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ty/i\. The constant is sharp in the cases 0 < q < w | and q = 1. The proof, which is postponed to the next section, uses inequality (2.7) for the case q = 1 and Theorem 2.5 for the case 0 < q < W | . For the remaining cases, an interpolation argument is used. Remark 2.8 The sharp constant in the above result is unknown in the case w | < q < 1. By investigating the form of the sharp constant in known cases, one is tempted to guess that
30
Multiplicative
Inequalities
of Carlson Type and
Interpolation
is the sharp constant whenever 0 < q < 1. However, this remains an open question. Problem 2 Find the sharp constant in Proposition 2.3 for
2.7.2
General p
The following result extends Theorem 2.5 in the sense that we introduce a new parameter r. More precisely, we consider the inequality S(0, l ) 2 r + 2 < C • S{p - q, r + l^-^^^Sip
+ q, r + I ^ + ^ - P ) ) / " . (2.13)
The result is the following. Theorem 2.6 Suppose that r > p. Then a necessary and sufficient condition for the existence of a finite constant C such that the inequality (2.13) holds is that q>r-p.
(2-14)
If r-p0 is defined by „2
_ . . / P ( P + Q MP + r)-r\ max 4(r+1)
<7°-
\^Tr'
j.
then we may choose
C = ^ ) = <(iB(^,^)f, and this constant is sharp. In the case Q>Qo,
we may choose C =
C(p,q0,r).
Some Extensions
and Complements
of Carlson's Inequalities
31
Remark 2.9 If we put r = p in Theorem 2.6, we get an extended version of Theorem 2.5. If, in addition, 1 P<~2, then there are cases which give the best constant but which are not covered by earlier results. Indeed, let r =p=q in Theorem 2.6. We then get the inequality
+i
(£^<^<^))> gH)V for all non-zero sequences of non-negative numbers, and the constant is sharp. In particular, when q = | , we obtain the inequality (2.11) with the sharp constant
as announced. Remark 2.10 Let us mention that the sharp constant in Theorem 2.6 is unknown in the case q > qo. Problem 3 Find the sharp constant in Theorem 2.6 for q > q0.
2.8
Proofs
For the proof of Theorem 2.6, we need two lemmas. We apply a convexity argument similar to that used for Theorem 2.5. This is stated in the first lemma below. Its proof is elementary, and is not included here. See, however, Lemma 2 in [53]. Lemma 2.2 Let *M=
(yP-O+yP+iy/r*
V
> °"
32
Multiplicative
Inequalities
of Carlson Type and
Interpolation
If p, q and r are such that „2^ fp(p + r) 4p(p + r)-r\ -m&X\^TT> ( r + l) /' 4
q
(2 15)
-
then the function / is convex. Our second lemma is another type of convexity result, in the guise of an interpolation inequality. This is what we need to prove Propostion 2.3. This was inspired by the Kjellberg Principle (see Proposition 3.2 of Chapter 3). Lemma 2.3 Let S(a,r) be as defined by (2.12). If 0 < 9 < 1 and <7i,Tj, i = 0,1 are real numbers, then S((l - 6)a0 + 0*i, (1 - 6)T0 + On) < Proof.
Sivo^-'SitTuTi)".
The Holder-Rogers inequality with exponents l-6>
and
6
yields the desired inequality.
•
We can now prove our extensions of the Landau and Levin-Steckin results. Proof of Proposition 2.3. The case 0 < q2 < | follows from Theorem 2.5 if we put p = 1. Suppose that < q2 < 1, and let 1
i n -^-\ T
2
2?V8
2
l + \ l = (l-9)(l-q)
+
l - \ l
+ v(^ + q),
V— 1
Then, since
J
and
« i 6=—I
7
i
\ —. 2g V 8
e(l+q)
and = (l-v)Ci--q)
Some Extensions and Complements
of Carlson's Inequalities
33
it follows from Lemma 2.3 and the case q = »/|, that 5(0,1) 4 < C-s(l - y 1 - z W l + y ^ , 2 < C - 5 ( l - g , 2 ) 1 - " 5 ( l + g,2)" S(l-q,2)1-eS0. + q,2)e = C-S(l-q,2)S(l + q,2), where
c-c(.7;)=?.». The case q = 1 is precisely (2.7). Finally, for the case g > 1, let 9=q-±± 2?
and
„=«fl. ' 2q
Then 0 = (1 - 77)(1 - g) + 7?(1 + 9) and 2 = {l-9)(l-q)
+ 0{l+q),
so again by Lemma 2.3, 5(0, l) 4 <7r 2 5(0,2)5(2,2) < TT2[S(1 - q, 2) 1 ""5(1 + q, 2)"S(1 -
•
Proof of Theorem 2.6. Suppose first that r-p<
q < q0,
and let A > 0. It is clear that if g is defined by g(y) = \-%
fiX-V'y)
= (\yP-1 + ±yP+0^) \
y>0,
where / is as in Lemma 2.2, then g is convex whenever / i s . As in the proof of Theorem 2.5, for any values of the parameters p, q, and r for which g is
34
Multiplicative Inequalities of Carlson Type and Interpolation
convex, we can apply Hadamard's inequality so that for any k it holds that 9
\ ~
2)
<
9 dy
/
^ -
It follows, then, from the Holder-Rogers inequality with exponents r + 1
and
r + 1,
that 00
S(0,l)=5>fc k=l 00
r
P+Q\
-£4>-\r+k*-\
+l
k=l
P-Q
*<*-;r4H)
p+g\ r + 1 ak
P q
~ Uk-±V**
<
7+T
r+1 fc
\S(p-q,r
+ l) + -S(p + q,r + l)
/•OO
/
<
\S(p-q,r
g{y) dy
+ l) + jS(p + q,r + l)
Jo
In the integral above, we can subsitute X1/qt1/2q x), which yields f00 / \ „ «
1
D+aV*
,
for y, then put £ = x/(l
A00 « + (r-,) . _
A^ 2q
N
! (it
r— p
A r«
•
f\l-x)
~27 Jo Ar * -B Df9-(r-p) 2q
2rq
q-(r-p)
<7+(r-p) rr
2rq
(fa
2rg
(1 — x)x
q+(r-pY '
2rq
Some Extensions and Complements of Carlson's Inequalities
35
Note that the integral converges, since by assumption p — q < r < p + q. Letting A
_
jS(p + q,r + l) S{p-q,r
+ l)
we get (2.13) with 1r
1q
\
2rq
2rq
as desired. Furthermore, by comparing the cases of equality in the application of the Holder-Rogers inequality above, we find that this constant is sharp. Suppose next that q > qo, and define --« + * 2q
q qo ~ 2q
and
V=
Thus p + qo = (l-6)(p-q)
+ 6(p + q)
and p-q0
= (1-V)(P-Q)+V(P
+ <1),
and hence by Lemma 2.3, together with what has been proved above, it follows that S(0, l ) 2 r + 2 < C(p, q0,r)S(p - q0,r + l)aa=^S(p < C(p,qo,r)[S(p-q,r
+ q0, r + l ) *
+ l)1~r'S(p +q,r + 1)"]
[S(p - q,r + l)1-°S(p + q,r + = C{P,q0,r)S(p-q,r
+ l)3^ir£lS(p
+ q,r +
2 1
^
«o
l)6}3**^ l)3±^£l,
which is (2.13) with C =
C(p,q0,r).
Thus the condition (2.14) is sufficient for the inequality (2.13) to hold for some constant C.
36
Multiplicative
Inequalities
of Carlson Type and
Interpolation
Suppose now that (2.13) holds for some constant C, and assume that q = r — p. Then the inequality under consideration reads /OO
\P+9+l
(5>)
OO
1/2
,
.p+g
fc
<^ E( 4) <+q+1- (2^)
Define a f c = 0 , fc = l,2 and 1\, /. 1 «fe=( ( f c - - ) l o g [ f c - - ) )
-l
,
fc>3.
Then
y^ afc = oo, fc=i
while p+q
+,
_oo_
/
,\-l
/
/
n
\ \
-p-g-1
£B)'> -x:H)>H))'
<
OO.
It follows that (2.16) does not hold, which means that (2.13) does not hold in general. Moreover, it cannot hold for q < r — p either, since then by what has been proved above together with Lemma 2.3, it would hold for q = i— p as well. The proof is complete. •
2.9
Levin—Godunova
In 1965, Levin extended his Theorem 2.2 in cooperation with E. K. Godunova [57]. By basically the same method as Levin used for Theorem 2.2, they were able to overcome the restriction of using an odd number of factors on the right-hand side. Also, their inequality allows more flexibility in parameters, although the sharp constant is not found in the most general case. We recover the results below. In the proofs of the two main results, two different lemmas are used. However, the methods are similar, and we therefore give the details for the first theorem only. Throughout this section, m is some fixed positive integer.
Some Extensions and Complements of Carlson's Inequalities Lemma 2.4 Let Cj > 0 and X3 > 0, j = 1 , . . . , m, and put
s
1 1
f2
m
=m— £ ** —s
where s is some integer between 0 and m. Suppose, moreover, that the \j are such that r 2 > r\. Then
(
f Jo
\
-l/mp
where 1 1 nl fn — s s \ ,„ , „ . min — —.—B\ ,— . (2.17) i< s <m-i r 2 - r i [ s * ( m - s ) m _ a ] 1 / m P \ mp mpj Remark 2.11 In the above lemma, we need to assume that the Xj are so that r 2 > ri for any choice of s. This can be achieved, for instance, by arranging the numbers in increasing order: Ai < • • • < A m . This requires the Aj to be distinct. This is not a major restriction, however, since if two of the Aj are equal, we may replace them by a single A by adding the corresponding Cj. A=
Proof of Lemma
(
2.4- Define £>i and D 2 by \ 1^s
TT Cj J 3=1 J
(
and
m
x
!/("»-*)
D<2 = (m — s) I TT Cj \j=s+l
Thus, by the arithmetic-geometric mean inequality Al x > s(cia; •••c— sx ') cix Al + . . . + csxx° = s— = s(ci • • • c s ) 1 / s x ( A l + - + A s ) / s =
Dixri,
37
38
Multiplicative
Inequalities of Carlson Type and
Interpolation
and similarly cs+ixx°+1
+...+ cmxXm
D2xr\
>
so that />00
I<
dx D2xr*y/p'
JQ (flu" +
This integral can be evaluated to /r>P-r2 7 - 1 n-p\ 1 /p( T "2-'-i) (^1 ^2 ) fi/
p - r j _
r2-r!
r
- p
2
\p(r2-n)'p(r-2-n)
Note that m
1
-
m
P - r2 = — y ^ A,
y^
j= l •m. ^-—' J' TO'—' I
A,
J=S + 1 \\m m — «s
s
ml ml
^—' ^L—'
J
m
i
A l £ A . _£._!_ V A j m s ^-^
mm —
]=1
s.^-^,
.7=3 + 1
—(n -r2), m or p-r2 T2 — r\
m
and in the same way we get r\ — p r2 — r\
m—s m
It follows that I <—-—s-s'mv(Cl
r2-n
•••
ca)-l'mp
x (m - s)-^~*Vm»(ca+l
• • • cJ-VwB \
( Since this can be done for any s, the assertion follows.
(^—^, mp \
—) mpj —l/mp
TT -l
j
•
Some Extensions and Complements of Carlson's Inequalities
39
The first theorem from [57] goes as follows. Theorem 2.7 (Levin-Godunova, 1965) Suppose that Xj > 0, j 1 , . . . , m, and define 1
m
TO . J= l
Then for any non-zero sequence ai,a,2,... inequality / oo
a
\ (P+l)"»
E * \fc=l
m
of non-negative numbers, the oo
/
(2-18)
J=lfc=l
holds, where we can choose C = Apmm"\ where A is defined by (2.17). Remark 2.12 If m = 2 M + 1 is odd and if Xj = j — 1 in the above theorem, then 2M+1
= _ _ — - y (j - 1 ) = M . ; ^ 2M + 1 f-' w P
This reduces (2.18) to the inequality (2.3) of Theorem 2.2. However, Theorem 2.7 might not give the sharp constant. Remark 2.13 As opposed to Theorem 2.2, Theorem 2.7 implies Carlson's inequality (1.1). Indeed, if m = 2, Ai = p — q, and A2 = p + q, then Theorem 2.7 reduces to Theorem 2.1 (with sharp constant). In particular, we get (1.1) if we put m = 2, Ai = 0, and A2 = 2. Proof of Theorem 2.7. For j = 1,...,TO,let 00
and put
(
771
l/m
40
Multiplicative
Inequalities
of Carlson Type and
Interpolation
Thus
(
l/"i
\
j=i
j
The Holder-Rogers inequality implies, then, that / oo
\ (P+!)m
X> \fc=l
/ oo
- I> /
ifcAl +
+c
--- ™
fcAm -1/p
)
\fe=l /
> N
OO
A
x l^(Cl^+...+cmfc -)aP
0
>
+1
j
(ax>* +... + cmx^yiv)
JUL i
We apply Lemma 2.4 and note that the choice of Cj implies that m
whereupon the desired result follows.
D
The second result is yet another variation on Carlson's inequality. The proof is almost identical to that of Theorem 2.7, and also of Theorem 2.2, and we therefore omit it. However, we state the corresponding lemma needed for the proof. L e m m a 2.5 Suppose that po = 1 and pj > 0, j = 1 , . . . , m. Put m , ( m *i = - + h \ 3 - - ) , where h > 0. Then
(
\ -2/m
/
with equality if and only if for some p, Pj =p>, j =
v -2/m(m+l)
l,...,m.
Some Extensions and Complements of Carlson's Inequalities
41
Theorem 2.8 (Levin-Godunova, 1965) Let Xj be as in the above lemma. Then / oo
\ (fe+l)(fc+2)/2
EH \fc=l
m
^
/
j=Ofe=l
where
c=h-^m+i^(m+i)m+i
n (m)
is the sharp constant. 2.10
More About Finite Sums
The following two results by L. Larsson, Z. Pales and L.-E. Persson [52] concern finite sums rather than infinite series, and extend the discussion in Section 1.4 of Chapter 2. For a > 0, /? > 0 and 0 < t < 1, we define the truncated Beta function by B(a,P;t)=
f (l-s)a-1s0-1ds= Jo
[ (1 - s)as'3 — Jo (1
ds s)s'
Suppose that m is a positive integer. Then the inequality in (a) of Theorem 2.4 holds, with the same constant (or even with Levin's sharp constant — see (4.9) in Chapter 4), for sequences { a f c } ^ f° r which Ofc = 0 whenever k > m. However, if we consider fixed-length sequences only, the constant can be taken strictly smaller. Theorem 2.9 Suppose that m is a positive integer, and that r > 0, and > Ci\ > 0. If a i , . . . , am are non-negative numbers, not all zero, then
OL
X>*ai °* +1 )
£fcaX+1
, (2-19)
where we may choose C = 2^+T
1 / a2-r -ai+r a.2 — ot\ \(c*2 — oc\)r' (0:2 — Q!i)r' 1 + m a 2 ~ a i (2.20)
42
Multiplicative
Inequalities of Carlson Type and
Interpolation
Remark 2.14 The constant C defined by (2.20), which depends on m, tends to the sharp constant in (2.5) when m —» oo. In the special case r = 1, a i = 0, Q.1 = 2, we have „ / l 1 m2 \ „ . m -"( o> 2 ' 2o'' 1i + m 9V ) ~ 2arcsin V I + m 2 = 2arctanm, so that the inequality (1.8) follows as a special case of Theorem 2.9. When m —> oo, this constant tends to the sharp constant sfH in (1.1). This special case is just Proposition 1.1. Proof of Theorem 2.9. Let m
m a
= J2k
S
'ark+1
and
^2ka2ark+1,
T =
k=l
fe=l
and let A be any positive number. If we write ak = (Xka> + ifc a2 )-?TT . (AfcQl +
jka*)^iak,
we get by the Holder-Rogers inequality / m
a
E0
\r+\
/ m
A a, + A lfcaa
<(D *
\
r
/ _m_
~ )~M E(
^(Xk^+x^k^yr)
Afcai+A_lfca2 +1
\
K
{XS + X^T). (2.21)
Since the function x^(Xxai
+X~1xa2)-r
is non-increasing for positive x, the last sum in (2.21) can be estimated by an integral. In fact, by making the substitution
Some Extensions
and Complements
of Carlson's Inequalities
43
we find that m
rm
1
J
Yixk^+x- ^)-^ < /
—
r
Jo (Ai-i+A- 1 !**)* 2r-(g +o ) °2-°l
£=t
1
\
2
m
2 /"A /•A2 +m°2-ai °Q2-'2-r +m a 2 -Ol / (1 — s j t,a2-a1)r
(«2-»l)''
«2 - «1
- ° li + r 7 g{a2-ai)
n-
Jo Jo
ds
(1 - s ) s
Qj +<*2
A 2 Z' 0.1 — r — cei+r m' — -B c*2 — cx\ \(tt2 — oi\)r' (a2 cti)r' A2 + m" 2- ™ 1 (2.22) The function t H-> B(a,/3;t)
is obviously increasing, and hence m
a,/?;
A M B
a
2
-ai
A2 + m a 2 - a i
is decreasing. Moreover, we always have S < T, so if we put
A=
vl'
then
B(a,P;™
\2
~)
+ ma2-aiJ-
y >r<
™
i+mai-ai
Hence, in view of (2.21) and (2.22), it follows that
+1 (±aX <2\(*2 (-^—B( \t^[ J — ai
a >~\, ( \{a2-oii)r
r(ai+\-> ^ ^ - a\)r l + m
a2_ai
2
«2- f —ai+r •S a2-<»l J 1 " 2 - i l .
This yields the inequality (2.19) with C given by (2.20).
•
As in Proposition 1.1, the constant given in Theorem 2.9 is not sharp for finite m. Problem 4 Find a formula for the sharp constant ^
in the inequality (2.19).
=
^m,r,ai,Q2
44
Multiplicative
Inequalities
of Carlson Type and
Interpolation
By combining these ideas with those of Levin and Godunova, the above theorem can, in some cases, be extended to hold for more than two factors on the right-hand side of the inequality. Theorem 2.10 Let N > 2 be an integer, and suppose that a\ < ... < OLNMoreover, let
r
=^£>-
Then, if a\,...,
am are non-negative numbers, not all zero, it holds that m
N
/ m
\
JV(r+i)
^KCn^XH fc=i
j=i \k=i
•
(2-23)
I
We may choose r
C= min (AN^Bl-?-,^-^;—^—
)
+
,
(2.24)
where A = '-*&—)-***
(2.25)
(r2~ri)r
and Ti — ri(s) are defined by 1 1 s — = -V^a,1
1 1 and — = — A
i=i
N
V ^ a,-.
