (G) is the Orlicz-Sobolev space of order m with weights /?, and W^f(G] D W^+1'*(G) where the norms satisfy N?£(f] < A% +1 ' $ (/). o Moreover, these are Banach spaces. Their closed subspaces W™^ (G) have the property (from the standard structure theory of Orlicz spaces) that o (WflJ* (G))* = H^5*(G) where ($, #) and (0, £) are complementary Npairs. We replace m by — m for dual spaces, since as m increases the W™g (G) decrease and hence their dual spaces satisfy (W™ff (G))* C With this property in mind, we may define W^ (G} and W^' (G) quite generally as follows. Let 6 = {9a, a\ — 0,1,2, • • • } and $ = 0,1, 2, • • •} be two sequences of Young functions and G C lRn be a domain. Now we set

where W|,,(/) = inf {It >0 : £ »„ I " '."'*•' ] fi(a) < 1 } . I

(32)

H=°

0

Then WQ£ (G) is obtained in (31) by taking those / 6 CC°°(G) with o Daf QG = 0 for a | > 0. One can verify that both Wg£(G) and Wj?f (G) are Banach spaces, but they can be trivial, i'.e., may have only "0" element. To have a nontrivial theory, one must find a condition in order that there are non-zero elements. The following result, extending the case of &a(u) = u\pa, Pa > 1) given in Dubinskij [1], p. 19, was established in our earlier volume (Rao and Ren [1], p.387), which we state here for reference. Proposition 10. The space (W0°g$(G), NQ g(-)) is nontrivial if there is a point x = (xi, • • • ,x n ) 6 G C IRn such that XQ — mini<j< n (|x;|) > 0 and oo

~l)P(a) < oo

(33)

for some KQ > 0. In particular, if 9a are uniformly (in a) ^ and (33) holds, then the space is nontrivial. [When the space is nontrivial, it is generally infinite dimensional]

9.1 Orlicz-Sobolev spaces for PDEs

343

The Dirichlet problem (1) can be restated for these spaces as follows. Suppose that 0 for some C > 2 uniformly in a\ > 0), 0a = 0 and that the norms {Nf$(f), \a > 0} o are uniformly bounded by some Kf < oo, depending only on /. Then (W™p* (G}}* can be identified with W^™'9(G). The latter is given by W^(G] = {/ : D°f where t h e norm N

€

L*-(G), **,(/) < oo},

(34)

- i s defined a s (35)

in which ^ = {\I/Q, |a:| > 0} with (3> Q ,\I> Q ), (0, £) being the complementary o A/'-functions. Then an element / eW™p® (G} is called a (weak) solution of L(/) = g, with Daf\QG — 0 as boundary conditions and g e W^0'*^)' ^ o for each /i 6l/F^'* (G) we have (L(f),h)

= (g,h),

(36)

o with <-, •) denoting the duality pairing. Thus L :W$* (G) ->- W^ (G) is a (not necessarily linear) differential operator, and the existence of / satisfying (36) for any h, and given g is obtained using the Orlicz-Sobolev embedding theorem. This work allows one to generalize most of the analysis appearing in Dubinskij [1]. Note that for a nontrivial solution, it is necessary that o W™p® (G) and hence W^'^(G) be nontrivial, and Proposition 10 gives a reasonable sufficient condition for this to happen. It is clear from the above work that an explicit expression for the solution of a PDE can not be expected. After the existence and uniqueness results, we would like to find its (continuous) dependence on the initial data. One establishes that the solution belongs to a function space such as L$((7), and its properties should be studied. On the other hand, the mapping L of equation (1) can be considered, and one finds that it generates a class of (linear) operators forming a semi-group (or an "evolutionary" family) on the space L®(G) which is typically positive. The latter property of the semigroup is intimately related to Markov processes and thus to stochastic theory, more particularly to diffusions. This connection leads to an extensive analysis

344

IX. Nonlinear PDEs and Orlicz Spaces

when both these areas are combined. Here we indicate how this relation gives an "explicit" representation of solutions in terms of expectations of certain functional of diffusion processes, thereby showing other possibilities of this method with connections to some work in the preceding chapter. We consider the abstract Cauchy problem given for functions u : [0, oo) x Mk —> M, satisfying .

,

N

J^ , , . du

0P'* = 2 4^ aij^ gx.dx (*> X) + L kfo)foTfox> + c(x)u(^ = (Lu)(t,x) + c(x)u(t,x),

(say), t>0,xelRk,

x

) (37)

with initial condition u(Q,x) — f ( x } . Here the (k x fc)-matrix of coefficients A(x] = [ajj(rr)] is assumed to be positive definite and has constant rank (i.e., is independent of a;), and c(-) is bounded and continuous satisfying a Lipschitz condition. It is desired to obtain a unique solution in a class of functions u, when /(•) is a bounded continuous function. The probabilistic method is to associate a stochastic differential equation (SDE) to (37) whose solution is a Markov process, thus relating it to a class of continuous semi-groups whose generators are the operators L of (37). This will produce essentially the desired solution. We now describe this method precisely but briefly. Since A(x) is positive definite, it can be expressed as A(x) = a(x]a(x}* and now associate an SDE:

dXf = b(X?)dt + ff(X?)dBt, X% = x e lRk,

(38)

where {Bt,t > 0} is a Brownian motion (BM), and 6(-),cr(-) are assumed to satisfy a Lipschitz condition of the type MX) - a(y}\\ + \\b(x) - b(y)\\ < KQ\\x - y||,

x,y € JR\

(39)

for an absolute constant KQ > 0 and || • || is an appropriate (matrix or vector) norm. Under these conditions and the boundedness of the components of b(-) and cr(-), (38) has a unique solution {Xf,t > 0}. Since {Bt,t > 0} is BM, the Xf-process is also well-known to be Markovian starting at time t = 0, in "state" x, (cf., e.g., Rao [23], Theorem 2, on p. 249, where this is studied somewhat more generally). The solution u of (37) in a class of functions that grow no more than exponentially, i.e., in the space /C where 1C — {u(-, •) : sup \u(t, x)\ < ceat,t > 0, a, c < oo}, x£Rk

one can establish the desired result as follows.

9.2 Removable singularities, PDE and Orlicz spaces

345

Theorem 11. If{Xf,t > 0} is the unique solution of the SDE (38), governed by the probability space (J7, £,P X ) where PX(X0 = x) = I , then the function u(-, •) given by u(t, x) = Ex lf(Xf) I

exp [ /* c(Xf)d8\ } , t > 0, x € 12*, l/o JJ

(40)

is a solution of the PDE (37), where EX(X) = f^XdPx is the expectation under Px. [Hence Px may be replaced by a "regular" conditional measure P( • \XQ) and then Ex becomes the conditional expectation.] Moreover u £ /C, as introduced above. An interesting part of this method is its represention given by (40) which is also known as the Feynman-Kac formula. Details and extensions of this result may be found in the leisurely treatment of the subject given by Freidlin [1]. If in (37) or (38) L is replaced by L+eL1 or a by (a+^a'} where e \ 0, then one can study the stability properties of the solutions u£ or X£, and the work leads to the large derivations analysis of Section 8.1 with their rate functions [namely vector Young functions] playing an important part. This is also discussed in Freidlin [1]. One can moreover ask whether the class /C above, of solutions of the Cauchy problem, may be replaced by an Orlicz space L*(Hk}. These questions lead to separate, nontrivial parts of the subject and it connects with many deeper aspects of PDE analysis. Instead of the Cauchy problem, the Dirichlet equation can be studied similarly. Without going to PDE, but concentrating on the semi-group part, these equations (both Cauchy and others) were discussed by Goldstein [1], and a comparison of both approaches will be rewarding. We shall now proceed to some other applications of PDE and Orlicz spaces.

9.2 Removable singularities, PDE and Orlicz spaces A well-known result in complex function theory, due to Riemann, states that a holomorphic function / on the punctured disc with the property that \imz-+ozf(z) = 0 is holomorphic through out the disc. So the singularity at the origin is removable. Bochner has generalized this as: if G C JRn is an open domain and C(G] is a class of functions, A C G is closed, such that for any / G £(G), f(x)[d(x, A)]q is small uniformly for x in compact subsets ofG,q> 0, d(x, A) being the distance of x from A, then any singularity of the /'s on A can be removed, i.e., A is a "removable singular set" for C(G}. A generalized version of this concept is as follows. Let P(x,D) = Yl\a\<maa(x)Da be a partial differential operator with a formal adjoint denoted as P*(rc, D) given

346

IX. Nonlinear PDEs and Orlicz Spaces

by P*(x,D)f = £ (-l)H£>> a /), / € £ ( G ) , |a|<m

where aQ 6 G°°(G). Then a closed set A C G is said to be removable for £(G) if for each / e £(G) such that P ( x , D } f ( x ] = 0 for a; e G \ A, one has P ( x , D ) f ( x ) = 0 for all x e G. If C C JRn and e > 0, let G£ be an ^-neighborhood of G, i.e., Ge = {x : d(x,C} < e}. Then a consequence of Bochner's (by now classical) result is that if £(G) is the set of locally pth power (p > 1) //-integrable functions (//— Leb. meas.) and / e £(G) satisfies P(x, D}f = 0 on G \ A as well as

where d = n — mp', p' = p/(p — 1) and C£ is a neighborhood of C for any compact C C A, then A is a removable set. Here m is the order of the operator P(x,D). This result immediately raises the question that the Lebesgue measure JJL may be replaced by a (sharper) metric measure on IRn, such as the Minkowski content or the Hausdorff measure and obtain a more precise characterization. In fact, Bochner has also obtained in the 1950's a result replacing n(C£] by upper Minkowski content. Then these sharper assertions were established in a refined form for (weakly) pih power integrable functions, using Hausdorff measure by Harvey and Polking [1]. Their result was extended to Orlicz spaces by Shapiro [1] whose theorem will be presented below after recalling the concept of a Hausdorff (metric) measure, since it is a somewhat specialized notion which however is a basic component of geometric measure theory and is also needed in this study. Definition 1. Let G C JRn be an open set, <E> an N-function and A C G & closed set. If P(x, D) is a partial differential operator of degree m, recalled above, then A is termed a removable set for P(x,D) relative to Z/*(G), if P(x, D)f = 0 weakly in G \ A in the sense that

/ P(x, D}f(x] • g(x)d^x) = I P*(x, D)g(x] • f(x}d^(x] = 0 JG\A JG\A for / 6 L*(G) and g 6 CC°°(G), implies that P(x, D}f = 0 weakly on G, i.e., the integral is 0 on G itself, with // again as the Lebesgue measure on Mn. Let h : 1R+ —>• IR+ be a nondecreasing (left continuous) function with h(t} — 0 iff t — 0, and let (M, d} be a metric space with distance function d.

9.2 Removable singularities, PDE and Orlicz spaces

347

For each e > 0 and A c M, define f °° 1 H£h(A) = inf Y^ ^[diam(^)] : diam(A) < e, A C U^ A{ c M L

(1)

=i

where diam(^) = sup{d(o;, y) : x, y e ^} is the diameter of ^. Then it is seen that Hh(A) = lim^o H^(A) exists, and is additive on Borel sets that are a positive distance apart, i.e., Hh(A U B) = Hh(A) + Hh(B] if d(A, B) = inf{d(x, y) : x € A, y e B} > 0. Moreover, Hh(-} is regular in the sense that for each A c M there is a GS set B D ^4 such that Hh(A) = Hh(B}. [For a quick view of this result and related properties, see e.g., Rao [20], Sec. 2.5. p.86ff.] If h(t] = T, a > 0, then Hh(-] is denoted Ha(-) and called the Hausdorff a- dimensional (outer) measure, and we call Hh(-) a generalized Hausdorff metric (outer) measure or simply an h-measure. If Ha(-} is the Q;-dimensional (outer) measure, then a set B C M is said to have Hausdorff dimension 0, defined as ft = ft(B) = sup{a : Ha(B) = +00}.

(2)

To appreciate the sharpness of these concepts, one notes that the Cantor set C C [0,1], with the usual metric, has Lebesgue measure zero, but its Hausdorff dimension ft(C) = Iog2/log3. Thus //(C) = 0, but HQ(C) > 0 for a ^ 1 implying the sharpness of Ha. If a. = 1, M = M, then HI(-) = //(•) (the Lebesgue outer measure) and if M = Mn, then Ha(A) — 0 for A C M, satisfying a > ft (A), and ft (A) < n; and if a = n, then Hn(-} = Knn(-}, where KI = 1 and for n > 2, Kn = (^)tp(l + |) > 1, which is a nontrivial result. However, every countable set C in IRn has ft(C) = 0. These facts will be used in what follows. A readable account (with proofs of the above statements), completely devoted to Hausdorff measures, is the book by Rogers [1] which is a helpful source. We now present Shapiro's [1] result on removable sets in G C Mn, stated in some generality obtainable from his proof. Theorem 2. Let (, ^) be a complementary pair of N-functions and h(t) = tnty(t~m}, 1 < ra < n— 1 for n > 2, be nondecreasing. Suppose that G C lRn is an open set and A C G a closed a-compact a-finite subset for Hh(-}, i.e., A = \J?=lAk, Ak compact and Hh(Ak] < oo. If P(x,D] = Y,\a\<mca(x}Da is a partial differential operator of order m with ca G Cm(G) (i.e., m-times continuously differentiate functions), then A is a removable set for P(x, D) relative to the class L®(G}, the latter being the closed separable subspace generated by C0°°(G) in L*(G). Remark. This result extends the classical case that $(t) = \t\p, 1 < p < oo, since h(t) = c\t\n~^ for 1 < m < n - 1, n > 2 and n - ^ > 0, is

348

IX. Nonlinear PDEs and Orlicz Spaces

considered with P(x, D) as a Laplacian. But it can be more general than the power function since for instance $(t) defined by

(eP

+ „c2,

+

(n _ m) 2

^i . < oo,

< f

is an N-function, and one can verify after a straightforward but tedious computation that the related h(t) is strictly increasing and /i(0) = 0. Since both P(x, D) and 4> can be more general, the result applies to many more situations than the Lebesgue theory. We shall sketch the details of proof extending and using some results from Harvey and Polking [1]. Proof. We first consider the case that Hh(A) < oo and later, using the regularity of the metric measure Hh(-}, extend the result to the general case, as given in the theorem. Then we approximate Hh(A) with values of Hh(') on dyadic cubes, following standard procedures in measure theory. Note that a compact set need not have finite Hh(-) measure (Cantor set for Ha(-) being an example) and a set C with Ha(C] < oo may not be precompact, so we need care in the desired approximations. The argument is presented in steps. (I) Under the hypothesis that Hh(A) < oo, A is closed and P(x, D)f — 0 weakly on G \ A, relative to / 6 L%(G], we assert that P(x, D)f = 0 on G itself weakly. For the following argument one needs a result on the "partition of identity" subordinate to a covering, in a special form given by Harvey and Polking ([1], Lemma 3.1) which is needed for our work. This is stated for the space filling collection of closed dyadic nonoverlapping cubes Qj (i.e., of disjoint interiors) of length 2~ fc with vertices having coordinates of the form p2~k, p being an integer. These are used in approximating an cn-dimensional Hausdorff measure Ha of a set A C Rn, by La(A) = lim^o L£a(A) where = l t

,

j=i

with Sj denoting the (side) length of the cube Qj. Then one can verify that there are constants CQ, Ca > 0 such that caLa(A) < Hh(A) < CaLa(A),

A C Mn.

A dyadic cube of (side) length s is contained in a ball of radius ^s in Mn, and each ball of radius r can be covered by 3n dyadic cubes of length s = 2~k where 2~^ +1 ) < r < 2~k. These facts are needed in working with Hh(-). We let |Q to be the cube, which is a magnification of a cube Q with the same

9.2 Removable singularities, PDE and Orlicz spaces

349

center x (also denote Q(x,s)), of length Is to |(2s). The desired partition of unity is given by the following crucial Lemma. Let {Qi, 1 < i < N} be nonintersecting (finite) collection of cubes of side length Sj. Then for each i, there exists a gi 6 C£°(lRn), supp(gi) C |Qi, such that Y,i=\9i(x) = 1 for all x e (jfL-^Qi. Further, with a multiple index (3, there is an absolute constant cp such that \D^9i\(x} < cps-W,

l
(4)

Proof of Lemma. By relabeling, if necessary, we may assume Si > s-2 > • • • > SAT, and consider 6 € C^(lRn} with the property that 9 = 1 on the unit cube, i.e., {x € ]Rn : \xi\ < 1, 1 < i < n}, and 0(x) = 0 for \Xi\ > | for some z, and set 9k : x i-» g( 2 ( x ~ x '} where xk is the center of Qk. Now we define the i sequence inductively as g\ = 61, and for k > I , gk+i = Ok+i n^=i(l — #7) so that gk € C^(JRn] and supp(gk) C supp(0 fc ) C \Qk, 1 < k < N. Then it is verified (by induction) that E*=i #j = 1 — IT)=i(l — #7) and Y!j=\9j(x) = 1 for x € UJ^Qj, as the desired partition of unity. To establish the key inequality (4) of \Dagi\(x), consider fk = X^Li 9j — 1 — IT)=i(l — Qj}- Using the ordering s; > si+1, we shall obtain the bound with fk, which in turn gives a bound on gk. So if 1 < v^ < k, Vi integers, define auxiliary functions, for r > 1, JO, 9vi,-,vr = j n^,,!,...,«,.,i<*(l - 0i)»

if Vi = Vj for some i ^ j vt are distinct.

if

Then there exist absolute constants Kait...tar depending only on the suffixes, with (using the Leibnitz type differential formula) k

E

(5)

the summation extending over all sets of multiindexes, on\ > 1, \a\ — a\ + \-ar. Distributing the inner sum in (5) to each suffix v^, we get an upper bound. Consider a typical sum £j=1 |DQ°^|(a;). Since supp(DQ°^) C \QV, for x 6 \QV we have \Da°6v\(x] < Kaos~\a°\ for a0 C a, and some absolute constant Kao > 0. So for a constant Kao > 0 depending on Kao, one has: k

\( \ < £> 1-^ C'Dl^J !^ -"-QQ E \r)aof) T

11=1

o V^ Sq-laol r /r r/Q 2^i v 5 ^ t pVv Sv>S

350

IX. Nonlinear PDEs and Orlicz Spaces

But for each integer p > 1, there are no more that 2" dyadic cubes Qv of length sv(= sky] with x G \QV. Consequently k

oo

£ \D*ev\(x) < TKao $>,2TM < KQ0s~kM. v=l

(6)

p=0

Substituting this in (5), we get D«fk\(x)
< Kas-kla(

(7)

for some absolute constant Ka > 0. In view of the earlier comment this gives the desired bound on \Dagi\(x), 1 < i < TV, and hence the lemma follows. (II) We restate this result in a convenient form for our proof. For the set Qj above whose center is a/7 and radius TJ (2rj — Sj), where 2"1 > r\ > • • • > TN, Qj = Q ( x ^ , r j ) , define the disjunctification |J Q f x f c , | r f c ) , z / k=j+i ^

j = 1,2, • • • , TV.

(8)

Then there exist 9j e C%°(lRn) with supp(^-)Q(^\ | r j)> such that if ° = EjLi Oj, then we have (a) 9(x) = 1 on Uf = i Q(xj, ^), and (b) for 0 < a\ < m < n — 1, \DQ9 (x) < Kmr~m, x e TJ, for some absolute constant Km > 0, j = 1, 2, • • • , N. In (8), take T/v = Q(xN , fr^), the union there being empty. To prove the assertion about / € L*(G), we need to establish that for each h e C£°(G), fG f(P*(x,Da)h)dn = 0, or equivalently for any £ > 0 and any such /i, the above integral is bounded by E which is shown in this form using the fact that if A* = A n A~h (A-h =supp(/i)) a compact set, then Hh(A*} < Hh(A] < oo where h(-) is the increasing function determined by $, as in the statement of the theorem. Now since P*(x,D}8 is a finite linear combination of Da9j, property (b) above may be expressed as: with h € Cn^R"), if J] G C°°(G), T C G, 0 < p < \ and \Dar]\(x] < Kmp~m, x G T, a\ < /3, then we have P*(x,D)(hr,)(x)
z€T,

(9)

for an absolute constant Km^ depending on Km and h only. (Ill) Proceeding with the proof, let / 6 LQ(G). Since the norm on Lf(G) is absolutely continuous, ||/XB||(*) -> 0 as n(B] —> 0. Hence given e > 0 there is a 6 > 0 such that n(B] < 6 implies Kmll}]-1.

