This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
0/(s) = i f (  s ) ,
0},
s>0,
s< 0.
Recall Young's inequality ts<$(t)
+ V{s),
t,s>0.
The classical concept is that of the Orlicz spaces: A function
= inf{ A > 0 ; /
$(f(x)/\)dx
and this normed linear space is called the Orlicz space. The Luxemburg norm is equivalent to the Orlicz norm sup < / f(x)g(x)
dx ; / V(g(x))
dx<\
R"
KR»
where & is the complementary (Young) function with respect to 4>. 1
2
Chapter 1. Integral operators A more general class can be introduced:
Definition 1.1.2. L e t # G $ , 0 < A < n . Then we define the class $\(L) = { / measurable ; \f\#t\ =
SU
P r" r>0n 
z£R
1
J/
OrliczMoirey
$(f(y))dy < oo}.
B(z,r)
Clearly, $0(L) = $(L). If only 0 G $, we can define the Orlicz space L$ as the linear hull of <£(£) with a locally convex topology of the quasinorm inf{A>0;y#(/(y)/A)(fy
Definition 1.1.3. A locally integrable a.e. positive function g : Rn —> [0,oo) is said to be a weight function. Starting with Chapter 2 we will work with weighted Orlicz classes and weighted Orlicz spaces: Let Q be a weight on Rn and 0 G $■ Then the weighted Orlicz class is the set of all functions / for which / $(f(x))e(x)
dx < oo
and the weighted Orlicz space is its linear hull. We will adopt the notation L$(Q) for the latter set. If # is a Young function, one can define the weighted Luxemburg norm and the weighted Orlicz norm in analogy with the above nonweighted norms. The following concept is one of the most frequently occuring in the theory of the Orlicz spaces and classes. Definition 1.1.4. A function 4> G $ satisfies the (global) A2 condition if there exists c > 0 such that $(2t) < c$(t), t > 0. If
1.1. Preliminaries 3 Definition 1.1.5. Let $ € $ satisfy the A 2 condition. Put ft#(A) = sup  ^ ,
A> 0 ,
and define the iower index of$ by
,(#) = hm l^iM^i = A0+ and the upper index of$ by m
=
log A
i im iSlM^i A — oo
log A
=
sup l^gMA) log A 0
l°gA#(A)
inf l
log A
The existence of the above limits follows from the theory of submultiplicative functions and the details can be found in the papers by Matuszewska and Orlicz [1], Gustavson and Peetre [1], Boyd [1], Butzer and Feher [1], Maligranda [1], and the treatment on the submultiplicative functions (the subadditive ver sion) in the monograph by Hille and Phillips [1]. The indices enable among others to estimate the growth of a function 4> £ $,<£ G A2, by means of power functions, namely, given e > 0, there is a constant Cc > 0 such that 0(Xt) < Ce max (x*W— , A 7 W + £ ) *(t),
A, t > 0 ,
and 0(\t)
> Cc min ( A ' ( * )  £ , A 7 (*) +f ) #(t),
A,t > 0 .
It is clear that # € A2 if and only if I($) < 00. As a matter of fact, there will be no use of special properties of Orlicz spaces or classes throughout the book, however, the following references might be useful: The classical theory of the Orlicz spaces is the subject of the monograph by Krasnosel'skii and Rutitskii [1], Zygmund [1], Zaanen [1], and for the more general concept (modular spaces) we refer to Luxemburg [1], Musielak [1]. The contemporary theory of Orlicz spaces is surveyed in Rao and Ren [1]. Next we are going to introduce and investigate the special class of the quasiconvex functions. Definition 1.1.6. A function $ : [0,oo) —► R1 is said to be quasiconvex on [0,00) if there exist a convex function w and a constant c > 0 such that (1.1.1)
u(t)<$(t)<cu(ct),
t€[0,oo).