(2.26)
j=s+i
Remark 2.15 We see that when N = 2, this reduces to the special case of Theorem 2.9 where r = p. However, for N > 2, the constant does not necessarily approach the sharp constant for the corresponding infinite series inequality when m —> oo. Proof of Theorem 2.10. Let m
Si = ^2ka*ark+1,
i=
l,...,N.
fc=i
If A i , . . . , XN are any positive numbers, we write ak = (Aifc"1 + . . . + XNkaN)'^
• (A^"1 + ... +
XNkaN)^ak
Some Extensions
and Complements
of Carlson's Inequalities
45
and apply the Holder-Rogers inequality with parameters and
r + 1,
r which yields m
\
fc=i
/
(
r + 1
r
/ m
\
Vfe=i
/
N
i=i
The first sum on the right-hand side can be estimated by the integral (Aix a i +... + XNxaN)
1= Jo
r
dx.
Let 1 < s < N, and let r\ and r2 be as defined by (2.26). Moreover, put Di = s(\i • • • \3)i
and
D2 = (N - s)(A s + 1 • • •
XN)^.
By the arithmetic-geometric mean inequality, it holds that X1xai +... +
XNxaN
= (XlXai +... + Xsxa°) + (Xs+ixa°+* +... + XNxa") XlXa> +... + Xsxa° Xs+1xa°+l +... + XNxa» —s
\- (N - s) s ai
>s(Xix
—
' a
•••Xsx 'y
N-s a
1
•••XNxaN)1^
+ (N - s)(Xs+1x °+
=DlXri +D2xr\ and therefore, also using the relations r2 — r s , r2 — ri N and
r — ri v2—r\
N —s N
we get
I<< /I (D (DlXri+D2xr2)--dx Jo
n7
r2—r\ (D'^D;^)' r2-n
/ a \Nr'
N~s Nr
r2 — v r — r\ m \r(r2 — r i ) ' r(r2 - n ) ' g i + m r 2 - n mr2"ri \ ' go.+mr2-ri /
S~JV?(jV — S)""N>"(AI • • • Ajv) _77?
r2— T\
T>(
s
" \ Nr'
mr2-ri
N —s
Nr
gl
+
m
\ r2-rj-
46
Multiplicative
Inequalities of Carlson Type and
Interpolation
Now, choose the numbers A., so that '•£•
\jSj
— y
\ *
Jj5i)
(2.27)
, j = l,...,N.
Thus Ai • • • Ajv = 1,
and if A is defined by (2.25) it follows that
(X>*)
( s
N-s Nr
mr2_ri ' Sx + m r 2 - r i
N
\ TT
n*
Now, we note that £>i
.
A^~
7V
AT
_s(A1---As)^-).
Since S, increases with j , it follows from (2.27) that \j decreases with j , and since the product of all of them equals 1, it must hold that Ai---As>l. Thus
Since this can be done for any s, the conclusion follows upon taking ( r + l ) t h roots. • As was mentioned prior to Theorem 2.10, the constant is not sharp when N > 2, even if we let m tend to infinity. P r o b l e m 5 Find the sharp constant ^ —
^m,r,N,ai,...,ctN
in the inequality (2.23). R e m a r k 2.16 There are also continuous versions of the above two theorems, i.e. where the sums are replaced by integrals over bounded intervals. This will be discussed in Chapter 3.
Chapter 3
The Continuous Case
Implicit in Carlson's original paper [23], we find the following integral version of Theorem 1.1. Theorem 3.1 If / is a non-negative, measurable function on (0, oo), then
a
oo
\ 4
f(x)dx)
»oo
/>oo
f(x)dx
X2f(x)dx.
(3.1)
2
Here, as before, the constant n is the best possible. Equality in (3.1) is acheived precisely when / has the form
where a and b are constants. Below we give some different proofs of this theorem. We will also show how the discrete inequality (1.1) follows from the continuous version (3.1). We thus have some alternate proofs of (1.1), in addition to those presented in Section 1.3 of Chapter 1. Our first proof of Theorem 3.1 is analogous to Hardy's first proof of (1.1). Proof. P u t /»oo
/-oo
J'
f2{x)dx and V= x2f2(x)dx. o Jo We may assume that both U and V are finite. Let a and /? be positive 47
48
Multiplicative
Inequalities of Carlson Type and
Interpolation
numbers. It then follows by the Schwarz inequality that
(jf/(.)*)"- ( f ^Lpv^VM«x) IT
1
2 v¥
(aU + pV)
With a = V and /3 = U, the rightmost expression is TrVt^V, which, upon squaring both sides, yields the desired inequality. Equality in the application of the Schwarz inequality is obtained precisely when for almost all x
f(x) =
A
a + fix1
for some constant A or
as claimed.
•
Our second proof is based on Calculus of Variations. It is, in fact, somewhat similar to Carlson's original proof of (1.1). Proof by Calculus of Variations. /•OO
Suppose that i-OO
/ f2(x)dx = a2 and / x2f{x)dx = b2, (3.2) Jo Jo where a and b are some non-zero numbers. We want to find the infimum of /•OO /•OO
7 m dx Jo
subject to the constraints (3.2). The Lagrange function for this problem is given by L(A0, Ai, A2)
f
Jo
Xof(x) + X1f2(x)
+
X2x2f2(x) dx.
(3.3)
The Continuous Case
49
Let us denote by / the minimizer of the integrand in (3.3). If Ao = 0, we must have / = 0 which cannot satisfy (3.2). By dividing through by Ao we may therefore assume that AQ = 1. Thus, let us consider h(u) = — u + X\u + X2x u . Then h'(u) = - 1 + 2(Ai +
X2x2)u,
so that h'{u) = 0 when U
= 2(X1lx2x>) = I{X)-
We have -I
/•OO
/»00
dx + X2x2
IT
4\/AlA2 ' dx + A2x2)2
i * * = i i (XT TV
I6A1VA1A2
and -I
/*oo
rOO 00
™ 22 X dx + A2x2)2
7T
I6A2VA1A2 Then, in terms of a and 6, we get
/(*)
2 a3/263/2 7r 62 + a 2 x 2 '
50
Multiplicative
Inequalities
of Carlson Type and
Interpolation
It follows that the sought constant is given by
a
oo _
\ 4
f(x)dx) _(v^) =7r2_
/ „oo
J
4
/-oo
p{x)dx
x2p(x)dx
J
D Remark 3.1 The main advantage of the method used in the above proof is that it automatically gives the sharp constant and extremizing functions. We get this, however, on the expense of more tedious calculations. B. Kjellberg [45] mentioned that the corresponding calculations can be made for a more general case (see Chapter 4). R. Bellman [9] proved, in addition to Theorem 2.4, the corresponding continuous versions (see Theorem 3.2 below). Bellman's basic trick was to split the integral on the left-hand side into a sum of two integrals. By streamlining his proof, using specific values of the parameters, we arrive at the following proof of (3.1) with n2 replaced by 16. Proof by "Partition
of Unity" 1. Since i y i = -1+2/ + i +y y '
it follows by the Schwarz inequality that if g is any positive function, then /»0O
-I
/»00
/ 9{y) dy= Jo Jo
+
-I
/»(X>
•7——g{y) dy+ i + y io
r——yg{y) dy i+ j
( f (irrt**),(f»vw*)1 °° 0
92{y)dy)
"\ ^ /
/ f°° +[ / V0
Let S be any positive number, and put g(y) = f(y/S) tuting x for y/S, this yields 6
f(x)dx
f2(x)dxj
+ U
3
/
y2g2{y)dy
^^
above. After substi-
x2f{x)dx\
The Continuous Case
51
or f
f2(x) dx) * +6^ (f°°
f(x) dx<5-i([°°
x2f2(x)
dx)
If we put
( CP(x)dxy \j~x*P{x)dx)
'
we get r°o J
/ r°°
\ i / r°° (J
f2(x)dx)
f(x)dx<2ll
x2f2(x)dx
which, after taking 4th powers, is (3.1) with 16 in place of TV2.
•
Another way to write the constant function 1 as a sum of two functions is used in the following proof. Proof by "Partition of Unity" 2. Let S be any positive number. The Schwarz inequality then implies that / Jo
f(x)dx=
/ f(x)dx+ Jo =
/ Js
f(x)dx
1 • f(x) dx +
+
—xf(x) dx
{ (x)dx
{r>yuy '
<6i f j°° f2(x) dx)
2
+ < H ( f°° x2f2(x)
dx
With
(f0°°x2f2(x)dxy { S~P{x)dx )
'
also taking 4th powers, we arrive at /•oo
/ Jo
\ 4
f(x)dx)
J
poo
< 16 / Jo
poo
f2(x)dx
Jo
x2f2(x)dx,
Multiplicative
52
Inequalities of Carlson Type and
Interpolation
i.e. the same conclusion as in the previous proof.
•
Remark 3.2 The two preceding proofs have not given the best constant in (3.1), as 7r2 is replaced by 16. However, the best constant can be achieved also with this idea of proof. This fact is shown in our next example, which can also be seen as a more general and unified approach to the idea used in the two proofs above. Proof by "Partition of Unity" 3. Let k be any measurable function on (0, oo) taking values in [0,1]. Then, since 1 = (1 — k(x)) + k(x), it follows by the Schwarz inequality that f°° f(x) dx = / " " ( l - k(x))f(x) Jo Jo °°
a+
dx+ [ ^-xf(x) x Jo \ 2 / r°°
(1 - k(x))2 dx)
(
dx \ J
f(x) dx
f ^ ) * (/>*««* 1
2
Prom here, we may proceed e.g. as in any of the following three cases. (1) Replace f(x) by f(x/6) and put x = Sy in the integrals involving / . Rename y as x and divide through by 6. This yields /
(l-k(x))2dx)
f(x)dx<8-l(
+ 4
f2(x)dx
(
,
,w
* (rw*) (f**' *
If we choose 5=
ffil-kWfdxfffWdx^ J0 (k(x)/x)2
dx J0
x2f2{x)dx
and take 4th powers, we get
i r f{X)dx\
/•OO
x / Jo
/"
f(x)dx f2(x)dx / Jo
x'r(x)dx.
The Continuous
Case
53
We are now free to make any choice of the function k of x. If we let k(x)
x
1+x'
then the two integrals involving k(x) both evaluate to 1, and we essentially have the first "Partition of Unity" proof above. (2) Let &=
X(6-i,oo)-
Then
(i - k(x)f dx = r 1
/ Jo
and
f(?)
2
dx = 5.
With s =
(
/o°° / 2 ( x ) dx_ Jo x2f2(x)dxj
this is essentially the second "Partition of Unity" proof. Hence, after taking 4th powers, we get the desired inequality with constant 16. (3) In the two applications of the Schwarz inequality, we obtain equality if for some constants A and /i we have
f(x) = A(l - k{x)f and J-2/„\ x„22f{x) =
,,/
x
k( ) n(^'^
respectively. Solving for A; as a function of x yields k(x) = for some S > 0. Thus
S2x2 1 + S2x2
54
Multiplicative
Inequalities
of Carlson Type and
Interpolation
and 2
f(«)-f^-i' If we put 5=
Jo°° P(x)dx x2f2(x)dxj
Jo
this yields J"
f{x) dx < 2 J |
(J°° f2(x) dx\
/ Uo
<7r2/ f2(x)dx Jo
4
(J™ x2f2(x)
dx
or f(x)dx)
J
Jo
x2f2(x)dx.
Thus, in the last case, the function k is chosen optimally. Moreover, any case of equality can be traced, by solving for f(x) once k(x) is determined. We find that equality occurs if and only if / has the form 7
/(*)
1 + 62x2'
n
We conclude this section by proving that the continuous version of Carlson's inequality indeed implies the discrete version, thus obtaining some new alternate ways of proving the discrete inequality. Proposition 3.1 The discrete inequality (1.1) follows from its integral analogue (3.1). Proof. Suppose that (3.1) holds, and let {a^l^Lj be a non-zero sequence of non-negative numbers. Fix the positive integer m for the moment, and define m
fm(x) = ^afcX[fc-i,fc)Oz), fc=l letting xi denote the characteristic function of the interval I. Since at least one Ofe is non-zero, the function fm is non-zero for m sufficiently large. Thus, for such m, it is not of the form
f{x) =
irb
The Continuous Case
55
for any constants a and b, and hence we have strict inequality in (3.1) for this function. Since X[k-i,k){x)X[j-i,j)(x) if k ^ j and, moreover, xj — Xi f° r
'
m
Ylak)
\4
anv
= 0
interval / , it follows that
(m f°° V = afe ( X l / X[k-i,k)dx\ t
4
fm{x)dx o /•OO
2
<7r / Jo
(-00
x2fl{x)dx
fl(x)dx Jo
™
r-k
= 7r2 2_,afc /
m
k 1
k=\ m
•' ~ m
fc=i
it=i
,k
dx Y j a2 / fc=l
^
x2 dx fe
-!
This result follows by letting m —> oo.
3.1
•
Beurling
In 1938, when studying Fourier transforms, A. Beurling [15] proved that -i
=
/-oo
/
/
\f(x)\dx<(
/-oo
/>oo
\g(t)\2dt
\
\g'(t)\2dt)
1/4
,
(3.4)
where / is the Fourier transform of g, i.e. 1 f°° g{t) = - = e**f(x)dx. V27T 7_oo
(3.5)
Another application of this to Fourier transforms can be found in P. Brenner and V. Thomee [16]. R e m a r k 3.3 The remark on absolute values when considering infinite series also applies in the continuous case, i.e. we may put absolute values on the functions appearing in the integrals, thus also covering the case of complex-valued functions. In Beurling's inequality, however, the absolute
56
Multiplicative
Inequalities
of Carlson Type and
Interpolation
values are necessary, since the Fourier transform of / may not be real-valued even though / is. Beurling used the same ideas as Hardy for his proof. By applying the Parseval identity to (3.4), we arrive at an inequality similar to (3.1), namely oo
/ /-oo
/-oo
/
\f(x)\2dx
\f(x)\dx
\ 1/4
x 2 |/(x)| 2 cte
\J — oo
J—oo
.
(3.6)
J
In fact, (3.1) and (3.6) are equivalent. To see this, note that
/ 1
\ V 2 , !
\i=0
/
/
vl/4 ,
\i=0 /
1
\i=0
1 4
=v 2(a 0 +a 1 ) / (&o + &i)
1/4
,1/4
( 3 7 )
I
.
Assume first that (3.1) holds. It is clear that we can apply Carlson's inequality to the functions /o and / i , defined by /o(x) = f{x),
x > 0,
h{x) = f(-x),
x < 0,
which yields 1/4 /
J00
f{x) dx
f{x) dx)
(
/•oo
/
(J
r0
x 1/4
X2f(x)
\ 1/4 / »oo
f2(x)dx\
(I
dx \ 1/4"
x2f2(x)dx)
so that by (3.7) with the proper choices of a^ and bi we get (3.6). Assume, conversely, that (3.6) holds. Any function defined on (0, oo) can be extended to, say, an even function on (—00,00), and the value of the integral of the extended function is then, of course, twice the value of the integral over
The Continuous Case
57
(0, oo). To any such function, (3.6) can be applied, and we get -I
/•OO
/
/.OO
f(x) dx = - / < \fc
=
f(x) dx (J°° x2f(x)
(|_°° f\x) dx^j
2 /2(x) dx
v? ( r
a
2 x2f2{x) dx
) (r
oo
dx)
\ 1/4
/
-oo
)
f2(x)dx\ IJ x2f2(x)dx In Remark 4.2 of Chapter 4, a more general version of this equivalence is pointed out. Remark 3.4 Note that the inequality (3.4) can be thought of as an uncertainty inequality.
3.2
Kjellberg
B. Kjellberg [44] published in 1946 a paper with a slightly different viewpoint. Below, the main thoughts from this paper are reconstructed. Consider Carlson's inequality (1.1) and its continuous counterpart (3.1). They state that if the series and integrals, respectively, on the right-hand sides are finite, then so are those on the left-hand sides. This is the essence of Kjellberg's way of looking at the inequalities. Note, however, that in (1.1), the fmiteness of the series ^ a2, is implied by that of the series J^ k2a\, while in (3.1) the convergence of J f2(x)dx and Jx2f2(x)dx are of equal importance when asserting the convergence of J f(x)dx. Furthermore, if the fmiteness of ^ ^2flfc is known, then directly by the Schwarz inequality OO
°°
k=l
1
fe=l OO
^
OO
< \
fc=l /
fc=l oo
x
1/2
58
Multiplicative
Inequalities of Carlson Type and
Interpolation
Now, since ^
r-
the inequality (3.8) may be numerically better than (1.1). For these reasons, Kjellberg found it more interesting to study the continuous case. The basic Kjellberg idea is as follows. If a and /? are real numbers, let rOO
I{a,/3)=
xaf(x)dx.
/ Jo
Thus, for instance, we can rewrite (3.1) as /(0,l)4<7r2J(0,2)/(2,2). Moreover, if a = (1 — 9)a\ + 6a2 and
p = (1 - 6)pt + 6(32
for some 6 £ (0,1), i.e. if the point (a,/3) is situated on the straight line segment between the two points (ai, /3j) and («2, ^2) in the a/3-plane, then, by applying the Holder-Rogers inequality with the conjugate exponents
and
ih
i
we arrive at I(a,p) 2, I{aj,Pj)
j = in the
I(a,@)
< 00.
l,...,m convex
hull of the
points
The Continuous
Case
59
(a 4 ,/3 4 ) Fig. 3.1 If the integrals corresponding to the points (aj,0j), j = 1,2,3,4 converge, then so does the integral corresponding t o each point in their convex hull.
Example 3.1 Assume that
f
f2(x)dx
< oo
and
Jo / oo x f (x)dx < oo, Jo
as suggested by the inequality (3.1). Suppose that 0 < T < oo and write /•oo
/ Jo
pT
xaf{x)dx=
roo
/ xaf(x)dx+ Jo
/
xaf(x)dx.
JT
By assumption, the two integrals on the right-hand side are finite (3 = 2 and a — 0,2. Thus, by the Kjellberg Principle, also for (3 = a any number between 0 and 2. To see this, consider functions / vanish off (0,T) and off (T, oo), respectively. Furthermore, for any a rT a
x f°(x)dx / Jo
< oo,
so again by the Kjellberg Principle rT
xafl3(x)dx / ./o
< oo
when 2 and which > —1
60
Multiplicative
Inequalities of Carlson Type and
Interpolation
whenever (a, /?) is in the shaded region of the upper diagram in Figure 3.2.
Fig. 3.2 The shaded region in the upper diagram represents the points (a, /?) for which /o xafP(x)dx is finite, under the assumption that / 0 f2(x)dx and / 0 x2f2(x)dx are both finite. The shaded region in the lower diagram represents the corresponding points for integrals over (T, oo).