(10)

9.2 Removable singularities, PDE and Orlicz spaces

351

Also by hypothesis on h, h(t) = tn^(t~m}, 1 < m < n — 1, increases to 00 with t and -^ = ty(t~m) —>• oo as t —>• 0, since \& is an N-function. So tn/h(t) —»• 0 as t —> 0 and hence for the same 5 > 0 of (10) we can find a t0 > 0 such that 0 < t < 3t0 implies

The definition of h(-) also implies for any a > 1 and t > 0, by the convexity of *, that h(at) < antna~m^(t~m) = an~mh(t) < anh(t). (12) Since A* is compact, we can find a finite covering by open balls {B(yk,pk}, 1 < k < NQ\ of radius pk, 0 < pk < min(to, j), such that No

No

A* C \J B(yk,pk), and ^h(pk) < Hh(A*} + 1. k=l

k=l

Also by the above lemma, each B(yk,pk) [with 2~(l+1) < pk < 2~*j can be covered by the interiors of the union of 3n dyadic cubes of side length 1~l. Using these two properties we may draw the following consequences: there exist nonoverlapping cubes {Q(xi, TJ), 1 < j < N} satisfying the four properties listed below: (i) min(£ 0 , |) > ^i > r2 > • • • > rN; (ii) A" C intQ - \J^=lmtQ(x^rj)] (iii) Ef=i h(rj) < 3n(Hh(A*) + 1); and Then a partition of unity for this Q-covering is given by 6(x) = Y^jLi ^j(x}i with all the properties of the lemma, restated as (a) and (b) at the beginning of Step (II) above. Since A C A* U (A-h)c, we find from Part (b), that h(l-0] £C$°(G\A). Hence / f ( x ) P * ( x , D}[h(l - 0}}(x}d^i(x) = 0,

(13)

since P(x, D)f = 0 weakly on G \ A. Thus we only need to show the same property with hO on G: f Jo IG

For this, let Q* = \J?=1 Q(xj, fr^) and let 7}, j = 1, 2, • • • N be their disjunctification. Thus Q* = UjLi Tj and 0(x] = 0 on JRn\Q*- Hence (14) becomes,

IX. Nonlinear PDEs and Orlicz Spaces

352

on replacing G by Q* there, the following: N

r

3=1

(15)

JT

i

We now consider each of the integrals in (15), and estimate them. Using Young's inequality, for x G Tj, (P*(x,D)(6~h)) -(*) $

\fXQ'

and integrating and summing on j, the left side of (15) is bounded by

/

/

\

}

» r

where the two terms are denoted as /i and J2. Thus (15) will follow if we show that 7j < |, z = 1, 2. Now noting that by (10), and (i) above, <

<

+ !)(! + X^))"1 ^ /i(3r,.)-

(17)

Taking a — 3 in (12), with (iii) we get (J,(Q*) < 5. Using the definition of the gauge norm we conclude from (10) that

(18) \

( ijT—TTJ—) dp, < 1, it results that Regarding /2, we use the bound on P*(x,D)(Oh) given by (9), where one takes p = TJ there. Thus \P*(x,D)(dh)\(x) < Km^m. But since Tj C Q(xj^r-j), we have

(19)

9.2 Removable singularities, PDE and Orlicz spaces

353

However, ^(Km-hr^m) < \P((1 + Km-h)mr^m), since h(-) is increasing and h(i+K ) — M r j)- Hence as before /2 < II/XQ- !!(*)(! + ^)"[9B(^(^) + 1)] < \. Thus the theorem is proved in case Hh(A) < oo. (IV) To complete the argument, we have to remove the restriction that Hh(A) < oo. Thus let P(x, D}f = 0 weakly on G \ A, f G Lf (G). Consider Co = {y € G : P(x,D)f = 0 weakly on B ( y , p ) for some p > 0} where B(y,p) is again a ball centered at y and radius p. Then the set Co(c G) is open and so F = G \ CQ is (relatively) closed. We now assert that F = 0 so that G = CQ obtains. Our hypothesis implies that G \ A = CQ, so F C A. Hence by cr-compactness of A, we have oo

F = AHF=\jAk,

(20)

k=l

where Ak is compact for each k, and FDAkis closed, hence a G,j-set (cf., e.g., Sims [1], p. 39), so that F is a G§. Now if F is nonempty, then by the classical Baire category theorem (cf., Sims [1], p. 126), it is nonmeagre; and hence there exist some a;0 and r0 > 0 such that x° G F, B(x°, 2r0) C G and for some integer j, FC\Aj is dense in B(x°, r 0 )DF. Thus Fr\Aj being a closed subset of G, from the last property above, we have that Hh(B(x°, r0}C\F} < oo. So by the special case proved above, P(x, D)f = 0 weakly in B(x°, rQ)\B(x°, r0)r\F and since B(x°, ro)flF is relatively closed, P(x, D}f = 0 weakly on B(x°, r 0 ). By definition, then XQ G C0 and so x0 G F. But C0 PI F = 0 and we have a contradiction. Thus F = 0 must be true, and consequently the original assertion must hold. D Although the basic properties of Orlicz spaces in the form of /i(-) and Hausdorff measure are important at crucial points, the small details are numerous, and the validity depends on the careful verification of all these subresults. The special property, given as a lemma in the proof, is also essential. We now present another result for comparison, based again on PDE analysis in Orlicz spaces, also due to Shapiro [2], without a detailed argument, and the reader is referred to the original paper. This is called a (weak) comparison principle. In certain problems of mathematical physics involving divergence operators, it is of interest to find the pointwise order relations between pairs of functions and similar relations between the divergence operators on them. Thus let G C IRn be a bounded open set (or domain) and consider the

354

IX. Nonlinear PDEs and Orlicz Spaces

generalized divergence operator Q given by Qf = div$'(\Df\)\Df\-lDf

+ F(x, /),

(21)

where divg, for a differentiable vector g = (gi, • • • , gn], is defined as usual by

Hence in (21)

9l

= V(\Df\}\Df\~1^

or = 0,

accordingly as \ D f \ 2 =

Er=i(f^) 2 > 0 or = 0, i = l , 2 , - - - , n . The function F : G x IR -+ 1R is Lebesgue measurable and satisfies the growth condition \F(x,s)
(22)

where C is a positive constant and u(x] > 0 such that $ ( u ( - ) ) e Ll(G) with (4>, \I>) as complementary A^-pair and lim ^4^ = 0. The operator Q s—>0 I ! of (21) is quite important in PDE theory since for instance, Qf = 0 is a nonlinear Dirichlet problem and its solution (existence and uniqueness) will be of importance. Here we describe the (weak) comparison principle which has a key role in the uniqueness of Dirichlet's problem, among others. (See, e.g., Douglas, Dupont and Serrin [1], as well as Trudinger [1].) Such a Q in (21) is said to be of divergence form. Let u, v € Wl^(G}. Then Q is said to satisfy a weak comparison principle to <&, if u < v (for <$) on the boundary s

o

dG, to mean that the positive part (u — v)+ of u — v is in W1'® (G), and Qu < Qv (for <J>) in G, whenever Q(u, v. h) < 0, where Q(u,v,h) =

<

f {[(g(Du) - g(Dv}} • Dh](x) - [F(x, u(x}} - F(x,

JG

v(x)]h(x)}dfj.

(23)

0,

for all 0 < h e C%(G), and where g(Du) = (gi(Du), • • • , gn(Du)) with .

9l(Du)

,

®(\Du\) du - - V ' U • —, Du

Du > 0,

and Qi(Du) = 0 if \Du\ = 0. Note that u < v is not necessarily a pointwise ordering. A general discussion and comparison of maximum and co-relations

9.3 Strong and weak type inequalities in Orlicz spaces

355

principles may be found in the book by Gilbarg and Trudinger ([1], Ch.10) where one also sees results on its use in establishing existence and nonexistence of solutions of PDEs. The following assertion, also from Shapiro [2], is an Orlicz space version of the weak comparison principle, discussed by others for C2(G), LP(G) and related spaces (e.g., of bounded variation spaces etc.) Theorem 3. Let G C Mn be a bounded domain, $ an N-function which is twice continuously differentiate on (0, oo) with $"(t) > 0. Suppose the following hypotheses on and the divergence operator Q hold: (i) Q"(t) is either nonincreasing, or nondecreasing for 0 < t < oo, or else $"(t) > 0 and there is 0 < ti < oo such that &"(t) is nonincreasing for ti M defined in (21), is such that F(x, •) is nonincreasing and satisfies (22); (Hi) the functions u, v e W1)*(G) satisfy u < v on dG for <& and for the divergence operator Q of (21), implies Q(u) < Q(v] (for $) in G, the orderings are as defined above. Then we have u < v a.e. in G (for the Lebesgue measure). Remark. All the conditions are satisfied if, for instance, $1 (t) = (1 + ty[log(I + i)]2,

1 < p < oo;

The proof of the theorem depends on a verification of some key inequalities. In particular, one establishes and uses the inequalities on g as (using the dot product and | • | the metric of IRn}\

[g(y) - g(x}} • (y - x)

i *i(\y-x\

> \y-x\ max [&(\x\ + \y- x \ ) - $'(|z|)],

\y- x\) - $'

z| + |y - x

where C$ is a constant depending only on <$. We omit the details referring the reader to the original paper. We devote the next section for applications of certain inequalities.

356

/X. Nonlinear PDEs and Orlicz Spaces

9.3 Strong and weak type inequalities in Orlicz spaces As seen in Section 6.2, if T : Lp(^i) -> Lq(^2) is quasi-linear, i.e., (a) |TCA + /2)| < CdT/xl + |T/2|), (b) \T(af] = \ a \ \ T f \ ,

(1)

for all a G JR and /i, /2 G L P (/UI), 0 < C < oo, [C = 1 gives sublinearity of T] then T is said to be of strong (p, q)-type if there is an absolute constant K > 0 (depending on p, q) such that \\Tf\\q < K \ \ f \ \ p , f € L p (^), p > 1, q > 1,

(2)

and T is of weak type (p, 0 (depending on p, q) and for allt > 0 one has 9

, / G Lp(vi).

(3)

Note that by Markov's inequality, (2) implies (3) so that the weak type is weaker than the strong type (p,q). The examples and work on conjugate functions / for the / G £ p (/^i) show that T : / ^ / is just sublinear and of weak (1,1) type but not strong (1,1) type. Another familiar operator in Trigonometric Series is the 'maximal' mapping. The importance of these types is already seen in the study of interpolation of Marcinkiewicz's theorem (cf., e.g., Rao and Ren [1], p.247). Motivated by the analysis of Section 6.3 and other problems in Fourier transforms, here we discuss some sharp extensions of the above inequalities for Orlicz spaces. The smallest number K (or K] of (2) (or of (3)) is termed the strong (or weak) (p, q)-norm of the quasi-linear operator T between Lp(pi] and Z/ 9 (/u 2 ). An important example of a sublinear operator is the Hardy-Littlewood maximal fractional operator Ma defined as Ma : f i—>• M Q (/), (M a /)(x) = sup

*

S3x HSj]

/ \f(y)\dy,

xG

tf»,

(4)

n Js

where 5 is a ball containing x, f locally integrable and 0 < a < n. If / € Lp(p), p > I and a = 0, written M for MQ, [called the Hardy-Littlewood maximal operator] then the classical analysis implies that Mf £ Lp(^) satisfying (2) with p = q, so that the sublinear (hence quasi-linear and not linear) operator M is of strong type (p,p), but if p = 1, then it is merely of weak type (1,1) so only (3) holds. Thus M is always of weak (p, p)-type for p > I . These, in the context of Orlicz spaces, are as follows.

9.3 Strong and weak type inequalities in Orlicz spaces

357

Let ($i, \I/i), i = 1,2, be pairs of complementary Young functions. Then a quasi-linear operator T : L $1 (/^i) —>• L $2 (// 2 ) is of strong ($i,$ 2 )-type if there is an absolute constant K > 0, depending only on $1} $2, such that

P7 ||(*a)<

(5)

and it is of weak ($1, <£>2)-type if there is such a K > 0 for which t] <

(6)

for all t > 0. Since evidently (5) is an extension of (2), it is not obvious that (6) is a generalization of (3). Let us verify that it is obtained from (5) just as simply, using again Markov's inequality. Indeed with the definition of (1), we have for t > 0 and |/| ^ 0, a.e.,

2

.

>$ 2

2

$5

Vn:

$,

\Tf\ Vn-

\\Tf\\w

—1

(7)

Thus (7) is just (6). In the general case one takes this inequality as the definition of weak type. Also recall that for any Young functions $1, $2, one says that $1 is stronger than <$2 (globally), denoted <3>i >- $2 (cf. Rao and Ren [1], p. 16), if there are b > 0, UQ > 0 su7ch that 0, (w0 = 0).

(8)

The strong and weak type results for classical sublinear operators of the types of Hardy-Littlewood maximal, singular integral, fractional integral and others were studied in a series of important papers by Cianchi ([l]-[3]), with some extensions also to the quasilinear case. Here we include a few of her

358

IX. Nonlinear PDEs and Orlicz Spaces

results with an indication of their use in PDE. They are taken from the above articles which illustrate the need for important additional analysis. We now have the following result on a Hardy-Littlewood (or H-L for short) maximal fractional operator Ma. Theorem 1. Let Ma : L®l(G) —> L®2(G) be a maximal fractional operator, where !, $2 ore Young functions, 0 < a < n, G C Rn with JJL as the Lebesgue measure. Then (i) Ma is of weak type ($i,$ 2 ) iff $1 >~ Qi globally for p(G] — oo, and locally for n(G) < oo; where Q% (depending on $2, & and n) is the inverse of Q-l(u)=un$-l(u). (9) (ii) Ma is of strong type (<E>i, $2) iff$i >- $2, globally, where l>2 (depending on &2iQ:in) is the complementary function of (it is obviously a Young function) V2(v] = fVt^-l(g^(t^)}^dt (10) Jo with ru &„(+]

9i(u] = / -~^dt < oo, u > 0. JO

ti+n-Q

(11)

Proof. Later we also discuss the case that a = 0, so that the precise conditions for weak and strong type of H-L maximal operator M are recorded. Part (i) on weak type (^i,^) is direct, but Part (ii) needs several auxiliary computations. So we detail the simple result, and then make remarks on the strong type assertion later. (i) Sufficiency. Suppose $ >- Q2 globally and G — 1RH. Then by definition (8), there is a b > 0 such that Q?,(u} < <3>i(6w), u > 0, or equivalently, by (9) ®il(u) < bQ^l(u) = bu-^l(u], u > 0.

(12)

If we take / G Z/ $1 (G), satisfying / H ^ ) = 1, so that the modular 1, then by Jensen's inequality, for all ( nontrivial) balls 5

s Letting M =

n

s

/ s $i( f(y)\)dy in (12) and using (13), we get

(is)

9.3 Strong and weak type inequalities in Orlicz spaces

< b

1

359

>,-'

and

We have from the definitions of MQ, M and (14), the relation (Maf)(x)

< b^l((M^(f))(x}},

x e JRn.

(15)

Hence (15) implies

(16) But now we observe that the classical H-L maximal operator M is of weak type (1,1), i.e., M : Ll(G] —> L°(G], the space of measurable (not necessary integrable) functions on G. This is also nontrivial. In case n = 1, it is given in the classical volume "Trigonometric Series" by Zygmund [1], p. 33, and for general n, it is detailed in the text nook by Wheeden and Zygmund [1], p. 105. The result from the latter states that there exists an absolute constant C > 0 such that ti[x : (Mg)(x) > A] < ^ j^ \g(y)\dy, A > 0.

(17)

We take this classical fact here, referring the reader to the last reference for details. Thus, by letting g = $i(/) and A = $ 2 (f) in (17), we get from (16) H\x : (Maf)(x)

>t]< -5

-

t

~

t

"

(18)

since ||/||($l) - 1 and £$ 2 (f) > $ 2 (^) = $2(|), if C > 1, and K = Cb. Thus MQ is of weak type ($1, $ 2 )) by (18) and the definition given by (6). Necessity. Let Ma be of weak type ($ l 5 $ 2 ), i.e., (18) holds for some K > 0. For an arbitrarily fixed ball 50 C Mn, define a step function / = $ i " 1 ( ) X 5 o so that Il/H^o = 1 and for x £ S0 one has

w=* L

360

IX. Nonlinear PDEs and Orlicz Spaces

But by definition, 50 C [x : (Maf)(x) > t0] ift0 = [ y u(5 0 )]n$r 1 (^)- Hence by the weak (^1,^2) inequality (18) with H / H ^ ) = 1, : (Maf)(x)

> t0] <

Writing u = [H(SQ)]~I, this becomes on expansion, K

< w,

u > 0,

proving $il(u) < Kun^l(u) - KQ^l(u} by (9), i.e., $! >- Q2 globally. This establishes Part (i) of the theorem. D The strong type inequality depends on a different line of attack, and so we refer the details in the original paper of Cianchi [2] . Here are some comments. Recall that in obtaining an extension of the Hausdorff- Young theorem for Orlicz spaces, in Section 6.4, we noted that the classical method of the Lp-spaces, which conveniently uses the Risez convexity theorem, has to be significantly changed since interpolation between Lebesgue spaces will not take us outside the //-structure. No (non Lp) intermediate Orlicz spaces are covered. On the other hand, if we have the inequalities on (sub) linear operators between Orlicz spaces, then our account in Chapter 6 of the companion volume (cf. Rao and Ren [1], Sees. 6.3 and 6.4) applies to a wide class of intermediate (Orlicz) spaces. The problems in Section 6.4 (of the present book) showed that one has to obtain a priori strong type inequalities between Orlicz spaces themselves, involving hard work. Thus the proof of Part (ii) proceeds through such interpolation theorems. The Sobolev embedding is one such key example. Others, with interpolation ideas using the Lp-theory, will now be extended which complements the general work of the previous volume; but now it is somewhat tailored to the PDE analysis. To begin with, we observe that the H-L maximal operator M is of weak (1,1) type as well as strong (00,00) type. The latter means that M : LGO(Rn) -> L°°(Rn) is bounded, and in fact \\Mf\\oo < ||/||oo- Also M is sublinear. The methods we use here work for quasilinear operators as well. So let us start with the following simple case, which is an extension of the operator M to Orlicz spaces. It is due to Gallardo [1]. Theorem 2. Let (f2j, £j, //j) ; i = 1, 2 be a-finite measure triples and i = 1,2 be Orlicz spaces where $ is an N-function. Let T : Z/*(/^i) -> L* (/^2) be a quasilinear operator which is of weak (1,1) and strong (00,00) -type on these measure spaces. If $ € V 2; then T is of strong ($,$)-type so that \\Tf\\(*}
/eL*(^),

(19)

9.3 Strong and weak type inequalities in Orlicz spaces

361

for some absolute constant K > 0 depending only on <£ and on the norms of (1,1) and (00,00) -types. Proof. By hypothesis one has for A > 0

\ / G L1^);

A

(20)

< call/Hoc, / 6 L°°(/ii), and

|T(/ + g)\(y) < cs[|T/|(y) + |T0|(y)], a.a. y G Q 2 ,

(21)

for absolute constants Q > 0, i = 1,2,3. Consider the truncation of / at A>0:

A = / x > _ A _ , /A Then /A G Ll(^} and / A G L°°(A*I). If $(u) = ft (j>(t)dt, u > 0, by the image measure theorem, (20), (21), and the Fubini-Tonelli theorem, we have

o

A f /.2cica|/(i)| rt!,(A) 1

/(x)| /

^dX\d^(x}.

|_vo

A

J

(22)

To simplify the right side of (22), consider first the inner integral for u > 0, using integration by parts,

rdx=r<

Jo

A

Jo

A

=

u

+ Jor dA x 2

.

(23)

inf ^-- > 1 (cf. Theorem 1.1.2). So So there t However $ G V2 iff inf exists some > 1 such that t(f)(t) >p$(t),t> 0. Hence for 0 < A < u,

Consequently

A"-2

362

IX. Nonlinear PDEs and Orlicz Spaces

Substituting this in (23) gives

r

A)

u

A

u13 Jo

(3 — 1 \ u

(24)

Taking u = 2cic 3 |/(x)| in (24) and simplifying (23) with it, we get n2 Setting Ki=/3/(fi-l)

- • - • - •

/? — l Jsii

'

(25)

"

and K2 = 2ciC3, we have from (25) /A

which implies (19) with A" = A^/G. D We now show that the condition $ 6 V 2 for the validity of (19) is also necessary in the case of the H-L maximal operator M. Thus it is the best possible hypothesis on M to be strong (<£>, <£)-type, which is also due to Gallardo [1]. Theorem 3. Let $ be an N-function and M be the H-L maximal operator on L*(lRn) -» L*(JR n ), with Lebesgue measure. Then (19) holds for T = M Proof. Sufficiency. Letting (Jlj,Ej,^j) = (Mn,Y^,^) with // as the Lebesgue measure and T — M, if $ 6 V 2 we have immediately from Theorem 2, for some absolute constant K > 0,

Necessity. Let (26) hold for some K > 0. In particular, / = XA, A e S (so //(^4) < oo, since/ 6 L®(lRn)), then

Let B(x,r) = {y £ iRn : x — y| < r}, a ball in IRn of radius r with center rr, and let an be the volume of the unit ball JB(0,1). Set y4(w,s) = B(0, (a n ^s)"n) where v > 0 and s > 1, and let C(v) = B(0, (anv}~n). Then x ^ A(v, s) =>• A(t», s) c J3(x, 2 x|) as the latter is a bigger ball including the origin and \x > 1, where x\ — Y!i-i(\Xi 2 ) ^ if x = ( x z , • • • ,xn) e lRn. Hence without taking the supremum on the balls, one has

1

r 'xeA(v,s)]

. , , , _ _

, „, . 1

^

9.3 Strong and weak type inequalities in Orlicz spaces

363

If g(x) = $~1(v)xc(v)(x}, then the modular p$(g) = 1 where ^ is the complementary function to $. Hence (28) gives 2||MxA(w,s)||($) > ||-^XA(v,s)||$> (the Orlicz norm) JC(v)

~ n

> (2 anvs}~l^f~1(v}

I

L

L

J[(anvs)~ n <|x|<(a n v)~n]

\x\~ndn(x)

= (TvsYl^-l(v}\ogs.

(29) l

l

However, the Orlicz norm ||XA(«,S)||$ = (vs]~ ^~ (vs). Hence (27) and (29) imply 2-n(vs)-ly-1(v)\ogs < 2K(vs}-ly~l(vs), v > 0, s > 1, which gives on taking 5 =exp(/f2 n+2 ), 2$-1(v} < ^(v exp(K2n+2)).

(30)

Now setting v = *(w), (30) becomes *(2u) < K0^/(u) with KQ = 2 n+2 ) > 2 so that ^ € A 2 or equivalently $ e V 2 , as asserted. D Reexamining the strong ($, )-type inequality, which is based on the weak (1,1) and strong (00,00) inequalities, one can ask whether similar results are obtainable if we have (p, p) and (00,00) instead. More generally, can we consider weak type (p, q) and (r, s) and find intermediate Orlicz space bounds for ($1,^2)? There are considerable difficulties here. We present a couple of results on weak (p, p) and (oo, oo) types for 1 < p < oo, or (l,p'), and (p, oo)-types for 1 < p < oo with p' = -^-, based on the work of Cianchi's ([l]-[3]). It will be seen that the (nearly optimal) methods are somewhat different from those in the preceding two results, but now they can be given more generally for quasilinear operators. Their specialization for H-L operators is possible, perhaps indicating the necessity of the conditions used, but will not be considered here. We also sketch an application via Theorem 1, to a Dirichlet problem, on the key role played by these inequalities. Let (fij,£j,/^) be a pair of a-finite measure triples, and T is of strong (p, q)-type if

\\Tf\\q < AH/11,, / e z / M ,

(31)

for some absolute constant K > 0. The operator T is of weak (p, t]
-\\f\\~q , t > 0, / 6 L^),

(32)

364

IX. Nonlinear PDEs and Orlicz Spaces

where It is seen that || • ||~ is a norm for p > 1, and (LI(HI), || • ||~) is a Banach space. In (32) K > 0 is an absolute constant depending on p, q > 1. Evidently strong type implies weak type. We have the following interpolation result for Orlicz spaces L*^/^), i = 1,2, from certain weak and strong types extending the L $P (/UI) to I/ $ (/^2) spaces, where &p is determined by $ and p. Theorem 4. Suppose (J7j, Ej, //i), i = 1, 2 are a pazr of a -finite diffuse measure spaces, and T is a quasilinear operator of weak type (p, p) and (strong) type (00,00), as defined above, where 1 < p < oo. //$i,^>2 are N-functions, then T : L $1 (^) -+ L* 2 (/z 2 ) saft's/ies ,

(33)

an absolute constant C > 0, depending only on the weak and strong type norm constants, where <E>i and $2 satisfy the inequality relations: (i) either p = I , and there is a constant K\ > 0 such that s

Jo

-~dt< $i(#is), s > 0 ; z t

(34)

(ii) or 1 < p < oo, and there is a constant K^ > 0 swc/i fs

/ °

< > 0, ~ ^2, 5 ~

(35)

where p1 = -1— . Finally, if ^2(^2) < CXD, t/ien we can demand that (34) and (35) be valid only for large s and then (33) holds with C depending also on Remark. It should be noted that $1 and $2 are not any N-functions but must satisfy (34) or (35), and the quasilinear operator for weak (p, p) should be defined not on Lp(p,i), the Lebesgue space, but on (Lf(//i), || • ||~), the space related to the Lorentz's class (cf. e.g., Rao and Ren [1], p. 421). Proof. In establishing (33), we may suppose || /(!($!) = | because of the homogeneity of the equation. Then (33) follows if we show

9.3 Strong and weak type inequalities in Orlicz spaces

365

Thus if H/ll^) = \, we first observe that Tf is well defined. In fact, let ft = fx[\f\ 0. Then ft is bounded and we assert that ||/*||~ < oo. To see this, by the Holder inequality with p' — -^-, we have

/

.

°° °°

<

/

(37)

(• 'PI

Since ^f^ < 0i(2s), 5 > 0, if $,(w) - /0H &(s)ds, i = 1, 2, using the image law, we have the second factor on the right side of (37) as /• /o

since H/H^) = \. Regarding the first factor of (37), by the || • ||oo norm for p = 1; p1 — oo, this factor has the value

' where 9<(s) For 1 < p < oo, the first factor of (37) is finite by the assumed finiteness of (35). Thus ||/*||~ < oo and \\ft\\oo < oo, so that Tf is well defined for To establish (36), by the image law again, we have for any c\ > 0,

/•

=

/ Jo /-cx>

2(t)iJ,2[\Tft\>clt]dt,

(38)

JO

where f = ft + f* and CQ is the constant in the definition of quasilinearity. We now show that the right side of (38) is bounded by 1 if GI is a suitably chosen (large enough) number using the strong and weak type norms or bounds. Let c2 > 0 be the strong (oo, oo)-type bound (or norm) of T. Then H^Ytlloo < c2 1 1 /t|| oo < c2£, t > 0. Hence if Ci > c2, the first integral on the right side of (38) vanishes. It is for Tf1 we need to estimate the bound using

IX. Nonlinear PDEs and Orlicz Spaces

366

the weak (p, p)-type. Thus by this hypothesis, there exists c% > 0 (depending only on p) such that

, t>Q.