4
Chapter 1. Integral operators
L E M M A 1.1.1. Let $ G $. Then the following statements are equivalent: (i) 4> is quasiconvex on [0, oo); (ii) the inequality 0(txi + (1  t)x2) < CI(<
t)$(ax2))
holds for all xltx2 G [0, oo) and all t G (0,1) with a constant cx independent ofxi, x2, t; (iii) for all f G Lloc(Rn) and all bounded intervals I,
dx) <
±j$(cf(x))dx
Ii
Ii
with a constant c independent of f; (iv) there exists a > 1 such that $(a:i) xx
_
a$(az 2 ) z2
for all 0 < Zi < x2. Proof: The implications (i)=>(ii), (i)=>(iii) follow easily from (1.1.1) and the corresponding properties of convex functions. We prove (i)«=>(iv). Let
and a constant Cj > 0 exist such that u(t) < 0(t) < ciw(cit),
t G Ri.
As the function 1i—► ui(t)/t is nondecreasing, we have
^i)
0
'2
Thus ^(
<
ClU)(ci
CiU)(cit 2 )
_ tl — tx and (i)=^(iv) is proved. Conversely, let
*(*i) . r *(£i*a) —:—< c i —  — , *1
'2
t2
ci^(cit2)
<2
n
^ , „ , Q
1.2. Maximal functions in $ (L) classes
5
Consider the function Jt // c ,
u(t) = 1 [ sup ^ds. sup *(r)
Then w is convex on [0, oo) and u(t) <  j C
sup  ^ < #(*),
l 0
^
2<
2t
2ciw(2ci<) = 2 / sup ^ 
>
T
J 0
T
=*(*).
Whence (iv)=>(i). We prove the implication (iii)=>(ii). Let J be an arbitrary interval of the unit length. For a given tt 0 < t < 1 let us write J = I\ U h where 7i = t, 1^21 = 1 — t and set in (iii)
■{
/(«) =
*1 «2
if u Gil, if u eh
Clearly, we get (ii). Now we prove (ii)=>(iv). Obviously, in virtue of (ii) we have for t\ and t2 with 0 < ti < t2,
*(,,) = * (. „) = * (1. „ + (,  ji) o) < which is (iv). Q
1.2 Maximal functions in &(L) classes Let / : Rn —* iZ1 be a locally integrable function and (1.2.1)
Mf(x,t) = suVj^J\f(y)\dy,
(x, t) € Rn x i?,
6
Chapter 1. Integral operators
be the generalized maximal function, the supremum being taken over all balls B C Rn with x e B and r a d £ > 2 _ 1 i . For t  0 we get the classical HardyLittlewoodWiener maximal function Mf(x). We will frequently work with Mf(x) defined with cubes replacing balls, clearly, M/(x)~sup—r / Q9* \Q\ J
\f(y)\dy.
Q
As well known, M is bounded in Lp for 1 < p < oo. We are going to give a characterization of those functions 4> 6 $ for which (1.2.2)
f$(Mf(x))dx
[$(cf(x))dx,
feLloc,
ii"
with a constant c independent of / . The inequality (1.2.2) will sometimes be referred to as a modular inequality (the nonweighted case). The symbol /? will stand for a positive measure on R++1 = Rn x R+ , B(x,r) will be the ball centered at x € Rn and with radius r, B(x,r) will be the cylinder B(x,r) x [0,2r). First, we give equivalent conditions for (1.2.2). T H E O R E M 1.2.1. I e t # 6 $ . Then the following statements are equivalent: (i) there exists a positive constant c\ such that the inequality (1.2.2) with Ci replacing c is true for all f € L\oc; (ii) the function <Pa is quasiconvex for some a € (0,1); (iii) there exists a positive constant c^ such that
(1.2.3)
jmdt<S^Lt
0
<.
o
(iv) there exists a positive constant C3 such that for t > 0
«..«,
/
)
*
<
c 3
1.2. Maximal functions in $(L)
classes 7
(v) there exists a constant a > 1 such that (12.5)
4>(t) < i  * ( a * ) , 2a
*>0.