The Continuous
Case
61
Similarly, since / ;
xafu(x)dx
< oo
whenever a < — 1, our assumptions and the Kjellberg principle imply that xafl3(x)dx
< oo
/ ;
for all (a,/3) in the lower shaded region in Figure 3.2. Together, these two observations imply that poo
xaf0(x)dx
< oo
Jo
for all (a, (3) in the region shown in Figure 3.3. Note, in particular, that the point (0,1), corresponding to the integral on the left-hand side of (3.1), is contained in this region.
Fig. 3.3 T h e point corresponding to the integral on the left-hand side of Carlson's integral inequality is in the region of convergence, provided that the integrals on the right-hand side of the same inequality converge.
It is clear that this method can be extended to involve any finite number of points corresponding to convergent integrals. Although we lose track of best constants when forming inequalities of the form (3.1), the method is
62
Multiplicative
Inequalities of Carlson Type and
Interpolation
indeed very powerful, and the geometric nature gives intuition a great deal of help when dealing with problems of this kind. In Chapter 5, some more of Kjellberg's results will be discussed. 3.3
Bellman
Theorem 2.4 was shown by Bellman [9] to hold also for integrals. Theorem 3.2 (Bellman, 1943) (a) Suppose that p, q > 1 and A, /i > 0. Then there is a constant C such that
(
\ PM+9A
roo
J
f(x)dx)
/ /-co
\ J
xP-1-xfP(x)dx\
(
(3.9)
q 1+li 9
f°°x -
f (x)dx
holds for all non-negative functions / . (b) If a,/? > 1, then there is a constant C such that
a
oo
\a(3+a—/3
f(x)dx\
„oo
/ /-oo
fa(x)dx(J
\ a— 1
x^f(x)dx\
.
Remark 3.5 Obviously, with p — q = 2 and A = /i = 1 in part (a), and with a = P = 2 in part (b), we get (3.1). We will reconstruct Bellman's proof of part (a) of this theorem, which uses the next lemma. However, in order to be able to track the constant in (3.9), we state the lemma in a more precise form than the original one. Lemma 3.1 If a, b, c > 0 and v > u > 0 are such that bpu
+ apv,
p>0,
then c»-«au
> "«(t,-ury
-
Proof.
vv
The statement follows if we put
_ (^£\l/(v~u) P
~\av)
'
n
The Continuous Case Proof
of Theorem
63
3.2 (a). Note t h a t for any r, s we can write rr/p
1 =
xs/q
+ X"'«
r
X /P{\ + X)
[1+ i
T h u s , by applying the Holder-Rogers inequality and letting r = p — 1 — A, s — q — 1 + JU, we find t h a t / Jo
f(x)dx
= / Jo
—J-T-/(x)dx + / xr/r(l+x)Jy J0
a
<
I/P'
dx ajr/(p-l)(l+x)p'
\
A
J V
^f(x)dx ' \
oo
1/9
dx Jo
a a
+
1/p
x7P(x)dxj
/
+
T J l
x S /
I«/(9-l)[l + I
'
\
oo
1/9
iV*(a:)da;J ^i/p
1/9
+j
w
w /,(i,,b
'(^i-^T) (r*" * I/P
\ i/«
xp-1-A/p(a;)da;)
=:•&(/
.
+ M ( /
x ^ 1 " ^ / 9 ^ ) dx)
Replacing / ( x ) b y f(x/p) and making the substitution x = py, this yields, upon switching back from y to x /
fOO
P /
f(x) dx
l-OO
I /
a
\
1/P
x p - 1 - A / p ( x ) dx J \
oo
1/9
x9-1+"/9(x)dx) . Now, we can apply t h e convexity inequality ( a + /3)' < 2 t _ 1 ( a * + / ? ' ) (which is valid for all a , f3 > 0 whenever £ > 1) with £ = pq and divide t h r o u g h by p9(p-A)5 t 0 g e t t h a t
a
oo
\ pq
/(x) dx)
pqX < 2pq~1LpqPq
+
2pq-1MpqQppPfx+gX,
64
Multiplicative
Inequalities of Carlson Type and
Interpolation
where />oo
P=
/ Jo
/»oo
xp-1-xfp(x)
dx
and
Q=
xq-1+»fq(x)dx.
\ Jo
Thus Lemma 3.1 with 2pq~1MpqQp,
a=
pq
b=U
f(x)dx' 2pq-1LpqPq,
c= u = qX
and
v = pfj, + qX yields
a
oo
where
\ P?(PM+9A) pq(PH+qA)
f(x)dxj l/pq
2(P9-l)(PM+9^)/P9
V (p/i)P"(gA)^ J \ Kp—\'p
\\(P-1)M
— lJ
/
\ (9-l)A
(3.10)
\q — Vq — l
The desired inequality thus follows if we take pqth roots.
•
Remark 3.6 This method of proof is obviously based on one of Hardy's original ideas, but also on the genuinely new idea of proving a multiplicative inequality by going via an additive inequality. This method was later pushed to perfection by Levin [56] (see Chapter 4). It should be mentioned that the constant obtained implicitly in the above proof is not sharp, except possibly in some special cases. For instance, in the Carlson case p = q = 2 and A = // = 1, Bellman's proof yields the constant 16 in place of n2. The sharp constant in the general case was found by Levin. Again, we refer to Chapter 4 for details concerning this improvement. Remark 3.7 In his book on Sobolev spaces, V. G. Maz'ja [65] stated and proved part (a) of Bellman's theorem as a lemma. This was used in connection with embeddings of Sobolev spaces.
The Continuous
3.4
Case
65
Sz. Nagy
Beurling's inequality (3.4) and the definition (3.5) imply that for any s e t / /-oo
l$(*)l<
/
/-oo
W)?dt
\ 1/4
\g'(t)\*dt)
\J—oo
J—oo
,
(3.11)
/
so that, translated into the language of L p -norms,
NlooR is absolutely continuous and let a > 0 and p > 1. Put (3.12)
p' + a where p' is the exponent conj ugate to p:
p
p'
(we put p' = oo when p — 1). Then
II/IL
- (26»)e "^" a
6
||/"11° lip
(3.13)
Remark 3.8 Note that by (3.12), we always have 0 < 9 < 1. Note also that if we put a = p = 2, then (3.13) reduces to (3.11). Proof of Theorem 3.3. Suppose first that p = 1. The inequality to prove is then ll/lloo<^ll/'lli-
(3-14)
We may assume that the right-hand side is finite, in which case
66
Multiplicative
Inequalities of Carlson Type and
Interpolation
l i m x ^ ± 0 0 f(x) = 0. For any £ e R /•£
OO
/»00
/ \f'(x)\dx
=
-oo
\f'(x)\dx
+
J— oo £
\f'(x)\dx J£ /-oo
/
f'(x)dxT/'(a;)fifo / / ' W ^ f'(x)dx ± lim -oo ( / ./£ + / = ±2/(0Thus, for all f, it holds that
l/(0l<5ll/'lli, which implies (3.14). Assume now that p > 1. In view of the Holder-Rogers inequality oo
/
/ /.oo
(
\ l / p ' / ,-oo
^ | / ( x ) | 0 / " |/(x)|da; < ( J ^ | / ( a : ) | a d x j 1 0)/9
= ll/lli "
(j_
\ l/p
l/'MrdxJ
||/'ll P -
It remains to prove that oo
/
\f(x)\^'\f(x)\dx>29\\f\\1Jie.
(3.15)
-oo
It holds that oo
/
/
|/(x)|a/"'|/'(x)|da:> -oo
/>0
/ \J — oo
/>oo \
- /
sgn^)!/^)!^'/'^)^ JO
/
= « lim [|/(0)|"« - | / ( - T ) | " » - |/(T)| 1 "> + 1/(0)1'"] i —>oo
= 20|/(O)| 1 / e .
The Continuous
Case
67
Now, it is clear that this estimate remains true if / is replaced by any translate f^(x) = f(x + £), which shows that for all £ we have oo
/
\f(x)\a^'\f'(x)\dx,
-oo
and, hence, (3.15) holds and the proof is complete.
•
The second result from [82], the proof of which we will omit here, allows more flexibility in parameters. If u and v are real numbers, let „, , H(u,v) =
uuvv
T(l+u
+ v)
u v
(u + v) + r ( i + «)r(i + v)'
where T(-) denotes the Gamma function. Theorem 3.4 (Sz. Nagy, 1941) Suppose that / : R -» R is absolutely continuous and let a, f3 > 0 and p > 1. Let V =
P P' a + /3 p' + a
and p> Then II , fl\\V ii/iL + ^
where 0/q(a+0)
c 3.5
(3 p1
Klefsjo
In the exercise section of Elementa 54 (1971) No. 2, B. Klefsjo [42] posed the following problem, with accompanying hint (which more or less gives the entire solution to the problem). EXERCISE. Show that if / is a non-negative, integrable function, then /•oo / f(x)dx<2U
/ r°° y/Zf(x)dx)
\ i / r°° U (f(x))2dxj
\ 4 .
68
Multiplicative
Inequalities of Carlson Type and
Interpolation
Hint: Estimate with the Schwarz inequality /»oo
/
/ Jo
poo
/>oo
\g{x)\2dx-
g(x)h(x)dx<(
\h(x)\2 dx
\Jo
Jo
in easiest possible way, f0 f{x) dx and / t °° f(x) dx by integrals on the righthand side of the formula which is to be shown. For each t > 0, we have
f f(x)dx<
(f
IdxY
f2(x)dxY
(j
and -I
/•OO
/ Jt
-I
/»00
f(x) dx < —~ / yt Jt
/»00
y/xf(x) dx < —= I yt Jo
y/xf(x)
dx.
From this, it follows that rt rl
/*00
/ Jo
f(x) dx =
Jo
rOO
f(x) dx +
f(x) dx
Jt
(3.16)
2 x dx
f ( ) Y +T= I Vx~f{x) dx
for each t > 0. By studying the function of t on the rightmost side of (3.16), one finds that it attains the value OO
2[
\
4
/
/-OO
2c f2(x)dx\
U
y/x~f(x)dx
I ( j
Jx-f{x)dx
for t=
j
f2{x)dx
Remark 3.9 We leave it as a problem to the reader to examine (e.g. by comparing to other results in this book) whether the constant in Klefsjo's inequality is sharp.
3.6
Hu
In 1993, K. Hu [36] proved some multiplicative inequalities, generalizing another result by Sz. Nagy [81].
The Continuous Case
69
Theorem 3.5 (Hu, 1993) Suppose that —oo < a < 6 < oo, a > 0 and p > 1, and put a
q= l + -. P' Suppose, moreover, that the function e : [a, b] —> R is such that 1 - e(x) + e{y) > 0,
x,y€[a,b]
and that / is absolutely continuous with / ' € L\{a, b) and that / vanishes at some point in [a, b]. (a) If 1 < p < 2, then \ 2/P-I \f(b)\«
\f(a)\« +
i/V
[b\f'(x)\Pdx
[b\f(x)\adx Ja
i Jo
(b) I f p > 2, then l-2/p
6
a
|/(a)|* + |/(&)|*< 9 Q \f(x)\ dx l/2p b
b
[ \f'(x)\'dx
a
f \f(x)\ dx]
-B
2
In both cases, we define B= /
\f'(x)\Pdx
Ja
[ \f(x)\ae(x)dxJa
f Ja
\f'(x)\'e(x)dx
f
\f(x)\adx.
Ja
Remark 3.10 For other results regarding integrals over bounded intervals, see Section 3.9 below. 3.7
Yang-Fang
G.-S. Yang and J.-C. Fang [84] gave another generalization of Carlson's inequalities. Their paper contains the inequality in both the discrete and continuous cases. We only consider the continuous version here.
70
Multiplicative
Inequalities of Carlson Type and
Interpolation
Theorem 3.6 (Yang-Fang, 1999) Let f,g : [0,oo) -> K be Lebesgue measurable functions and g be continuously differentiable with g(0) = 0, linix-^oo g(x) = oo and 0< M =
inf
g'(x) < oo.
z€[0,oo)
Suppose that p, a and r are real numbers such that p > 2 and a > 0. Then
QTi/cioida) P <(~)2 (l*V-a(s)i/(s)ip(1+2r-r,,)
a
\ 2(p-2)
51+a(x)|/(x)|p(1+2r-rp)da;)(
/
l/Wr^i (3.17)
R e m a r k 3.11 Theorem 3.6 generalizes a result by S. Barza and E. C. Popa [7]. See also the Ph.D. thesis [4]. A multi-dimensional extension of Theorem 3.6 will be discussed in Chapter 5. 3.8
A Continuous Landau Type Inequality
In 1998, S. Barza, J. Pecaric and L.-E. Persson [6] considered the continuous version of Landau's inequality (2.7), i.e. an inequality involving integrals with weights of the form x — a. We begin by presenting the following result. T h e o r e m 3.7 Suppose that a > 0. Then, for every non-negative, measurable function / on R+, it holds that f(x)dx\
<47r2/
{x-a)2f{x)dx,
f(x)dx
(3.18)
and the constant 47r2 is sharp. R e m a r k 3.12 Note that by Carlson's Theorem 3.1, the constant in the above inequality is not sharp if we were allowed to put a = 0. Proof of Theorem Thus, let
3.7. We proceed as in the proof of Theorem 3.1.
/•OO
U=
Jo
/.OO
f2(x)dx
and
V=
/ (x Jo
a)2f2(x)dx.
The Continuous
Case
71
Assume, without loss of generality, that U < oo and V < oo. Let a and B be any positive numbers, and write f
=
W
/ A t V« +
.2y/<* + B(x-a)2
f3{x-a)*f(x).
Thus by the Schwarz inequality
Wr
/•oo
M
/•<»
7nr«n—^
{a + p{x-af)f\x)dx
Jo oc + p{x - ay J0 = —= ( J + arctan (a\ -)) (aS + 0T) y/ocp \2 V or / =
- + a r c t a n (o J ^ ))l
-S+JZ-T
).
Since arctan£ < | for any (finite) £ > 0 (see also Remark 3.13 below), by choosing a = T and P = S, we obtain
(/"
/(i)dij
which, after squaring, yields the desired inequality. To prove that the constant is sharp, we consider the functions 1 + 72(a; — a ) J We have /
/ 7 (x) dx = — I — + arctan(7a) I,
and the corresponding integrals on the right-hand side are
^ = ^(? + a r c t a n ( 7 a ) + rrw) and 1 fir . . 7
72
Multiplicative
Inequalities of Carlson Type and
Interpolation
Therefore,
(/Ato**)4
Ihn
7—oo j f%(x) dx J(x - a)2f2(x) dx
= 47r2
is a lower bound for the constant in (3.18). But we have already shown that 47r2 is an upper bound, and thus this is, indeed, the sharp constant.
• Remark 3.13 Since arctani is strictly smaller than ^ for any choice of the positive number t, we do, in fact, have strict inequality in (3.18) whenever / is a non-zero function such that the integrals on the right-hand side converge. Remark 3.14 Suppose instead that a < 0. Since then, trivially, x < x — a, it follows from Carlson's inequality (3.1) that (3.18) may be complemented by the inequality
a
oo
\ 4
f(x)dx)
/.oo
<7T2/
/-oo
f2(x)dx
(x-a)2f(x)dx
(3.19)
2 in this case. In fact, as can be seen1 by + Aconsidering (x - a) 2 the functions 2 X and letting A —• 0, the constant also when a < 0 (cf. Section fx{x) 7r is sharp 3.9 below). Note, in particular, the remarkable jump in the sharp constant over the parameter value a = 0 in (3.18) and (3.19).
We will state a two-dimensional, extended variant of Theorem 3.7 in Chapter 5.
3.9
Integrals on Bounded Intervals
The results in Section 2.10 have continuous analogues. Here, we will present some recent results of L. Larsson, Z. Pales, and L.-E. Persson [52]. This gives perhaps the most straightforward way of considering inequalities of Carlson type for integrals over a bounded interval rather than on the whole real line. We cannot, however, just transfer the summation from 1 to m to integration over (0,m), and expect to get a smaller constant than in the
73
The Continuous Case
case of integration over (0, oo). Indeed, consider e.g. the case where the parameters are as in Carlson's inequality /
f(x)dx\
x2f{x)dx.
f(x)dx
fx(x) =
A2 + x2'
(3.20)
Let A
Then
/
Jo
/
m A
A(x) ax = arctan —,
fx (x) dx = — I arctan y +
mA A2 + m 2
and /
m = — I arctan A
x2f\(x)dx
mA A2 + w?
Letting A —> 0 yields ( arctan y j C> fc ( arctan ^ + ^
)
A(
arctan f
_ _^L^
4 ( arctan :^y J ,2
2
f arctan y_A /j
_
2
m A2 2 2 2
(A +m )
Thus, the sharp constant is IT2 also in the inequality (3.20). In words, this can be explained by saying that there are maximizing functions for the inequality (3.1) with their mass concentrated arbitrarily close to 0 (i.e. the functions f\ as A —> 0). Moreover, the upper bound 7r2 for the constant in (3.20) can be found as in the discrete case. We thus have the following.
74
Multiplicative Inequalities of Carlson Type and Interpolation
Proposition 3.3 If / is a non-zero, non-negative function on (0, m), 0 < m < co, then
a
m
\ 4
pm
<7T2/
f(x)dx) The constant n2 is sharp.
pm
f2(x)dx
x2f2(x)dx.
In view of the above proposition, we alter the interval of integration slightly. Also, for this purpose, we introduce the following truncated Beta function for 0 < u < t < 1. B{a,f3;u,t)=
f (1 -
s)asp
Ju
ds (l-s)s-
Theorem 3.8 Suppose that m > 1 and a2 > r > a\ > 0. Then the inequality rm
pm
/
/
xai fr+1 (x) dx
f(x)dx
(a 2 -
1
J m~
\Jrn~
r
x f (x)dx
holds for all non-negative, measurable functions / on (m choose C
(3.21)
-*M
(a2-ai)(r+l)
a2 r+1
1
, m ) . We may
a.2 — r r — a\ ;u,t -B ot2 — ai \ r ( a 2 — a i ) ' r ( a 2 — a i )
(3.22)
where m2(a2-r)
_ i
u = —— ^ m% 2 - a i ) _ I
m2(a2-ai)
and
t =—
m
_
m2(r-ai)
2(a 2 -ai) _ l
Remark 3.15 As in the discrete case, the constant C = C m , r , a i , a 2 given in (3.22) tends to the sharp constant in part (a) of Theorem 3.2 with p = q = r + 1, X = r — ai, and fi = r + a2 (see also Theorem 4.1 in Chapter 4). Proof of Theorem
3.8. For any A > 0, we have by the Holder-Rogers
The Continuous
Case
75
inequality I
(\xai+\-1xa2)-^{\xai
f(x)dx= - 1
+\-1xa2)^if(x)dx
1
Jm "
Jm"
,
dx i (Ax
ai
\ ^ 1
+A- x
a2
)r
(Xxai + \-1xa2)fr+1(x)
( f 1
\
r + 1
/
2r-(ai+o 2 )
A -» J -i
-B
«2 — a i
dx) \ r+1
(ASi + A _ 1 5 2 )
where St=
/ xa*fr+1(x)dx, 1 Jm-
i = l,2,
and .n°2~°l
B
( 1 — s) r("2-°,l> S r '(a2- a l) •
/
J
ro-(°2
—1) V
A2 + m-( a 2-ai)
'
(I"*)*
Now, we can minimize this truncated Beta function with respect to A, simply by differentiating. We find that the minimum is „ „ , a2-r r-ai m2^-^ - 1 m 2(«2-«i) _ m 2 ( r - « 0 B = B r(a - a i ) ' r ( a - " i ) ' m2(a -a ) - 1' v 2 2 2 x m2(<»2-ai) _ i We may now put
x-M V*i
which yields (3.21) with C given by (3.22). • R e m a r k 3.16 In the next chapter, we will focus our interest on a famous result of V. I. Levin (see Theorem 4.1). This result suggests that we should look for an inequality of the type / 7m"
xp-1-xfp(x)dx)
f{x)dx
\Jrn~
1
xq-1+>1fg(x)dx
( / J
\Jrn-
1
. J
(3.23) The proof of Theorem 3.8 cannot be directly extended to cover also this case.