(39)

Using inequality (39), for the second integral on the right side of (38), one has r

i

Mi/I

JQ

> s)]->ds

dt. (40)

The boundedness of (40) will be seen with the conditions (34) and (35). Since <£> 2 (t) < tfa(t) < $ 2 (2t), t > 0, for p = 1 one has from (34)

t

IQ

(41)

Jo

and for 1 < p < oo we have by (35), and ^^ <

/' Jo

(r)

< 1.

(42)

These inequalities on the Lp'-spaces can be estimated. There is a key classical result on weighted Hardy inequalities, for which simplified proofs were given by Muckenhoupt [1], using a clever use of interchange of integrals with Fubini's theorem to establish (for measurable U and V): rx

U(x)

Jo

p

f(t)dt

i-

dx\ J

r roo


\V(x)f(x)\pdx

(43)

holds for some 0 < C < oo iff = sup /

\U(x)\pdx \ \

r>0 Ur

\

\V(x)\~pdx


(44)

I/O

with B < C < pp (p1) p' B for 1 < p < oo, and JB = C i f p = l o r p = +00. This result with U(x) = [(j)2(x)]p/x and V(x) = 4>i(x)/a;, implies that the right side of (40) is bounded by f°°

cpc3c^p / Jo

$1

//^l/l > s] —

ds

9.3 Strong and weak type inequalities in Orlicz spaces

367

with cp = 1 for p = 1 and cp = p(p'}^-1 for 1 < p < oo. Consequently, the second term of (38) is bounded by cpc3c^p. Hence if we set Ci > i i max(c 2 ,C3Cp), then (38) is bounded by 1. Taking C > 2c0c1; we get (36), and hence the main result. If ^2(^2) < oo, we can take $2 as the principal part of an N-function that satisfies the hypothesis of the above result, and then the same result holds with (possibly) changed constants, but the conclusion is unaltered since we get equivalent norms on L $2 (/z 2 ). n Similar interpolation results are available for weak (p, p)-type, 1 < p < oo, and (oo,oo)-type but <JE>i = <£2, and "joint-weak" and "joint-strong" types with different (diffuse) measures, considered by Cianchi in the earlier noted papers. The condition on the TV-functions are complicated and are given as technical results needed for (Sobolev) embedding type inequalities to be used in the PDE analysis. So we shall omit further discussion of these specialized results. It is observed that there is no general Riesz-Thorin type interpolation applicable to large classes of Young functions, than what we have given in our earlier monograph. We end this discussion with the following "concrete" application to a Dirichlet problem, also due to Cianchi [3] where a detailed proof may be found. Theorem 5. Consider the Dirichlet problem "

d f

du(x)\

"dMx)

.

- E or K-to-ar- + £ -sir- = °. m G> i,j=l

UX

i

\

UX

3

)

t=l

(45)

OX

i

with the boundary condition u = 0 on dG where G C lRn, n > 2, is a bounded domain. Assume f — ( / i , - - - , / n ) is a vector with fi e L2(G), and the coefficients a^(-) are bounded measurable and satisfy the (strongly elliptic) condition Eij=iOij(x)^j > |£|2(= ££=i |&|2A for a.a. (x). Let $ be a Young function such that $(£2) is convex, and n is defined as (with

,

(46)

where ^ is the complementary Young function to <&. Then the weak solution u of (45) satisfies the embedding relation IMI ( * B ) < A-||/ll(*),

(47)

368

IX. Nonlinear PDEs and Orlicz Spaces

where \f\ = [Y%=i \fi\2]*, for an absolute constant K > 0 depending only on $,n and the volume ofG. As noted clearly, the complete details of proof use the interpolation of the type given by Theorem 4 and several other estimates. The details may be found in the above reference. We next turn to another type of application for composition and Nemitsky (nonlinear) operators in the final section of this chapter. 9.4 Composition and Nemitsky operators in Orlicz spaces So far we have considered quasi (and semi)-linear operators on Orlicz spaces, but there are important applications in which more general classes of nonlinear operators appear. Consequently we discuss briefly in this section two classes of nonlinear mappings namely composition (also called substitution) and more inclusively Nemitsky (or superposition) operators, introduced as follows. Definition 1. Let G C Mn be a domain (i.e., a nonempty open set), and F : Gx]Rm —> M be a mapping such that F ( - , •) is a Caratheodory function in the sense that (a) for each y e lRm, F ( - , y ) is //-measurable on G (p, =Lebesgue measure), and (b) F(x, •) is continuous for a.e. (x). Then (I) the mapping TF : (/i, • • • , / m ) ^ F ( . , (/!,...,/„)(•))

(1)

is called a Nemitsky operator, and (II) if m = 1, F is independent of G (i.e., F : M —> M), and is continuous, then it is termed a composition operator, so that Tp : / i—>• F(f), is well-defined for / in a suitable function space. Although the concepts appear simple, they include vast areas of analysis. We consider special classes of these operators to illustrate their wide applicability. Let us first discuss simple examples of (linear) compositions induced by measurable transformations on (general) measure spaces. Thus let (£7, E, v] be a cr-finite triple and S : Q —> Q a point mapping such that S-1(E) C E, i.e., S is measurable. Then the linear mapping T : L®(v) —> L®(i>) given by (Tf)(uj) — (f o S)(uj) = f(S(ui)), defines a measurable function as a composition for / 6 L*(z/), an Orlicz space. But T has a somewhat complicated structure, depending on the range of 5. We first discuss a couple of structural results in terms of the properties of S, essentially based on a recent paper by Kumar [1]. Proposition 2. Let 3> be a continuous Young function such that <&(w) = 0 iff u = 0, and (fi, S, v} a a-finite measure space. Suppose S : f2 —> (7

9.4 Composition and Nemitsky operators in Orlicz spaces

369

is a measurable transformation, then it induces a contractive composition operator T iff v(S~l(E}} < v(E], E E E. The same characterization holds for T bounded by M > I , if $ E V (globally) instead, so that the condition becomes iffv(S~l(E}} < Mv(E], E E S (M = M(M, 3>) > 0 is a constant). Further, the contractivity of T is equivalent to the nonexpansiveness of S. [Recall that a (perhaps nonlinear) mapping H : X —>• y (X, y Banach spaces) is nonexpansive if \\H(x) — H(y)\\y < \\x-y\\x-] Proof. In one direction let T be a contractive linear operator induced by a measurable 5 : fJ —>• Jl. If E1 E E such that XE € L*(v) one has ||TXE||($) < •M"||X£;||($) < ||xs||($) since T is contractive, so that M < 1. But TXE — XE ° S — Xs~l(E)- So we have (cf., e.g., Rao and Ren [1], p.78) )] ' = \\TXE\\W < \\XE\\W = f^1 (^y)] ' • (2) The continuous $ is strictly increasing on [0, oo), and 3?"1 is concave increasing. Thus v(S~1(E)} < v(E) as desired. In case M > 1, and if $ E V so that $(u}$(v) < $(buv), u, v > 0 for some 6 > 0 (cf., Rao and Ren [1], p.28), one has r

/ i \ i—1

\(s>^ = M or \

\

\

/ / /

*M'

v-l

Thus

l

\MbJ v(E]

1

and v(S~ (E}} < Mv(E} with M = [fc^)]" . The converse procedure is the same in both cases in which one can take M > 1. By the Radon-Nikodym theorem, if voS'1 «v (i.e., i^(S~1(E)) < Mv(E] as above), then dv o S~l C = —-: < M < oo, a.e. \Jils

We now take M > I . For 0 ^ / E L (z/), one has ,(

Tf

(3)

IX. Nonlinear PDEs and Orlicz Spaces

370

/ M\\f\\(*)

dv o S~ -dv dv

W f

<

M- \||/ (*)J

Thus ||T/ ($) < M||/||($) and the linear operator T is bounded by M. d A further property of such induced compositions can be given as follows. Proposition 3. Let T be an induced composition operator by a measurable mapping S : Q —>• O in L®(v] as above. Then T is injective iff S is a surjective mapping outside of a v-null set. Proof. Let T be injective and consider the set £1Q = {u; £ Q : ^^—(a;) = 0} and the subspace L $ (f7o, E(^o)> i'o) = L®(v0) of L?(v] whose elements are supported in 00 with VQ = t>|E(£7n)- It is then clear that

dv o dv

=0

(4)

supp(f)

where supp(/) = {cu € : /(a;) ^ 0} e E is the support of /. Then is the null space of T. This is seen as follows. If T(L*(z/o)) = 0, a.e., L $ ( = 1. Since (v o S-I)±VQ by (4), 0 7^ / e Z/*(V 0 ), then we may take ||/ we have n0 5 dv

Hence T/ = 0 a.e. and I/*(^0) C A/r, the null space of T. On the other hand, let / € A/r, so that / o 5 = 0 a.e. (v}. Hence 0 =

}dv = f Jfi

dv

This implies that ^^—^ supp(f) = 0, a.e. and so / e L^(VQ). Consequently A/T = L?(VQ). Now assume that T is injective, i.e., A/r = {0} = L®(VQ). Then (y o 5"1)J_zxo and z^(^o) = 0. Hence S is surjective on f2 outside of a iA-null set (zx-equivalent to QQ). Conversely, if v($l — S(£l}} = 0, then L?(VQ) = {0} in the notation above and A/r = {0} = L?(VQ), so that T is injective. d The above computation shows that, in case the measure v is diffuse, then v(S($l}} > 0 implies that L?(VQ] is infinite dimensional, and thus for

9.4 Composition and Nemitsky operators in Orlicz spaces

371

noninjective composition operators T induced by S1, their null spaces are also infinite dimensional. Such induced operators are a subclass of linear operators on Banach function spaces, and a great deal of specialized (and sharper) results can be established by considering different mappings 5* and triples (fi, E, v}. IfticC and L*(Q) is replaced by analytic function classes, a detailed analysis of compositions is given in the recent book by Cowen and MacCluer [1], and other types are detailed in Singh and Manhas [1]. Here is an example of a nonlinear composition, leading to a different type of analysis, where we consider Sobolev spaces for treatment. When the operator is nonlinear, its boundedness and continuity are distinct, and this already shows a need for additional work on the problem. The embedding results of the preceding sections are a certain special aspect of compositions. We now elaborate the general problem. The following result, due to Bourdaud [1], explains the issues. Theorem 4. Let Wm'p(M), m>2, be a Sobolev space that can be embedded continuously into L°°(]R). Let F : 1R —> M be a continuous mapping with F(0) = 0. Consider the nonlinear composition operator Tp on Wm'p(]R} into M(1R}, the space of real /^-measurable functions, where /z is the Lebesgue measure. Then TF(Wm>p(M)) C Wm'p(JR) and the inclusion is a continuous embedding iffF& W^P(]R), i.e., F is in this space on every compact subset of M andTF is thus a "differentiable composition operator". The details of proof are many, and the reader is referred to the original paper, or alternatively to the recent book by Runst and Sickel [1], p.268. The result shows real differences between the induced linear and the nonlinear composition operators generated by functions, such as F. An extension to Orlicz-Sobolev spaces Wm'^(JR) is possible. A more inclusive case in the context of Nemitsky operators will be discussed below. Recall that, in Definition 1, there is a function F : G x lRm —> ]R generating the Nemitsky operator TF on various spaces of functions. This is a Caratheodory function which, for achieving some generality and for applications in PDE, is usually assumed to be: F ( - , t ) is (Lebesgue) //-measurable on G C JRn for t = (ti, • • • , tm) e Mm, and F(x, ti, • • • , tm) is separately continuous in tk, k — 1, • • • , m for each x = (xi, • • • , xn) e G C JRn. In Theorem 6 below, we consider TF(f)(x)

= 7>(/i, • • • ,/ ra )(x) = F(xJ1(x)J2(x)1

• ••/,»(*)),

where / = (/i, /2, • • • , fm) e xf=lW^ (G). If m = 1, /(= A) € Wk*(G), one should define the kth derivative | of / in some weak sense in each

372

IX. Nonlinear PDEs and Orlicz Spaces

coordinate Xi, and this presents some complications because there will be implicit differentiation together with chain rules in this context. Thus a reasonable assumption is to consider / to be locally, (meaning on compact sets) absolutely continuous in each coordinate, i.e., on almost every line L parallel to a coordinate axis, / is assumed to be locally absolutely continuous on L n G, as well as the function F ( - , t ) . But then the derivatives belong to equivalent classes and for further work (i.e., to iterate the procedure for higher derivatives) it is necessary to select members from these equivalence classes. Hence some type of a "lifting operation" is needed, and such an operation is only possible if these classes are essentially bounded (cf., e.g., Rao [20], Chapter 8 on the lifting map). Several additional conditions are needed to simplify the problem. We discuss it here first for k = I . Thus if A(G] denotes the class of functions locally absolutely continuous on L n G for almost every line L parallel to the coordinate axis, let A'(G) be the class of functions / such that dxj exists for each z, and / coincides a.e. with an element / G A(G) . Hence each equivalence class of the Lebesgue differential dxif of / € A(G), agrees a.e. with dxj of / G A'(G). To proceed further, one restricts G to have a general cone property (cf. Thm. 1.1) in that for each s > 0, G = (J^Gj and Gj = \JX£Bj(x + Qj), diam(Sj) < e with Qj as a parallelepiped with a vertex at 0 (Bj C Gj). Thus G is a "good" domain. Under these conditions a classical theorem of E. Gagliardo has been suitably modified and then a fundamental embedding theorem with Nemitsky operators on Sobolev spaces Wl'p(G) —>• Wl'p(G) has been established by Marcus and Mizel [1]. It is extended to Orlicz spaces by Grahame Hardy [2] which we now present as follows. Proposition 5. Let G C Mn be a bounded domain with the (above noted) cone property, and <£> an N-function. Suppose f G A'(G) and its derivative dxj G L*(G), i = 1,2, • • • , n. Then f G L*(G) itself. We thus present an Orlicz-Sobolev space version of a result on Nemitsky operators, where the various conditions are abstracted from the Lebesgue case (i.e., of Sobolev spaces). Theorem 6.(G. Hardy [2]) Let G C Mn be a bounded domain having the cone property, and F : G x lRm —> 1R be a locally absolutely continuous Caratheodory function. Suppose Q, Mk and Qk, k = l , 2 , - - - , m are Nfunctions satisfying the following sets of conditions [recall that the (generalized) Sobolev conjugate function Mf. of Mk is well defined and is again an N-function, as seen in Proposition 1.5]: (I) $ 6 A 2 (oo) n V 2 (oo), $ -< Mk, Mk G A 2 (oo) [and hence Q^1 o M* G

9.4 Composition and Nemitsky operators in Orlicz spaces

373

A 2 (oo), $~1 o Mk € A2(oo)/; where 1 < k, j < m; (II) there exist complementary pairs (4>jt, $k) such that $o$k^Mk

and

$otyk-^Qk,

k = 1, 2, • • • , m;

(HI) there exist nonnegative functions a,b,ak,bkj such that \dXiF(x,t)\
+ b(t) a.e. on G, i = l , 2 , - - - , n

and

/or a/mos* all x and all t, ( x , t ) 6 G x l^"1, w/iere a e L*(G), 6 : lRm ->• .ZR+ and fejfcj : JR —>• JR+ are continuous and Borel functions respectively; ak € LQk(G), bkk are continuous, and finally define the composition operators Tb : LMi (G)x- • -xLM™(G) -> L*(G) and similarly Tbk. : L M /(G) ->• L«*(G), T/ien i/ie Nemitsky operator Tp induced by F, maps x™=1Wl'Mk(G) into W '^(G), and is bounded as well as weakly continuous, i.e., (/[, f£ , • • • , /^) -> (/i, f t , -', fm) asr -too in norm implies F(; /[, • • • , frm] -)• F(-, /i, •, /m) weakly in Wl^(G}. [This continuity is stronger than weak continuity, and is also termed in the literature, "demi- continuity". Here Tp is defined on the tensor product space x^_lW1'Mk(G) equipped with the product norm, or equivalently the maximum of the individual norms as in Section 1. 4-] Further if we enlarge the range space of Tp to L®(G), from the subspace Wl'®(G], then TF is actually continuous relative to the norms in both spaces. 1

The details of proof of this result are many and have to be studied first in the Lebesgue-Sobolev case which is due to Marcus and Mizel [I]. Then the above version to Orlicz-Sobolev spaces is extended by G. Hardy [2]. Here we present the statement to show how several specialized conditions have to be imposed in order to cope up with different types of nonlinearities. However, the following examples show that Theorem 6 includes a substantial part (not all) of the Marcus and Mizel theorem, and also contains others that can not be obtained through the corresponding Lebesgue-Sobolev results. Example 7. Let n > 2, K p < qk < n and q'k = ^_ (j.e.; JL + _^ = I;. u) = \U\P and $k(u) = £|u| % #*(*>) = JH^, e A 2 . Choose

Mk(u) = *[**(«)] = (£} |u|«, Qk(u] =

then

^ e A 2 H V2;

374

IX. Nonlinear PDEs and Orlicz Spaces

so that M£(u) = c\u qk, where c is a constant and q^ is the Sobolev conjugate of qit, i.e., ql = ^±~- These N-functions satisfy conditions (I) and (II) of Theorem 6. Thus Theorem 6 applies to a large set of Lebesgue-Sobolev spaces. To present the second example, we need the following lemma, due essentially to G. Hardy [2], which is of some independent interest. It is also used to simplify condition (I) of Theorem 6 in some applications. Lemma 8. Let <£(u) = JQ (f>(t)dt which defines an N-function and let $ e A2(oo). Suppose that for n > 2 B$ = limsup

< n.

(5)

Then $* € A 2 (oo) and B*. = lim sup —f-f < - - , $*(u) n - J5$ u->oo

, .

where $* is the Sobolev conjugate of$, i.e., its inverse is given by

[The additional property $~1(1) < oo may always be assumed by passing, if necessary, to an equivalent N-function $ ~ $ having the same behavior as $ for large values.] Proof. Let e > 0 and /3 = B$ + £ satisfy /3 < n. Then there is a UQ > 0 such that ^^ < J3 for u > u 0 , and so ,^ ., ' < j 3 ,

t[^~i(t)\

t>vQ = $(w0),

(8)

where ($~1) is the right derivative of $-1. By (8), one has v

and hence

dt

,

( v\D

6

9.4 Composition and Nemitsky operators in Orlicz spaces [It should be noted that (9) implies that follows from (8) that l

375

= oo since ft < n] It

r$ (t

Jv0

tl+n

tn

^0

=

ft

$~ (Vni 1

&~ [ v ] 1 Vn

vn

+

ft\r

m

n JVQ

*-'t'+i(*)

Letting /?* = ^^, we have from (7) and the above inequality: (10) "1

^

1

Since [^^v)]' = $- (^)^" ~", (10) yields

^—1 / \

i

(11)

+

But (9) implies that li

(v)} = 0, since ft < n. Hence

limsup

or

by letting v = $* (u) . This gives (6) . n Remark 9. If $ 6 A2(oo) fl V2(oo) and 1 < A$ < B
n — A^

(12)

where ^1$, = lim inf "^ Ay - In particular, if A$ = B$ = C$ (say), where = U—lim >OO

*(,UJ

, 1 < C$ < n, then = lim

= 5$, = C$ where >( (u)

(13)

For example, if <J?(w) = [(1 + \u\) log(l + |w|) - \u\]p, 1 < p < n, then C$ = p, and its Sobolev conjugate TV-function <&* has quantitative index

376

IX. Nonlinear PDEs and Orlicz Spaces

C$t — C$(= ^-), the Sobolev conjugate number of p. Lemma 8 and the remark show that for a bounded domain G C Rn, if L*(G) (W*'*(G)) is separable (reflexive), then L**(G) (Wk'®*(G)) is also separable (reflexive). Example 10. Let n > 2, 1 < p < n, $(u) = |w|p, $ fc (w) = (1 + u\) log(l + |w|) - |w| and * fc (u) = e |v l - |v| - 1. Then Mk = $ o 3>k e A2(oo) since lim "M vv = p. By Lemma 8, M£ e A2(oo). Thus the conditions of Theorem 6 are satisfied. But LMk(G) is not a Lebesgue space and W 1>Mfe (G) is not a Sobolev space. However Theorem 6 covers these spaces that are not included in the classical work. Thus far we considered only the first order Orlicz-Sobolev spaces, and there is naturally a kth order version. It involves, as may be anticipated from Theorem 6, kth order Sobolev conjugate TV-functions, and we shall briefly discuss it for completeness. Recall that for an A7"- function <E> and integer n > 2, the Sobolev conjugate <3>* is the inverse of ^J1 given by (7), and $* is an TV-function if ^^(l) < oo but $7^00) = oo (cf. Proposition 1.5). If $1 — <£* and inductively <££ is defined as in Section 1, all of which depend on n, then there exists the smallest J0 > 1 such that $}0+1 is not an TV-function (I < JQ < n), since <£}o+1(oo) < oo. Then a Nemitsky operator in W fci$ (G) can again be considered, and in fact the following result for iRm, m — 1, is obtained by J. Appell and G. Hardy [1]. Theorem 11. Let G C lRn be a bounded domain with the cone property. Let $ be an N-function, $ e A 2 (oo) and F : G x ]R —> 1R be a Ck -function such that k > JQ + 1 where JQ is the exponent of the maximal Sobolev conjugate given above. Then the Nemitsky operator Tp is defined on Wk^(G) into itself, and moreover is bounded and continuous (in their norm topologies). We refer the reader to the original paper for a proof which uses some properties of a class of spaces, called "multiplicator" Orlicz spaces of independent interest. Briefly, if <3?i, $2 are TV-functions and $1 -< <£2 for large values, or that $1 -<;-< $2 (i.e., lim^oo ^-ifl = 0) we define ($! : <E>2) by the *1 W relation v>0

and if lim sup ^^ < oo for some A > 0, then we set ($1 : <& 2 )(u) = 0 for u—voo

2 u

^>

u\ < 1, = oo for \u\ > 1. It is seen that ($1 : $2) is a Young function. Then one finds L^(G) »L* 2 (G) = L^ i: * 2 )(G) when the left side space is endowed

9.4 Composition and Nemitsky operators in Orlicz spaces

377

with the (norm) functional < 1}.