We split the proof into several lemmas. The key to the theorem is the following lemma. L E M M A 1.2.2. Let (1.2.5) hold. Then constants a 6 (0,1) and ax > 1 exist such that (1.2.6)
*«(*)< J  * " ^ * ) ,
t€(0,oo).
Proof: For any a € (0,1), o v ; < 7^V* v *"(0 ~ (2a) a (a0 ' •
3a If loS2<. y <
a
< 1.
then
2a 0a(t) < T6a* {at). Consequently, . #"(0
which is nothing but (1.2.6) with ai = a 2 .
2
□
L E M M A 1.2.3. The condition (1.2.5) implies the quasiconvexity of the func tion
£ M < £i*fcl*2) f tl
12
whenever
0 < t, < t 2 .
8
Chapter 1. Integra] operators
Let 0 < ti < <2 a n d ti < at\. As $ is increasing in [0, oo) it is 0(t a )
>
4>(h)
therefore * ( ' l ) ^ a#(<2) ^ a$(a<2) <1 *1
_
"
*2 *s
_
"
<2 *2
Now let 0 < h < <2, h > ati. Then ([•] denotes the integral part) «P(ta) = $ ^2 . A
=
<£ ( a «og.« a /tx tl ) > * (aDog.(«a/«i)]tl)
> (2o)~ 1+log  ft « / * i) #(*i) > 2  1 + l o g  ( t 2 / ' l ) a  1 + l o g ° ( t 2 / < l ) # ( * i ) >
<2
.
—a
■^(ti),
h whence tfjfr) <1
—
aff(<2)
a<^(a<2)
<2
*2
We conclude that (1.2.7) holds whenever 0 < ti < t2.
D
Definition 1.2.1. Let # £ $ and T : L$ —* £$ a sublinear operator. We will say that the operator T is of weak type {4>, $) if a constant c > 0 exists so that #(A){ i G f l " ; T/(x) > A} < c / *(c/(x)) dx
for all / G 0(L) and all A > 0. The above definition is a natural generalization of the concept of the weak type (p,p), 1 < p < oo; if T : Lp —* Lp, 1 < p < oo, then T is said to be of weak type (p,p) if there is a c > 0 such that
{ x e Rn ; \Tf(x)\ > A } < cA""  /(*)*
Due to the nonhomogeneity of 5? G $ another variant of the weak type (#,
1.2. Maximal functions in $(L) classes 9 in details in Chapter 2 in a more general setting (weighted inequalities; see Section 2.2). The next lemma describes those functions from $ for which the inequality of weak type (<£, 0 exists such that the inequality (1.2.8)
*(A) \{xERn;Mf(x)
> A } < c /$(c/(ar))dx
is valid for all / 6 Ljoc and all A > 0 if and only if the function # is quasiconvex. Proof: If
< cu{cMf{x))
< cM(u(cf(x)))
<
cM($(cf(x))).
The function w is increasing, therefore *(A) { x e Rn ; Mf{x)
> A } = *(A) { x € Rn ; $(Mf(x))
< ^ % *
>
e i T ; M ( * ( c / ( i ) ) ) > #(A)/c}
c
< c / #(c/(z))efo;
the last inequality follows from the weak type (1,1) of the operator M (the celebrated fact due to Wiener [1]). The sufficiency is proved. Let (1.2.8) be true, we show that 4> is quasiconvex. Let 0 < t\ < t2,
I = { x = ( n , . . . , *„) e Rn; 0 < xt < {h/t2)1/n ,
i = l,... ,n}
and put / ( x ) = t 2 x / ( x ) . Easily, for any x £ (0,1)", we have Mf(x)>t2\I\=h
.
Thus {I€iJn;M/(s;)>ii}>l
10
Chapter 1. Integral operators
and (1.2.8) implies that 4>(h)
$(ct2),
I
i.e. 0 is quasiconvex according to Lemma 1.1.1.