76
Multiplicative
Inequalities of Carlson Type and
Interpolation
Problem 6 Prove the inequality (3.23) with a constant strictly smaller than the sharp constant given by (4.3) for the corresponding parameters, preferrably the sharp constant, for each m > 1.
Chapter 4
Levin's Theorem
In this chapter, we will present and discuss a remarkable result of V. I. Levin [56]. In 1948, he found the following generalization of (3.1), which is, in a way, the most general inequality of its type one can hope for (see Remark 4.5 below). Theorem 4.1 (Levin, 1948) Suppose that p > l , g > l , s > 0 and t > 0, and that A and /z are any real numbers. If s=
^—pfi + qX
and
t=
-, pn + qX
(4.1)
then /
f(x)dx
xp-1-xfp(x)dx)
f /
xq-l+>1fq{x)dx\
(4.2)
for all non-negative functions / , where the best constant C is given by
^fe)'(s)'(^B(r^iT^))'"'"'' <43) Conversely, in order for the existence of a constant C such that (4.2) holds, it is necessary that s and t are defined by (4.1). As will be seen below, Levin's proof is a genious refinement of Bellman's proof of Theorem 3.2. Proof.
Suppose that (4.2) holds for some C. First, let f(x) 77
= pg(x),
78
Multiplicative
Inequalities
of Carlson Type and
Interpolation
where p > 0 and g is some non-zero function. Then (4.2) becomes /•OO
pi-ps-qt
/
/•OO
/
\
xp-1-xg(x)dxJ
g^dx
xq~1+flg(x)dx\
x ( f
8
.
If we let p —> 0 or p —> oo, we see that we must have ps + qt = l.
(4.4)
Moreover, by replacing x by px in (4.2), we get /
/•oo
^1_(p_A)s_(g+M)t /
\ s
/-oo
f^jdx
xp-l-\f(xjdx\
x ( f
xq-1+»f{x)
dx)
.
Thus {p-X)s
+ (q + p)t = l.
(4.5)
The unique solution (s, £) of the system of linear equations (4.4) and (4.5) is precisely (4.1). Suppose now that (4.1) holds. Let i.p-i
\ i
,(l-fc) 9 _ 1
^
(4.6)
where 6=
(p-l)p+(q-l)X.
Then z, regarded as a function of k, is continuous on (0,1) and increases strictly from 0 to oo in this interval. Therefore, we can define its inverse k by the relation (4.6). Let y be any positive number. We write p-l-A
f(x) = (1 - k(yx))x
p x
y-l-X
»
f(x)
+ k(yx)x~ i x i f(x) in the integral on the left-hand side of (4.2). Thus, if P and Q are defined by /'OO
pp= / Jo
/-OO
xp-1~xfp(x)dx
and
Qq =
Jo
xq-1+T(x)dx,
Levin's
79
Theorem
it follows by the Holder-Rogers inequality that
I
(l-k(yx))px^—\
f(x)dx<(
P
= Q f V *(*))'** *)*»-**> = y~vUP + y^VQ.
Here, p' and q' are the exponents conjugate to p and q, respectively. The integrals U and V can be calculated explicitly, as follows. By the change of variables (4.6), we get
f -i(0,-i)
/>00
_dz (l-k(z)fz^
UP' = / Jo
up— l
\ip-i
x ( ( p - l ) ( l - f c ) + («-!)*)
'-rjy**
+
A
"~x
ii,
dk (1 - k)k
A
Jo
(1 - * ) *
v y 0 ( i - f c ) p T'-ifcT+(r=*)* 9
—
1
f
i«
nn'.ild.A + l
^
80
Multiplicative
P~l
Inequalities of Carlson Type and
f ,-> , . ^ + 2 . 4 '**-
•JoA l - * ) * H
dk (l-k)k
p — \ fi + S fi <5 ^ + A + <5/i + A Pfi
(p/i + qA)(/i + A)
B
((i
Interpolation
(n X^
'(?•!)
A
(S-i)
M+A Vl-s-t'
1-s-t/'
Similarly, qt V«' = * "
( B( -s fi + X \ l - s - t ' /J
t
-
1
1-s-t/'
Now, put qt
UP\:^+^
TsVQ, Thus rOO
/
Jo
f(x)
^mr^my^ " + ( £ ) } Up3V<>tPpi>Qit
= j fSl\
1 \(p_1^/
l V / l V / ^psj /
\qtj s
\fx + XJ t
1 \ (9-1)*
\/U + Ay
\ (p-l)a+(g-l)t
which is (4.2) with C given by (4.3). By considering possible simultaneous cases of equality in the applications of the Holder-Rogers inequality above,
Levin's
Theorem
81
we find that equality holds in (4.2) with the constant (4.3) precisely when f(x) is a multiple of y~~ i-1 k(yx)~«zri
x~ G - i - 1
for some y > 0. Here, k is the function defined by (4.6).
•
Remark 4.1 In 1955, B. Kjellberg [45] explained how to prove the inequality (4.2) with the method of calculus of variations (see also the proofs at the beginning of Chapter 3). Remark 4.2 (Cf. [5]) We prove that Levin's inequality (4.2) is equivalent to a similar inequality, where the integrals involved are taken over the whole real line. Thus, in particular, Carlson's inequality (3.1) is equivalent to (3.6), and thus to Beurling's inequality (3.4), as was shown separately in Section 3.1 of Chapter 3. The inequality under consideration is oo
/
/
f{x) dx < 21~s-tC
\ i
roo
I / x (/
\x\p-1~xfp(x) dx J \x\q-1+^fq(x)dx\
(4.7)
where C is the constant given by (4.3). If / is defined on (0, oo), extend / to an even function on (—oo, oo) and apply (4.7). Since OO
/-OO
/
g(x) dx = 2 / g(x) dx 2 2 -oo if g(x) = f(x), g(x) = f2(x) or g(x) = xJO f (x), we get (4.2). In the other direction, we apply the elementary inequality ofbi + a56| < 2 1 - s - ' ( a 1 + a 2 ) s (6i + b2)\
(4.8)
valid for all non-negative numbers
= 2 1 - s " t (a 1 + a2)s(h
+b2)t.
82
Multiplicative
Inequalities
of Carlson Type and
Interpolation
Let O ai
/
i-oo
\x\p-l-xfp(x)dx, -oo 0
/
\x\p-1-xfp(x)dx,
a2= J0 /-oo
IxIfl-i+M/?^)^
\x\9-1+ltfq(x)dx.
b2=
-oo
JO
Then, by (4.2) and (4.8) OO
/
rOO
rO
f(x)dx= -oo
/ a
J— oo
f{x)dx
+ I
f(x)dx
JO
1 s t
< C{a\b\ + a 2b\) < 2 - ~ C(a1 + a 2 ) s (6i + 6 2 )' |x| p - 1 - A / p (x)da;J ( / |a;| < '- 1+ ' 1 / 9 (x)da;J . Remark 4.3 The inequality (4.2) is really just the first part of Bellman's Theorem 3.2. However, as stated in Theorem 4.1, Levin found the sharp constant in this inequality. By comparing the constant (4.3) with the constant (3.10) found by examining the proof of Theorem 3.2, we see that Bellman's constant is in general not sharp. Remark 4.4 It is clear that the discrete inequality corresponding to (4.2), namely oo
/ oo
\
1
]>>
\fc=l
s
/ o o
Q> /
\/c=l
\ '
-1+
a
" 0 ,
(4-9)
/
with the same constant, can be proved to hold under the same conditions on s and t, by following exactly the same method as when proving the continuous version. In fact, we even have strict inequality in (4.9), unless all the at are zero. The special case of (4.9) where q = p and /J, = A = 5 is just Gabriel's Theorem 2.1. Moreover, the inequalities (4.2) and (4.9) were proved by Bellman in Theorems 3.2 and 2.4, respectively, although he achieved the best constant only in the special cases q = p and \x = A. Remark 4.5 In Levin's inequality (4.2), put f{x) =
\g(x)xd\r-. x
Levin's
Theorem
83
Moreover, put Po = i"P do = d
an
d
and
Pi = rq, d\ = d H
Po
Pi
and 0 =
Pi A Pofi + Pi\
The inequality then, after taking rth roots, becomes r°°
1 r
d rj„\
/
(J~\9(x)* \ ^)
^C1^ r1r\e/pi
j _ \ ( l - 9 ) / P o / />oo
a; y
\7o
'"N '
'
(4-10)
^ J
The conditions (4.1) together with the requirements s,t > 0 can now be translated to d = (1 - 6)d0 + 6di
and
d0 ± d\-
(4.11)
Moreover, since p, q > 1, we need to have r < po,Pi. We thus have the following reformulation of Levin's theorem. Recall that we use L* to denote the Lebesgue spaces formed when using homogeneous measure. Theorem 4.2 Suppose that po,pi > r > 0 and 0 < 9 < 1. Suppose, moreover, that w(x) = xd, w0(x) = xdo, wi (x) = xdl,
0 < x < oo.
Then the inequality II.HIL;(R+)
< CH/^o||l,: o 9 ( R+ ) H/willi; i( M + )
holds for some constant C if and only if the conditions (4.11) hold. In Chapter 5, we will give a generalization of Theorem 4.2, and thus of Theorem 4.1, to infinite cones in K".
Chapter 5
Some Multi-dimensional Generalizations and Variations 5.1
Some Preliminaries
In this chapter, we will investigate some multi-dimensional generalizations of Carlson's inequality. Some guiding questions are the following. • How should a straightforward multi-dimensional generalization of Carlson's integral inequality (3.1) look like? • How can Levin's theorem (Theorem 4.1) be generalized to higher dimensions? In order to introduce the results we aim to prove, we give the following two-dimensional examples. E x a m p l e 5.1 One may start by asking whether there is a constant C for which the inequality ( //
f(x,y)dxdy)
\JjRl
f2(x,y)dxdy
J
xyf2(x,y)dxdy
JJR
2
JJRI
+
(5.1) holds for all measurable, non-negative functions / on R+. In fact, this is not the case, which can be seen by considering the characteristic functions / R of the sets {(x,y) €R2+;0<x2
+ y2
For in this case, we have // J Jul
fii(x,y)dxdy=
f%(x,y)dxdy JJRI 85
-R2
= 4
86
Multiplicative
Inequalities of Carlson Type and
Interpolation
and //
-R4.
xyf%(x, y) dx dy =
A lower bound for a possible constant in (5.1) is therefore —R 2 , R > 0. 8 By letting R —> oo, we see that the existence of such a constant is impossible. Example 5.2 We consider now the possibility of finding a constant C such that the inequality
(ff
f(x,y)dxdy)
f2(x,y)dxdy
\JjRl
J
ff
JJw.1
x2y2f2(x,y)dxdy
2
J Jn + (5.2)
holds for all / . For R > 1, we let x
fn(x,y)=
l+x2y2X*(
>y)'
where we denote by \R the characteristic function of the set
{(x,y)£R2+;Ky
ER = We then have
/fl(x y)dx dy R
IL+ '
=i ( r TTW *")
=
2y-dy=2l°*R'
A
f2 y)dxdy
iiK ^
dy
dx dy
=[
{r^w ) Ay
4
Some Multi-dimensional Generalizations and Variations
87
and
ff Wft{x,y)dxdy = Ji
^^^—d^dy
— dV Ay
= /
=
~ l o § R4
Thus, any constant C in (5.2) must satisfy
4
flogfl' ^=7T2(l0gi?)2
C > ^
flogU
If we let R —> oo, we see that this is not possible.
Example 5.3 Our final example will concern the following inequality.
// J
JR%
f(x,y)dxdy)
f(x,y)dxdy J JR*_ • !f J Jul
(5.3) 2
2 2
(x +y ) f(x,y)dxdy.
Let A > 0, and define the function k : E l —> R+ by
*(.,*)=
A +
( x^2 + 2 / 2^) 2 -
88
Multiplicative Inequalities of Carlson Type and
Interpolation
By the Schwarz inequality
//
f(x,y)dxdy =
J JR2+
(1 -
k(x,y))f(x,y)dxdy
J JR2+ JJR2+
x
+y
(JJ2{l-k(x,y))2dxdy)
<
•lfj^f\x,y)dxdy\
•(jj^{x2+y2)2f{x,y)dxdy\
.
As the reader may check, with the above choice of the function k, we have
Jj2(l-k(x,y))2dxdy
= ^VX
and ff k2(x,y) j 2 JJK(x +y2)2dXdy-l6S\-
_f_J_
Thus, if we let _nR2+(x2+y2)2f2(x,y)dxdy 2 IIR JR2+
f2(x,y)dxdy
we get
II2 f(x,y)dxdy<
| I //
f2(x,y)dxdy\
(jj^(x2+y2)2f2(x,y)dxdy\
,
Some Multi-dimensional
Generalizations
and
Variations
89
which, after taking 4th powers, shows that (5.3) in fact holds, with
16 In the next section, we prove a result (Theorem 5.1) which explains and put these examples into a more general frame. We finally mention the following two-dimensional result by Carlson [23] (see (B.ll) in Appendix B). Proposition 5.1 (Carlson, 1935) If y> and tp are non-negative functions, then
a
oo
/-oo
\ 2
J{x)dx) We have equality when
o
/-oo
/-oo
I
¥>(*)= V(z) =
v\x)ip2(y)(x2 + y2)dxdy.
Y^i-
This, however, only covers the case where / can be written as a product f(x,y) ={y).
5.2
A Sharp Inequality for Cones in R n
Let S be a measurable subset of the unit sphere in R™, and define the infinite cone CI by Cl = ix € R n ; 0 < |z| < oo, ^
e S \.
(5.4)
Suppose that the positive, measurable functions w, wo and w\, defined on fl, are homogeneous of degrees 7, 70 and 71, respectively (we say that v : CI —> R + is homogeneous of degree a if, for all t > 0 and all x £ CI it holds that v(tx) = tav(x)). Suppose that 0 < p < po,Pi < 00, and fix 6 e (0,1). Define d = -y + P and IT
di = ji-\
Pi
,
i = 0,1,
90
Multiplicative
Inequalities of Carlson Type and
Interpolation
and define q by the relation 1 _ 1 q p
1-61
6
Po
Pi
We then have the following result by S. Barza, V. I. Burenkov, J. Pecaric and L.-E. Persson [5], which extends Levin's theorem to a multi-dimensional setting. It is very convenient to work on a cone of the type (5.4), since we can transfer a great deal of the problem to (0, oo) by switching to polar coordinates. T h e o r e m 5.1 Let 0 < p < po,Pi < oo. Then the Carlson type inequality l l / W l l L p ( n , d x ) — C \\fW0
\\fwl
II LPQ(n,dx)
II LP1(Q,dx)
(5.5)
holds for some constant C if and only if d = (I - 6)do + 6di, do ^ di,
(5.6) (5.7)
and ,i-
(5.8)
&Lq(S,a).
Here, a is used to denote surface area measure on S. In (5.5), we may use
C =
V
(\-6)~~^6~~i 1
1
P
Q
Po
' Pi / I
PoPi|do-di| J
<~6<
(5.9)
Lq{S,a)
and this constant is sharp. Equality in (5.5) holds with the constant given by (5.9) if and only if / satisfies
\f(x)\=Hf(rx) for almost every x, for some H > 0, r > 0, where P
\ V(PO-P)
(5.10)
Some Multi-dimensional
Generalizations
and
Variations
91
and k is defined by the implicit relation
where
1 = 1 - 1 , * = 0,1. n
P
Pi
Remark 5.1 By applying an interpolation technique, the inequality (5.5) can, assuming that the conditions (5.6)-(5.8) hold, be proved for all p satisfying 1
1-61
- > P
9
+ —• Po
Pi
We will, however, postpone the details until Chapter 6, where we deal with general measure spaces. Remark 5.2 Of course, this is a strict extension of Levin's Theorem 4.1, but we can also get for example Beurling's inequality (3.6) from Theorem 5.1 by letting S = {—1,1} in the case n = 1. Remark 5.3 The if and only if nature of Theorem 5.1 gives an explanation of why the inequality (5.1) in Example 5.1 fails. Indeed, in that case we have d = 2,
do = 1,
and
d\ = 2.
Thus, there is no 6 € (0,1) such that (5.6) holds. Remark 5.4 In Example 5.2, we replaced xy by x2y2 in the last integral on the right-hand side. Thus, in this case, we have w(x,y) = l,
w0(x,y) = l,
and
wx{x,y) = xy,
so that d ~ 2,
do = 1,
and
d\ = 3.
Thus (5.6) holds with 9 = \, and we clearly also have (5.7). However, since 1_1 q 1
5 2
2_1 2 2'
92
Multiplicative
Inequalities of Carlson Type and
Interpolation
or q = 2, and therefore \q _ 1 xy'
w{x,y) e ^- (x,y)wf(x,y)J
Therefore, the condition (5.8) translates to
i: l0
1
dip < oo,
cos ip sm
which is clearly not the case. Thus, in view of Theorem 5.1, it is impossible to find a constant C such that (5.2) holds. Remark 5.5 In Example 5.1 above, we used w(x,y) = l,wQ(x,y)
= 1,
and
w1(x,y) = x2 + y1.