(15)

If $i(w) = \u\pl/pi, $ 2 (w) = HP2/P2, then $! -<-< <3>2 iff Pi < P2, and hence

r

r

pl

p2

These spaces are of interest in the work on (nonlinear) integral operators, in addition to Nemitsky (or superposition) operators. For an extended discussion of this topic we refer the reader to the work by Appell and Zabrejko ([1], p.62) where L$1 • L*2 is denoted by L$1/L$2 which is not a quotient. To note the difference between the linear integral operators and of the nonlinear kind, we also include a brief discussion, arising from a key application. A classical problem raised by Gelfand and Vilenkin ([1], p. 275) can be stated as follows. Let C£°(JR} be the Schwartz space of infinitely differentiable functions on ]R with compact supports. If /,/„ G C?(M), then / „ - » / € provided {/„, n > 1, /} all vanish outside a compact set and f^(x) — uniformly as n -» oo, for k > 0. This defines a topology under which C^°(M) becomes a complete locally convex linear topological space. Let M : C~CR) -> LP(P) (or L*(P)) on a probability space (ft,£,P), be a continuous linear map, i.e., / „ — > • / in C^°(JR) implies M(fn) —>• M(f) in LP(P), or only in probability. Such an M(-) is called a generalized random functional. It is said to have independent values, if M(/i) and M(/ 2 ) are independent random variables whenever /i • /2 = 0, i.e., /i,/ 2 have disjoint supports. The structure of M can be studied by considering its Fourier transform L(f) = The independence hypothesis implies immediately that for /r/ 2 = 0, L(afi + 6/2) = L(a/i)L(6/2) for all a, 6 e JR, /!,/2 G C?(M). Such an L(-) : C%°(]R) —>• C is termed a local functional by Gelfand and Vilenkin who raised the problem of their characterization. For instance L(f) = eM^ where M(-) is a superposition functional defined by M(f) = f^ F(x, /(ar), /'(*), • • • , f ™ ( x ) ) d x ,

f € CC°°(J2)

is of interest here and L = eTF. If C™(M) is replaced by C^(JR), or CC(G), where G is a locally compact group (or by an Orlicz space), each of the

378

IX. Nonlinear PDEs and Orlicz Spaces

characterizations is nontrivial. For the case that G is a locally compact space and M : CC(G) —> -K, which is additive on functions of disjoint supports, a characterization is given in the book (Rao [20], p.517 ff). If M is defined on Lebesgue spaces and having the same property, then Marcus and Mizel [2] have characterized M. It will be of interest to extend these results to L?(p} also, based on the above ideas. There are many other possibilities and several different spaces to consider. We have to leave the subject at this point. Finally it is of interest to remark that the study of Nemitsky operators on Orlicz-Sobolev spaces of infinite order is also useful. There are some new problems here since the generalized Sobolev conjugates are not available any more, and projective and injective limits of spaces play key roles, as noted in Dubinskij [1]. This is largely an unexplored area. Bibliographical notes. We have given most of the references in the text, and it suffices to make some brief remarks here. Section 1 presents a broad framework for the PDE analysis in the context of Orlicz spaces exhibiting a flexibility afforded by it in applications. The basic motivation for the existence and uniqueness in the context of Dirichlet's problem is followed, adopting a readable presentation of Mikhailov [1] which shows a clear role of functional analysis in this subject leading to a use of weighted Orlicz-Sobolev spaces of finite order. The case of (generalized or higher order) Sobolev conjugate TV-functions and the corresponding embedding results are based on Donaldson and Trudinger [1] and its extension by Adams [1], the latter (generalized) result (Theorem 1.7) appears perhaps for the first time only in this book. The results on infinite order Orlicz-Sobolev spaces are treated briefly, partly for space reasons (although they have an interesting role in certain stochastic differential equations) and partly because the subject is not yet well-developed. Our discussion may help motivate researchers to explore this area further. Section 2 discuss some key results on removable singularities in the generality introduced by Bochner and further developed by Harvey and Polking [1] in the Lebesgue context. The main result (Theorem 2.2) is due to Shapiro [1] in its extension to Orlicz spaces. This indicates an interesting widening of the area of applications. Similarly the weak comparison principle plays a key role for the uniqueness results in PDE, and we briefly included a result (Theorem 2.3) which also is due to Shapiro [2] to emphasize this fact. We then consider several results based on strong and weak type inequalities, and consequent interpolation analysis. They complement and sharpen similar ones given in the companion volume (cf. Rao and Ren [1], Chapters V and VI). Many of these results involve intricate computations using various

9.4 Composition and Nemitsky operators in Orlicz spaces

379

classifications and order relations of Young functions. This work is recent and Theorem 1.9 as well as most of Section 3 is taken from the papers of Cianchi's ([l]-[3]). Theorems 3.2 and 3.3 are due to Gallardo [1], which show the optimality of these conditions. We thank Dr. Cianchi for sending us her interesting work prior to publication. In the final Section 4 of the chapter we consider composition and (superposition or) Nemitsky operators. The former has both linear and nonlinear components. The linear part specialized the structural analysis of operators using methods of linear functional analysis, which are better understood. We include a brief account of these, with Propositions 4.2 and 4.3 from Kumar [1], and indicating other types of compositions on analytic or other function spaces. We then turn to nonlinear analysis in the rest of the section. The material on nonlinear superposition or Nemitsky operators in the Lebesgue case is studied in several papers by Marcus and Mizel of which we refer to some basic results in [1]. They also treat local functionals on various spaces. Some of their results have been extended by G. Hardy [2] in several parts. The Gelfand-Vilenkin problem has deep probabilistic significance, and some aspects of it have been considered in Rao [15] on the continuous function spaces (giving also earlier references to some other works), and on Lebesgue spaces by Mizel [1]. The methods in various spaces differ considerably showing the intricate nature of the subject. Applications to nonlinear analysis and PDE of Nemitsky operators have been summarized in Runst and Sickel [1], and a general survey of these operators on different spaces is in Appell and Zabrejko [1]. We thus conclude the work at this point, but also outlined numerous problems of interest for future investigations.

Chapter X Miscellaneous Applications

This final chapter is devoted to various applications that are raised by the analysis in some of the preceding works but could not be included there. First we treat the (new) class of Beurling-Orlicz algebras to the extent known. The topics considered in the other sections are as follows. Studying another geometric invariant, called the Riesz angle of Banach lattices, the second section contains an analysis as specialized and sharpened to Orlicz spaces. Section 3 complements the earlier work on embedding properties of Orlicz spaces restricted to discrete measure, hence sequence spaces. The material here is due to Ren and it sharpens aspects of Chapter I. Section 4 is devoted to results on uniform Gateaux differentiability of norms when the underlying measure space is restricted to be just diffuse or discrete. The final section is devoted to generalized Orlicz spaces, when the measure is finite, but the 0-function is allowed to be concave. Most of the work here is recent and complements that of the earlier general study, containing refinements.

10.1 Beurling-Orlicz algebras It was shown in Section 1.3, in particular in Propositions 1.3.13 and 14, that L*(JR) is closed under convolution if $ is a Young function such that $'(0) > 0. Hence L^(JR) is an algebra under convolution. It is possible that $ G A 2 but some times $ ^ A 2 . In these studies n is the Lebesgue measure. However, there are other convolution algebras with more general p, that is absolutely continuous relative to Lebesgue measure. Such were introduced by Beurling in the context of Lebesgue spaces and we present their extensions to Orlicz spaces, motivated by the preceding L$(JR) algebras. The basic idea is to consider positive upper semicontinuous submultiplicative weight functions as densities of the new measures and then study the structure of the resulting classes, to be called Beurling-Orlicz algebras. More precisely,

380

10.1 Beurling- Orlicz algebras

381

let w : M —> M+ be a mapping, termed a weight function, satisfying: (i) 1 < w(x) < oo,x 6 1R, (ii) w(x + y] < w(x}w(y},x,y G M, and (iii) w(-} is Borel measurable. For instance, w(x) = (1 + |X|) Q ,Q! > 0 satisfies these conditions. Since logw is subadditive, it (and hence w] is bounded on compact intervals, although not necessarily continuous, but can and will be assumed upper semicontinuous by replacing it with w(x) = limsupw(x if necessary. [For a discussion of subadditive Borel functions, one may consult Hille and Phillips [1], Chapter VII.] We first present an easy extension of Proposition 1.3.14 as: Proposition 1. Let $ be a continuous Young function with $'(0) > 0 where <3>' is the right derivative of$. If ft : A i—>• fA w dp, A C JR is a Borel set and // the Lebesgue measure, then L®(ji) on (J7, S,ju), is a convolution algebra in that, '* ' denoting convolution,

ll/*0ll*<0>||/||*y|*, f,geL*(j}),

(l)

where || • ||$ is the Orlicz norm, 0 < CQ < oo an absolute constant depending on $, and w(-) is a weight function introduced above. Proof. The argument is a modification of the earlier one, and can be outlined. Let 0 ^ g 6 L*(Ji} and K = \\g\\9) so that Jm$(-L}dJi < 1. Hence we get from the inequality |x|$'(0) < $(x) and the L^/^-norm,

Then for /, g e L®(ji), consider with Fubini's theorem,

< sup { / \g(y)\w(y) dfj.(y) I \f(x - y)w(x - y}h( (JR Jm. w

l*. since w(x) < w(x - y}w(y], (3)

Hence / * g 6 L^(]l} and the (Banach algebra) norm inequality is obtained. D

382

X. Miscellaneous Applications

The convolution algebras Ll(Ji} were introduced by Beurling [1], and are often called Beurling algebras for the weights w (cf. Reiter [1], p. 14). The corresponding L^(J1) can (and will) be termed Beurling- Orlicz algebras. These objects admit a further interesting and deep extensions as noted by Beurling himself in 1955, but published only in 1964 in [2], for the Lebesgue spaces. In the latter paper, he introduced certain subspaces of Ll(IR) of the following type which we present with an Orlicz space extension, again with weights. Let $ be a Young function satisfying a A' condition globally. Suppose its complementary function ^ is continuous and has the property ^(xy) < byty(x),x > 0 where by > I for each y > 0. This is a form of the A2-condition. Recall that <3> 6 A' means that $(xy) < C$(x)$(y),x, y > 0 for some C > 0, and replacing $ by ^, we may (and do) take C = 1 for convenience. Let W be a set of weights satisfying W — {w > 0 : fKwdp, < N(w) < 00} where w is a measurable (weight) function, and for each w G W and A > 0, Xw G W . Here N(-} is a norm with the property that N(wi*W2) < N(wi}N(w2) where V denotes convolution. Suppose moreover that {W, N ( - ) } is complete in the 00

00

sense that [wn,n > 1} C W, £) N(wn) < oo implies w = £ wn 6 W and ra=l

oo

n=l

N(w) < £ N(wn). An example of N(-) is || • ||i or || • |$ of Proposition 1, n=l

where <E> is suitably normalized which is possible. Next consider a subset WQ of W given by WQ = {w G W : N(w) = 1}, so that it is the positive spherical shell in W. Suppose that $(1) = 1, for convenience, by a redefinition if necessary. Then we can obtain a modified (or weighted) Holder inequality as follows: Proposition 2. //$ € A' and ^ 6 A 2 , for w £ WQ we define the weighted gauge norm \\f \WiW = inf A > 0 :

K*

j wd/z < 1 ,

(4)

and similarly ||g||(*)x is defined, then one has (with ^ as Lebesgue measure)

JR.

/(/|dA*<2||/||(*)>ll,|M|(*)x

(5)

where w' = b (^-L^) > 0, with y t-> by — &(•) > 0, as the A 2 constant of $ above. Proof. The argument is a simple modification of the classical one, and can be outlined as follows. By the A' condition of $ and the A 2 condition of ^

10.1 Beurling-Orlicz algebras

383

so that b ($-ifw)) > I, we have for &i, &2 > 0

Jm

*1

Jm.

&2

<

f

/

\

=

/

^

I $ I —$

1

/

\

\

,

(iy) \ dy+

^

f * JK \ki

f

-r

( 9

\I> —

/

\

^

/ c^Li,

m.

(6)

where w' = b ($^TT^T) > 1- Let fci = ||/||($),w and k? = ||^||(*)x m (^)> then (4) and similarly ||^||(*)x with w' in place of w imply the weighted Holder inequality (5). D Next consider (L^(IR), \\ • ||($),w) and (L*/(JR),|| • ||(*)x) as the corresponding Orlicz spaces, and define a pair of new spaces A®, B®\

: w G W0},

A* = A*(M, W0) =

(7) (8)

with w' determined by w as in (6). Define the norms in these spaces as: II/IU* = sup{||/||w,w : w G W0}, f € ^

(9)

and

w

WQ},

(10)

where w' is again determined in term of w, i.e., as in (6). In case $(x) = \x\p, 1 < p < oo, then Beurling [2] has shown that (A®, \\ • \\A*} is a Banach space dual to (B*, \\ • ||B*) and the latter is a convolution algebra. Moreover, every continuous linear functional on B® is representable as an integral (relative to n} of a unique element of ^4*. [Note that in this particular case both 3>, ^ satisfy a A'- condition.] The corresponding results for Beurling-Orlicz algebras are of concern here, and are naturally more intricate. All these considerations are meaningful if JR is replaced by a locally compact abelian group with p, as a Haar measure. The following discussion is relevant in connection with the preceding result, regarding subalgebras of Ll(G).

384

X. Miscellaneous Applications

Discussion 3. Taking w(-) to be also bounded in the above proposition, it is clear that L®(p,} is a Banach algebra contained properly in L 1 ^), the classical convolution algebra. Other subalgebras can be studied. An interesting such class was considered abstractly by Segal as follows. Let Ll(G] be the convolution algebra on a locally compact group G with a (left) Haar measure H on it. Then a set S C Ll (G) is called a Segal algebra if there is a norm || • \\s under which it becomes a Banach algebra, and || • ||5 has the properties:(i) ll r a/lls = \\f\\s where (raf)(x) = /(ax), a € G, (i.e., ra is a translation operator), and (ii) a (->• ||ra/||s is continuous in the sense that for each £ > 0 there is a neighborhood U of the identity of G, such that \\raf — f \ \ s < e for all a E U. Thus with w(x) — I (or a constant), and $'(0) > 0 , then L*(G) C Ll(G] and is a Banach algebra under convolution of Proposition 1, but the mapping a i—>• \\raf ($) will be continuous iff <3? G A2- Thus L^(IJL) is still a convolution (Banach) algebra even if <£ ^ A 2 (but $'(0) > 0), so that there are examples of convolution algebras L$ (G} C Ll (G} which are both Segal as well as non-Segal algebras. It will thus be interesting to extend the analysis of Segal algebras, given by Reiter [2] to Beurling-Orlicz algebras yielding new properties of the latter. This studt is not available at present. In the rest of the chapter we consider other geometric properties of potential interest in applications complementing the earlier analysis in this book.

10.2 Riesz angles of Orlicz spaces A new geometric invariant, called the Riesz angle, was introduced by Borwein and Sims [1] for certain Banach lattices, and then established some fixed point properties. This invariant will now be analyzed for (real) Orlicz spaces, which are Banach lattices, if the underlying measures are either purely discrete or diffuse. Let us first state the concept of a Riesz angle. Definition 1. The Riesz angle a(X] of a Banach lattice X is given by (V=max or sup) a(X) = sup{||(|*| V li/DH : |jz|| < 1, ||y|| < l , x , y 6 X}.

(1)

Thus 1 < a(X] < 2. A Banach lattice X is said to have the weak orthogonality property if xn E X, and xn —> 0 weakly implies liminfliminf \\(\xn\ A |x m |)|| = 0, (A — min). 771—>OO n—>OO

(2)

Then Borwein and Sims [1] established that if a Banach lattice X has the weak orthogonality property and a(X] < 2, then X has the weak fixed point

10.2 Riesz angles of Orlicz spaces

385

property in the sense that every nonexpansive self-mapping of a nonempty weakly compact convex subset of X has a fixed point. This is useful in applications, and hence we estimate the Riesz angle a(/(*)) with the gauge norm which has some independent interest, and is also related to the Kottman constant studied in Chapter V. Theorem 2. For an N-function $ and the Orlicz space l^ we have -> -V)<^W)< ^ C4/ 0!$

(3)

where a$ = liminf ^AA, and —

inf '*= <**= * *=i,2,- ITTm. $-ii *

inf

Proof. Note that a% was already seen in Section I.I. We consider the left side of (3). By definition of aj, we canfind1 > un \ 0, lim^ *"1("n) - ~° Then for any 0 < e < 1 there is n0 > 1 such that un < | and a| + e, n > n0. Thus if UQ — uno, then ,_if

x

$-1(uo)

Let A;0 be the integer part of ^-, so that /c0 < ^~ < ^o + 1 and hence 1 < 2u0(k0 + 1) or ^
i=l

i=k0+l

so that /0 and ^0 have disjoint supports, e; being the vector with unity at the ith component and zero elsewhere, we get ll/o||(*) = ||0o||(«) = Hence by (5) IKI/o|vM)||(*)

=

||/o

X. Miscellaneous Applications

386

l-2uo

establishing the desired inequality for a| of (3), since 5 is arbitrary. The corresponding result for a'$ is obtained by a similar procedure . In fact, let Zk and Xk be fc-vectors where Zk has zeros and Xk = $ -1 (^)(l, 1, • • •, 1). Define fk and #fc by fk ~

9k =

so that (7)

since \fk\ V |pfc| = ( X k , X k , Zk, Zk, • • •)• The arbitrariness of k in (7) shows that —1 9 .• tfci,z,

> --

1

.

^

Thus the left side inequality of (3) is established. Regarding the last half of (3), it follows from the definition of 5$ that (8)

Let /,^ e /(*) be such that ||/| ($) < 1, ||^||w < 1, so 1 for all i > 1. Let u be £$(|/(i)|) and |$(|^(i)|) in (8). Then ^(5*I/(OI) <

( /(Ol),

(9)

This gives for the modular p$:

(10)

10. 2 Riesz angles of Orlicz spaces

387

(10) implies ||(|/| V M)||($) < (a*)"1 so that (3) holds. D From this and some earlier work we deduce the following result obtained differently in Borwein and Sims [1] and Chen ([1], p. 118). Corollary 3. a(/<*>) <2iff$e

V 2 (0) .

Proof. If $ e V 2 (0) then by Theorem 1.1.7, a| > \ so that 5$ > | because £^gj > I for each 0 < u < \. Hence a(/<*>) < 2 by (3). Conversely, $ g V 2 (0) => aj = I so that a(/<*>) > 2 by (3) showing that o(/<*>) = 2, cf. (1). D Because of this corollary, for a(l^) one can restrict to $ 6 V 2 (0). Using Theorem V.2.1 on a Kottman constant K(l^), we obtain: Theorem 4. For an N -function <& 6 V2(0) with (f> as its right derivative, one has the following conclusions: (i) IfF*(t) = f is increasing on (O,^-^!)], then =2

= #(/*),

(11)

where C£ = limF*(«). * t^-o v y ^ If F$(t) is decreasing on (0, ^^(l)] (so C$ exists in M ): (A) either if C$ < oo so that $ € A 2 (0),

V2

or « C = oo «.e. $

Proo/. (i) If F*(i) /* on (O,^-^!)], then Cg exists and is finite. Hence by Lemma V.2.5, G$(u) = $-1$} *s a^so mcreasmg on (0> 2 ]j so that

and the first equality in (11) follows by Theorem 2. The second part of (11) is a consequence of Theorem V.2.6(i).

388

X. Miscellaneous Applications

(ii) If F<s>(t) \ on (0, ^"^(l)], then so is G$(w) on (0, \] and C$ < oo exists, so that *2-

Hence a(f<*>) = |I^I -m either of the cases (12) and (13). Since $ e V 2 (0), \2 '

so that /(*) is reflexive or not according as C$ < oo or C$ = oo, the result on K(l^) follows immediately from Theorem V.2.6(ii) and Remark V.I.7 (ii) respectively. D Let us illustrate these properties in the following. Example 5. Consider $(w) = e\u\ — |w| — 1 and its complementary function *, namely ^(v) = (1 + v|)log(l + \v ) - \v\. Then F$(t) = *|^- and Fy(s) — Sy y respectively satisfy condition (i) and (ii) (A) of Theorem 4 (cf. also Example V.2.7). Hence we have « 1.487. ^

\2^

We observe that in Example III. 3. 6

= \/2. The normal structure coefficients N(l^) and N(l^) for this pair have not been obtained. Example 6. Consider the TV-function $ given by <J?(u) = w 2 e ~ H ,

u

& M,

studied by Gribanov [1]. It satisfies condition (ii) (B) of Theorem 4. Indeed,

which is decreasing on (Q,$~l(l)],C$ = oo so that $ E V2(0). However, $ £ A 2 (0). It follows from (13) that a(/W) = 5-Jll « 1.282 < 2 = #(/<*>). *

12''

We also note that WCS(l^) = N(l^) = l(cf. Corollary III.3.3).

10.2 Riesz angles of Orlicz spaces

389

Remark. Chen ([1], p.119) shows that if $ 6 V 2 (0), then m<*> has the weak fixed point property, where m^ is the closed separable subspace of l^. It is seen that a(m^) = a(l^) for every <J>. We next consider Riesz angles for diffuse measures, especially if £7 = [0, oo) or [0, 1] and $ an TV-function.

Theorem 7. Let a$ = inf *-i$), as in Section I.I. Then a(L<*>[0,oo)) = -L. a$

(14)

Proof. By definition of a$, for any e > 0 there exists UQ(= WQ(^) > 0) such that $~l(u0) < (a$ -f £)$~ 1 (2w 0 ). Consider /0 = $~ 1 (2w 0 )xroL '2u -M' 0

#o = ^~1(2uo)xr-^ ,-L), so that ||/o||(*) = ||^o||(*) = 1 and i2ug «0

IK/o v a,)!!,., = •-'( Thus a(L^[0, oo)) > (a^)~l, since e is arbitrary. On the other hand, a$$~1(2u) < ^-^u) so that $[a$$~1(2w)] < u,0 < w < oo. If v = $-1(2u), one has < -$(u), 0 < u < oo. (15) If ||/i|l(*) ^ 1» * = 1> 2, so that p(fi) < 1 , one has from (15)

+

(/2) < 1,

so that ||(|/i| V |/2|)||(*) < (a^)"1. Hence a(LW(,K + )) < (a^)"1. This, with the earlier (opposite) inequality, establishes (14). n Corollary 8. a(L(M+)) < 2 iff® € V 2 . Proof. Since $ e V2 iffa$ > | (cf. Theorem 1.1.7), the result follows from (14). D Let us consider again a couple of illustrations. Example 9. Let $ r (u) = e^r -1, 1< r < oo. Then a(/^) = a( and the common value is 2^. In fact C$r = r and the function G$>r given by:

390

X. Miscellaneous Applications

is increasing on (0, oo). It follows from Corollary 1.1.9 that — —7!—

a^r = a$T = o£p = 2 **~r = 2~r . The assertion is a consequence of Theorems 2 and 7. Example 10. Consider the complementary pair ($, \&) as in Example 5. Then $ 6 V 2 ,* i V 2 and

Hence by Theorem 7 and Corollary 8, we find = 2.

It may be observed that Theorem 7 and Corollary 8 hold for general diffuse infinite measure spaces. The next result holds for finite measure spaces, but for simplicity it is stated for Z/*)[0, 1]. Theorem 11. For an N -function <3>; one has -i- < a(L<*>[0, 1]) < min ( ^-, 1 + -M ,

\Q;$

a$

(16)

2o;$/

where a$ zs defined in Theorem 7, and

Proof. For the left most inequality of (16), with u0 > 1 consider /i - $- 1 (2u 0 )X[1- 0'2u ,^), /2 = ^ 1 (2w 0 )X[^-,i), ' I2u ' U Q ' 0

0

so that ||/j||($) = l,i = 1,2 and a(L^[0, 1]) > ||/! V /2||w = $-1(2

o

V"0/

This gives the desired (left) inequality since UQ > 1 is arbitrary. Similarly a(L ($) [0, 1]) < (a<s>)~1 is established (cf. Theorem 7). Regarding the inequality with a$, by (17) one has a$$~ 1 (2w) < $~ 1 (w) if | < u < oo so that (taking v = $~l(2u)) < -$(v), $"x -

< u < oo.