□
P r o o f of T h e o r e m 1.2.1: We prove (i)=^(iii)=^(v)=Kii)=Ki) and (iii)«(iv). Observe that (v)=^(ii) follows from Lemmas 1.2.2 and 1.2.3 Now, it is easy to see that (ii)=>(i). Indeed, let a G (0,1) be such that 3>a is quasiconvex; making use of the Jensen inequality, we get I$(Mf(x))dx
j(^a{Mf{x)))lladx
= fl»
R"
f{M{<Pa(cf{x)))fladx
dx
R"
= c /
We show (i)=j>(iii)=>(iv). For x G Rn, r > 0, let B(x,r) denote the ball {yERn; \xy\
* \B(x\\x\)\
j
XB(o,i)(y)tdy > ^ J L 
B(x,2\x\)
which yields
J x>l
 ) dx < c\$(cit).
*V2"x\nJ
~
Whence 2""<
ci«>
$(s) ,
0
The condition (1.2.3) with the constant c2 = 2"c follows now from (1.2.9).
1.2. Maximal functions in <&(L) classes
11
Let us prove (iii)=>(v). First we show that assuming (iii), there is a real number b > 1 such that *(«i) b4>(bs2) < , si s2
0 < si < s2.
Indeed, we have 2JI
2S 2
*A<2[mdt<2rmdt<^(^).
*(«l)
l
J
t1

J
t2

s2
It suffices to put b = 2c2. Let d be some constant greater than 2 n (whose value will be determined later). The estimate (1.2.9) implies
rmdsssm,
J1
Id
s
t
therefore i*(^)log^<6c*(ct). If we put 2(
{be? (log 2" J
<
1 2'
then
*(r)<2^*(tedr) which is (1.2.5) with a = 6cd. Now we prove (iv)=>(ii). Lemma 1.2.2 guarantees existence of a £ (0,1) and ai > 1 such that *a(r)<^
r>0.
12
Chapter 1. Integral operators
Invoking Lemma 1.2.3, we have that $a is quasiconvex. It remains to prove that (ih)'Q'(iv). We start with (iii)=^(iv). Integrating by parts we get } d$(s)
$(t)
+
} *(«)
# ( t ) c2$(c2t)
ds
c3$(c3t)
+
J ~r  ~ r J — ^ — ~T~ * —1~ ■ 0
0
As to (iv)=J»(iii) we first show that under the assumption (iv) the function ${t)/t is quasiincreasing, i.e. if 0 < Si < s2, then $(si) c3&(c3s2) Si
s2

for some c3 independent of Si, s2. Indeed, we have
*M
1?
=
si
si J o
< ?rf^) < 7d*(u)  J u  j u o o
c3$(c3S2) s2
Whence / d#(«) . c2$(c2t) /*!*(«) W2 d, s^<#(cs0 ! M +, raw cMf
J0
s
t
J
s

t
The theorem is proved. □ Let us point out one particular consequence of Theorem 1.2.1. COROLLARY 1.2.5. Let $ € $ and assume t/iat there is a constant c > 0 such that (12.2) holds for all f G L\oc. Then there are constants /? G (0,1) and ci > 0 such that (1.2.10)
j &>(Mf(x)) dx
J &>(cxf{x)) dx,
f G L\QC.
R"
Indeed, according to Theorem 1.2.1, $ a is quasiconvex for some a G (0,1). Whence the function ($) 0 '/' 3 is quasiconvex for all j3 G (a, 1) and applying Theorem 1.2.1 we get (1.2.10). T H E O R E M 1.2.6. Let 0 G $• A positive constant c exists such that (1.2.11)
n n sup$(\)\{x€R ;cMf(x)> A }  < o o , swp
A>0
if and only if the function 4> is quasiconvex.
/ G #(L),
1.2. Maximal functions in$(L)
classes 13
Proof: Let (1.2.11) hold and suppose that, at the same time, <2> is not quasiconvex. Then according to Lemma 1.1.1 there exist positive t'k,t'k\ t'k < t'k' such that
W) . **mi ,k «W) m>i
(i.2.i2) Let
OO
ifc'X/t(z) .(*) =£4*<' E 4t'lxi,
/(*) == /(*)
fc=i fc=l
where Ik = {x Jk {* = ((Xl*,..,, ! , . „ , «xn»);) ; adfck < Xi < ak+1, i = <*i'<«*+l» = 1,2,..., l » 2 , . . . ,n» }] , ak ai
fci e—l
= (2^(4^V))"1/" . = ^£(2>*(4**y))i/"
Then / G #(X) for .