With d = 2, d0 = 1, dx = 3, and 6 = •§, we see that (5.6) and (5.7) hold. Moreover, the quotient w <~e< is constant on the part of the unit circle contained in M^_. Thus (5.8) holds as well. We leave it to the reader to find out whether the constant 16 found in Example 5.3 is sharp. Proof of Theorem 5.1. We note first that, when switching to polar coordinates x = pa, where x p = \x\ and (7= 7 7 , \x\ we get
= [ Js Jo
r\f{pa)w{pa)\^pn-1dpdcT
= [ wV(a) Js Jo
r\f(pa)\"pr'+n-1dpda
= [ w?(a) r'i/(H/r-^, Js
Jo
P
Some Multi-dimensional
Generalizations
and Variations
93
and similarly for the other norms, and hence we can apply Theorem 4.2 from the previous chapter on the inner integral. More precisely, if the conditions (5.6)-(5.8) hold, then for any a
r \f(P*)Pd\p ^
P
\f(Po)pdr° -
\Jo
(l-0)p/po
P
j)*'*1
x( r\f(p*)^r 0
and hence by the Holder-Rogers inequality on the measure space (S, a) with the three parameters Q P ° p'(l-0)p
and
51, 6p'
it follows that dpX
(1-»)P/PO
IPO
P 6p/p
Cp
r(
"fro
y
Js\wl-9{
Ifip^woipaW^p^dp' o
a
J
oo
\ dp/px
\f{pa)wl{pa)rpn-ldpyj w(a) \q
/
r
/-CO
r
da
\ (1—A)p/PO
as
\jsL /
\p/q
w
<j po n d da
\f(p > ^p ^ p ^ P ) roo
\ 0p/pi
w
\lsL
\^p^ ^p )\ p ~ P ) p
CP
wl~ew{
I/^OIILPOSU) L„(S,d
H/^1 HlPpi(n,dx)-
Thus the conditions are sufficient. Suppose that / is defined by (5.10). It can be shown, by using the definition of the function k together with the
94
Multiplicative
Inequalities of Carlson Type and
Interpolation
homogeniety of the weights w*, that for such / , we have exactly
ll/HIp WfwoWl^Wfwtl f .Ipo
w
MJ
= C,
Mlp!
where C is given by (5.9). Suppose now that (5.5) holds for some constant C. Theorem 4.2 then implies that the conditions (5.6) and (5.7) hold. We need to show that also (5.8) is true. We may assume that T = do — di > 0. Define the function h : R + —> (0,1) by the implicit relation Mz) p / ( p i ~ p )
PZ
~ (1 -
(5.12)
h(z))P/(Po-p)'
where r
ppo
ppi
P=
, qpo-ppi-p
and define u on S by w u.ti-\"-(^r^ Wo J \ W J Moreover, let
My) = (1 - M^)*)) ro/pr °
l
7 ,ro ,
wo{y)ro
where y = ax G fl. By (5.12) it follows that r(l-h(.(a)X))^0f^iyP^da \w0(ax)J x
/ | / o ( v M v ) | > ^n = / Jci \y\ JsJo = / ( w— I Js \ oJ = I (—) Js \woJ
/ h(w(a)x)ri/r'Uuj(a)x)px^-d^rop~ Jo
x
da
" /"h{uj{a)x) r ^<(uj{a)x) p x- e T r ^— da. Jo x
In the x-integral, we make the substitution t = oj(a)x, which yields
f (—} ° uj-6TToPda r h(t)ri/rHp-eTroP
= (CT [
Js Wo
(-^Yda, W V
Some Multi-dimensional
Generalizations
and
Variations
95
where the constant C" may be calculated explicitly in terms of the Beta function. In a similar manner, we obtain
It follows that if (5.5) holds, then
C f ( / l - r ^e - j ) '' Us W0- w$
w w0
da
—
w1
r
The proof is complete. 5.3
•
Some Variations on the Multi-dimensional Theme
Various multi-dimensional versions of inequalities of Carlson type have been proved by several authors over the years. We collect some of the results below. 5.3.1
Kjellberg
Revisited
We recall that the inequality (3.6) implies that if two integrals of the form OO
/
x2a\f(x)\2dx
are finite (in the present case with so is the integral of |/|. This way generalized in 1942 by B. Kjellberg which the finiteness of a number of /•OO OO
/
/*00
1 / ( 1 1 , • • • ,Xn)\2xfn
• •• / -OO
a = 0 and a = 1, respectively), then of looking at Carlson's inequality was [43], who studied the conditions under moments Mj, defined by
J
—OO
• --X2^"
dX! • • •
dxn,
96
Multiplicative
Inequalities of Carlson Type and
Interpolation
j = l,...,m, implies the integrability of | / | on R™. The main theorem is the following. Theorem 5.2 (Kjellberg, 1942) A necessary and sufficient condition for the finiteness of the moments Mj, j = 1 , . . . , m, to imply the integrability of / on R" is that
G i)«* is an interior point of the convex hull of the points (ai\...,ajn),
j =
l,...,m.
Suppose that S = {Qi,..., Q m } , where each Qj is a point {otji,. • •, a.jn) giving rise to a moment Mj. Let P(x) = J2T=ix2Qj> w n e r e standard multi-index notation is used to define x2®. By integrating the inequality (ab)1/2 < \{a + b) applied to a = P _ 1 and b = \f\2P, Kjellberg noted that
/j/i-4(/„>g4 Thus, in order for the finiteness of the moments to imply that of / R „ | / | dx, it is sufficient that P _ 1 is integrable. If P _ 1 is not integrable, however, there is / for which V • M? = 1 but J \f\ = oo. In one dimension, P _ 1 is integrable precisely when P contains two terms x2a and x2b, where a < \ < b. Kjellberg's result thus says that this intuitive description is true also in higher dimensions. We mention that Kjellberg's theorem is, apart from Proposition 5.1, the first multi-dimensional result of its kind.
5.3.2
Andrianov
In 1967, F. I. Andrianov [2] gave a proof of a quite straightforward generalization of Levin's theorem, based on Hardy's method. Suppose that
Some Multi-dimensional Generalizations and Variations
Q = Q(x\,...,
97
xn) is homogeneous of degree n. Let x\ = rcos
COS<^2,
£„_! = rsin2 ---sin i sin ip2 • • • sin
n-i and put Q(ipi,...,
=
—•
We assume that T
_
/"r/2
r / 2 s i n " - 2
JO
JO
Q{
is finite. We let x = (xi,...,
xn), and write x > 0 for Xj >0,j
Theorem 5.3 (Andrianov, 1967) Suppose that f(x) p, q,\,n> 0. Then /
Q-1-X(x)r(x)dx)
f{x)dx
JR™
J
\JR™
( f
(5.13) l,...,n.
=
> 0, x > 0, and
Q"-1+"(x)f'1(x)dx)
J
\JRI
,
where
"
pfi + qX1
t=
X
pn + qX
and C=
u 1 W l \ * {I 1 „[ s t -^—B \qtj \nX + fi \l — s — t 1 — s — t KpsJ
l
l
t
(B(-, •) denotes the Beta function). The constant C is sharp. Remark 5.6 Of course, we need to make sure that the arguments to the Beta function above are positive. This is the case if p, q > 1. Remark 5.7 Note that the constant C depends on the integral I defined by (5.13), which is merely assumed to be finite and might be difficult to find the numerical value of. However, Andrianov gave an explicit expression
98
Multiplicative
Inequalities of Carlson Type and
Interpolation
for the best constant in the special case where Q is the nth power of a homogeneous polynomial of degree 1. R e m a r k 5.8 Theorem 5.3 is really a special case of Theorem 5.1. Indeed, R™ can be written as a cone fi as in (5.4) if we choose S to be the subset of the unit sphere consisting of the points all of whose coordinates are positive. Andrianov also proved the following result, with more than two factors appearing on the right-hand side, which is a continuous, multi-dimensional version of Theorem 2.2. T h e o r e m 5.4 (Andrianov, 1967) Suppose that f{x) > 0, x > 0, and q > 0, k is a positive integer. Then
0
N (fc+l)(2fc+l)
r
f(x)dx) x>0
2k
.
Qk+«-V* (x)fk+1 (x) dx,
j=0Jx>0
where the constant , j v fc(2fc+l) 2fe
,
,,
is sharp. R e m a r k 5.9 Here, as well, the constant depends on the integral I. In [2], this was calculated in the special case where Q is a linear combination of monomials of degree n. 5.3.3
Pigolkin
G. M. Pigolkin [74] also used the Hardy method to prove his multidimensional version of the inequality. Here, we consider functions / = f(x\,..., xn-i) of n — 1 variables and a non-degenerate (n — 1) x (n — 1) matrix (/Xjj) with determinant
T h e o r e m 5.5 (Pigolkin, 1970) Suppose (Hij)"1 = (vij) satisfies 71-1
i=l
that
the
inverse
matrix
Some Multi-dimensional
Generalizations
and
Variations
99
I f O < 7 < 1, then n-l
n-l
1
\xi l"y dx ) • ( 5 J 4 )
fdxY "'
j=l
If <5 =
1
1-7' n-l
^i
7
1 — 7 cr ' n-l
then
_
?TT^7
r f ^ a ) n-lr
n (^)
. (n-D-> fl*
V
| G , | r ( 2 ) i=i
/
is the sharp constant, and equality holds in (5.14) with this constant if and only if / has the form
m=^\T{cH+^i[\xJryj for some ao > 0, a» > 0, i = 1 , . . . , n — 1. 5.3.4
Bertolo-Femandez
By using an induction argument, J. I. Bertolo and D. L. Fernandez [14] proved a multi-dimensional version involving mixed Lp-norms, which was used in connection with the multi-dimensional Mellin transform (see [13]). We recover the main result below. Suppose that P = (pi,... ,pn), where 1 < Pj < oo, j = 1 , . . . , n. The space L*P is determined by the nniteness of norms, defined inductively as
100
Multiplicative
Inequalities
of Carlson Type and
Interpolation
follows. If n = 1, we put
ii/iuMK+) = (^ 0 0 |/(x 1 )r^) 1 / P 1 . If n > 1, and ZA(R™_1) has been defined with P = (pi,... P = (P,pn).
,p„_i), let
Then Lp(K+) is denned via the norm
ii/iUM^) = (/ 0 °ii/(^«)ii P 4 (Rrl) ff) 1/P "Moreover, if k — (fci,..., fcn), we define n
Q(k) = l[e(kj), where
Theorem 5.6 (Bertolo-Fernandez, 1984) Let 6 = (6\,... , #„), where 0 < 6j < 1, and let P0 = (PO>---.PO) a n d -Pi = b i i - • • >Pi")i where 1 < p\ < oo, i = 0,1, j = 1 , . . . , n. If k = (fci,..., fcn) is such that each fc, is either 0 or 1, put Pk = (p^,••• ,p£ n )- Then
II/IILKRP^CIIII^^/II^U)
(5,15)
k
for all measurable functions / on R™. The product on the right-hand side of (5.15) is taken over all possible k with elements 0 and 1. We may choose
i=i
where each C(p30,p31,6j) is any constant for which the corresponding onedimensional inequality holds (see e.g. Theorem 4.1). 5.3.5
Barza
et al.
In 1998, S. Barza, J. Pecaric, and L.-E. Persson [6] also discussed the possibility to prove some multi-dimensional version of Theorem 3.7 (with weights involving factors of the type (x — a)a). In particular, they obtained the following result.
Some Multi-dimensional
Generalizations
and
Variations
101
Theorem 5.7 Suppose that / is a measurable function on R+, and let o-1^2,03 be any real numbers, and 71,72,73 > 1. Then 2 ^ -
//
\f(x,y)\dxdy
J Jw\
\J JR2+
f(x,y)dxdy J
(JJ2 \x-ai\^f2(x,y)dxdy ( [[
\y-a3\^f2(x,y)dxdy
2^T1
2T3T2
2T2T3
We may choose n _ r^-hz ^-
O — 0 7 l , 0 i <-/72,a2 ° 7 3 , a 3 i
where
^i,ai=2-(7i-l)"i
sin(7r/7j)'
with mj = 0 if a; < 0 and m* = 1 if Oj > 0, i = 1,2,3. 5.3.6
Kamaly
In 2000, A. Kamaly [40] (see also [38]) proved a discrete, multi-dimensional version of Beurling's inequality, which he used to get a sharp, local version of the Hausdorff-Young inequality (see Chapter 9). For a related result, see also G. W. Hedstrom [35]. Let T™ = R n / Z " denote the n-dimensional torus, and define the fcth Fourier coefficient f(k) of an integrable function / : Tn —> C by /(fc)= f
f{x)e~^ik-x
dx,
k£Zn.
^4(T") is, by definition, the vector space of integrable functions having absolutely convergent Fourier series. We equip A(T n ) with the norm
ll/IU(T») = £ n l/>)|, fcez
102
Multiplicative
Inequalities
of Carlson Type and
Interpolation
under which it becomes a Banach space. Note that the discrete version
0
/*27T
kez,
'
/»27T
\f(t)\2dt
^
\f'(t)fdt
J of Beurling's inequality (3.4) 0can, with a JOsuitable normalization of the Fourier transform, be written as
ll/IUm
• • • &xln '
where |-y I = 7i + • • • + j n . T h e o r e m 5.8 (Kamaly, 2000) Let / £ A(Tn). Suppose that 1 < r < 2 and that the positive integer a is such that a > ^. If /(0) = 0, then
V|7l=« In the general case, we get
ll/IU(T")
E
H^7/ll'
\h|=c In both cases, the constant C depends only on a, n, and r.
5.4
Some F u r t h e r Generalizations
In this section, we will point out the fact that, by using some ideas presented in this book, we can further generalize some of the results from the previous chapters in a multi-dimensional setting.
Some Multi-dimensional
5.4.1
A Multi-dimensional
Generalizations
Extension
and
Variations
of Theorem
103
3.6
We prove the following multi-dimensional extension of Theorem 3.6 (see [49]). In particular, we obtain the Yang-Fang result (see [84]), except that we do not give any estimates of the constant, upon letting n = 1, p = q, r = s and 7 = 1 if we consider functions / supported on R+. Theorem 5.9 Let n be a positive integer. Suppose that p, q > 2, 0 < a < n, and r,s £ R. Suppose, moreover, that for some positive constants M and 7, the function g : R n —> (0, 00) satisfies g(x) > M|a;| 7 .
(5.16)
Then there is a constant C, independent of M, a and 7, such that
(/ R J /wl
g{x){n-a)h\f{x)\p{l+2r-rp)dx
x(7 imrdxY
[
5 (z)
( +a)/7
"
|/(a;)| 9 ( 1 + 2 5 - s , 3 )
(f \f(x)rdx)g . (5.17)
We give the following simple example to illustrate the usefulness of the weaker assumptions on the function g in Theorem 5.9 in comparison with those of Theorem 3.6. Example 5.4 Let a and b be positive real numbers, and define g : [0, 00) —> (0,00) by / \ _ / a'
0 < x < b, < 00.
Then g is not differentiate (not even continuous at the point x = b) and g(0) = a ^ 0 . Thus two of the conditions on g in the hypotheses of Theorem 3.6 fail. Nevertheless, g is an admissible function for Theorem 5.9, since (5.16) holds with 7 = \ and M = ^ . Remark 5.10 Although our proof shows that we can get away with somewhat weaker assumptions on the function g than those imposed in Theorem 3.6, the condition lim x _ 0 0 5(x) = 00 cannot be relaxed too much. More precisely, we cannot let the function g be essentially bounded. To see this,
104
Multiplicative
Inequalities
of Carlson Type and
Interpolation
we consider the one-dimensional case, and assume that 0 < g{x) < G almost everywhere on [0, oo). For R > 0, let fR be the characteristic function of the interval [0, R). Then for any t
I
oo
o
\fR(x)\tdx
= R,
Jo
and /•OO
/ Jo
s ( z ) ( l T a ) / 7 | / « ( x ) | t dx <
RG^a)/t.
Hence
U\fR\dx)p+q (j\fRrdx)'(p-2)
J5(l-«)/7|/ij|p(l+2r-rp)
dx
(j\fR\°*dx)-^-2)
J £ , ( l + a ) / 7 | / i j | g ( l + 2 S - ^ ) dx R2
RP+i
-
jRp-2^-2i?G(l-a)/7i?G(l+a)/7
2
~ Qh '
which tends to infinity as R —> oo, and thus the inequality cannot hold for any finite constant. It is clear that this also shows failure of (5.17) in each multi-dimensional case. To prove Theorem 5.9, we apply Theorem 6.1 of Chapter 6, which we quote below in a special case, suitable for our present needs. Lemma 5.1 Let (fi, n) be a measure space on which weights w > 0, too > 0 and w\ > 0 are defined. Suppose that po,pi > 1 and 0 < 6 < 1. Suppose also that there is a constant A such that M (jw; 2m < ^ M
< 2m+l \)
meZ
(5.18)
and that w -T-r^eL00(n,fi).
(5.19)
Then there is a constant C such that
ll/^IU^CIIMIli^HMII^. The constant C can be chosen to have the form C = CoA1^1-^-0/"1, where CQ does not depend on A.
(5.20)
Some Multi-dimensional Generalizations and Variations Proof
of Theorem
5.9.
105
Let fi = W1 a n d define t h e measure fi on Q. by
where dx denotes Lebesgue measure in R n . Define, on fi, w(x) = WQ(X) = \x\n~a/v and w\(x) = \x\n+a/q, and let p0 = p ' , p i = 9' and
\x\n,
P+Q Then ieL
<^X
0
so (5.19) is satisfied. Moreover, wo(x) wi(x)
_Q(i + i)
Let
T
= a ( - + -)>0. vP 9,
T h u s U ; 0 ( Z ) M ( : E ) G [ 2 m , 2 m + 1 ) if and only if 2-(m+l)/r
TmlT.
< |a;| <
Hence, using polar coordinates, we get n — m/T
2
J
-
•• log 2
where w„ denotes the surface area of the unit sphere in R n , which shows t h a t (5.18) holds with
a
106
Multiplicative
Inequalities of Carlson Type and
Interpolation
Thus (5.20) implies /
\f{x)\dx
= f
\f(x)w(x)\dn(x) (l-fl)/po
I /n
\f(x)w0(x)rdti(x)'
x ( / \f(x)Wl(x)r
6/P1
In
\x\(n-a)^p-V\f(x)\p'
= c(f
dx)
(p-l)/(p+g)
(<J-I)/(P+«)
\x\ln+a)/i''-1)\f(x)\q'dx\
x f f or
) x(
P+Q
\x\in-a)/(p-l)\f(x)\p'
f
n
\Jlg.
x\{n-a)/{p-1)\f{x)\p'
dx)
\x\ln+a)/i''-1)\f(x)\9'dx)
x ( [ We write
. J
= ('| x |("- Q )/(P- 1 )|/(x)| p ' ( 1 + 2 r _ r p ) ) | / ( x ) | p V ( p _ 2 )
and | x | ( n + a ) / ( 9 - D | / ( a ; ) | 9 ' = (\x\(n+a)/(q-l)
|^^|^(l+2.-«,)^
^^g's(q-2)^
and apply the Holder-Rogers inequality with the pairwise conjugate exponents p — 1 and (p — l)/(p — 2) in the first integral, and the exponents q — 1 and (q — l)/(q — 2) in the second integral, respectively. This gives
( f \f(x)\dx) x f
\x\n-a\f{x)\p^+2r-rpUx
[
\x\n+a\f(x)\^1+2s's^dx
x(7 \mrdxY (f i/wn^Y .