(18)

391

10. 2 Riesz angles of Orlicz spaces -l

If 6 = 1 + d:$, since 0;$ < 1 < b we have by (18)

and

flf

o

2

Hence

It follows that a(L^[0, !])-00 * \' ) iu

that a$ = i, since a* > o£ > £. Hence a(L^[0, 1]) = 2 by Theorem 11. If $ 6 V2(oo), then 0:$ > | and 0;$ > | because ^AA > | for each u > |. Hence a(L<*>[0, 1]) < (1 + (2d$)-1) < 2. D Example 13. For the TV-function defined by $p,r(w) = |w|p[log(l + |w|)]r, with 1 < p < oo and 0 < r < oo, the function F$p r given by

is decreasing on (0, oo). Thus C

= limF$ pr (i) — p + r and C$pr = 2w

limF$ piP (t) = p. By Lemma IH.3.4, G^r(u] = decreasing on (0,oo), and hence

=2

= 2~p.

is also

392

X. Miscellaneous Applications

Then by Theorems 7 and 11, one gets a(L(*^[0,1]) = 2* = a(L^ (fi). But -£v£\ = 2~*,0 < w < oo. Hence the 2, the symbol $1 >~ <3>2 at 0 denotes that <$2 is weaker than $1 for small values, and this signifies that there are UQ > 0 and b > 0 such that $2(w)<$i(H> ° < w < w o ,

(1)

and $1 ~ $2 at 0 (i.e., equivalent for small values) if $1 >- $2 and $2 >~ $1 at 0. It is seen that $1 ~ $2 at 0 iff \I>i ~ ^2 at 0, ^; being complementary to $;, i = 1, 2. We record a known result. Proposition 2. The following assertions are equivalent: (i) $! >- $2 at 0; (ii) there is a constant C > 0, such that

\\f\\(*,}
/e/*1;

(2)

10.3 Embedding theorems for sequence spaces

393

(Hi) /$1 C I*2 ; (iv) m$1 C /$2 where m$1 is the closed separable subspace of I®1; (v) m$1 C m $2 . [In (iii)-(v), the inclusions are continuous.] Proof. We establish (i)1^ (ii), and omit the proofs of the others which use similar reasoning . By (i) there are UQ > 0 and b > 0 satisfying (1), and set K = m a x l , * f c . If 0 ^ / = (/(l),/(2), • • •) 6 /*>, then ^ -

l and hence

. -

) -

L>

Kb

From (3) and (1) we obtain

which implies (2) with C = bK. The remaining implications are left to the reader. D Corollary 3. The following assertions are equivalent, (i) <$i ~ $2 atO, i.e., there exist UQ > 0 and 0 < a < b < oo such that <3>i(em) < $i(u) < $i(bu) for 0 < u < UQ, or equivalently ,.

. $2l(u)

,.

^Htt)

0n < hmmff .I,; ; < limsup.:.!; : < oo. u->0

$1 (U)

u-+0

*! (U)

(ii) There are constants 0 < C\ < Ci < oo swc/i that < ll/ll(* a ) < < Example 4. Let 3>(w) = e|w| - |u| - 1 and *(?;) - (1 + \v\) log(l + |u|) - \v\. Then I* = l2 = l*. (4) In fact, let $ 2 (w) = £. One has /*2 = / 2 and ||/||($2) =

(i=iE |/WI / 00

2

\ 2

• We then have for 0 < u < 1 7/ 2

oo ,.n-2

.2

/

c

(«) < *(u) = y + u2 E ^- < |- + (e - I

||/||2, where

394

X. Miscellaneous Applications

which proves that $ ~ <3>2 at 0, and hence I® = I2. Further, one can verify that ^||/||2 < ||/||($) < ||/||2. Since # ~ #2 at 0, where $2(v) = ^, the right side of (4) follows again from Corollary 3. Example 5. Let ($, ^) be a pair of complementary TV-functions and let $(w) = e®^ — 1, and ^ be the complementary TV-function of $. Then we have /* - /* and /* = /*. (5) Indeed, one has for 0 < u < ^"^l),

Hence $ ~ $ at 0 and the first one of (5) follows from Corollary 3. In particular, if <3>(w) = u\p, I < p < oo, then $(u) = e\u\P — I and J* = p,/* = /?,(! + I = 1). Furthermore, in this case we have ||/||p < ||/||(5) < 2?| /||p. [Compare with Theorem VVV.3.1 and its consequences.] Problem 6. Does there exist such an TV-pair (<£, \?) that satisfies I1 C I* C H lp C U lp C J* C c0. P>I p>i

(6)

Note that in (6) the first, the third and the last inclusions are always true. An answer is given by the following: Proposition 7. An N-pair (3>,^) satisfies (6) iff for any p > 1 there exist UQ = u0(p) > 0 and b — b(p) > 0 such that up < $(6w), 0 < u < UQ.

(7)

In particular, (<£, \I>) satisfies (6) if lim

, , — 1,

v( o;)

where (j> is the right derivative o/$. Proof. Let $ p (u) = p-^u p and * p (u) = g~> 9 with i + ^ = 1. If (7) holds for any p > 1, then by Proposition 2 one has /* C lp so that /* C f| lpP>I Since $ >- $p at 0 iff $>Q X- ^ at 0, we have / 9 C /* for any q > 1 so that

10.3 Embedding theorems for sequence spaces

395

U lq C /*. Thus (, #) satisfies (6). It is seen that the converse is also i true. In particular, if $ satisfies (8), then for any given p > 1 we can find a UQ = UQ(P) > 0 such that ^ < p for 0 < t < UQ. Hence for 0 < u < UQ

i.e, up < Wo[$(wo)]~ 1< ^ ) (w),0 < u < UQ, which proves that (7) holds. D Example 8. Let $ be complementary to the ^-function \I> where

\ < \V\ < 00,

(due to Kaminska [2]). Then the pair ($,^) satisfies (6). In fact, Cl = lim y = +00 so that C£ = lim & = 1 by the formula 1 1 + (C^)" = 1 (cf. Corollary 1.1.3). The assertion then follows from Proportion 7. Definition 9. We say that 3>2 is essentially weaker than $1 for small values, in symbols $1 >->- $2 at 0, if for any e > 0 there is a WQ = u0(e) > 0 such that $2(w) < $i(ew) for 0 < u < w 0 . It is seen that $1 >->- $2 at 0 iff lim |t44 = 0, iff ^2 ^^ *i at 0, where v->0 *2 ( v ) • ^i is the complementary JV-function to $j, (i = 1, 2). We recall Lions' lemma (cf. Berger [1], p. 35). Let Af,^ and Z be Banach spaces satisfying X c 3^ C Z. Suppose that the embedding X c y is (conditionally) compact (i.e., the identity operator I maps every bounded set S of X into a compact set 7(5) in y\ then for any given 5 > 0 there is a K = K(e] > 0 such that Y

£

x

z,

feX.

(9)

This lemma is useful in nonlinear analysis. Problem 10. If 1 < pi < p2 < oo, then 1PI C 1P2 C c0. The identity operator / : 1PI —> 1P2 obviously fails to be compact. Is it true in this particular case that for any given £ > 0 there is a K = K(e] > 0 satisfying \P2 ^

(10)

X. Miscellaneous Applications

396

The answer is an immediate consequence of the following theorem since \u\pl >->- \u\p2 at 0 iffpi ->- $2 at 0; (ii) for any given £ > 0 there exists K — K(e] > 0 such that the (Lions) inequality holds:

l l / l k < £ | | / l k + # l / I U /e/* 1 ;

(11)

(Hi) if fn € / $1 ,sup \\fn \$l
Proof. (!)==>. (ii). By (i), l^ C I*1 C c0 since $1 x>- $2 =>• $1 >- ^2 (at 0), and ^2 ^^ ^i at 0, where \I>j is complementary to 3>j,z = 1,2. For any given e > 0, by Definition 9 there is a v0 = fo(^) > 0 such that ,

(12)

0 < V < VQ.

For each g = (0(1), 0(2), • • •) € /*2 with p* 2 (^) < 1, let Wi = {i 6 W = {1, 2, • • •} : |#(z)| < vQ}, JVf = IN\JNi and /^ be the counting measure. It is clear that (Wf being a finite set and n({i}) — 1)

By (12) one has

Let AT = |^| . Then if / = (/(I), • • •) € Z* 1 , we have from (14) and (13)

E \g(i)

<

By taking "sup" over p^2(g] < 1, we obtain (11).

10. 3 Embedding theorems for sequence spaces

397

(ii)=^(iii). Suppose that H/nll^ < C and ||/n||oo —* 0 as n -> oo. For any given e > 0, let E\ = ^. Then (ii) implies that a K = K(EI} > 0 can be found such that ||/n||*2 ^

11/nlUi

Choose no > 1 such that ||/n||oo < ^ for n > n0. Thus ||/n||$2 < £ if n > no, so that 1 1 /n|| $2 —>• 0 as n —>• oo since e is arbitrary. (iii)^=^(i). Assume that (i) fails. By Definition 9 there exist £Q > 0 and \ > un \ 0 such that ^r 1 K)>£o^ 2 " 1 K), n > l .

(15)

Let kn = f[_ — 1 , the integer part of — , so that kn < — < kn + 1 and Un J Un Un -
(16)

Define a sequence /„, n > 1 as /„ = $f 1 (w n )(l, 1, • • • , 1, 0, 0, • • •) with "dimfn" = kn,n> 1. Then p* x (/ n ) = knun < 1 so that \\fn\\*i < 2||/n||(*1) < 2 and ||/n||oo = ^r 1 ( w n) —>• 0 as n -» CXD. On the other hand, from (16) and (15) we obtain

which shows that (iii) also fails. D Definition 12. We say that $2 is completely weaker than <3>i for small values, in symbols $1 >-| <£2 at 0, if for any given e > 0 there are UQ — u0(e} > 0 and K = K(e] > 0 such that $2

-

< K$i(u),

0
(17)

\ c. /

It is seen that $ 6 A 2 (0) iff $ H $ at 0, and ($1 ^^ $2) =>• ($! H $2) => ($1 >- $2) (at 0). A result of Ando [1] can be adapted to the case of Orlicz sequence spaces, from its continuous measure case. Theorem 13. Let <£>i, $2 be N -functions. Then the following assertions are equivalent: (i) $1 >-| $2 at 0; (ii) /$1 C m*2; (iii) for any given e > 0 there is a 5 — 5(e] > 0 such that p^1(f) < 5 => ||/||($2) < £> i-e-> l/ll(*a)=0;

(18)

X. Miscellaneous Applications

398

(iv) each \\ • H^^-bounded set in I®1 is p$ 2 (-)~ bounded in /* 2 ; i.e.,

(19)

Proof. (i)=>(ii). If 0 ^ f = ( / ( I ) , - - - ) E J*1, we have to show that P* 2 (A/) < °° f°r any fixed A > 0. Let e = (AH/H^))" 1 , then e > 0 and by (i), there exist UQ > 0 and K > 0 such that ,

0
0

.

(20)

Let JNi = {i G W = {1,2, • • • } : riu IK^I r -) < u0} and W2 = W\W l5 then $1 ^(iV2) < oo since / C c0, where // is the counting measure. It follows from (20) that

E + E *2(A|/(t)l)

(i)^=^(iii). For any given e > 0, by (i) there are UQ > 0 and K > 0 satisfying (17). Let 6 = min^M, K"1}. If / = ( / ( I ) , - - - ) e J*1 and P*i(/) < <J, then $i(|/(z) ) < 5 < $I(MO) so that \f(i)\ < UQ for all i > I . Thus we get from (17)

I/WI proving ||/||($2) < e. (i)^=^(iv). Let S be a norm bounded set in / $1 , sup-fH/H^) : / G 5} — a < oo. By (i), for £ = ^ there are MO > 0 and K\ > 0 such that $2 (aw) = $2 (7) < -^i^i(w),0 < M < u0. We assume MO < $il(l) and let ^2 = maxl^g^ : u0 < u < ^r^ 1 )}- Hence $ 2 (aw) < 6$i(u),0 < w < $\l(1-}, where 6 = max(J R f 1 , K2}. Since / = ( / ( ! ) , - • • ) e S implies ™) < P*i (f) < 1, and ^ < $^(1) for all i > 1, we obtain for / e 5°

l/tt

/(Ol

10.3 Embedding theorems for sequence spaces

399

The remainder of the proof is left to the reader. D Letting $! = $2 — $ (say) in the above theorem, we obtain the following known result. Corollary 14. Let $ be an N -function. Then the following assertions are equivalent: (i) $ 6 A 2 (0); (ii) m$ = I®; (Hi) lim ||/||($) = 0; and (iv) p*(/)->o

i™ II /Ik*) = °°-

p*(/)->oo

Definition 15. For TV-functions $1 and 3>2, we say that <3>2 decreases more rapidly than $1 for small values, in symbols $1-^ $2 at 0, if for any given e > 0 there are 6 = S(e) > 0 and u0 — uo(e) > 0 such that 7$2(<5w) < e$i(u), 0 < u < w 0 . (21) o It is seen that $ e V 2 (0) iff $-»- $ at 0, and that $1-^- $>2 at 0 iff \I>2 x| *i at 0, where ^ is the complementary function of $i, i = 1, 2. Also, >2 ==> $x >- $2 (at 0). Theorem 16. Let ($j,^),i = 1,2 6e iwo pairs o/ complementary Nfunctions. Then the following assertions are equivalent: (i) $!->- $2 ai 0; (ii) It is true that lim

£?^-=0;

(22)

(Hi) Each \\ • \\^^-bounded set in /$1 is a(/*2,/*2)-sequentially compact in I®2, i.e., the embedding /$1 C I®2 is cr(/ $2 ,/* 2 ) — compact. Proof. (i)=> (ii). Suppose that for any e > 0 there are 6 > 0 and w0 > 0 such that (21) holds. Let r — min(<5, <3>i(5w 0 ), 1). We assert that if / = (/(I), • • •) G /$1 with 0 < l l / l l ^ ) < r , then p$ 2 (/) < ell/H^), proving (22). Note that 0 < r < 1. So $i(|/(i)l) < p 9 l ( f ) < \\f\\(^) 1. Since H/H^) < r < 5, we have from (21)

6 6 P*!

, g

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X. Miscellaneous Applications

< ell (ii)=>(iii). For a bounded set S C I®1 with supdl/H^) : / e 5} = a < oo, and if £n \ 0, then 0 < ||fn/||(*i) < Cn^ —>• 0, 0 ^ / € S. Thus we have from (22)

,. P$ 2 (£n/) n <. a sup hm — ——— = 0, f€Sn-+°°

||6,/|k

so that 5 is a(/* 2 , /*2)-compact in /*2 in view of Theorem 1.2.15. (iii)=>(i). Suppose to the contrary, (i) fails, i.e., there are e0 > 0 and l $ 2 ( z ) > un\Q such that ), n > l .

(23)

Let kn be the integer part of g , 1 . , so that | < 1 - $i(u n ) < A; n $i(u n ) < 1. Define a set 5 = {/n, n > 1} with

Clearly 5 is bounded in /*J since H/nll^) = un [$Tl f^-)] n > 1. On the other hand, if £n = ^ then (23) implies

< 1 f°r

n

which shows that S fails to be a(/ 4>2 , /*2)-compact in ^ 2 , by Theorem 1.2.15. Thus the original implication holds, n Letting $x = $2 = $ (say) in the above theorem we get the following. Corollary 17. For an N -function <$, i/ie following assertions are equivalent: (i) $ € V2(0); (M,) /* /ias the property iim

n .I,

— U;

(^mj each norm bounded set in I* is a (I*,!®) -compact, where $ is the complementary function to <E>.

10.3 Embedding theorems for sequence spaces

401

Combining Theorem 16 with Theorem 13 we can prove the following weakly compact embedding result. Theorem 18. Let $i,$2 be N -functions. Then the embedding mapping T from Z*1 into /*2 is weakly sequentially compact iff (a) $1 >\ $2 at 0, and (b) $2 at 0. Corollary 19. Let $ be an N -function. Then /$ is reflexive iff$£ V 2 (0).

A 2 (0)D

We conclude this section with some geometrical invariants of the sequence space /* endowed with the Orlicz norm || • ||$. Let <£(u) = /Ju' <j>(t)dt and \&(v) = Jri' i/)(s)ds be a pair of complementary TV-functions. For 0 ^ f € /$, a constant k = k(f) > 0 satisfies

*= iff k <=[k*(f),k* *(/)], where **(/) = inf{A; > 0 : p»[^(A;|/|)] > 1}; fc"(/) = sup{fc > 0 : p»[0(*|/|)] < 1}, (cf. Theorem 1.2.16 for the function space case). For the sequence space /*, we set q* = inf Ik > 1 : ||/||* = i[l + /!*(*/)]} (24) ll/ll*=i I A; ) and Q $ = sup *>1:||/||* = [1 + /!>•(*/)]. (25) The following estimates supplement Lemma V.2.11. Theorem 20. For an N-function <£ and the corresponding space /*, the geometric invariants q$ and Q$>, defined by (24) and (25), satisfy T < <7* < Q$ < „.

7;

$

(26)

,

where a$ = inf < and

fMm

: 0 < £ < i/'f^"^!)] > ,

(27)

1 (28)

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X. Miscellaneous Applications

Proof. The right side of (26) has been proved in Lemma V.2.11 under the assumption that $ e A 2 (0) n V 2 (0). If $ ^ V 2 (0), then a| = 1 so that the right side of (26) holds. On the other hand, if $ ^ A 2 (0), then b\ = oo so that the left side of (26) is also true. If $ 6 A 2 (0), one can easily prove that Remark 21. Inequalities (26) can be written in the from 4 < q* < Q* < &*,

(29)

where

and h*, Uy ~~ — bup <\

—— •. nu < ^ f < Aj <\T \VI / ~ {!) i 1 i >f .

(^01; SI I

In fact, we have (a|)-1 + (by)~l = 1 so that ^-j- = by, and the right side of (29) follows from (26). Similarly, < = 6|(6| - I)-1. Example 22. Let $(w) = e|u| - w| - 1 and *(u) = (1 + \v\) log(l + H) - |u|. Then ^(s) = log(l + s) and ^-^l) = e - 1. However F$(t) = ^ is increasing on (0, ^[^"H1)]] = (°» !] so that

On the other hand, Fy(s) = ^f^ is decreasing on (0, ^"Hl)] = (0. e - 1] so that < = **(*)| s=e _i = e - 1, ^ = Ji_m^*(s) = 2Finally, we have from (26) and (29) 1. / Io ~ 6

1 = flx, = T—

b* —\

'cf) ^ t / $

a* -1

10.4 Differentiability properties of Orlicz spaces

In this section, we specialize some abstract Banach space concepts and present necessary and sufficient conditions under which the norm of an Orlicz

10.4 Differentiability properties of Orlicz spaces

403

space is uniformly Gateaux differentiable. These results largely complement those of Section 7.2 of our companion volume [1]. We recall the concept. Definition 1. Let (X, \\ • ||) be a Banach space with S(X) denoting its unit sphere, S(X) = {x G X : \\x\\ — 1}. Then the norm || • || is (a)Gdteaux differentiable if for every x G S ( X ) , y G S(X) .

j|1

(1)

exists, and is uniformly Gateaux differentiable if the limit is uniform in x G S(X) for each fixed y G S ( X ) . (b)Frechet differentiable if the limit in (1) exists uniformly in y G S(X) for each fixed x G S ( X ) , and is uniformly Frechet differentiable, if the limit in (1) exists uniformly for the pair ( x , y ) G S(X) x S(X). If the norm of X is Gateaux differentiable, we also say for short, as in Day [1], that X is Gateaux differentiable and denote X G (G). Similarly, for the other corresponding concepts, we denote X G (UG), X G (F) and X G (UF) respectively. It is seen that X £ (G) \R X € (S) (smooth), X G (F) iff X G (55) (strongly smooth) and that X G (UF) iff X G (US) (uniformly smooth). It is also clear that X G (UF) =» X G (F) =» X G (G) and * G (UF) => X e (UG) =* * G (G) , but X G (F) =^ * G (C/G) and # G (UG) =^> X G (F) in general. Before we study (UG) -differentiability of Orlicz spaces first recall a basic result on (G) differentiability of the closed separable subspace M^ of L^ endowed with the gauge norm and M* of L* endowed with the Orlicz norm, on a finite measure space (Q, £,//), with ^ nonatomic. Theorem 2. Let <&(u) = /d (t)dt define an N -function. Then the following are equivalent: (i) (j) is continuous on (t), oo); (ii) M^ G (G); (in) M* G (G). Proof. (i)=£>(ii). This was essentially proved in Rao [2] (cf. Rao and Ren [1], p.278). The proof of (ii)=^(i) was given by Wang and Chen [1] (cf. Rao and Ren [1], p. 280). That (i)«=»(iii), was noted in Chen ([3], Theorem 2). But the proof for (iii)=^(i) seems less easy, and recently Ren established it as follows. [Remember that jj, is nonatomic.] (iii)=>(i). Suppose that (f> fails to be continuous. Then there are UQ > 0 and vi, v2 satisfying 0 < v\ = (f>-(u0) < <j>(u0) = v^ < oo, with 0_ as the left derivative of $. Equivalently, ^(^i) = ^(^2) = WQ, where ^f(v) = JQ ip(s)ds defines the complementary function to <£. Choose G\,Gi G E such that = 0, 0 < ^(Gi) = ti(G2) < //(n) and

404

X. Miscellaneous Applications

Further, choose c > UQ such that 0 is continuous at c and *[0(c)]//{lA(Gi U G 2 )} > 1 - VfrMGJ

- *(i/2)/i(G 2 ).

Thus a subset G3 c fi\(Gi U G 2 ) can be found (by nonatomicity) satisfying *M/i(Gi) + *M/z(G 2 ) + *[0(c)MG3) = 1.

(2)

Next define / = u0XGi + woXc 2 + CXG3 and set

Then / € M* and |^||(*) = 1 since p*(#) = l,i = 1,2 by (2). We claim that 11/11* = 1 + P*(/) = 1 + *M»(Gi U G 2 ) + $(c)MG3). (3) Note that l l / l l * = i[l + M*/)]

(4)

iff A; € [A;i(/),A; 2 (/)], where /d(/) = inf{a > 0 : p*[0(a|/|)] > 1} and jfe 2 (/) = sup{a > 0 : p*[0(a|/|)] < 1} (cf. Theorem 1.2.16). If k > 1, one has from (2)

> On the other hand, if k < 1, again by (2) we have

Thus &i(/) = ^(/) = 1 and (3) follows from (4) with k = 1. We also observe that (the Young inequality now becomes equality, cf, Section I.I) ]

(5)

and for i = 1 , 2 ViU0 = Vi^(Vi) = V(Vi) + $[^(Vi)] = V(Vi) + $(U 0 )-

It follows from (6), (5), (2) and (3) that j fgidn

=

(6)

10.4 Differentiability properties of Orlicz spaces

405

= \\f\\*.

(7)

= / fgid/j,. Jn

(8)

Since n(Gi) = /^(G 2 ), the first equality in (7) implies that / Jn

Finally, by taking /0 = TTT-, we get from (7) and (8)

which proves that the linear functional induced by /o has two support functionals g\ ^ g2 , so that M$ ^ (G). (i)=>(iii). This can be found in Rao and Ren ([l],p.282). D By means of the above result, Wang and Chen [1], and Chen [3] proved the following. Theorem 3. Let <$ be an N -function and let L^L? be the Orlicz spaces, with the two norms , on a finite nonatomic measure space (ft, £,//). Then L($) e (G) iffL* e (G), i/f $ e A2(oo) and $' is continuous on (0,oo). To simplify the proofs of the results on (UG) differentiability of the Orlicz spaces, we again need to recall some concepts and results on abstract Banach spaces. [Again we follow Day [1] for the most part.] Definition 4. (a) We say that a Banach space X is weakly uniformly convex, denoted X e (WUR), if rcn, yn e S(X) and ||o;n+yn|| -> 2 imply (xn-yn) -+ 0 weakly, (b) The dual space X* of X is said to be weak* uniformly convex, in symbols X* 6 (WUR), if fn,gn € S(X*) and ||/n + gn\\x* -> 2 imply (fn — 9n) -^ 0 weak*, i.e., lim (/„ — gn}(x) — 0 for each x G X. Lemma 5. (Smulyan [1]) Let X be a Banach space and X* be its dual space. We have (i) X* e (UG) iff X e (WUR); (ii) X e (UG) iff X* 6 ( Theorem 6. Let $(u) = JQ (f>(t) and W(v) = /J" ip(s)ds define a complementary pair of N -functions and let L® = L*(£l,E,/j,) with ^Q,E,//j being nonatomic, //(ft) < oo. Then the following assertions are equivalent. (i) L* 6 (UG); (ii) Af<*> e (V^t/^); f«»; L<*) e (WC/^); ^ ^ 0 is continuous on (Q,oo)[i.e., *& is strictly convex on [0, oo)/, (b) $ € A2(oo), (c) $ € V2(oo). Proof. (iii)=^(iv). It was proved by Kaminska and Kurc [1], and Wang, Wang and Li [1] independently (see also Chen [1] p. 92). The fact that (i)^=^(ii) follows immediately from Lemma 5 (i) since L$ = (M^)*. Clearly,

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X. Miscellaneous Applications

(iii)=>(ii). Now we show (ii)=>(iii), which will complete the proof of the theorem. Suppose that (ii) holds. We assert that ^ 6 A2(oo). If not, then f3y — lim sup

,

. = 1.