OO
/U(f(x))dx $(f(x))dx = J>(4V ')K +1+i aak k) )nn J2^kfct'k)(^ Rn
k=l OO
=£ fc=l k=l
,
9*
< OO.
Choose an integer k such that c4fc > 1. Then
*(*i)K*
n #(t' t ){ x €efl"fl" ; cW/(*) ; c4 fc 4'M XX/ kk(x) > «i eM/(i) > :> t'«ik } > #(*i)[{ #(«»)[{ x* e R i?"; t'k } }
Ux = (x1,x2,,..,xn)iaia
> ##(4){x ( 4 )  { x eG R" i?n ;; t'lMxM HlMxiM
>t'k}\. >t'k)\
the cube
n J/*k = {xeR ;ak<Xi
then Ik C Jk and
Mx/fc(x)>
_*1
l*T
l
k
, ii = l l, ,22,,.. ••,«}, ..,n},
14
Chapter 1. Integral operators
Our assumption (1.2.12) yields
*(Wi k W)\{ *ZRn> cMf(x) > t'k } > p ^ i ^> 2 for large k and this is a contradiction. The converse implication is a part of Lemma 1.2.4.
□
THEOREM 1.2.7. Let 0 £ $ . Then there exists a positive constant c such that cMf e $(L) for all f £ $(L) if and only if 1
(1.2.13)
y
* £ > < «**(**> , S2
o
t>0,
£
for some constant c\ independent of t. Proof: Suppose
(1.2.14)
J
d${s) s
ds >
2*#(2*
o Let
CO
f{x):
"tkXld*) fc=i
where J* = {* = (*l,.,*n) £ R"; afc < z, < a f c + 1 } , kl
ak = £(2^(2^))" i/„ Clearly / £ # ( i ) , as DO
/
&{f(x))dx
< £>(2*<*)(a f c + 1  a * ) "
_ f> «(2*«t) A 1 Z . 2*#(2*
<
1.3. Vectorvalued maximal Junctions 15 Further, oo
= [\{xeRn;
J
n
M(cf)(x)
> A }<#(A).
0
Using the converse inequality for the maximal function (see, e.g. de Guzman [1]), we get OO
J \{x € Rn ; M(cf)(x) > \}\d$(\) o
><]( 0
J
W(*)>M
oo
■.
_ y> 2 kt k £r[2*
lrtM*)«2 = ./l/Wl( '/ /
c2t<
o
)
c *"
rfg(A) dt A #(2*tfc)
J o
\
dx
2
*
/•
fl" *"
A
f diP(A) 7 A ■ o
For & such that c2k > 1, we have by (1.2.14) f
$(cMf(x))dx>2k,
R«
which contradicts our assumption. The converse follows from Theorem 1.2.1. Q Notice that if # is a Young function, then conditions (ii), (hi), (iv) of The orem 1.2.1 are equivalent to the A2 condition for the function complementary to#.
1.3 Vector—valued m a x i m a l functions in $(L)
classes
In this paragraph we will point out the substantial difference between be haviour of the scalar and the vectorvalued maximal function. It turns out that the set of # G $ for which the scalar maximal function is of weak type ( # , # )
16
Chapter 1. Integral operators
contains as a proper subset those
1 and b > 1 exist such that (1 .3.2)
^ ) < ^ i )
for0
Proof: Assume that 0 <
g(2fi)
gfti)
i2
ij
ij
with the constant c from the A 2 condition for 0. With no loss of generality we will assume that c > 2. If 0 < 2ti, then ([•] denotes the integral part) = 0 (2Io"»(*a^**>*i) < # (2 [logs (W<0]+if ^ < c [log 2 (t 2 /< ^+10(h)
<2>(t2) = 0
<
feti}
c log 2 (<s/<
•) +1 <£(*l)
or
0(h).