(5.21)
Some Multi-dimensional
Generalizations
and
Variations
107
Since 1-0 Po
6 _ 2 Pi P + q'
we can choose 2/( P +g)
2
C=
C0A /^=Co(^X
so that (jp+q
_
D 9 >
where D does not depend on a. In view of (5.16), we get
U K ' ' *M" " * Now, using this inequality to estimate the factors | x | " _ a and |a;|" +a in the integrals on the right-hand side of (5.21) yields (5.17), and the proof is complete. • 5.4.2
An Extension
of Theorem
5.8
We show how Theorem 6.1, in the shape of Lemma 5.1, can be applied to get the following generalized versions of Theorem 5.8. We give a detailed proof of the continuous version only — the discrete version is proved similarly (see also [50]). We define the Fourier transform / of an integrable function / on R" by )= f
f(x)e-iuxdx,
weR".
JM.n
Theorem 5.10 Suppose that 1 < ro,ri integer satisfying
< 2, and that a is a positive
a > —.
(5.22)
Define
108
Multiplicative
Inequalities
of Carlson Type and
Interpolation
Then there is a constant C such that
II/IILI(R»)
P7/II^(R»)
< c ( J2
)
\|7|=a
Proof.
ll/ll l(K ")-
/
Define, on K", the measure ^v '
\u)\n
and the weights w{w) = |w|",
1«0(W) = | w | " / p 0
^
|u^|,
|7|=«
and Wl{uj)
Put p = 1, po = rg
an
d Pi
= r
\uj\n'rK
=
i - Moreover, let W(LJ)
W(w)
-(1-0)
\ITI="
where q is defined by 1 _ q
1-fl
61
PO
pi
Then W is homogeneous of degree --(l-9)a
= 0,
and thus constant along rays from the origin. In particular, since W is continuous, as is easily seen, W is bounded on R n . This gives the condition
Some Multi-dimensional
Generalizations
and Variations
109
(5.19) of Lemma 5.1. Moreover, if we define V by V » =
w0(w)
i w r ( n-^) E
iw7i>
|7|=a
then V is homogeneous of degree r := a + nl and T > 0 by the assumption (5.22). Thus, there is a constant Bo such that H({2m
< V < 2 m + 1 }) < — , m G Z , T
which is (5.18). Therefore, by Lemma 5.1
ll/IUi(R") /
l/wol^dM)
/
/
=
C
r,
w7
|M| P l d/x)
\(l-<>)/r'0
r
(
Pl
( ^ n ( E l/>) l) ° ^ I
(^ n l/HI dw)
=c (JRn ( E I ^ / H I ) " 0 ^ I
(jf n i/nr'1 <*")
= G|i;i7l=a|5r/||^
II/IIJ
||/|lir,(R-),
\|7|=a
||DV||Lr,(H»)) °
n)
/
where we have used the triangle inequality. Now, applying the HausdorffYoung inequality (this is where we need the assumption 1 < r* < 2) to each term in the sum and to the rightmost norm, we get
ll/lkcR-) < C I ] T \\D-*f\\Lro{&n) ) which is what we wanted to prove.
||/||i r i ( R B ) l •
110
Multiplicative
Inequalities
of Carlson Type and
Interpolation
In the discrete case, we need to treat the 0th Fourier coefficient with a little extra care. Theorem 5.8 follows if we put TQ = r\ = r below. Theorem 5.11 Suppose that 1 < ro,r\ < 2, and that a is a positive integer such that n a > —. ro Define 6 as in Theorem 5.10. Then there is a constant C such that
ll/IU(T») < |/(0)| +C l ^ \l7l=«
PVUWT") ]
ll/lliP1(T»).
/
Sketch of proof. Assume first that /(0) = 0. We can then proceed as in the proof of Theorem 5.10, but define the weights w» and the functions W and V only on the integer lattice. The general case follows easily from this.
•
Chapter 6
Some Carlson Type Inequalities for Weighted Lebesgue Spaces with General Measures The setup for a Carlson type inequality can be generalized to a general measure space. This fact will be stated and proved in Section 6.1 (see Theorem 6.1). The most general result if this type when we have a product measure with n factors (for any n s Z + ) is stated in Section 6.3 (see Theorem 6.4). In order to avoid unnecessary technical details, we state and prove our results of this type in the two-factor case in Section 6.2. 6.1
The Basic Case
Suppose that (f2,/x) is a measure space, on which positive, measurable weight functions w, WQ and W\ are defined. We would like to translate the conditions (5.6) and (5.7) to this abstract setting. Suppose, for simplicity, that n = 1 and S = {(1,0)}. Thus O = (0, oo). With y, defined by dfi(x) = — , x the weights in Theorem 5.1 are w(x) = xd, Wi(x)=xdi
i = 0,1.
The condition (5.6) could then be interpreted as w(x) wl- {x)w\{x) e
= 1,
Thus the quotient w w\~ewi 111
0 < x < oo.
112
Multiplicative
Inequalities
of Carlson Type and
Interpolation
is in Loo(n, /x). In the general case, we may allow a slightly weaker condition (see Theorem 6.1 below). The condition (5.7) means that the quotient w0(x) WI(X)
is not constant. The corresponding condition in Theorem 6.1 below says that the weights wo and wi may not be too close to each other on arbitrarily large sets (in a measure theoretic sense). We may also prove the inequality for a wider range of the parameter p, by using an interpolation technique. The theorem we have in mind is the following result by L. Larsson [50] from 2003. Theorem 6.1 Suppose that p,po,Pi € (0, oo] and 9 G (0,1) are such that -:= Q
+ — >0. P
Po
(6.1)
Pi
Suppose, moreover, that there is an s € [q, oo] for which W
tirf-V
eL.(n,n),
(6.2)
and that for some constant B we have [i(luen-,2m <'^^<2m+1Y\
(6.3)
Then there is a constant C such that the Carlson type inequality
holds for all measurable functions / . Remark 6.1 As the proof of Theorem 6.1 will show, the condition (6.3) is not necessary if (6.2) holds with s = q, where q is defined by (6.1). However, there are examples showing that (6.3) is needed if we only assume that w/wQ~ewf is in Ls for some s > q (see Remark 6.5 below). Remark 6.2 The proof will also show that the constant C can be chosen to have the form C =
C0B1/q-1/s
w L (Q,n) rf-V 3
where CQ does not depend on w, WQ, or w\.
General
Measures
113
Remark 6.3 Theorem 6.1 applied with p = 1 and s = oo, together with Remark 6.1, gives the crucial Lemma 5.1 of Chapter 5. Let us give an example of an elementary but non-trivial inequality, where Theorem 6.1 needs to be applied with s strictly between q and oo. Proposition 6.1 Let po,pi > 1 and 6 S (0,1). Define q by
q
Po
Pi'
Moreover, let a be a non-zero real number and t > 1. Then there is a constant Co, which is independent of a and t, such that for all Lebesgue measurable functions / on (0, oo) it holds that
y 1 x1/
Jo xv*9
^pd^max{l,(t-l)-V*
Proof.
Let
a
oo
}
\ ( l - 0 ) / p o / too
\f{x)e6ax\Po
"
| / ( z ) e - ( 1 - 0 > a x | P l dx)
(
. j r 1 / ' 9 , o < x < l,
, , ( I )
dx)
\ 9/Pi
=
< T - l / « , X > 1.
Put w 0 (x) = eBax and wi(x) = e - ( 1 - e ) a : c . If |_B| denotes the Lebesgue measure of the set E, then 2m < ^ ° < 2 m + 1 =|{x;2 m <eax < 2 m + 1 } | = I i - l o g 2 . \ci\
This is (6.3) with
Since wQ
w^ = 1, we have w}r0vfi
W
'
114
Multiplicative
Inequalities
of Carlson Type and
Interpolation
and w £ Ls if and only if q < s < tq. Thus Theorem 6.1 applies. If s
1+t
= —^—
then 2(t + l)\ 2 /< t + 1 >« This behaves like 4l/q(t — l ) - 1 / 9 when t is close to 1 and tends to 1 as t —> oo. By Remark 6.2, we can write CQBl'q-1/s
C =
w wk-"w!
Ls
and with the choice of s made above we have 1 1_ lt-1 q s qt + V and hence the desired result follows.
•
Remark 6.4 We show by an example that the condition
i>i^
+
l
(6.5)
P Po Pi of Theorem 6.1 cannot, in general, be relaxed. Consider Q — [0,1] and dn(x) = dx. Fix 6 G (0,1) and Po,Pi > 0. Let w = WQ — w\ = 1. Since the measure is finite, the condition (6.3) holds trivially. Moreover, w W 1, 9
wl- w{
and hence W € Ls for any s. Let e > 0, and let f£ be the characteristic function of the interval [0, e]. Then, for to* = 1 and any p , > 0,
/
\few*\p' d[i = e,
Jn and hence
ll/HI li/^oiii;!ii/e«iiiL
i = gp
1-8
e
PO
PI
.
Thus, if (6.5) is violated, this quotient can be made as large as we want by choosing e small, i.e. there is no finite constant C such that (6.4) holds in this setting.
General
Measures
115
Let us show that the condition (6.3) cannot be removed in the case s > q in order for (6.4) to hold in general. We consider ft = (1, oo) and /x defined by dfi(x) = dx. Let w(x) = x~p and
Wi(x) = x~p*,
i = 0,1,
so that w(x) =
X
-1/9
w™(x)w°{x) This quotient is in L s precisely when s > q. For R > 1, let /R be the characteristic function of the interval (1,12). Then, for any w*(x) = x~l/p' and any p* > 0, we have
/' \fRw.\»d» = [
Jo.
Ji
—=\ogR, x
and thus
1 Iff. -.. =(*«)•*• ll/fi^o|| ll/fi^il Lpo
When 12 —» oo, this clearly tends to infinity and the claim follows. Proof of Theorem 6.1. We may assume, without loss of generality, that Po < Pi- Moreover, we can assume that all parameters are > 1 (this is needed when we apply the Riesz-Thorin interpolation theorem below). Consider the diagram in Figure 6.1. The strategy is to prove (6.4) first in the case when s = oo and p < po,Pi, after that on the "diagonal" s = q. We then apply an interpolation argument in order to get the inequality (6.4) in the convex hull of these sets (i.e. in the shaded region of the diagram in Figure 6.1). This will complete the proof. Thus, suppose first that w wl0-6w{ G Loo (ft,/x) and p < po- For i = 0,1, define r* by
1 - I _ J_ n
P
Pi
116
Multiplicative
Inequalities
of Carlson Type and
Interpolation
Fig. 6.1 T h e diagram shows the region of admissible parameters for a Carlson type inequality on a general measure space.
It follows by the Holder-Rogers inequality that if E is any measurable subset of fi, then
= KII/»ollE„(0^ + Mf||Mlli pi(nirt . Let S be any positive number. We want to choose the set E so as to obtain
General Measures the estimates M0
and
< JVKH1-^,
Mx
where NQ and iVi are some constants. For if we let 'i_0y/pWillMllL *
(n>M)
^oll/^o||Lpo(n,M):
/
this yields (6.4) with
c
'((i-ey-oeoy/p'
For m £ Z, we define fim t o be the subset of fi on which 2™ < }Hl < 2 m + 1 . Moreover, we let E be the set where Wi ~ where the integer mo will be specified later on. It follows t h a t
Jn\E V^o
W
\wiJ
V
J^JsimWo-e<) <
w;
u^X w w0-ewt
W
r0
mp —1 m+1 er ^ ) o 2(
5 •^oo(^2,M)
771= —OO
20moro B 1,00(0,^)
1 _ 2-*»-o '
and similarly t h a t 2-(l-6)m 0 ri
^"X Let
Loo(n,M)
5 1_
2-(i-e)n
118
Multiplicative Inequalities of Carlson Type and
Interpolation
and suppose that the constants No and Ni are chosen so that Wo
JVl
( l _ 2 - ( '0rr oO )\ 1/0T-O
(1-2- - ( i - 0 ) r M - i / ( i - e ) n (6.9)
We can then choose mo so that i N l
\
^
WBV**J
i
Cl _ 0 - ( l - e ) T - 1 N - I / C I - ^ T - ! < ? ! ! ! . < f [ S
'
- 6
N
°
Y
{wBVr°)
U
l
C\_2-0rO\i/ero
)
<
which together with (6.7) and (6.8) gives (6.6). It is readily shown that (6.9) is equivalent to 2«(i-»>WB*
Atf-X = (1
- 2 - e r o ) ( l - 0 ) / r o ( l _ 2-(l-«)»'i)0/r1
Thus, in view of the discussions above, we have proved that (6.4) holds with w C = <-*<
ioo(^,M)
2 e ( 1 - 0 )Bi ((1 _ 0)i-00«)£(i - 2- 9 r o)(i-e)/r 0 (i _ 2-(i-e)ri)e/ri
For later purposes, we write C0 =
2»( 1 -°)BJ ((i _ ey-eee)v{\
- 2-er°)(l-e)/r°{i
- 2-( l - e ) r i) e M
(6.10)
so that (6.4) holds with C = C0
w
urf-V
LooCn,^)
Suppose next that s = q. This can be written as p s
|
p(l - g) p0
|
pg = Pi
t
and, hence, the Holder-Rogers inequality can be applied with three factors, using the exponents
£
P°
?!
p'p(l-/9)'p0'
General
Measures
119
which yields
w wl~6w\
<
II r
MP(l-e)
L,(nlM)
HMlC(n, M )'
This is precisely (6.4) with C =
w ,i-e,„e
z,,(n,M)
It remains to prove the inequality for (^, i ) in the shaded region in Figure 6.1. Fix the function / , the weights wo and w\, and the parameters po and pi. Define the linear operator T : LS(Q, fi) ^> Lp(a,») by
For any w € I/ s (0, /x), we can write
iitf-X for the correct choice of u;. The previously shown cases of (6.4) now state that Tw
llLp(n,M) -
C
° 11/™°IIi-poV/0 HMIIip^n,/.) HwllLoo(n,M)
and Tw
11-0
llL p (n, M ) ^ H/^o|| L p o ( n , M ) H/«'i|lLP1(n,/i) IMIi,,(n,/i)
respectively, where Co is as defined in (6.10) with any admissible po a n d P\. In other words, T is a bounded linear operator L^ —> Lp with norm at most Co||/^o|| Lpo( n, M )ll/^illL P1 (n, M ) and Lq —> Lp with norm not exceeding II/^OIIITVUIIIM"' LPI iL P0 (n,M)
(n,fi) •
120
Multiplicative
Inequalities
of Carlson Type and
Interpolation
Thus, if 0 < a < 1, then the Riesz-Thorin interpolation theorem (see e.g. [12] or [18]) implies the boundedness of T : LaLp, where sa is defined by the relation 1 sa
1 — cr a oo q
a q'
with norm at most Co
a
,1-0
ll/«'o||iPO(n,M) WfwihPl(n,»)
•
By translating this fact back to the original situation, this shows that (6.4) holds also in the remaining cases, completing the proof. • 6.2
The Product Measure Case - Two Factors
Once we have Theorem 6.1, a corresponding Carlson type inequality for product measure spaces can be proved. Before we present our most general result (see Theorem 6.4 in Section 6.3), we will in this section clarify our ideas by stating and proving two versions of the two-factor case. As before, we define q by 1
1
1-0
9
q V Vo and we assume that this quantity is > 0.
Pi
(6.11)
Theorem 6.2 Suppose, in addition to the assumptions of Theorem 6.1, that (E, v) is a measure space with weights v, i>o, and v\, and assume that -^r~e
€ Lq{y).
(6.12)
Let X = fl x S and d,K = dfj, x dv, and put u(w,£) and
=w(u)v(£)
v,i(w,£) =Wi(u))vi(£), i = 0,l,
w£f!,(6S.
Then \\MLAX,K)
< C H/uolli^V,*) IIMIIw*,*)
holds, where we may choose a constant of the form
(6-13)
General Measures
121
where Co is independent of w, v, WQ, VO, WI and v\. Proof.
We assume that p,Po,Pi < oo. Then (6.11) can be written as p
|
(1 - 6)p
q
|
6p
po
=
1
pi
and, therefore, the Holder-Rogers inequality with three factors can be applied, using the exponents £
Po
Pj_
p (1 -ff)p
Qp
Assuming that (6.4) of Theorem 6.1 holds with C = C0
wuln-ew<> u/j LB(fl,fi) 0
we thus get, also using Pubini's theorem
\\M\PLvM = JsVPJQ\fMPd^du
(l-0)p/po
I \fv0w0\P° dn) q
L U-v [ f
U
\fvlWl\^dfiJ
\ P/
dv (l-0)p/po
\fv0w0\podndu
JE JU \ Op/pi
J J \fvlWl\Pl dfiduj Cp
dp
vl-%<{ £ „ ( E , I > ) II/«OII^,-)II/«IIIL W ( X .« )•
Taking pth roots, we get the desired inequality with
dis
122
Multiplicative
Inequalities
of Carlson Type and
Interpolation
If p = oo, then we must also have po = Pi — q = oo, and
l/«l<
v y,i-«.,e
o
m
ioo(S^)
^V
|/uo«o|1_e|/«i«i|'. •tooCn.M)
By taking suprema, the desired result follows (with Co = 1). If p < oo but po or pi is finite, the inequality follows similarly. • Theorem 6.2 is not symmetric, in the sense that we have different conditions on the respective measure spaces. If we impose a condition corresponding to (6.3) also on the second measure space factor, then we can loosen the condition (6.12) slightly, and we get a symmetric version of this two-dimensional result. T h e o r e m 6.3 Suppose that the hypotheses of Theorem 6.1 hold with s = S\. Suppose, moreover, that u({2m < vo/v! < 2 m + 1 })
m e Z,
and that € L S 2 (-,!/), v0
Vl
where 1 1 0< — < - , si q
1 1 1 1 1 0< — < - , —+ —>-. s2 q si s2 q
(6.14)
Then the inequality (6.13) holds with C' = C0
i-<
w,
,i-«„e
w\
L.2{S,u)
Proof. Theorem 6.2 applied first as it is and then with the two measure space factors interchanged, gives the result when the point
J_ 1_ si's2 is situated on two of the edges of the triangle shown in Figure 6.2. To prove the inequality for the remaining points under consideration (see the shaded area in the diagram of Figure 6.2), we use bilinear interpolation. We now fix everything except the weights w and v, and define a bilinear operator T : L32 (£, v) x LS1 (Q, /i) -* LP(X,
K)
General
Measures
Fig. 6.2 For points ( s j - 1 , s^1) on the right and top edges of the shaded triangle in the diagram above, t h e conclusion of Theorem 6.3 is implied directly by Theorem 6.2 by letting the two factors switch roles. In the convex hull, then, the inequality follows by applying bilinear interpolation.
by T(W,V)
l(fw0v0y-8(fw1v1)e}WV.