(cf. Theorem 1.1.7), so that there exist [p,(Q}}~1 < vn /* oo such that (9)

Choose r^0 C f) with //(i7 0 ) = fM^) and Gn C Q\Q0 satisfying //(G n ) = 2^,n > 1. Let c - tf-1 (2^5) and define

Then fn,gn € 5(M^) since p*(/ n ) = p9(gn) = ^(c)^(^ 0 ) + vnn(Gn} = l,n > 1. By (9), we have

2(1-1) \ n'

2vnfj.(Gn) = 1,

showing that 2 = ||/n||(*) +||0n||(*) > ll/n + pn||(*) > 2(1-J) and Ji ^n||(*) — 2. On the other hand, fn — gn = 2cxn 0 ) which does not converge weakly to zero, giving a contradiction to (ii). Thus ^ G A2(oo) so that LW = Af(*) 6 (l^C/J?), proving (ii)=>(iii). D For the gauge norm the corresponding result can be obtained from known facts. Theorem 7. Let (<&, ^) be a pair of complementary N-functions and let /,(*) = Z/*)(Q, E, IJL) with (T], E, fj,) being as in Theorem 3. Then the following assertions are equivalent. (i) L^ e (UG); (ii) M(*) € (C7G); (Hi) L* - (M( $ ))* € (W*UR); (iv) (a) ty is strictly convex on (t), oo), and (b) ^ is uniformly convex for large values. Proof. (iii)4=>(iv). This was proved by Wang, Wu and Zhang [1] (cf. Chen [1], Theorem 2.47 ). The fact that (ii)4=>(iii) follows from Lemma 5 (ii). It is trivial that (i)=>(ii). Finally we show that (ii)=^(i). Suppose M^ 6 (UG). Since (ii)4=>(iv) and (iv) (b) implies * e V 2 (oo), i.e., $ e A 2 (oo), it follows that I/*) = MW, and L™ e (UG). D

10.4 Differentiability properties of Orlicz spaces

407

In comparing (UG) differentiability with (F) differentiability, we restate the following. Proposition 8. (Chen[l], p. 100) Let $ and (Ti,E,/zj be as in Theorem 6. Then the following assertions are equivalent: (i) L* G (F); (ii) L^ G (F); (Hi) (a) $' is continuous on (Q,oo), (b) ® G A 2 (oo) and (c) ® G V 2 (oo). Remark 9. By Theorem 6 and Proposition 8, we see that for the space L* equipped with the Orlicz norm, L* 6 (UG) iff L* G (F). The following examples show that for the gauge norm there is no relation between L^ G (UG) and L™ G (F). Example 10. Let ®(u) = (I + |w|)log(l + \u\) — \u\ . Its complementary TV-function given by ty(v) — e\v\ — \v\ — 1 , is strictly convex on [0, oo) and is uniformly convex for large v (in fact ^/ is uniformly convex on [0, oo), see Example VII.4.11). It follows from Theorem 7 that L<*) G (UG). On the other hand, since ® £ V2(oo), L^ £ (F) in view of Proposition 8. Example 11. Consider the TV-pair ($, xfr) given in Milnes ([1], p.1483). Let 2,3, • • • . Join the points (ui,vi) to (w 2 ,t> 2 ); (wn,v n ) to (uj,,v{,) ; and (w^ to (un+i,vn+i) by straight line segments . Let this function be t/j. Then

as n —> oo so that ^(v) = f^ i/j(s)ds is not uniformly convex for large v, although \P G A 2 (cxD). The inverse ^ of ^ is continuous and satisfies 0(2u) < 4(f>(u),u > 0 . Thus $(u) = /d"1 ^(*)d* defines $ in A 2 (oo). By Theorem 7 and Proposition 8, L™ i (UG) and L^*) G (F). Now we turn to the sequence spaces /$ (with the Orlicz norm) and l^ (with the gauge norm), and their closed separable subspaces m*,m(*). We record the following special result, which can be found in Chen ([1], p. 90-100, and Theorem 2.57). Proposition 12. Let ($, fy) be a pair of complementary N-functions. (i) m* G (G) iffty is strictly convex on [0, ^-1(|)]. (ii) /* G (G) iff3? is strictly convex on [0, ^~H|)] and $ € A 2 (0). (Hi) /* G (F) iff^ is strictly convex on [0, ^(j)] and® G A 2 (0)nV 2 (0). (i)' m.W G (G) iff& is continuous on (O,^-^!)). (ii)' / ($) G (G) iff®' is continuous on (0, $~l(l)) and $ G A 2 (0). (Hi)' /(*) G (F) iff® is continuous on (O,^-^!)) and® G A 2 (0)nV 2 (0).

408

X. Miscellaneous Applications

Theorem 13. Let ($, ty) be a pair of complementary N-functions. Then the following assertions for the sequence spaces are equivalent. (i) /* e (UG); (ii) m™ € (WUR}; (Hi) lw € (WUR}; (iv) (a) V is strictly convex on [0, ^ ~ 1 ( ^ } ] , (b) $ G A 2 (0) n V 2 (0), and (c) ^ is strictly convex on [^/-1(|), ^""^(l)] or ^ is uniformly convex for small values. Proof. (iii)^=^(iv) This was noted by Li [1] (for a proof, see Chen [1], Theorem 2.44). Since (m^)* - /*, (i)«=>(ii) follows from Lemma 5(i). It is trivial that (iii)=^(ii) since m^ is a subspace of /(*). To complete the proof of the theorem, we need only show that (ii)=>(iii). Suppose m^ e (WUR}. We assert ^ G A 2 (0) . If not, then ^ = limsup ^i"l = 1 (cf. Theorem ™.

vA

V

/

1.1.7 ). Thus a sequence | > vn \ 0 can be found such that ^ /o.. \Vn>\ >^,1-1- ~ _' n„ ^> i.i ,T,_I

Let kn = ^- , the integer part of ^, then ^ - 1 < kn < 1 3 2 < 4 ~ Vn

<

(-\o\ ^iu;

so that

3 ^^ - 4'

Let an = ^" x (l — knvn} and define two sequences {/„}, {gn} in m^^ as /„ = (a n , *~ 1 (w n ), • • • , ^-^VH), 0, 0, • • •), gn - (-a n> *"!K), • • • , ^'Hun), 0, 0, • • •)• with dimfn = dimgn - l + kn. Clearly, /?*(/„) = pv(gn] - *(«n) + knvn - 1 so that |/n||(*) = l^nl^*) = l,n > 1. By (10) and (11), we have fn + 9n

proving that 2 = | fn\\m + \\gn\\(v) > ||/n + 5n||(*) > 2(1 - £) andj lim ) ||/ n + gn\\w - 2. On the other hand, since an > ^ / ~ l ( l - |) = V (\) > 0 for all n > 1, the sequence (fn - gn} — (2a n ,0, 0, • • •) does not converge weakly to zero, which contradicts the hypothesis. Thus ^ G A 2 (0) and then /(*) = m^\ proving (ii)=^(iii). D Theorem 14. Let ^3>, ty) be a pair of complementary N-functions. the following assertions are equivalent:

Then

10.4 Differentiability properties of Orlicz spaces

409

(i) f<*> € (UG); (ii) m<*> e (UG); (Hi) /* = (m<*>)* € (W*UR); (iv) (a) <£' is continuous on (0, $~1(1)), and ^ ^ is uniformly convex for small values. Proof. The main part, namely (iii)4=>(iv), has been proved by Li and Wang [1] (see also Chen [1], Theorem 2.47). Lemma 5(ii) implies that (ii)^=^(iii). It is trivial that (i)=>(ii). Finally, since (ii)^=^(iv) and (iv) (b)=> $ e V 2 (0), i.e., $ <E A 2 (0), one has that / ($) = m ($) , proving (ii)=»(i). D Remark 15. By Theorem 13 and Proposition 12 (iii), we conclude that /$ e (UG) => I® € (F)— but not necessary conversely. The following two examples show that there is no inclusion relation between (UG) and (F) diffentiability for the space l^ endowed with the gauge norm. Example 16. Let

f

v=o

o,

if,(v) = { v-2e~v, ve (0,|] ~2v, v € [|,oo). Then v= |V| f °' ° V(v)= I" ^(s}ds=\ e -R, He(0,J] «/0 I j O O i i r1 \ I 4e~V, |r;|e[i,oo),

defines an A^-function (due to Kaminska [2], p. 304). Let $ be the complementary A^-function to ^. Since ip is strictly increasing on [0, oo), $' is continuous on (0, oo). We now show that ^ is uniformly convex for small values. For a given £ > 0, let Ke - e2e(l + s)~2 and v£ = [2(1 + e)]"1, then Ke > 1 and for 0 < v < v£

Thus /($) 6 (UG) by Theorem 14. On the other hand, since 2 rlim fr( ,. e^ -i- — +00, , _ . ^). = lim

v-*0

ty(v)

v^O

one has ^ ^ A 2 (0),$ ^ V 2 (0). It follows from Proposition 12 (iii)' that /<*> ^ (F).

410

X. Miscellaneous Applications

Example 17. Let v =Q

f 0,

Then ty(v) — /0 tp(s)ds is strictly convex on [0, oo) since if) is strictly increasing on [O.oo). Let $ be the complementary function to \&. Then $' is continuous on (0,oo). We claim that $ G A 2 (0)nV 2 (0). Ifv € [2~( n+1 ), 2~ n ), then ^v^ < 2 rc ^ < 4, n > 2. Hence we have for 0 < i; < ^ < 64

i.e., * G A 2 (0) so that $ G V 2 (0). If vn G [2~( n+1 ), 2~ n ), then 3vn > 2~ n and VvJ"/ —> 3 as n —•> oo. Thus there is an no > 1 such that 2i^(v) < ift(3v) for 0 < v < 2~n° so that $'(2w) < 3$'(u) if 0 < u < UQ = V(2~ n °), which proves that $ G A 2 (0). By Proposition 12 (hi)', /(*) G (F). On the other hand, if vn = 2-( n+1 ), then 2^n+^ < (1 + \}vn < 2~n and 1hm —L-T7—--- = n, lim

n->oo

1/j(Vn)

^°° [

. 1 . + — -—

! =

1,

2(n+l)J

showing that ^f fails to be uniformly convex for small values. Thus /(*) ^ (UG) in view of Theorem 14. Remark 18. Example 17 is due to T. F. Wang (communicated to Ren), which is analogous to Example 11 given by Milnes [1]. It implies that there exist TV-functions ^ G V 2 (0) which fail to be uniformly convex for small values, while Example 11 shows that there are TV-functions ^ E V2(oo) that fail to be uniformly convex for large values. For a measure space (fi, S, //) with //(fi) = oo and p, nonatomic, we may consider 17 = [0, oo) = IR+ with the Lebesgue measure and record a couple of results of the same type for reference as follows. Theorem 19. For an N -function $, the following are equivalent: (i) L*[0,oo) 6 (F); (ii) L(*)[0,oo) € (F); (Hi) L*[0,oo) € (UG); (iv) $' is continuous on (0, oo) and $ G A 2 H V 2 . Theorem 20. L^ $ ^[0,oo) G (UG) iff^ ^ is complementary to <&.

is uniformly convex on [0, oo), where

10.5 L? 1 -spaces and applications

411

Problem 21. As seen in Remark 15, /* G (UG) => /$ G (F) but not conversely. A natural question then is the following. Does there exist a class of Banach spaces X for which X G (F) =$• X G (f/G) but not necessarily conversely? We conclude this section with the (UF) differentiability of Orlicz spaces for completeness. The following result is variously proved by Luxemburg [1], Milnes [1], Akimovic [1] and Kaminska [1]. Some of the proofs can be found in Rao and Ren [1], Section 7.2, and in Chen [1]. By X G (UR) we note that the Banach space X is uniformly convex (or uniformly rotund). Theorem 22. Let ($, ty) be a pair of complementary N-functions, and let (0, E, IJL) be a measure space with /^(Q) < oo and /j, nonatomic. Then the following are equivalent: (i) L* G (UF); (ii) L<*> G (UF); (Hi) L<*> € (UR); (iv) L* G (UR); (v) ^ G A2(oo), ^ is strictly convex on [0, oo) and uniformly convex for large values. [If (j,(£l) = oo, (v) is replaced by (v)': ^ G A2 and \£ is uniformly convex on [0, oo)./ Theorem 23. (Kaminska [l])For an N-pair ($, \£), the following are equivalent: (i) /* G (UF); (ii) /(*> G (UR); (in) (a) * is strictly convex on [O,*-1^)], (b) \£ G A2(0), and (c) ^ is uniformly convex for small values. [Note that (Hi) (a) with (Hi) (c) imply that ^ is uniformly convex on [0, ^~1(^)]JTheorem 24. (Tao [1]) For an N-pair (<$, $/), the following are equivalent: (i) /(*) G (UF); (ii) I9 G (UR); (Hi) (a) $' is continuous on (0, ^(l)), (b) $ G A 2 (0), and (c) ^ is uniformly convex for small values.

10.5 L* -spaces and applications Recall that in Section VI.5, a generalized Orlicz space L^(fi) on (0, E, //) is defined by a 0-function (f> : JR —> M+, that is increasing, continuous, (f)(—u) = (w),0(w) = 0 iff u = 0, and u—>oo lim 6(u) = +00. The 0-functions, include convex and concave functions, as (j>p(u) = \u\p, 0 < p < oo, are examples so that we have just L°°(Q) C L*(Sl) C L°(Q), since L°°(fi) C LP(O)(0 < p < oo) C L°(fi), when n($l) < oo and L°(fi) = {/ : / is fj. measurable and is finite a.e. on £1}. For the classical spaces L p (f2), 0 < p < 1, one defines the F-norm by \\f\\p = Jn \f\pdfj,, and for 1 < pi < p2 < oo one has L°° C LP2 c IS1 C L1 c L^T C L^ c L°, (I) in which all the spaces are defined on the same finite measure space (Q, E, p)

412

X. Miscellaneous Applications

and Lpi are the usual Lebesgue spaces, i — 1,2. On the other hand, A. C. Zaanen (MR 34(1967), #6434) observed that a convex function 0 need not satisfy the Lesniewicz condition /J30 ^- < oo (see Theorem VI.5.4), since

4t

(2t)'

t < I

—— du —> 0 as t —>• oo,

i.e., lim ^ = 0, and I/^O, 2yr) C Z/^[0, 2?r). In this section which is in the form of an addendum to Chapter VI, we establish the theory of Z/$ -space defined by the inverse $-1 of an TV-function , which is adapted from a recent doctoral thesis by J. C. Wang [1]. We present this in 5 subsections, consisting of the basic structure, somewhat different from our volume ([l],pp. 405-411), and application Fourier analysis. Subsection 1. The space L*"1 Definition 1. Let $~1 : 1R+ —>• M+ be the inverse of an TV-function <3> with $~ 1 (—u) = Q~l(u) for u < 0. On a measure space (17, E,/Lt) we define L* (£7) = {/ : / is //-measurable and ||/| ($-n < 00}, where (2)

i.e., it is defined to be the modular p$-i(-). The sequence space is oo

^ = {/ = (/(I), Proposition 2. For an TV -function <£>, L* l(ti) is a linear space and ||-||($-i), defined by (2), is an F-norm. Moreover, L* ((7) is complete under the metric d(f,g) = \\f — g||($-i) ; i.e., it is an F-space. [see Remark after the proof.] Proof. Since $ is an TV-function, so that <&~ 1 is concave increasing, we have

(^(Ajwl) < $-1( u\) if 0 < A < 1, and $~l(X\u\) < \<&~l(\u\), A > 1.

(4)

It follows that L* (0) is a linear space and \f+g |($-i) < ||/ll($- 1 ) + llfi'll('i'- 1 )Also (4) implies that if A > 0 and |/n||($-i) —> 0, then ||A/ n ||($-i) —> 0;

10.5

L^ l -spaces and applications

413

moreover A n —>• 0 and ||/||($-i) < oo =>• ll^n/ll^- 1 ) —> 0 . Hence || • ||($-i) is an F-norm. The proof of completeness is mostly similar to that of Lp(&), 0 < p < 1 (cf., e.g., Megginson [1], p. 164). D Remark. There are two different definitions for the Frechet space. The first definition is that an F-space is a topologically complete linear metric space under the metric d(f, g) = \\f — g\\ with || • || being an F-norm, and a Frechet space is a locally convex F-space. By this definition, Lp[0, 1] and V with 0 < p < 1 are F-spaces that are not Frechet spaces. The second definition is that a Frechet space is an F-space as above, which is not necessarily locally convex (cf., e.g., Rao and Ren [1], p. 401). Example 3. If$-l(u) = |u| p ,0< p < 1, i.e., $ p (u) = \u p, then Z,**1^) = , and by (2), \\f\\(^ = ||/||p - Jn |/|^. We have L1^) C V(to) C if /i(ft) < oo and 1° C lp C ^(0 < p < 1), where 1° = {f = (/(I), • • •) : f ( i ) ^ 0 only for finitely many i}. For an JV-function
(5)

\

where u(il)' < oo \u is counting!. Indeed, since U—lim —-^ = 0, there is a KX> -1 MO > 0 such that $ (u) < u for u > u0. If / € ^(fi), i-e., ||/||i < CXD, let : |/| > u0} and Q2 = fi\fii, then by (2) l

v

J

M

ll/ll(*-i) < jfii / l/M/i + S'1(«o)/i(n2) < ll/lli + s-^uoMn) < oo, proving the first of (5). On the other hand, lim $~u ^ — +00 implies that u->0+

there is a t0 > 0 such that u < $ -1 (w) for 0 < u < t0. If / = (/(I), • • •) € _i °° / $ ~ , i.e., ||/||(*-i) = E ^"HI/WD < °°» tnen since ^"H^o) > 0, there is an i0 > I such that g ^-^(/(i)!) < ^"H^o) so that |/(i)| < t0 if i > «o- Thus —1

ll/lli < z=l E I/(OI + i=io E s-HI/COl) < i=l E I/(OI + ^(^o) < oo, which prove the second part of (5). Proposition 4. Lei $ 6e an N '-function. Then L®~1 (£1, E, /^) zs separable iff p, is separable. In particular, L? [0,1], L$ [0, oo) anrf / $ ~ are all separable, [since the Lebesgue measure and the counting measure are separable.]

414

X. Miscellaneous Applications

Proof. Since $- x (2u) < 2$>~l(u),u > 0 (so fc-1 € A 2 ), each / € L*'1 (u) has an absolutely continuous F-norm || • ||($-i), i.e., for any given e > 0 there is a 6 > 0 such that A < 6 =$> fXA\(*-i) = I ®~l J A

The rest is similar to that of Theorem 1 in Rao and Ren ([1], p. 87). D Subsection 2. Nonlocal convexity A linear topological space (X, T] is said to be locally convex if r has a basis consisting of convex sets. Theorem 5. Let <£ be an N-function, then L* [0,1], L* [0,oo) and Z* are all nonlocally convex. Proof. We prove the result for the space L* [0, 1] (or L* (fi) with oo). We claim that for any r > 0,co(f/ r ) = L*"1^,!], where Ur = {/ € L* [0, 1] : /H^-i) < r} and co(Ur) is the convex hull, i.e.,

i=0

i=0 1

If /GL^" )^,!], we set

where fi = [0, 1]. Then ^ n fi,- = 0(i ^ j) and E ^(fi,) = ^(0) = 1; here i=0

// is the Lebesgue measure. Since $~ 1 (2|/|) G I/^O, 1], for any given r > 0, there is a 6 > 0, and an integer HQ = HQ(S) > 1, such that n( \J f2j) < 5 and 00

U n<

Since U—>00 lim —-^ — 0 , there exists a WQ > 0 such that u

T

_i, ^ .

ru

(7)

10.5 L

ff\~~ 1

-spaces and applications

415

Choose an integer m > ^p^ and divide the set (J Qj into m parts: GI, GI, i=i • • • , G m , such that Gj n Gj = 0(i / j) and

Let us define functions /o, /i, • • • , fm as IJ

MX) =

X £

i t Q U(

U

; Tin —1

0,

0,

if

i-\

x^G,, j = l,2,...,m.

771

Then /(x) = ]£ c^/j, where CQ = \ and GI = c2 = • • • = cm = ^ so that j=o m ^2 Cj = 1. Finally we show that fj € Ur, i.e., ||/j||($-i) < r, j — 0, 1, • • • , m. j=o For /0, by (6), one has ||/o||(*-i) < r. For /j,j = 1,2, • • • , m, since Gj C n

o—1

U ft*, we have ^ < |/(x)| < n0 and 2m|/(a;)| > ^ > w0 if x 6 G,,. Thus

r 2 < — [2mn0/i(Gy] < -r < r.

Since / 6 L® [0, 1] is arbitrary, we have proved that co(Ur} = L*-1[0, 1] for any given r > 0, so that Z/$ [0, 1] is not locally convex, n Recall that for a linear topological space X, the dual space is denoted by X*, the space of all continuous linear functionals on it. Theorem 6. We have (i) (L*"1^,!])* = {0}, (/*- 1 )* = /;°°.

(L^O, oo))* = {0}. (U)

Proof, (i) Let / e (I/*'1 [0, 1])*; we assert that /(/) = 0 for each / € L*"1 [0, 1]. Since / is continuous, for any given e > 0 there is an r(= r ( e , l ) > 0), such that ||0||(*-i) < r => \l(g)\ < £. For each / E L*~l[Q, 1] and this r > 0, by

416

X. Miscellaneous Applications

the proof of Theorem 5, there are GJ > 0, /,, 0 < j < m, with £ Cj = 1 such 3=0 m that / — ]T cj/j and H/jll^- 1 ) < r- Hence

j=0

j=0

j=0

Since £ is arbitrary, we deduce that /(/) = 0 so that / = 0. Similarly we can verify the second part of (i). (ii) I f / e (J*' 1 )*, we set a, = /(e*) where e{ = (0, • • • ,0, 1, 0, • • •) € J*"1 i

whence ||ej||($-i) = $ -1 (l). An element / — ( / ( ! ) , • • • ) € /*"' since it has oo

the form that / — £) /(z)ej, and

ll/-E/(Oe i ||(*-i ) = i=l z= as n —> oo. The continuity of / implies

Let g — (ai, a 2 , • • •). We need to show that g € /°°, i.e., H^Hoo = sup |aj| < t>i oo. Suppose, if possible, that sup Oj| = oo. Then there is a subsequence of {tti}^, denoted again by {ai}^i suc^ ^at \di\ > i,i — 1,2, • • • . Let fi = (0,0, • • • ,0, Y , O , • • •), where | is the ith component. Then ||/i||($-i) = $~l(}) -^ 0 as i ->• oo. It follows from (8) that (/(/;) | = \\a{\ > l,i = 1, 2, • • •, which contradicts the continuity of /. On the other hand, if g = (0(1), 0(2), • • •) € /°°, with 0 ^ I^IU = sup \g(i) < oo, we define / = lg such that

(9) We assert that I is continuous. Indeed, there exists a UQ > 0 such that u for 0 < u < u0 since lim -^ = 0. For a given E > 0 let 5 = min(u 0 , I f / - ( / ( I ) , - - - , ) 6 i^and ||/||(*-i) < 5, then ^ ( I / W I ) <

10.5 L$ -spaces and applications

417

6 < u0 for alH > 1 so that |/(i)| = ^(^(I/WI)) < ^"'(I/WD- Finally, we have from (9) OO

CO

OO

< *=i EMOII/WI < IMIooEl/WI) < IMIooE 1=1 »=i Example 7. If 0 < p < 1, then (i) Lp[0, 1],LP[0, oo) and /p are not locally convex; and (ii) (L p (ft))* = {0}, where ft = [0,1] or [0, oo) but (/p)* = /°°, (cf., e.g., Megginson [1], Examples 2.2.6, 2.2.7 and Exercise 2.2.29). In fact, $ p (u) = \u\'(Q < p < 1) is an JV-function and ||/||($-i) = ||/||p. The conclusions follow from Theorems 5 and 6.