To get (1.3.2) it suffices to put p = log2 c. Q Proof of Theorem 1.3.1: Let us prove first (ii)=>(i). The quasiconvexity of # follows from the corresponding result for the scalar case. We show that 0 satisfies the A 2 condition. Set /*(*) = A X / k (x),
k=l,2,...,
where h = {xeRn;2k2<xi<2k1,
i = l,2,...n],
k =
l,....
18 Chapter 1. Integral operators It is
1/9
ll/(*)ll.= (£l/*(*)l')
=AXUk/k(x),
xeiT,
so that the right side of the inequality in (ii) is estimated by c#(A). Let now 1 < k < m < j where m will be fixed later and denote Ijk = {x£Rn;2j2
<Xi <2k~1
i=
l,...,n}.
If x G Ij, then it is easy to check that
Mfk(x) >
^
and therefore ,m
9
\i/»
M/(x)<>(£M/t(*) J v
t=i
A
1/0
>~,
whence, in accordance with the hypothesis,
m IM 'Am 1 /"
< c*(A).
j>m
Choosing m in such a manner that m1/e2 0.
" > 2, we get the A2 condition for
We prove (i)=^(h). Let /(x)» G <*>(£), x G Rn. Fix A > 0 and define for i = l,2,... , A(«) i f   / ( x )   f l > A , */*(*) 0 if/(x)fl
»f(x) = tfkix) / H ; \0
lf
H^)ll«< *>
if/(x)fl>A.
Let A/(X) = ( ^ ( x ) , . . . , ^ ( x ) , . . . ) , */(*) = ( A /i(x), A / 2 ( x ) , . . . , ) . It iis
Mfk(x)<Mxfk(z)
+ Mxfk(x),
xGiT,
1.3. Vectorvalued maximal functions 19 by Minkowski's inequality we have M/(x) 9 <M A /(x) 9 + M A /W S , thus $(\)\{xeR";\\Mf(x)\\e>\}\ <$(\)\{xeR«;\\M>f(x)\\e>\/2}\ r$(\)\{xeRn;\\Mxf(x)\\e>\/2}\. Theorem A gives #(A){x e Rn ; \\Mxf(x)\\, > A }  < £ ^ 1 1   V ( * ) M * R"
<*W A
>
nn,,,dx
{t/MIU>M
<«i/ll/(*)ll
*HI/(*)ll«) dx ll/(«)ll«
 "AT"
y
ll/(x)le dx c
{/(*)IL<M
which concludes the proof. D
 x/(x)? dx
 J ll/(x)lis fl"
H/(*);
dx
20
Chapter 1. Integral operators
Notice that (i)=>(ii) in the above proof can be obtained for any subadditive operator of types (1,1) and (p,p) for sufficiently large p. THEOREM 1.3.3. Let 0 6 $, 1 < 6 < oo. Then the following are equivalent:
statements
(i) $ £ A2 and $ " is quasiconvex for some a € (0,1); (ii) a constant c exists such that J*(\\Mf(x)\\e)dx R"
forallf
=
<
cj$(\\f(x)\\s)dx R"
(f1,f2,...)eLloc.
Proof: First we show (ii)=>(i). The inequality in (ii) implies the weak type ( # , # ) of M, i.e. 0(X)\{ x e Rn ; \\Mf(x)\\e
> A } < c J $(\\f(x)\\e)
dx .