=
Since w w0
w1 v0
v1
= fwv,
the inequality (6.13) with C' = C0
w\-6w<>
Lai((l,n)
vQ
vx
L,,(S,i/)
can (with the appropriate interpretation of the underlying measure spaces)
124
Multiplicative
Inequalities
of Carlson Type and
Interpolation
be written as w
V
WTWV^KCoWfwoVoW'-'Wfwwfn
1_
7; V
«2
wl-6w{
or, if we denote by || • ||(Sl,s2),p * n e n ° r m of the operator T, it holds that n(auq),P
-.(1)
1-0
\\fw0vo\\L-0a
\\fwlVl
'Pi
and
Urll(,,*a),P ^ Co2) l l / ^ o | | p ; 0 ||M«i 11^ whenever <TI,<72 € [9,00]. Suppose now that (si,S2) is an Y point on the triangle (6.14) off the right and top edges (cf. Figure 6.2). Then there are 01, (T2 6 [g, 00] and 77 € (0,1) such that _1_ si
1 — T? . »7 ^ + ^ and
,
^1
1
S2
1
_n+v_
Thus, by applying multi-linear interpolation (see e.g. Theorem 4.4.1 of [12]), it holds that
imi ( » 1 . S 2 ) . P
\ 1—tj
<
«WI
l/wiuil
Wo^i
ll/wi^illpj 1-0 1
•,(2) v
(cpy->(cn \\fwov0ca\\fwlv1 7
(l)xl-
which, in turn, can be written as (6.13) with w
V
e
<- < The proof is complete.
Si
v0
V-L
a
Remark 6.6 The triangle (6.14) is, at least in the case po = Pi, the largest possible region in which we have the inequality (6.13). It suffices to show failure on the diagonal si = 52 = s, s > 2q. To see this, consider the point P in Figure 6.3. By interpolation techniques, we know that the region in which we have the inequality (6.13) is convex. Thus, if we show that such an inequality does not hold on the dashed diagonal, then it cannot hold at the point P either, since we can draw a straight line segment joining P to the triangle denned by (6.14) crossing the diagonal. It only remains to prove that Theorem 6.3 fails in general on the dashed diagonal in the diagram in Figure 6.3.
General
Measures
Fig. 6.3 To show failure of the Carlson type inequality on the product measure space, it suffices to consider the diagonal, since the region in which inequality holds is necessarily convex, as is seen by interpolation.
To do this, we let fi = E = (2, oo) and consider the measures \i and v defined by d^iix) = — x
and
dv(y) = —. y
Define 1/po
w0 (x)=x
1+1/po
,
vo(y) = y1+1/po,
Wl(x)=x
- y „VPO v1(y) =
Also, let
w = gwl
e
wl
,
126
Multiplicative
Inequalities
of Carlson Type and
Interpolation
and
v = gvfvl where g(t) =
(logt)-^.
Then V
«tf-'t
and W
wk-'v*
are both in Ls if and only if s > 2q. The condition (6.3) is satisfied on both spaces, and hence Theorem 6.1 guarantees that the Carlson type inequality holds on both factors. Consider, however, the functions / # , defined on fi x 5! by
/«(*, V) = (9(z)9{v))9/po V^Ty^
-l-2/p,
KR(x, y),
where KR is the characteristic function of the set PR = {(*,») e (2,co) 2 ;r 0 < y/x2 + V2 < R, £ < arctan | < y
j
and Tr, =
8\/2 -v/2 — l"
It can be shown that there are constants d and D such that for all (x,y) 6 PR
d(log y/x2 + y2)2 < (logx)(logy) < £>(log yjx2 + y2)2. Thus if K denotes the product measure, we have
f\fRvw\PdK>cf J
Jra
dr rlogr
General Measures
127
and . (1-0)/Po
\fRVow0\Po dn l/Po
Ju^r^
WfRViWi" i iiip0
ujn
rR dr Y/q >H\ I — ->oo, Ir0 rlogrl which proves the statement.
(£
6.3
R-^oo,
The General Product Measure Case
We conclude this chapter with a brief discussion of Carlson type inequalities on product measure spaces with any finite number of factors. For j = l , . . . , n , let (fij,/x^) be a <7-finite measure space, on which weights w^\ WQ and w[3' are defined. We let fi = fii x . . . x fin and d\x = dfx\ x . . . x dfj,n, and define w on 0 by w(wi, ...,wn)
= ww(ui)
• • -w{n){iJn);
wo and w\ are defined analogously. Furthermore, we put W^0)=
j =
l,...,n.
(wP)i-°(wP)e We then have the following multi-dimensional extension of Theorem 6.1, whose proof can be found in [50].
128
Multiplicative
Inequalities
of Carlson Type and
Interpolation
T h e o r e m 6.4 Let k be an integer such that 0 < k < n. Suppose that
w^€L3j(nj,^),
j =
i,...,k,
where 0<-<-, s,
q
- + ... + - > — , si sk q
(6.15)
and W& e LgiQj,MW),
j =k+
l,...,n.
Suppose, moreover, that for j = 1 , . . . , k, there are constants B^ such that
AtW)({2m < w^/wP
< 2 m+1 }) < Bj,
m£Z.
Then the inequality holds for some constant C. Remark 6.7 The example in Remark 6.6 above can be generalized to n factors, but only to a smaller set than the complement of (6.15) if k > 2. It remains an open question whether the region defined by (6.15) is the largest possible set on which we can prove the inequality (6.16) in general.
Chapter 7
Carlson Type Inequalities and Real Interpolation Theory In the previous chapter, we saw that interpolation theory can be used to prove some new Carlson type inequalities. The aim of the present chapter is to initiate a discussion of a closer connection between Carlson type inequalities and real interpolation theory. More information on this theme can be found in Chapter 8. In order to make our text reasonably self-contained, we present in Sections 7.1 and 7.2 notation and basic facts concerning interpolation of normed spaces and the real interpolation method, respectively. In Section 7.3, we state and prove a recent result by L. Larsson [51], concerning embeddings of real interpolation spaces into weighted Lebesgue spaces. Throughout this chapter, we assume that AQ and A\ are normed spaces, which are compatible, in the sense that they are continuously embedded in a Hausdorff topological vector space V. We denote by A the couple (Ao, A{) of normed spaces. For notational simplicity, the norms on AQ and A\ will sometimes be denoted by ||-||0 and || -1| j_, respectively. Since the results of purely interpolation theoretical nature can be found in numerous texts on the matter (see e.g. [12] or [18]), the ones we will need are presented here without proofs. If the two spaces AQ and A\ are Banach spaces, we will call A a Banach couple.
7.1
Interpolation of Normed Spaces
If AQ and A\ are compatible, it makes sense to define the intersection and sum of A by A(A) = AQ n At 129
130
Multiplicative Inequalities of Carlson Type and Interpolation
and E(i5) = {/o + / i ; / i € A i , * = 0 , l } , respectively. Here, of course, + denotes addition in V. A(v4) and T,{A) are normed spaces under the norms defined by 1 1 / 1 1 ^ = maxdl/lloJI/UJ and ll/lls(A) = inf{||/o|| 0 + l l / i l l i ; / = fo + hJi
€Ai,i
= 0,1},
respectively. Moreover, they are Banach spaces if AQ and Ax are. Suppose that X is a normed subspace of V. X is then called an intermediate space between AQ and A\ if A(A) C I C Y,{A). Recall that this means that the inclusions are continuous. We write T : A —> A to denote a linear mapping T on T^(A) such that T\Ai:Ai^Ai,
» = 0,1;
thus we say that T is continuous on A if T maps the subspaces Ai continuously into themselves. An intermediate space X between AQ and A\ will be called an interpolation space between AQ and Ai if it has the additional property that T\X:X^X whenever T : A —> A.
7.2
T h e Real I n t e r p o l a t i o n M e t h o d
There are many known methods to produce interpolation spaces. In this section, we will present two of the most well-known methods (the if-method and the J-method), which turn out to be equivalent. Any of the two methods is then called the real interpolation method. There are also other equivalent characterizations of this method, which is very useful for several applications.
Carlson Type Inequalities and Real Interpolation
7.2.1
The
131
K-method
If t > 0, we define the Peetre K-functional K(t,f;A)
Theory
= mf{\\f0\\0+t\\f1\\1;f
on £(A) by
= f0 + fufre
Aui = 0,1}.
For any fixed t > 0, K(t, •;A) is a norm on T,(A), which is equivalent to the existing norm on T,(A) (note that the latter is K(l, •; A)). Definition 7.1 Suppose that 0 < 0 < 1 and 1 < p < oo. We define AgtP to be the space of / G 51(A) for which
llfll
(C(t~eK(t,f;A)rf)1/p,l
=I
' ""'P
\supt>0r*tf(i,/;A),
p = co
is finite. Proposition 7.1 For 0 < 9 < 1 and 1 < p < oo, the space Ag,p is an interpolation space between AQ and Ai. 7.2.2
The
J-method
If t > 0, we define the Peetre. J-functional on A(A) by J(i,/;i)=max{||/||0,t||/||1}. The existing norm on A(^4) is J ( l , •; A). Any other t > 0 gives an equivalent norm on A(A). Definition 7.2 Suppose that 0 < 0 < 1 and 1 < p < oo. We define AJ6 as the space of those / in T,(A) which can be written as r0 dt f ° , ,dt
(7.1)
for some strongly measurable function u : M+ —> A (A). We define a norm on AQ by taking the infimum of the expressions
f (/0VJ(tlU(t);^f)1/p,i0*
e
J(t,u(t);A),
p = oo,
where u ranges over all the functions for which (7.1) holds. Proposition 7.2 For 0 < 6 < 1 and 1 < p < oo, the space AJe interpolation space between AQ and A\.
is an
132
Multiplicative
7.2.3
Inequalities
The Equivalence
of Carlson Type and
Interpolation
Theorem
The K- and J-methods are equivalent, in the sense that they produce interpolation spaces which, as topological vector spaces, are equal. Proposition 7.3 (The Equivalence Theorem) Given the couple A = {AQ, AI) of normed spaces, 0 € (0,1) and p e [1, oo], we have •™-9,p
=
AgiP,
with equivalent norms. Any of the two equivalent K- and J-methods is referred to as the real interpolation method, and the resulting space is called a real interpolation space. 7.2.4
The Classes Cj and CK
Given the couple A and 6 € (0,1), an intermediate space X is said to be of class Cj = Cj{6; A) if there is a constant C such that for all / e A(^4) it holds that ||/||x
(7.2)
The following result relates the real interpolation method to Carlson type inequalities. Proposition 7.4 If X is a Banach space, then the following statements are equivalent: (i) X is of c l a s s e d ) . (ii) Ag,! C X. (iii) The Carlson type inequality
ll/llx^CII/lt'll/H^ holds for some constant C and all / € A(J[). To see, for instance, that (7.2) implies (7.3), we note that with
,
H/lk ll/IU
(7.3)
Carlson Type Inequalities and Real Interpolation
Theory
133
we have
\\f\\x
Suppose, conversely, that (7.3) holds, and let t > 0. Then \\f\\x
^Ct-^maxdl/IU^tH/IUJ^^maxdl/IU^ill/IUJ0 =
Ct-eJ(t,f;A).
An intermediate space X is said to belong to the class if for some constant C it holds that
CK
=
CK(0;A)
K(t,f;A)
Reiteration
The real interpolation method possesses a remarkable and useful stability property, which is described in the following result. Proposition 7.5 (The Reiteration Theorem) Consider the two couples A = (Ao,Ai) and X = (Xo,X\) of Banach spaces. Suppose that 0 < 8Q < 6i < 1, and put, for some 77 € (0,1), 0 = (l-»j)<9o+770i. (a) If Xi is of class
CJ(0J;
A), i — 0,1, then, for any q £ [1, oo],
(b) If Xi is of class Cxi^i', A), i = 0,1, then, for any q g [1, oo], Ae,q 3 XVtq.
134
7.2.6
Multiplicative
Interpolation
Inequalities
of Carlson Type and
of Weighted
Lebesgue
Interpolation
Spaces
In order to connect Carlson type inequalities with interpolation theory, we need to identify the real interpolation spaces produced from a couple of weighted Lebesgue spaces. We will only need the following special case of this type of results. Proposition 7.6 Let Ai = Lri(v? n),
i = 0,l
and suppose that 1 _ 1- 6 r r0
6_ n'
Ar,e =Lr(vr
fj,),
Then
where v = vl0-ev{. 7.3
Embeddings of Real Interpolation Spaces
In general, the scale Ae,q of real interpolation spaces is increasing with q. Thus, in particular, Ae,i c Ae,p for any p > 1. Suppose now that X =
LP{WPIJ,)
and Ai = LPi(w?in),
i = 0,1-
Theorem 6.1 then gives conditions on the weights w and Wi in order for the inequality (7.3) to hold. Moreover, according to Proposition 7.4, the same conditions also give the embedding Ae,i C X
Carlson Type Inequalities and Real Interpolation
Theory
135
of the real interpolation space AQ^ into X. By combining Theorem 6.1 with the results from interpolation theory presented in this chapter, we can, in fact, prove the following stronger theorem by L. Larsson [51]. Theorem 7.1 Suppose that the hypotheses of Theorem 6.1 hold. Then Ae,P C X. Proof.
Take 6t, i = 0,1, such that 0 < 60 < 6 < 6X < 1, and define 77 =
0-0o C7 0
71 -
For i = 0,1, let
Vi=w[
\W1J and define r, by 1
1 q
=-+
1 - Bi 6i -+ — po pi
Let Xi = L r i (u[ ; /x), i = 0,1. By assumption, 1
1-0,
n
po
6j _ 1 > pi
Q
q ~
and u» wjr^wf*
to «>*-9-
e £„(/*),
and hence it follows from Theorem 6.1 that there are constants Cj such that
WfWx^CiWfW^Wffl,
i = o,i.
In other words, Xi G 0 ( 0 , ; A), i = 0,1. By using part (a) of the Reiteration Theorem (Proposition 7.5), it follows that -<^0,p _
^n,P'
v^v?
=w
Now,
136
Multiplicative
Inequalities
of Carlson Type and
Interpolation
and
v_ = 1
1
P
so Proposition 7.6 implies that
= x.
ViP
This completes the proof.
u
Since the scale Ag,r increases with r, the conclusion of Theorem 7.1 holds true if p is replaced by any r € [l,p). However, the following partial converse shows that we cannot, in general, go beyond p. Proposition 7.7 Given p,Po,Pi € (0, oo] and 9 € (0,1) satisfying (6.1), for any r € (p, oo], there is a measure space (fi, /j.) and weights w, WQ and w\ satisfying (6.2) and (6.3) such that Ae,r % X. Proof.
Let 0 = (e, oo) and define the measure /i by dy.(x) = — , x
where dx, as usual, denotes Lebesgue measure. Define w(x) = 1, wo(x) = x6, and Wi(x) = x~^~6\ Then w{x)
-i,
wl-\x)w{{x)
xen,
so the condition (6.2) is satisfied with s = oo. Moreover, since w0(x) wi(x) we have
„ ( J2- <™ ^o ^ < „2m + i l"\
/"'
dx
= log2m+1-log2m=log2, so that (6.3) holds with B = log 2. However, consider the function /(X) =
(logx)1^'
Carlson Type Inequalities and Real Interpolation
If 0 < t < e, define / 0 (t) = 0 and f[t] = f. Then At)
Theory
0, while
ffi Ao
dx\l/pi
/ {<*> x-nV-V
137
n
and hence (t)
t-eK(t,f;A)
Ao
Dtl~",
(t)
+ tl
/}
0
If t > e, put /£> = / X ( e ,t) and /<*> = / X [ t i 0 o ) . Then /£> + /<*> = / for all t, and 9po xT°I dx (logx)P°/ p x
r'
r
<
~
t6po
£)P0
° (logt)Po/P
and similarly t~(i-e)Pl
[(fit)w1rdfi
(logt)Pi/p'
Jn Thus f(t)
(t)
Ao
/i
^ A) + -Pi A1 -
_
(iogty/p
It follows that
\\nlr
dt + iDo + DiYj^ t(iogty/p'
Since r > p, the last integral converges (and so does the first). We see that / € A~e,r- However, since
f(fw)"dn= f
Jn
Je
dx xlogx
CO,
it holds that f $ X. Thus Ae,r £ X, as claimed.
•
138
Multiplicative
Inequalities
of Carlson Type and
Interpolation
Remark 7.1 Although we will refrain from doing so here, it is possible to prove a multi-dimensional version of Theorem 7.1, using the corresponding multi-dimensional Theorem 6.4 (see [51]). Remark 7.2 We mention that the interpolation spaces we get from applying the real method to a couple of weighted Lebesgue spaces (in off-diagonal cases, i.e. when the relation 1 _ 1- 6 r r0
6_ n
in Proposition 7.6 fails) have been characterized by several authors, see e.g. L. Maligranda and L.-E. Persson [64]. By using such results we can obtain further refinements of some of the results in this chapter. Remark 7.3 (More general embeddings of interpolation spaces) The proof of Theorem 7.1 relies on the fact that we have a Carlson type inequality associated to the spaces involved, and that we can rewrite the space X as a real interpolation space and apply the Reiteration Theorem. We can state the conclusion in more general terms, namely, as soon as we can prove a Carlson type inequality for the auxiliary spaces XQ and X\, or equivalently, show that Xi € Cj(9i\ A), i = 0,1, we have the embedding
The spaces X and Ai, i = 0,1, need not necessarily be weighted Lp spaces, as long as there is a Carlson type inequality and a corresponding reiteration theorem. Much more general embeddings could be achieved by using the method used for the proof of Theorem 7.1.
Chapter 8
Further Connection to Interpolation Theory, the Peetre (•)<£ Method 8.1
Introduction
In order to prepare for the discussion that will follow, we present an inequality which, in a sense that will become clear below, is equivalent to Carlson's inequality. Proposition 8.1 The following two statements are equivalent. (A) There is a constant C\ such that for all sequences {ak}kLi °^ negative numbers, the inequality oo
\
fe=l
/
(
4
oo
non
"
oo
fc=l
fc=l
holds. (B) There is a constant Ci such that for all sequences {bi}fZo of nonnegative numbers, the inequality
(
oo
\
oo
i=0
/
1=0
,2
°°
1=0
holds. Proof.