Subsection 3. Embeddings For latter use we establish some other results. Propositions. Let$i,3>2 be N-functions and L*i (ft),L*z (ft) with ^(Q] < oo be the F-spaces as above. Then the following statements are equivalent. (i) $1 >- 2 (for large values), i.e., there are t0 > 0 and K < oo such that > t0, or equivalently, (u\u> u0 = $ 2 (t 0 ). (iiJL^M

(10)

C L*r l (fi). ^m; ||/||(*-i) -> 0 =» H/ll^-ij ^ 0. ^ 5 C

Z/^ft) anc? sup{||/||($-ij : / € 5} < a < oo z'rap/y t/io< supfd/H^-ij : / e 5"} < 6(a) < oo, w/iere 6(a) is a constant depending only on a and $i. Proof. (i)=»(ii) and (i)=^(iv). By (10), / € I/^'^ft) implies

so that / e I/$i (ft). From the above (iv) follows with 6 = $i1( (i)=^(iii). For any given £ > 0, choose A > 0 satisfying $f 1 (A)//(ft) < | and let K\ = max < LI M: : A < u < u0 >, /f2 = maxj/C, A!"i} with w0 and K {

9

2

(Ul

)

being as in (10). Thus fcf^u) < ^^J1^) for u > A. Let 5 = ^. Then / e LV (ft) and l l / l l - t < J imply

(iii)=>(i) and (iv)=»(i). Suppose that (i) fails to hold. Then there are un /• oo such that ^l(un] > 4 n $^" 1 (u n ),n > 1. Choose Gn C 0 such that

X. Miscellaneous Applications

418 Gn n Gm = 0 (n ^ m) and

Then for /„ = M n xc n , we have ||/n ^-ij = ^$2 X^)/^) ->• 0 as n -> oo, but IIMI^i) = $r 1 (u n )At(G n ) > 242-1(w1)/^) -^ oo as n -^ oo. The proof of (ii)=>(i) is left to the reader. D Corollary 9. The following assertions are equivalent, (i) <&i ~ 2 (for large values), i.e., 0 < liminf ^

< limsup *

<

< oo.

(n)

>-i) = Otf Jim ||/J(^

-i, : / 6 5} < oo for S C = 0. (iv) sup{ | tytfsupd/IU-v/e S} < oo. This result will be used in the proof of Theorem 24 in Subsection 4. For the sequence spaces we can verify the following. Proposition 10. Let $i,$ 2 be N-functions. Then the following assertions are equivalent, (i) $\ >- $2 for small values, i.e., there exist to > 0 and K < oo such that $i(i) < $\(Kt) for 0 < t < to, or equivalently 3>il(u) < ll/nll^- 1 ) ->• ° imPly ll/n ($-i) ~> 0. fwj supdl/nll^-i) : / € 5 C /*''} < a implies sup{||/ n ||^~i^ : / 6 S1} < b for some b = b(a) > 0. Corollary 11. The following assertions are equivalent, (i) <E>i ~ $2 (for small values), i.e., 0 < liminf

< limsup

u—>0

f e S C /*2 -1 } < oo i f f s u p { \ \ f

u^O

($ -i }

i . . < oo.

$2l(u]

(12)

: / € S1 C /*^} < oo.

Example 12. If $x >- $2 for large values, by Proposition 8 and a result for Orlicz function spaces (cf., Rao and Ren [1], p. 155) one has L°° C L*1 C

C

C L°,

(13)

10.5 L® -spaces and applications

419

where all the spaces are defined on a finite measure space (ft, E, /z) with p, nonatomic. Note that (13) is a generalization of (1). On the other hand, if $x >- $2 for small values, we have /° C I*'

C 1*1 C I1 C I*1 C I*2 C C0,

(14)

where /° is defined as in (5), / $ *(i=l,2) are the Orlicz sequence spaces and c o = {/ — ( / ( ! ) > • ' • ) : Jim /(«) = 0} with norm ||/||oo- In particular, If z—>oo 1 < Pi < P2 < oo, then /° C In c /« C /1 C /Pl C /P2 C CQ.

(15)

Example 13. Let $(u) = (1 + |u|) log(l + |u|) - |w| and V(v) = e|v| - |u| -1, then L°° C L* C D Lp C |J Lp C L$ C L1 C L*"1 C fl p>l

p>l

C

L?

0
U VCL^CL0,

(16)

0
where all the spaces are defined on (ft,E,//) as in Example 12, and L^,L* are Orlicz function spaces. Indeed, for any 1 > p > 0 we have that \u\p >$(u) for large u and by Proposition 8, L*~ C Lp so that L* C f| Lp. 0
Similarly we can verify the other inclusions in (16). Example 14. Let ($, #) be the TV-pair defined in Example 3.8. Then we have 1 : I /P(. c V—/*~ c \—I1 t cV c\_ C\ I/o c V— 7*I I I p fc cV_ I II I 0
0
/* C pi l C |J l C /* C C0.

(17)

Example 15. Consider the TV-function <$(«) = eu — 1. Clearly, 0(w) = $'(u) = 2we" for u > 0 and ^(v) = \og(l + v), v > 0 so that Q r(t)dt, where

for which r(0) = +00 and r(oo) = 0. This leads to the following.

420

X. Miscellaneous Applications

Lemma 16. Let $(u) = JQ (f>(t)dt, define an N-function. Then its inverse $-1 can be expressed in the form $-l(v) = [V r(t)dt,

v>0,

(18)

where r(t) is decreasing on (0, oo) with r(0) = +oo,r(t) > 0 if t > 0 and r(oo) = 0. Conversely, if r(i) satisfies these conditions with $-1(l) < oo and $-1(oo) = oo, then 3>~l defined by (18) is the inverse of an N-function Proof. Since $[$-l(t)] = t,t > 0, we have [$-l(t)]r(t) the right derivative of <£ -1 (t), so that

= 1, where r(*) is

r(t] =

which satisfies all the conditions stated in the lemma. The second part can be verified as the generalized Sobolev conjugate function $* of <$, (cf., Proposition IX. 1.5). D Recall that for Orlicz function spaces, we have L^Q) = \J{L*(ty : $ ranges over all ^-functions},

(19)

where /J.(£i) < oo (cf. Rao and Ren [1], p.50), and for the sequence spaces I1 = ("]{/* : $ ranges over all TV-functions},

(20)

(cf. Chen [I],p.l69). Similar to (19) and (20), one has the following. Theorem 17. Let (fi, S,//) be a finite measure space. Then Ll(£l) = P|{£* (^) : $ ranges over all N-functions},

(21)

and

I1 = [J{1®

'• 3> ranges over all N-functions}.

(22)

Proof. By the first part of (5), L X (Q) C the right side of (21). We now show that if / £ Ll(ty, then / ^ L^o^Q) for some TV-function $0. Let n n = {w G fi : n - 1< |/| < n}, n > 1. Then =00,

10.5 L® -spaces and applications

421

m—i since / ^ L1^). Choose HI > I such that X) ^A*(^n) > 1; n2 > ni such n=l

that X) ^M^n) > 2; • • •,njfc > n^_i with

X)

^A*(^n) > A;, • • •. Let an = I

if 1 < n < HI] — | if n\ < n < n2; • • • an = | if n^_i < n < n^, • • • , so that an —> 0 as n —> oo, and oo n\ — \ oo ifc~l /^ fln77.//(Qn) = ^ ^ Q,nn(J,(£ln) -\- 2_. / ^n^AH^n) n=l n=l fc=l n=7ifc-i

==

C^1-

(23)

Define

- / 7t'

0 < i <1

Then ^^^f) = /Ov r0(t)dt(v > 0) defines the inverse of the JV-function $0 by Lemma 16. Note that for u > 0 : ^ro(ti)

(24)

and X) ttn/X^n) < 53 A^(^n) = M^) < °°- Hence we have from (24) and n=l

n-1

(23) ,.

/ J ^(l/l)^

= E /jfln n-l n=l >

—

^

— V^ r) n

0 Z_^ Z n=l

i.e., / ^ L*» (Q), which proves (21). The proof of (22) is left to the reader. D

Theorem 18. Let <&i,
X. Miscellaneous Applications

422

(ii) Each \\ • \\,^-i Abounded set S C I/*2 (Q) has an equi-absolutely continuous F-norm \\ • $-i in I/*1 (Q,), i.e., sup||/|L>,$ -u < a < oo implies 1 2 ) = 0.

(Q),sup ||/n||$-i < a < oo and fn —> 0 in measure,

(Hi) If fn then

(25)

0.

Proof. (i)=^(ii). Let F(t>) = ^^[^(i?)], then F is strictly increasing on [0, CXD). By (i), letting v = $il(u), one has

.. F(v) .. hm —^- = hm

= +00.

For a set S C I/*2 (Q) with sup/ n <J>2 l(\f\)d/-i < a < oo, let 5 = {g = /es ^rXl/ ) : / e 5)}. Then fnF(g)dn < a for all 0 e S1. It follows from the Vallee-Possin theorem (cf., e.g., Rao and Ren [1], p.3) that S has equiabsolutely continuous || • ||i-norm in Z/ 1 ^), i.e., lim sup||px^lli = 0 so that sup / $r 1 (|/|)d/^ = 0.

lim

Ai(,4)->0 |_/6,5 7A

(ii)=»(iii) If / n ||/$-i)< a < oo and fn —>• 0 in measure, then by (ii), for any given e > 0 there is a 8 > 0 such that A G S, //(.A) < „ I .f ... II ^ £ Let T «j- O f , . ^- O . ^ /. .\ I ^ /T\ / £ M then ^-U«», . , / sup ttn = ' J n>l

0 as n —>• oo so that an integer n$ can be found such that (J>(£ln) < 6 for n > HQ. Hence we have for n > no ^(l/. x _i\n n. so that |/ n ||($-i) < ll/nXfinll^- 1 ) + ll/nXn\n n || ( $-i) < | + | = £• (iii)=^(i). Suppose that (i) fails. Then there are un /* oo and SQ > 0 such that &^l(un) > ^o^^H^n)) 71 - ^ 1 so that (26)

eo

10.5 L? -spaces and applications

423

We may assume $±l (HI) n(£i) > £Q. Define fn = $2[j-$Tl(un}]XGn, where GncG satisfying fj,(Gn) = Sfj f°(U ). -> 0 as n -> oo. Then H/nlL-h = 1 for \ 2' n all n > 1 and /n —>• 0 in measure. On the other hand, we have from (26) = £Q,

71 > 1. D

Example 19. Let ^(u) = e|u|Pi - 1, 1 < pt < oo,i = 1,2. Then /„ ->• 0 in (the Lebesgue) measure on [0, 1] and sup/01[log(l + |/n(^)|)]p2^M( :r ) < a < oo n>l

imply Urn jfWl + |/ n (x)|)]«d^( a; ) = 0 iff Pi > P2- In fact, $1 >->- $2 for large values iff lim -

($>~^(u)

-

J___p L

,; ( = lim [log(l + u)]"i /J - - L ov

2 = 0,

iff pi > p2- Thus the assertion follows from Theorem 18. Proposition 20. Let $ be an N-function and n(£t) < oo. Then a set S C L*"1^) is (conditionally) compact iff (i) S is "compact in ^-measure", i.e., for any sequence {/n}i° C 5 there exist a subsequence {fnk : k > 1} and a function f 6 L$ (0) such that fnk —> / in measure; and (ii) S has equi- absolutely continuous F-norm \\ • ||($-i). Proof. Sufficiency. Let 5 satisfy (i) and (ii). By (i), any {/„} C 5 has a subsequence, denoted by {/„} again for simplicity, and an element / e L $-1 (Q) such that fn -> / in measure. For a given e > 0, by (ii) there is a S = S(e) > 0 such that A € E and [J,(A) < 6 imply supH/nX^II^-i) < n>l

(27)

^

Choose A > 0 to satisfy $~ 1 (A)yu(J7) < | and put Gn,m = {uj G Q : |/n — /m| > A}. Then ^(G m , n ) ~> 0 as n, m —»• oo. Select an integer no such that < 8 if n,m> n0. It follows from (27) that for m, n > n0 ^ (i/n -

424

X. Miscellaneous Applications

Thus S is (conditionally) compact since L*"1 (J7) is complete under the metric

d(f,g) = \\f-9\\(*-i)

Necessity. Similar to that of Theorem 5.2.2 in Rao and Ren [1], p. 158. n From Theorem 18 and the above result we get the following.

Corollary 21. Suppose that $1 >->- $2 for large values and //(fi) < oo (in this case L$2 (f2) c L*1 (^)j. For a subset S C I/*2 (£2), i/ 5 zs compact in p,— measure and sup/ n ^J^j/Drf// < C < oo /or some C > 0, iAen 5 is (conditionally) compact in L*i (Q). Example 22. Let 0 < p < 1 and Lp[0, 1] be endowed with the F-norm \\f\\p = lo \f\pdn, V being the Lebesgue measure. Then a set S C LP[Q, 1] is compact iff (i) 5 is compact in measure, and (ii) S has equi-absolutely continuous F-norm || • \\p. Further, if 0 < pi < p2 < 1, S C L P2 [°> 1],5 is 'compact in //-measure' and sup ||/||P2 < C < oo, then 5 is similarly compact in L?1 [0,1].

/es

Subsection 4. Local boundedness and <1>~ ^variation Recall that a set 5 in an F-space ( X , \ \ - |) is metrically bounded if 5 C {/ G X : ||/ | < r} for some r > 0, and S is topologically bounded if for each neighborhood U of zero there is an r > 0 such that S C rU\ in other words for any an \ 0 and fn € S one has lim ||o;n/n|| = 0. These two concepts n—>oo need not coincide (cf., Example 23 below). Further, an F-space is locally bounded if it contains a topologically bounded neighborhood of zero. Example 23. If an TV-function $ ^ A 2 (oo), then there is a metrically bounded set 5 in I/*"1^, 1], which is not topologically bounded. Indeed, $ ^ A 2 (oo) implies that there exists a sequence un /* oo such that 3>(1un} > n$(u n ),n > 1. By letting vn = $(un) one has *-l(vn)>l*-l(nVn),

n>l.

(28)

We may assume <3>~ 1 (t> 1 ) > 1. Choose Gn C [0, 1] such that //(G n ) ( l>~ 1 ( nv n) = 1 where // is the Lebesgue measure. Define /„ = nvnxcn- Then S = {fn : n > 1} is metrically bounded since ||/n | |($-i) = 1 for all n > 1. On the other hand, if an = ^, then (28) implies that > -^~l(nvn}^(Gn] = -, proving that S is not topologically bounded.

10.5 L* -spaces and applications

425

Theorem 24. Let $ be an N -function and (£7, S, //) be a measure space, JJL being nonatomic. Then: (i) L$ (fi) with p(£l} < oo is locally bounded iff ^ £ A2(oo). (ii) The sequence space I® is locally bounded iff ^ G A2(0). (in) L*~l(£i) with n(£l) = oo is locally bounded i f f $ £ A 2 . Proof, (i) Sufficiency. Let 3> e A2(oo). We shall show that for any r > 0, Ur = {/ G L $-1 (fi) : ||/||($-i) < r} is topologically bounded. Note that $ e A2(oo) iff fo = limsupl^gy < 1, or iff j- = lirninf £» > 1. Now U— K3O

*

'

U—rOO

V /

a = |(^$1 + 1) implies that (3$l > a > 1 and we can find UQ > 0 such that <3>~1(2w) > a$~l(u) for w > w 0 - Let c = "^ffiy , then c > 1 and we define an JV-function

It is seen that $1 ~ $ for large values and <3>i € A 2 so that /3$1 = sup *h^\ < I (cf., Section I.I), and ^l(u} < ~/3~$il(2u),u> 0. Hence for (2u u>0 *i

>

any integer n > I one has

(^r^M «>o.

(29)

If an \ 0 and fn £ Ur, by Corollary 9 there is an R = R(r) > 0 such that l l / n l l - 1 < R f°r all n>l. We assert that (30)

Since ^$1 < 1, for any given e > 0 there is an n0 > 1 such that (/3$1)n. £,n > HQ. For this no there exists HI > I such that |an| < 2~n° if n > since an \ 0. It follows from (29) that for n > ni

which proves (30) since e is arbitrary. Again by Corollary 9, (30) implies ll^n/nll^- 1 ) —> 0 as n —> oo. Thus Ur is topologically bounded, and L*"1^) is locally bounded. Necessity. If $ ^ A2(oo), then there exist vn /*• oo such that (28) holds. For each neighborhood U of zero, there is an r > 0 such that Ur C U. We shall show that Ur is not topologically bounded so that U is also not bounded, proving that L*~l (J7) is not locally bounded. Since vn /* oo and

426

X. Miscellaneous Applications

< oo, we may assume that $~ 1 (v 1 )//(fi) > r. Choose Gn C G satisfying r and denne /" = nvnXGn- Clearly ||/n||($-i) = \ < r, i.e., n) = 2 $-i (,», n )» fn £ Ur. If an = i, then by (28) we have

proving that Ur is not topologically bounded. (ii) Sufficiency. Let $ e A 2 (0). We show that for fixed r > 0, Ur = {} € J*~ x : ||/||($-i) < r} is topologically bounded. Since $ € A 2 (0) iff /?$ = limsup ^ i"^- < 1, we have /3| < 1, where u->0

^ "^

Hence ^-^w) < ^~l(2u) for 0 < u < $(r) and for any n > 1 (^) n ^ 1 W, 0 < W < $ ( r ) .

(31)

If /n € t/ r , then ^-^l/nWI) < E ^ - H / n W I ) = IIMI($-i) < r so that 1=1 |/ n («)l < ^*(r) f°r a^ n,i>l. Let e: > 0 be given, and pick an n0 > 1 such that (/3|)n°r < £. If an \ 0 as n —>• oo, one can find n\ > I such that n \0tr. < 2 ° for n > ni. Finally, from (31) we get for n >

which proves that Ur is topologically bounded, and thus /* is locally bounded. Necessity. If $ ^ A 2 (0), then there exist un \ 0 such that <$(2u n ) > n<£(u n ),n > 1. By letting vn = $(un) one has \ fl /

r\

\

71 / )

—

'

\

/

We assert that for every r > 0, Ur — {/ 6 /* : ||/||($-i) < r} is not topologically bounded. Since vn \ 0 we may assume that 83>~l(vi) < r. Note that (32) implies r r>^R_1 /

T \

/I/R — 1 / „ .

T \

—

AA^ — I f . .

\

\^'~*1

10.5 L^ *-spaces and applications

427

Let kn be the integer part of the left side of (33), so that kn < —rn

7
(34)

Now we define /„ = (nvn, nvn, • • • , nvn, 0,0, • • •) with dim(fn) = kn. Clearly, ||/n||($-i) = kn<&~l(nvn] < I < r. On the other hand, if an — ^ one has from (32)-(34)

which proves that /* is not locally bounded. (iii) It is simpler than (i). n Example 25. Let 0 < p < 1. Then Lp([0, 1]) and lp are locally bounded. [cf., Megginson [1], p. 202]. In fact, if $~l(u) = \u\p, the TV-function $ p (w) = e A 2 . Thus the assertion follows from Theorem 24. Example 26. Let $ r (w) = e |u ' r - 1,1 < r < oo, i.e., ^"^i;) = [log(l + H)]r. Then $r € A 2 (0) since ^g = lim |^\ = 2~r < 1; and clearly r

It—>0 *T-

(^ U J

1

3>r ^A 2 (oo), $ ^ A 2 . By Theorem 24, I*; is locally bounded, but L*'"1^) (A*(fi) < oo),^ being nonatomic, is not locally bounded. Recall that for a 0-function 0, the class of functions with (j)—boundedvariation on the closed interval [a, b] C M is defined to be ( V*[a,b] = l f : f ( a ) = Q and I

in which = sup 2J (f>(\f(xk) a

— /(^fc-i)D)

(35)

* k=l

where the supremum is taken over all partition TT : a = x0 < x\ < • • • < xm — b b. It will be interesting to find conditions on 0 under which lim V(, <*»/) = 0 for each / G V^,[a, 6] so that V^[a, b] becomes an F-space (cf. Maligranda [2], p.12). In the case that 0 = $~1, the inverse of an TV-function <£, J. C. Wang [1] obtained a partial solution of the above. We restate (35) as follows. Definition 27. Let $-1 be the inverse of an TV-function <£. The space of ^"^variational bounded functions on [a, b] is defined to be 6

:/(a) = 0 and Y/^

1

> /) < °° t »

(36)

428

X. Miscellaneous Applications

where

6

m

VC*-1, /) = sup £ Q-^fW - /(z f c _i)|), w

a

(37)

jfc=l

in which sup is taken as in (35). 7T

6

By (3) and (4), V$-i[a, b] is seen to be a linear space, and V($ a 6

6

, •)

satisfies the following conditions: \/($~ 1 , /) = 0 iff / = 0 on [a, 6]; V($-1, /) = V^" 1 ,-/); and the triangle inequality VC^" 1 ,/ + g) < fl

O,

6

ft

6

,
,/n) = 0 implies that for a 6 JR and

b nlirn

Hence V$-i [a, 6] is a linear metric space under the metric d(/, #) = V($ > /~ #), which is also complete since $~l is continuous.

a

Theorem 28. // an N -junction $ e A 2 (0), i/ien we 6

(i) V(^~ 1 , •); defined by (37), is an F-norm and V$-i[a, &] is an F-space. a

b

(ii) The F-space (V$-i[a, &],V($ a

r)) ^s locally bounded.

Proof, (i) It is to be shown that for any o^ \ 0 and fixed / 6 V$-i[a, 6] one has lim V(*~ W) = 0, »

(38)

under the hypothesis $ e A2(0). For any given e > 0, let SQ = ^, where C = VC^" 1 , /) 7^ 0 (if C = 0, (38) holds automatically). Since $ e A 2 (0), for a

these £Q > 0 and C < oo, there exists X = /^(eo, C) > 1 such that $(^) < K$(u),Q < u < C. By letting v = $(w) we get ^-^v) < eo^"1^).0 < ti < $(C), and hence ^), 0 < t < K$(C).