R"
But then (by Theorem 1.3.1)
0 such that (1.3.4) 2t e e > k)\Ikm\ fclB  ${2h)
A/2} < ^ 1
0 and all 0 < /i < 1. Inserting r = #(<)> this becomes fi 0 and all r > 0. Now for \i and i/ linked together by \xav = 1, i.e. /j = j /  1 / a , we get 1 Q 1 ( P  ^ I / T ) < i/ / # (r) for all r > 0 and i/ > 1. But the infimum of such 1/a is the upper index of # . The next step will be getting an estimate of # _ 1 (Atf)/#  1 (t) in terms of J ( $  1 ) . The simple estimate from Section 1.1 (following Definition 1.1.4) is not satisfactory as I ( #  1 ) controls there the behaviour of $  1 near oo. Never theless, setting gi{n) = s u p ^  1 ^ ) / ^  1 ^ ) , and g2{(j) = sup Q1^)/®'1 (fit), <>o
rip(r) for all r > 0, therefore \E°\ *(2*v{Q)/\Q\) 0 " a 0, conse quently,
*(2tfc)>4**(<0.
fc
= 1,2,... .
a, = J ^ S f e ) )  1 ' " ■
* = 1,2,....
Let fci
Let m be a positive integer whose value will be determined later. Define for keN, hm = {x = (xi,...
,xn) e Rn ; ak < x,
,
i=
l,...,n]
and for km < j < (k + l)m, 7 i = {x = (*!,., . , * „ ) € Rn ; a* +
"fc+i  gfc ajb+i  aj; *I ■ * < x{ < aKk+ r
2m(jkm)
'
2m(jkm)l
i=
l,...,n}.
1.3. Vectorvalued maximal functions 23 Let / = ( / ! , / » , . . . ) where Ij(x)
= tkXij(x)
for mk<j<m(k
+ l),
k=
l,2,....
It is t+ l ) m  1
/*(ii/wii^=f;W'1fVii E j= km = fcm
fc=l
fin
oo oo
<2:^)(a S + 1 a,r <E*co(*+io^ *(**) = 1^ 2k$(t Jfc=l
*■ *'
oo
.
^
1 = y j)=^ < oo. 2^2T<°°' k fcssl = l
For x G Jfcm we have * m < j < (Jfe + l)m. km<j<(k
Mf ^ /j(x)>£, i(*)>^. Whence
1
=(EWiW)')
* ( 22*i 0, )I{{*« € e i/J?"";; c ^ M / ^ x ) ) " )
,m(fc 1 ( ,+* +l )! ) l
> **((22***t )  {{**€€/i22""; ;ccf (
53
>2f, >2*.}l k} . l/»
(Mfjix^y (Jlf//(*))')
>2t > 2 *k}\t } 
j—mk
>$(2tk)\{xelkm;^m1t$>2tk}\. If m is such that cm1/*/2n
> 2, then the last estimate and (1.3.4) imply
0{2tk)\{k)\{x xeR€n; Rn ; c\\Mf{x)\\ 2tkk}\>$(2t } > #(2i»)/
We arrive at >2t oo lim ^(2ijt){x *(2t fc )l{* € Rn j; cM/(z) 2tkk}\} = = oo efl" s > 4Mf(x)\\„
fc»oo fc»oo
24
Chapter 1. Integral operators
which contradicts with the assumption. The quasiconvexity of the function # follows from the scalar case (Theorem 1.2.6). The converse implication follows from Theorem 1.3.1.