Suppose that (8.1) holds. For I > 0, put 2l
a* = | , 139
140
Multiplicative
Inequalities
of Carlson
Type and
Interpolation
Then oo 2"*"1 —1 ,
oo
oo
,
oo
2 +1 2
E^ = E E | = E|( ' - ') = E^ fc=l
k=1l
1=0
1=0
1=0
and oo
oo 2 ! + 1 - l
2
/,
x
2
oo
,2
;+1
oo
(
.2
E«* = E E (|) =E|r(2 -2 ) = E ^ fc=i
k=2l
;=i
v
'
i=o
;=o
Moreover, 2,+1-l
£
fc2<2-23(
fc=2'
for all Z, and therefore oo 2 , + 1 - l
oo
/.
\2
oo
V
'
1=0
,2
2i+1-l
E^l = E E * © = E|r E *2 fc=l
(=0 oo
2
fc=2' ,2
k=2l
oo
^ E | r 2 - 2 3 / = 2 E 2 ' &2 1=0
(=0
Thus by (8.1)
<^E^Efc2^
EM = 5 > W=0
/
\fc=l
/
k=\
k=l
^^EI^E^^^E^E2'^ (=0
(=0
1=0
where C-z = 2Ci, and this is (8.2). Suppose, conversely, that (8.2) holds. Put (*+>-!
N
ka
b,= \ fc=2< J2 l]
^
, / = o,i,...
1=0
Farther Connection
to Interpolation
Theory
We note that by the Schwarz inequality 2,+ 1 - l
2< + i _ l
X
a
«= X fci72fcl/2afc
fc=2< fc=2<
\
fc=2'
\ fc=21
/
because 2'+1-l fc=2'
for all/ > 0. Furthermore, since
X ^ = X ^ X fcafc (=0
i=0
fc=2<
/=0 fc=2! oo 2 , + 1 - l
oo
< 2 £ £ -1 = ^4 1=0 k=2'
fc=l
and similarly
X 2 bj! <X f c 2 l
•n2
*:=i
1=0
it follows from (8.2) that 1
VI
V
00
VI ;=o
z
1=0
00
00
<2C 2 ^ai^fcV f c , fc=i fc=i
which is (8.1) with Cx = 2C 2 .
142
8.2
Multiplicative
Inequalities
of Carlson Type and Interpolation
Carlson Type Inequalities as Sharpenings of Jensen's Inequality
Let ip : R+ —> R+ be any concave function. Thenthen t
V{t)<^
f-f1\t-t1)
+ v(t1)
(8.3)
whenever t < t\ or t > t2. Assume that there are t\ < t2 such that(p(t2), say tpfa) -
- h)
for some S > 0. Then by (8.3) we have tp(t) <-8(t - h) + tpfa) < 0 for t sufficiently large. This is a contradiction, and hence ip is nondecreasing. Moreover, the function t is non-increasing. Assume, namely, that there are t\ < t2 such that
say ti
f(ti)},
where 6 > 0. By (8.3)
^)<*(«;>-*fr>t12 — t l
'
if t is sufficiently small. This contradiction shows that t H-> tp(t)/t is nonincreasing. Define the function tp of two variables by lK',t) = ( : V ( ' ) , a , ' n 0 , n ^v ' ; \0, s = 0 o r i = 0.
(8-4) V
;
Further Connection
to Interpolation
Theory
143
Note that ip so defined is non-decreasing in each variable separately. This can be seen by an argument similar to those used above to show the m o n o tonicity statements about (p. With
/ * oo
bl
1=0
bf
21
2>
(=0
1=0
or \\{bi}i\\h < C'TP
r^u
W)}« h
We can consider this inequality for any concave function
W(2<)h lp> l 2 V(2')J
\\{bi}i\\h
(8.5)
We will return to this inequality shortly. Let V denote the class of concave functions
t_»o°
and
lim
t-»oo r
'
We also define the following subclasses of V. Define s„(t) = s u p ^ ^ , s >o v(s)
t>0.
• P+ is the set of concave functions
t—0+
• P - is the set of convave functions
£
• Vo is the set of concave functions ip satisfying lim
and
lim ^ - = 0. *—00
t
(8.6)
144
Multiplicative
Inequalities
of Carlson Type and
Interpolation
Example 8.1 Let w = -^—
1 + t•
^
Then, as the reader may check, we have sv(t) = max{l,i}. Thus this gives an example of a function which is in VQ but not in V±. Suppose thatR+ as defined by (8.4) is concave, positively homogeneous of degree 1 with ^(0,0) = 0. Then Jensen's inequality m
/ m
m
\
holds for all (finite) sequences xi,..., xm, yi,... ,ym of non-negative numbers (see A.-P. Calderon [19], p. 162). In particular, for j^
i
1p(u,v) = M f l ) ? ,
with 1 < p < oo and P
P
we have the classical Holder-Rogers inequality m t_1 fe=l
1
s m
\ - , m
\ •
v - 1 U fe=l
'/
/
v L_! V
fc=l
We want to prove an estimate stronger than (8.7), namely f > ( a * , y * ) < cJ\\{xk}ZLi\\lp,
||{Wfc}r=i|| A
(8-8)
with p, q > 1 and C independent of m. The inequality (8.8) is not true in general. In fact, i f l < p < q < oo and a > 0, put Xk = yk = a, k = 1 , . . . , m. Then (8.8) means mtp(a,a) <
<
Cip(amp,amv)
Cmpip(a,a).
Further Connection
to Interpolation
Theory
145
This shows that C must depend m. Also, if A > 0 and x^ = Xy/., k = 1 , . . . , m, then (8.8) means m
m
fc=l fc=l
C s m
i m
p
\ i- i m
\
-\
\ -
~
2fcgfc
Qfc k
y{2 Y
Vk
~
where f{2k) = xp(2k,l), the inequality (8.8) becomes
s«HiU}ii{&i}iJ-
(89)
-
The question here is: for which p, q and tp does there exist a constant C independent of m such that the inequality (8.9) holds? The answer was given by J. Gustavsson and J. Peetre [31] (see also [61], pp. 143-145). The inequality (8.9) holds in the following four cases: 1° p = q = 1 andl &ndip£V-. 3° p > 1, q = 1 and ip G V+. 4° p>l,q>l and
W}Remark 8.1 If we take ip(t) = min{l,t}, then the inequality (8.9) is not true.
146
Multiplicative
Proof of Theorem equality
Inequalities
fe=l
2
ak fc
2 >«
(
< ( £ ^(2*)'
x
2 f c >u
or
<4( /
Interpolation
8.1. For u > 0 we have by the Holder-Rogers in-
5Z °fc = y^, ak+YI fc
of Carlson Type and
^
y(2 fcfc\\ ) 9 2*
m
2fcafc )
m
V(2fe)/fc=1
^ ' y 2kak}m fc U(2 )/ f c = 1 f
il /•
\ m
f 2fcqfc l m W(2 fe )/ fe=1
where £)*, i = 0,1 are constants depending on
[2kakYn
W(2fe)Jfc=1
^(2 f c )J f c = 1
we obtain (8.9) with C depending on p, q and
•
Remark 8.2 If1, o > 1, then the constant C = C(6,p,q) increases to oo when 9 —> 0 + or 0 —» 1~. To see this, we note that the inequality in this case means l-6»
X> fe
\2kak\m
W(2fc)Jfe=1
If we take ak — 1, k = 1 , . . . , m, (8.10) becomes
(J2 ~ m
m
\
2 ekp
k=i
)
J
P
/ m
"\
lj22(1~8)kq
Vfc=i
t
(8.10)
Farther Connection
to Interpolation
Theory
147
or /
m
nOpm _ e m
e
y2 P (2 P
i
2(\-0)q
-1)
2(1 — S)qm _ ] \ « 2(1-6)9 - 1
When 6 —> 0 + , the last estimate means m < Cmp, which is impossible for large m. When 6 —> 1 _ , the same estimate means m < Cm«, which is again impossible. Thus the constant depends on 6; it has the order
This observation shows that [78], Lemma 4.6, for q = 2 with C = 1 is false. The author there referred to Gustavsson-Peetre [31], where they proved the estimate (8.9) with C depending on
The Peetre Interpolation Method and Interpolation of Orlicz Spaces
Let A = (A0,Ai) be a compatible couple of (quasi-)Banach spaces, and suppose that
(i) a = 2_] ui ( m ^(-<4)) and i=—00
(ii) for some constant C it holds that W(
£6 P(2')
and
E^ 2
(
W(
V(2')
for all /imte sequences {£(};gF with |^;| < 1. Here, F is some finite subset of Z. On (^4)^,, we have the semi(quasi-)norm \{A)V
infC,
where the infimum is taken over all admissible C in (ii).
148
Multiplicative
Inequalities
of Carlson Type and
Interpolation
Remark 8.3 The construction of the space {A)v is often referred to as the ± (interpolation) method. This method has the following interpolation property (cf. Maligranda [61], p. 133). Theorem 8.2 Let A = (Ao,Ai) and B = (B0,Bi) be any couples of (quasi-)Banach spaces, andB be any continuous linear mapping, meaning that T maps S(A) into E(-B) and T\A< : Ai —> Bi, i = 0,1 (recall that this statement means that the mapping is continuous). Then T : (A)v ^
(B)v,
and i m U ^ < B ) v <max j||T|Uo^Bo,||T||Ai^Bi }. In 1977, Gustavsson-Peetre [31] used their Theorem 8.1 as a step in the proof of the following embedding result, where the Peetre ± method is applied to couples of Orlicz spaces. Let (fi, /x) be a measure space, and let i> : [0, oo) —> [0, oo) be an Orlicz function, i.e. an increasing, continuous, convex function. Then the Orlicz space L* = L*(fi,/u) is the spaces of (equivalence classes of) functions / : Q -* R or C such that for some A > 0 (depending on / ) it holds that $(A|/(w)|)dMM
11/11. inf{A>0;/$(J^Wa,)
* -1 = *oV(fS)-
(8-11)
Then (L*°,L*1)V=^*
(8-12)
Further Connection
to Interpolation
Theory
149
with equivalent norms. Remark 8.4 Ifthen we can prove the embedding L*C(L*°,L*% and the norm of this embedding does not exceed 2 (cf. Maligranda [61], Lemma 14.4). The reverse embedding for ip g V± has norm depending on tp (cf. Maligranda [61], Lemma 14.6) and it increases to oo when we go with ip "to the boundaries" (cf. Remark 8.2 above). We are now ready to state the following result concerning interpolation of Orlicz spaces. Theorem 8.4 (Gustavsson-Peetre, 1977; Shestakov, 1981) Let $ 0 ,
$i, ^o and \fi be Orlicz functions. (a) If T : (L*"(,i),L*i(/i)) -> ( L * » , L * > ) ) and -iJ^\
,T,-i_lT,-i.y*r1
i /'
where
*-^ 0 - V (_L), *--.„-„(£) where
< Cmax | ||T|| L , 0 _ L . 0 , ||T||Loo^Loo } where the constant C depends on
150
Multiplicative
Inequalities
of Carlson Type and
Interpolation
For detailed proofs of the above results, as well as a more extensive discussion on interpolation of Orlicz spaces, see the exposition by L. Maligranda [61], Chapter 14. Remark 8.5 As will be seen in the next section, it is, in fact, necessary that the function
fc=i
f Ki } M2fe)J
E(lP)
2k\uk\] <^(2fc) J
E(lP)
where ip is denned as in (8.4) and the constant C > 0 is independent of m e N. 8.4
A Carlson Type Inequality with Blocks
N. Ya. Kruglyak, L. Maligranda and L.-E. Persson [47] used the so-called Brudnyi-Kruglyak construction, (cf. [18]) which, given a function
< CoV'
{ E 4
. { E »>} (a,k,bk)€S„
Further Connection to Interpolation Theory
151
Fig. 8.1
The main result from [47], and of this section, has two parts. The first part is the important observation that in order for the inequality (8.5) to hold, it is also necessary that the function1 be fixed. Then there exists a decomposition {xk} of E+ into closed intervals Xk with disjoint interiors, having the following properties, where we denote by i2fc the left endpoint of Xk-
152
Multiplicative
Inequalities
of Carlson Type and
Interpolation
(a) For each k, we can choose £2fc+i € Xfc such that ¥>(*2fc+l) hk+i (b) If t£xk,
=
1
then
rmm{l,t/t2k+i}tp(t2k+i)
and (c) For all k, >(*2fc+2) = r(p(t2k+i)Proof. For any s > 0, let x« be the closed subinterval of R+ consisting of those t for which ip(t) < r(see Figure 8.2). We consider now the intervals /Ck
X*2fc+i»
where the tk are chosen so that the right endpoint *2fc+2 of the interval \k coincides with the left endpoint of the interval Xfc+i- The properties follow from this construction. • Remark 8.7 The construction of the sequence in Lemma 8.1 is sometimes also called the Oskolkov-Janson construction, and its inductive definition was given by the formula
to=1 and min
l"^T'-i^rJ =r>L
K. I. Oskolkov used this construction in the paper [69] published in 1977, S. Janson [37] in 1981, and Brudnyi-Kruglyak [17] in 1981. The following two basic facts are of importance below. The first one gives an equivalent characterization of a function in the class Vo in terms of the Brudnyi-Kruglyak construction associated to it, and the second shows that the intervals Xfc are, in a sense, optimal.
Further Connection
Fig. 8.2
to Interpolation
Theory
153
The diagram shows how the interval Xs is defined.
Lemma 8.2 Let
sup k
„
< oo. t2k
Proof. We assume throughout that r > 1 is fixed. If1 such that for all s tp(su)
1
or v( 5M ) su
iy(«) r s
With s = t2ki this, together with the property (a) of Lemma 8.1 and the
154
Multiplicative
fact that f(t)/t
Inequalities
of Carlson Type and
Interpolation
is non-increasing, yields that t2k+l
tlk
In a similar manner, if ip € V+, then there is v satisfying 0 < v < 1 such that ^2fc+2
^
hk+i
v
Thus, for any
„ U
—
< - < oo,
t2k
V
so the only if part of the lemma is proved. Assume now that < M < oo t2k
for all fc. Let s € (0, oo) and pick fco such that ^2fc0 <
s
< ^2fc0+2-
Then *2A:0+4 <
Mt2k0+2
so by part (a) of Lemma 8.1, also using that the function
=
1 y(*2fc0+2) < 1^(5) r t2k0+2 ~ r s
By iterating this process, we find that for any m we have
r-mM2mip(s),
from which it follows that sv{t)
t
0,
t —> oo
(because of the continuity of cp), i.e.
Farther Connection
to Interpolation
Theory
155
Proposition 8.3 Suppose that { f i f e } ^ is any decomposition of the set { 2 m } m e Z . Then the inequality (8.13) holds for some C with the Xk replaced by fifc if and only if there is a constant M such that card({m; O m n Xk ¥= 0}) < M,
keZ.
As a final preparation for Theorem 8.5 below, we mention that the following extended version of Proposition 8.2 holds. Proposition 8.4 Suppose that 1 < p, q < oo. Let ip G Vo, and let {tm}m be its associated Brudnyi-Kruglyak sequence, with the corresponding intervals {Xm}- Then the following two statements are equivalent. (a) There is a constant C\ such that
2*e*„
(b) There is a constant C<2 such that
5>KA)*)eT,
a
4
{ H
6
4
)•
(ofc.bfc)GTm
As mentioned above, the main result of this section, from [47], is divided into two parts, the first of which tells us that we cannot get the inequality (8.5) unless
X>k
y^
2fcafc (8.13)
156
Multiplicative
Inequalities
of Carlson Type and
Interpolation
The best constant C in (8.13) satisfies C < (1 + V2)2. Proof of Theorem 8.5. (a) Suppose that (8.5) holds for some C. Note that this inequality can be written as the block inequality (8.13), but with the blocks Xk replaced by just single-point sets 0^ = {2k}. Thus, if (f £ V±, then Lemma 8.2 implies that for any m, we can find k such that ^2fc+2
2 m
tlk
But this means that at least m — 1 of the Clk are contained in Xm, which contradicts Proposition 8.3. Thus (p € V±. (b) Since ||-||j < ||-||; for any finite r and since tp is non-decreasing in each variable, it suffices to prove (8.13) with p = q = oo. We consider sup
£ »^2ky a-k
2kak
^
and put
-
-
!
•
Let mo be the index for which M G Xm0
an
d make the decomposition
J2a*= J2 Yl ak+ J2 a* + Y Y akm<mo2'=exm
2 fc exm 0
m>m0 2 fc e X m
By part (b) of Lemma 8.1, it holds that max(<2mo-3) <
^
Further Connection
to Interpolation
Theory
157
etc. We therefore have
E E * - E E ^j**) r t
< A Y^
f( 2m+l)
m<mo
r
=
Aip{M)^—=^{A,B)^—. r—1 r —1
+
^ rlz + ...
In a similar manner, we obtain
m>m02feexm
As for the middle term, suppose first that
E °^ E 2'eXmo
2*€Xm 0
^,„..^ max -^ V(2 ) 2*e „
f e
fc
)
x
^ E ^ V(2 T ^fc )'^ m o + l) 2*
=rip(A,B).
If, on the other hand, tfi(M) < / i N max .
24r
24r
^ 2 f c ) 2 f c ^ - 2*
y^
2kak r
<
- ^
By combining these inequalities, we obtain
r —1
158
Multiplicative
Inequalities
of Carlson Type and
Interpolation
The infimum over r > 1 is attained at r = 1 + y/2 and we therefore get (8.13) with C = (1 + V2)2. The proof is complete. • As a final remark, concluding this chapter, we note that the following extended version of Proposition 8.1 holds. It is suggested by Levin's theorem and the function (p(t) = te, where 9*
p/j, + q\ Proposition 8.5 The following two statements are equivalent. (A) There is a constant C\ such that for all sequences {ak}k*Li or" negative numbers, the inequality pfi+q\
fc=l
;=o
2
a=o
non
~
pti+qX
I OO
holds. (B) There is a constant C?, such that for all sequences {&z}j^0 °f negative numbers, the inequality
E^c
"
\fe=l
\k=l
OO
non
E^
\
PH + q\
/
OO
\
p^+g-X
£2^6? w=o
or
•<*>••• H l { ^ } J J { $ r } , _jA
holds, where
T h e Calderon-Lozanovskii Lattices
Construction on Banach
The Carlson inequality with blocks in the previous section gives a comparison of the Peetre method with the Calderon-Lozanovskii construction on Banach lattices, to be described below, and the embedding constants are independent of the function tp in the construction. This gives the interpolation result with a universal constant (independent of
Further Connection
to Interpolation
Theory
159
of Orlicz spaces is easy to describe, we therefore also get the interpolation result for Orlicz spaces with a universal interpolation constant. Let X = (Xo, X\) be a Banach couple of lattices, i.e. a Banach couple of functional lattices defined on the same measure space (fi, fi). For0 and t > 0, and ip(s, t) = 0 if s = 0 or t = 0. We consider the Calderon-Lozanovskii space