(39)

Since Q;J \ 0, an integer IQ > 1 can be found such that |a;| < -^ for all i > IQ, and a partition ?r : a = x0 < x\ < • • • < xm = 6, 3>~l(\f(xk] - f ( x k - i ) \ ) <

10.5 L* -spaces and applications

429

V($-i,/) = C so that \f(xk) - /(xk-Jl <$(C) < K$(C),k = l , - - - , m . a

From (39) we finally have for i > i0 i /V

l/^)-/

£0C - £,

<

which proves (38) since e is arbitrary. The completeness and linearity are proved in the usual way. (ii) If $ e A 2 (0), we claim that for any fixed r > 0, Ur = {/ € V$>-\ [a, 6] : 6

V($ a

, /) < r} is topologically bounded so that V^-i [a, 6] is a locally bounded

F-space. Recall that for any n > 1 there is a /?$ < 1 such that (31) holds since $ G A2(0). Let £ > 0 be given. For a sequence an \, 0 as n —> oo, find no > 1 satisfying (/?|)n°r < £ and choose n\ > 1 such that |o!n| < 2~n° if n > n\. For any partition TT : a = x0 < x\ < • • • < xm = b and { f n } f C Ur, we have that \fn(xk) - fn(xk-i)\ -1

n > HI (since $

< ^(\f(^ljn)) a

< ^(r), and by (31) for

is increasing)

jfe=i

<

(^

ff

*=1

proving that Ur is topologically bounded since e is arbitrary. D Theorem 29. Let $ be an N -function and $~1 be its inverse. If the linear b metric space V$-i[a, 6] under the metric d(f,g) = V(^ > ~ 1 ) / — d) is locally bounded, then $ e A 2 (0). Proof. Suppose that $ ^ A 2 (0). Then there exist vn \

0 satisfying (32).

We show that for each r > 0, Or = {/ € V^-ifa, 6] : d(/, 0) = V( < ^ ) ~ 1 , /) < r}

430

X. Miscellaneous Applications

is not topologically bounded, which proves the theorem. Let kn be as in (34) with r satisfying (33). Divide the interval [a, b] in (kn + 1) parts of equal length -^, i.e., G, = [a, a + ^), G2 = [a + ^, a + f^), • • • , Gkn+1 = [a + fc -M ; fr]> and define /n = OxGl + nvnXG2 H

H (fcn - l)wnXc fcn + knnvnXGkn+l,

n = 1, 2, • • -. It follows from (34) that 6

r

f u(]}; — — W d J ™ 1 ' ./«/ f }— — A/ k n^ <&~^(nii < ^.— ^~ \Jn-> V fV^ \lliUn)} ^:

u /Vf

so that /n € Or. On the other hand, if an — ^, we have from (32)-(34)

proving that Or is not topologically bounded. D Subsection 5. Fourier analysis in L$ -space Since Ll[Q, 2?r) c I/*"1 [0, 2?r) for every A^-function $ (cf. (5)), we consider again the conjugate function / of / G Ll[Q, 2yr) defined by f(x) =

lim < / 2?T e->0+ {JQ

/(t)cot

dt + / 2

7x+e

f(t)cot

-dt>. 2

(40)

J

(cf. Section VI.5). To see that the space L* [0, 2yr) is nice for applications, we include, for convenience, a proof of the following result, which is a modification of Lesniewicz's method. Theorem 30. Let $ be an N-function. If

du

< oo,

(41)

then for each f £ Ll[Q, 2ir), its conjugate function / G L* [0, 2?r) and the linear operator T : f i—> f is continuous from Ll[0, 2?r) into L® [0, 2yr). Proof. For v > 0 and / E Ll[Q,2n), let E(v) = {x E [0, 2?r) : |/(x)| > v}, which is measurable, and g(v) = n ( E ( v ) ) , where // is the Lebesgue measure. Then g satisfies the weak-L1 inequality d(v) < - l l / l l i , i;

(42)

10.5 L® l -spaces and applications

431

for some c > 0 (cf. Lemma VI. 2. 4). For v > 1 we have $-l(\f(x)\)dx

/ J[0,2*)\E(v)

=

$-l(\f(x}\}dx

L J{\f\
l $-U (\f(x)\)dx M W

,

\n

— ^ - [* *-\u)dg(u\ Jo as n —> oo. Using (41) we get (with integration by parts): /

J[Q,2ir)\E(v)

-

Jo

^

3>-l(u)dg(u)

o

i - 9(1}}

i

u

(43) Hence the above computation and (41) imply

i /i (*-i) = r *-\\ J0

=

lim /

v

^°° J[0,2n)\E(v)

$-l(\f(x)\)dx (44)

proving that / e ^'^[0, 2?r) since /^_ 1 1

< oo if ^~l(l) < 1.

432

X. Miscellaneous Applications

Next^we show that for any given e > 0 there is a 6 > 0 such that ||/||i < 6 => ||/||($-i) < £, i.e., T is continuous, by using

= - r $>-l(u}dg(u) - f ^l JEQ

JO 1

where 0 < £0 < v is chosen such that 27r~ (£o) < f for the above given £ > 0. Analogous to (43) and (44) we get (45) J$-l(£0)

^(l)

Let o =

dt

It follows from (45) that ||/||i < 6 implies ||/||(*-i) < e. D Corollary 31 If an N-function $ € V 2 (oo), then $ satisfies (41) so that T : f i—> f is continuous from I/^O, 2yr) into L® [0, 2?r). Proo/. If $ e V 2 (oo), then there are u0 > 1, o; > 1 and K > 0 such that <3>(u) > /CwQ for U>UQ (cf. Rao and Ren [1], Corollaries 4 and 5, p. 26). Thus (41) holds because r°° duu Ji Q

~

r u° du Ji

r°° du

Remark 32. There exist ^-functions $ satisfying (41), but $ ^ V 2 (oo). For instance, if <£(u) = |u|[log(l + |w|)] 2 , then by letting t = log(l + u) we have r°° du _ r°° dt r°° 2dt h

3>(U)

=

/Iog2 (1 - e^t2

<

710S2 ~W

<

°°'

i.e., (41) holds for this $. But $ ^ V 2 (oo) since lim "^!ff = 1. In general, if 0 < r < oo, then $ r (w) = |w|[log(l + \u\)]r £ V 2 (oo), but $r satisfies (41) if r > 1. Example 33. Let $ p (u) = e iu|P - 1, 1 < p < oo. Then $p e V 2 (oo), and by Corollary 31, T : f i—>• / i s continuous from L^O, 2?r) into L^p^[0, 27r), i.e., lim \\fn\\i = 0 implies that —

10.5 L® -spaces and applications

433

Lemma 34 Let $ be an N-function, and fo(x] L*"1 [0, 27r) iff® satisfies (41).

= cot|.

Then fQ 6

Proof. Necessity. If cot | is 3>~l integrable, then as in Lemma VI.5.2, n /•27T

oo >

/ Vo

/

7-

S-

27

oo

sin

~ ^o

r°° 1

/ —

7<»->

27r,_1/ 3 . r°° du —$ ( — ) -|- / i 3 2?r 7$- (—) $(w)

, _-.. .. (u — $ (t))

which proves that $ satisfies condition (41). Sufficiency. Let $ satisfy (41). It follows (as in the earlier lemma) that /•27T

o

/

T \

3>-l(\cot-\}dx V 27

rtr

/rn«: — \

fir

/TT\

= 2/$-1[^ f ]dx Vo Vsinf/ < 2 / $-: ( - ) dx 7o Vrc/ OO

/

*-'(')" G)- c=;) /I \

77-

Thus cot I has the asserted property. D Theorem 35. I f T : f \—>• / is continuous from Ll[Q, 2?r) into L*"1^, 2?r), i/ien $ satisfies (41). Proof. Similar to the proof of Theorem VI. 5. 6, for £0 = 1, there exists 0 < 6 < 1 such that \\f\\i < 5 implies
Define fn(x) = n6x[o,±](x),

(46)

n>l. Then \\fn\\i = 5, and for x € (£, 27r),0 <

X. Miscellaneous Applications

434

£ < x — ^ we have from (40) •e

o

+

-r

fn(t) cot

2

f2ir

-dt + /

,/£+£

+

/ n (t) cot

7-

r—

dt = nS j

2

vo

+

cot

7*

2

dt,

so that n6 r$

t-x

dt.

Hence lim fn(x) — ^ cot |, x e (0, 2?r). Using Fatou's lemma we get

27v

COL

1

2 J

7

'

I ctx. since

2-jr

\ J-.

lim ^""^/n(a;)|)da; byLebesgue'sdiff.thm. Q

71—^OO

/•STT ~

n->oo

„

./o

—

'

i.e., cot | is ^~1 integrable, and $ satisfies (41) by Lemma 34. D Similar to the proof of Theorem VI.5.8, we can verify the following. Theorem 36. If an N-function $ satisfies (41), then for every f G Ll[0, 2?r) we have x] - f(X)\}dx

=0=

im

where Sn[f] and Sn[f] are the n-th partial sums of the Fourier series and the conjugate Fourier series of f respectively. Corollary 37. (Kolmogorov [1]) // 0 < p < I , then T : f i—> / is a continuous linear operator from Ll[0, 2?r) into Lp[0, 2?r). Further, for every / e ^[0,271) Jim

= 0 = Urn

Proof. Let an TV-function be defined by &p(u) L p [0,27r) = L*rl[Q,27r) and | /|| ( $-i) = /02^ |/ follow from Corollary 31 and Theorem 36. D

- f(x)\pdx. u\p € V2(cxo), then - The assertions

10.5 L® -spaces and applications

435

Remark 38. 1. We know the Lesniewicz condition ff° ^f-du < oo, for a ^-function (f> as in Section VI.5. In the case that (f) = $-1, the inverse of an ./V-function, we observe that

r°°$-l(u),

\

Ji

2^du < 00 iff

u

r°°

du

—-r < 00,

/

y*-i(i) 3>(u)

or iff $ satisfies (41). Indeed, by letting u — $(t) we have

00

dt dt

= ^(1)

We may then call (41) the Lesniewicz condition in the case that = $~l. 2. Since $-1 is a special 0-function, one can also define the "old" F-norm || • ||$-i as in Section VI.5 by

( r i f\f\\ 1 ll/IU-i = inf 0 : I/ — $~l \l — }du. « l — I 1 1 1 1

(47) ^ '

Our "new" F-norm || • ||($-i) defined by (2) is simpler than (47). For instance, if / = axG>G C n,0 < Ai(G) < oo, then ||/||(*-i) = $>~l (a) ii(G] , but ||/||*-i is the solution c of the equation $~ 1 (-)//(G) = c. Moreover, we have (i) H/IU-i = 1 iff ||/||(*-i) = 1, (ii) if 0 < 11/11*-! < 1, then \\f\\l_, < ||/||#-i; and (iii) if ||/||*_i > 1, then ||/||$-! < \\f\\(*-i) < \\f\\i-i. For, since the 0-function $-1 G A 2 ($-1(2u) < 2$ -1 (w),w > 0), we have

(48) (cf. Rao and Ren [1], p.402). Thus, (i) follows from (48). If 0 < ||/||*-i < 1, then by (48) •*-') = / ^(l/DdM < / Q-1 f

yn

yn

$-1 "^ n /-I!

V

436

X. Miscellaneous Applications

proving (ii). Similarly one can verify (iii). However, both are equivalent F-norms, the L* (Q) being complete in both metrics. Bibliographical notes. A class of subalgebras of the convolution algebra Ll(M), relative to certain weights were first introduced by Beurling in the late 1930's for which the classical Fourier analysis results such as the Wiener-Levy theorem and others were extended. This rich class based on the Lebesgue space Ll(]R] were analyzed and termed Beurling algebras by Reiter [1]. In the context of L* (]R), $'(Q) > 0, we find that there is a natural extension to L*(JR), and (appropriately) termed them Beurling-Orlicz algebras. These new (convolution) algebras have a good prospect of analysis, but so far only a rudimentary study is completed. We introduce this class in Section 1 to have it available for future research, especially to extend Beurling's [2] results on new convolution operators for these algebras. This as well as the remaining sections contain various complements and miscellaneous results. Thus Sections 2-5 are devoted to specialized, and sharp analysis of Orlicz spaces when the underlying measure space is either purely discrete or of (finite) diffuse type. They discuss the following classes, the exact references having been included in the text, we only indicate a view of the work covered. Thus Section 2 treats a geometric invariant, Riesz angles, of Orlicz spaces and relates it to certain other invariants such as Kottman constants and reflexivity properties studied earlier. This concept has interest in applications related to nonexpansive mappings and fixed point theory. Then Section 3 complements the work on embedding theorems considered in the earlier volume, by obtaining sharp results for the sequence spaces. This is largely worked out by Ren for this book. Similarly Section 4 contains several additions to the differentiability and smoothness of Orlicz sequence spaces or those based on diffuse measures, paying special attention to both the gauge and Orlicz norms. The final section on certain nonlocally convex (or a class of generalized) Orlicz spaces contains several extensions of known results for L*-spaces with $ an ./V-function, to L^-spaces,
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Notation Chapter I $: JV-function, 2 \£: complementary TV-function to $, 2 A2(oo), A2(0), A2, V2, (growth) conditions on Young functions, 2 A$,B<s>,A<s,,B$,AQ,BQ,C, indices, 2, 8

cty,/3*,cF*,/9"*,arJ,/3j,7$, indices for $, 5, 8 (fi, E, /Li): measure space 13 E£A): trace of S on A, 14 L*: Orlicz class, 13 p^(-): modular, 13 L*: Orlicz space, 14 Xs: characteristic function of the set E, 15 || • ||$: Orlicz norm, 14 II ' ||(«): gauge norm, 16 ^*,^*\m*,m^: Orlicz sequence spaces, 19 / * g: convolution, 32 $ S) 0 < s < 1: intermediate Young function, 36 , e): modulus of convexity of X, 42

Chapter II X: Banach space, 49 J(X): nonsquare constant of X in the sense of James, 50 g(X}\ nonsquare constant of X in the sense of Schaeffer, 50 CNJ(X}'. von Neumann- Jordan constantof X, 51

Chapter III N(X}: normal structure coefficient of X, 73 r(A): relative Chebyshev radius of ^4, 73 d(A): diameter of A, 73 N(X}\ self-Jung constant of X, 75 BS(X}\ bounded sequence coefficient of X, 75 WCS(X}\ weak convergent sequence coefficient of X, 76 yl({:ci}): asymptotic diameter of {x^}, 76 V(X): Maluta's coefficient o f X , 77

Chapter IV X: normed linear space, 117 457

458

JC(X}: Jung constant of X, 117 r ( A , X ) : Chebyshev radius of Ac X, 118 fin: Euclidean n-space, 118 H(n+i)xn: Hadamard matrix of (n + 1) x n-size, 121 Chapter V

P(X}\ packing constant of X , 147 K(X): Kottman constant of X , 148 Ri\ Radernacher function, 168 Chapter VI

Ck(f}: Fourier coefficient of /, 184 Sn(f)'- nth partial sum of Fourier series of /, 184 Dn(f): Dirichlet kernel of order n, 184 /: conjugate function of /, 191 m/(A): measure function of /, 192 \\T\\piq: strong (p,q) norm of T, 192 H|T|jp >q : weak (p,q) norm of T, 192 /*: decreasing rearrangement o f / , 207 H<2k(-): Haar function on (0, 1], 207 0 : 0-function, 227 (Z^(ft), || • |0): generalized Orlicz space, 227 A4*: subspace of L® or t® spanned by simple functions, 221 Sn(f)'- n tAl -partial sum of conjugate Fourier series of /, 232 Chapter VII M®N: direct sum, 237 P§: closed projection, 238 sgnx: signum function of x, 240 EB: conditional expectation, 241 PM: prediction operator, 245 PB: conditional probability, 251 RN: Radon-Nikodym, 261 (H): //"-property, 261 (G): Gateaux differentiable, 265 (F): Frechet differentiable, 265 LI(IJL, X}: Lebesgue space of X- valued /^-integrable functions, 266 , X}: Orlicz space of X- valued functions on (Q, E, /z), 266 Chapter VIII

/(•): rate function, 275

Notation

Notation

459

A(-): complementary Young( or Legendre transform) of A(-), 276, supp(ii): support of the measure /z, 280 LDP: Large deviations principle, 284 BM: Brownian motion process, 284 O.U.: Ornstein - Uhelenbeck process, 287 (rj a , E a ,P^, # ai/ g): projective syatem of probability spaces, 295 (LA0, X } , || • ||(A)): Fenchel - Orlicz space, 301 /A(<3): A-variation, 302 ||G||( A ): A-norm of additive G, 302 (Y\(lJ"> %}•, II • II) : Banach space of additive set functions of A-variation, 302 (A 2 ) A', V': (exponetial) growth conditions for principal parts of Young functions, 304 5(A p ,u; a) 0,oo): Besov - Orlicz space, 311 w a ,/3(') : modulous of continuity function, 311 B(X,y): Banach space of bounded linear mappings from X to y, 313 y**: second adjoint of 3>, 319 Chapter IX (Wm'p(G), ||-)IU, P ) Sobolev space on the set G,of order m, 328 GI CC G: GI a relatively compact subset of (7, 329 (W™f(G},N™/(-}}: Orlicz - Sobolev space, 331 X <—> y-. X is (continuously) embeded in 3^, 332 p*: Sobolev conjugate of p, 333 p'(— -^Y): conjugate exponent of p, 333 *: Sobolev conjugate of <£, 334 (w&/(G}>N&,/3('^: Orlicz ' Sobolev sPace of infinite order, 342 Ha(-): Hausdorff a-dimensional measure, 347 Hh(-): generalized Hausdorff metric measure, 347 div g: divergence of #, 353 Ma: Hardy-Littlewood maximal fractional operator, 356 Tp'- Nemitsky operator, 368 L^'*')(G): "multiplicator" Orlicz space, 376 M(-): superposition functional, 377 Chapter X ijj(-): weight function, 381 | • ||($), w ): weighted Orlicz space, 383 ): Beurling - Orlicz algebra, 384 S: Segal algebra, 384 a(X}: Riesz angle of a Banach lattice X, 384

460

Notation

I/I V |0|(|/| A |0|): max (min) of |/| and \g\, 384 <3>i >- (~)3>2 at 0: $2 is weaker than (equivalent to) ->- (n)$2 at 0: $2 is essentially (completely) weaker than <J?i for small values, 395 (397) 4>i->- $2 at 0: $2 decreases more rapidly than i for small values, 399 q®-, Q&'- geometric invariants of £*, 401 (UG)[(UF)]: uniformly Gateaux [Frechet] differentiable norm, 402 (WUR)[(W*UR}}\ weakly [weakly*] uniformly convex, 405 (UK): uniformly convex (or rotund), 411 L^ : F-normed space denned by < £~ 1 , 412 Us-i: F-norm, 412 -i[a.6]: space of $^1-variation bounded function space, 427 C^" 1 * /) : ®~ ^variation of /, 427

Index conjugate function, 191 Fourier series, 232 contraction principle, 277 Cramer's theorem, 284 cumulant generating function, 276

a.a. (=almost all), 242 ABK-theorem, 263 admissible mean, 287 annihilator, 245 approximate identity, 224 associated intermediate TV-function, 38 asymptotic diameter, 76

D

B

Banach lattice, 150 Banach-Mazur distance, 74 Besov-Orlicz space, 311 best predictor, 237 Beurling algebra, 382 Beurling-Orlicz algebra,382 bounded sequence coefficient, 75 Brownian motion, 273

Caratheodory function, 368 Cauchy's equation, 329 CBS-inequality, 213 Chebyshev center, 120 inequality, 288 radius, 117 set, 244 subspace, 246 Clarkson's inequality, 41 complementary TV-function, 2 subspace, 245 concave function, 411 conditional median, 253

decreasing rearrangement, 207 de la Vallee Poussin's theorem, 323 Dirichlet's equation, 327 kernel of order n, 185 disintegration of measures, 242 divergence operator, 353 E

elliptic differential operator, 327 embedding theorem for OrliczSobolev spaces, 341 empiric distribution, 276 measure, 276 equimeasurable, 207 exponential type TV-function, 273

F-norm, 227 -property, 261 Fenchel-Orlicz space, 301 finite representation, 52 Fourier series, 184 Franklin functions, 312 Frechet space, 227 differentiable norm, 261 fundamental function, 207 461

462

Index

G

gauge norm, 16 Gaussian dichotomy, 287 probability measure, 297 process, 286 Gauteaux differentiable function, 298 generalized Orlicz space, 227 random functional, 377 H

Haar measure, 202 system, 207 Hadamard matrix, 121 Hajek-Feldman theorem, 287 Hardy-Littlewood maximal (fractional) operator, 356 harmonic conjugate, 191 Hausdorff dimension, 347 metric measure, 347 Hausdorff-Young inequality, 211 Hilbert-Schmidt operator, 291 Holder's inequality, 15 homogeneous Banach space, 201 H-property, 261

K Kolmogorov's critetion, 304 fundamental theorem, 299 Kottman constant, 148 Kovalevskaya's theorem, 330

Laplace's principle, 285 transform, 276 large deviations, 274 Legendre transfoem, 276 Lesniewicz condition, 233 Lions' lemma, 395 Lipschitz mapping, 76 local boundedness, 395 functional, 377 locally convex vector space, 297 Lorentz's class, 364 loss function, 254 Luzin's conjucture, 199 M

Marcinkiewicz's theorem, 355 Markov inequality, 306 process, 343 martingale convergence theorem, 256 strong operator, 316 image measure, 277 measure function, 192 independent random variables, 273 measurable subspace, 237 intensity parameter, 308 median, 253 intermediate Young function, 36 metrically bounded set, 424 ltd SDE, 312 minimum (maximal) conditional median, 316 modular, 13 modulus of continuity, 42 smoothness, 42 moment generating function, 276 Jensen's inequality, 125 Morse-Transue space, 14 Jung constant, 117

Index

463

multiplicator Orlicz space, 376

N N- function, 1 Nemitsky operator, 368 nonexpansive mapping, 77 nonlinear prediction, 249 nonlocally convex space, 414 nonsquare constant (James), 50 normalized young pair, 27 action functional, 299 normal structure coefficient, 73 O

Orlicz norm, 14 space, 14 sequence space, 19 space form of the Riesz-Thorin theorem,36 -Sobolev space of order m (oo), 331 (342) Ornstein-Uhelnbeck process, 287

packing constant, 147 Paley-Wiener-Zygmund integral, 287 Parseval's formula, 214 p-convex space, 150 p(3)-convex, 176 •^-function, 227 Poisson integral, 199 random variable, 199 Polish measurable space, 285 projective limit, 294 system, 294 Prokhorov (e, K£) condition, 195

Q quantitative indices, 5 quasi-complement, 237 quasi-linear mapping, 193

R Rademacher functions, 78 random process, 236 Radon-Nikodym property, 260 rate function, 275 rearrangement invariant function space, 207 regular conditinal measure, 241 distribution, 254 relative Chebyshev radius, 73 removable singular set, 345 Riesz angle, 384 Riesz-Kothe function space, 244 rotund (=Strictly convex) space, 75

Schauder basis, 206 Schilder's theorem, 292 Schur property, 76 self-Jung constant, 75 semi-invariant function, 276 Sobolev conjugate function, 334 number, 333 embedding theorem, 337 space, 328 Stieltjes integral, 230 Stirling approximation, 275 stochastic process, 273 strictly convex TV-function, 269 strong (p, g)-type, 192 (p, g)-norm, 192 ($i,$2)-type, 356 convergence of cr-algebras, 256

Index

464

sublinear mapping, 207 super-reflexive space, 52 symmetric function space, 207

tensor product space, 37 topologically bounded set, 424 total subspace, 261 trace er-algebra, 14 twice .F-differentiable norm, 464 U

unconditional basis, 206 uniformly convex space, 42 TV-function for large (small) arguments, 208 Predict (Gateaux) differentiate norm, 403 integrable, 260 nonsquare space, 49 normal structure, 74 smooth space, 42 V

variation of a vector measure, 241 von Neumann-Jordan constant, 51 W

weak comparison principle, 353 convergent sequence coefficient, 76 -Ll-inequality, 229 orthogonality property, 384 (p, g)-norm, 192 (p,g)-type, 192

($i,$ 2 )-type,356 weakly uniformly convex, 405 weakly* uniformly convex, 405 weighted Holder inequality, 383 Orlicz space, 383 X

A'-valued integrable function, 265

Yosida-Hewitt decomposition, 317 Young's function, 28 inequality, 2