□
T H E O R E M 1.3.5. Let <2> G $ and 1 < 6 < oo. Then /(.)» >> Af/(.) tf is a mapping from $(L) into ${L) if and only if& satisfies the A 2 condition and $a is quasiconvex for some a G (0,1). Proof: The condition of the theorem is sufficient according to Theorem 1.3.3. Conversely, if ]/(*)ll* G ${L) and, consequently, Af/(a;)0 G $(L), then it is sup#(A){ x £Rn; \\Mf(x)\\» > A } < 00 and, therefore, 0 G A 2 by The\>o orem 1.3.4. To finish the proof, apply Theorems 1.2.7 and 1.2.1 to sequences of the f o r m / = ( / ! , 0 , 0 , . . . ) . □
1.4 Riesz transforms in $(L)
classes
Here we will study the behaviour of the Riesz transforms RjfW
= Cn J i / i y\r!+1 f(y) dy ,
j = l,...,7l,
where X(X1
X)
c
 r ( ( " + 1 )/ 2 )
Let 0 e (L) denote the set of those functions / G $(L) for which i?j exists, 3 — 1 , . . . , n. An analogous notation (the letter "e" added) will be used in the sequel for various singular integrals currently dealt with. We will not discuss conditions for the convergence of singular integrals considered and refer to Stein [4], Sadosky [1], Zhizhiashvili [1]. THEOREM 1.4.1. are equivalent:
Let 0 G $, 1 < 9 < 00. Tiien the following
(i) 4> is quasiconvex and satisfies the A 2 condition;
statements
1.4 Riesz transforms in&(L)
classes
25
(ii) a constant c > 0 exists such that (1.4.1)
*W\{zeRn;\Rif(x)\>\}\
cf$(f(x))dx,
j=l,...,n,
is true for aii A > 0 and aJi / G $ e (L). To prove Theorem 1.4.1 we need the following L E M M A 1.4.2. Let 1 < p < oo. Tnen there is a constant ci such that (1.4.2)
! { * € # * ;  * , / ( ; c )  > A }  < c x A  ' f \f(x)\" dx,
j =
l,...,n,
for all A > 0 and a77 / 6 LP(Rn). The proof of the lemma can be found e.g. in GarciaCuerva and J. Rubio de Francia [1, Chapt. II, Thm. 5.7]. P r o o f of T h e o r e m 1.4.1: Assume that (i) is true and let / G $(L). A > 0 and define K
"
r/w > l O
Fix
if/wiA,
and
xf(
)=
HX)
We have \Rjf(x)\
//W \0
< \Rj xf(x)\+
{x G fln ; \Rjf(x)\
^ 1/(^)1 > A, if/(*)
\R? */(*), whence
> A} <  { z G i?" ; \Rj >J(x)\ > A/2} \{xeRn;\Rjxf(x)\8>\/2}\,
+ and
*(A){* G i i " ; /2//(x)l > A} < *(A)j{i G JZ" ; \Rj yf(t)\ > A/2} + #(A){*Gfl";lV/(*)l>A/2}. Recall Lemma 1.1.1: There exists a constant a > 1 such that ** (( ** li ))
*i *1
a ^ ( ^22)) ^ a#(at ~
<2 *2
26
Chapter 1. Integral operators
This together with Lemma 1.5.2 (see the next section) and the A2 condition yields
J\xf(x)\dx ft
c*(A) = ^ P A
a x / \f{x)\dx< dx Cl< c\ / $(J${af{x))dx f( )) dx l/(*)l / R" {/(r)>X} fi" (l/(*)l>M
< c2 I $(f(x))dx
.
R"
On the other hand, as <£ € A2, then in virtue of Lemma 1.3.2 there is b > 0 and p > 1 such that ■P
< ^p—>
0 < t i < *2
Employing Lemma 1.5.2, we get * ( » ) « > £ K" ; flf/WI > A} < ^ f f l y c*(A) A?
/
I/OOI' dx
lA/(x)'dx
cj\f{x)f
<
{/(*)
fl»
l/(*)lp
di
< c I
We prove (ii)=^(i). First we show that, assuming (ii), $ is quasiconvex. Let 0 < ti < t2 and I = {x = ( « ! , . . . , *„) 6 Rn ; 0 < x{ < {h/h)1'", fx(x) dt2Xi(x)
i=
l,...,n},
,
where d is a positive constant. If x = (xi,...
ie;/W ='Cndt"
,xn) £ (2, 3)", then
I \x f{y)dy> ■j/
n + 1
cnd 3n
+ l n (n(l)/2
'2
1.4 Riesz transforms in0(L)
classes 27
Putting d = 3 n + 1 n ( n + 1 ) / 2 c  1 , we get \{xeRn;\Rjf(x)\>t1}\>l. Consequently, * ( ' i ) < e [${f(x))dx J
= ^
< max(c,d) 
I
The function 4> is quasiconvex. Finally, we show that 0 satisfies the A2 condition. Let g(x) = x € Rn, where I = (0,1)". Inserting g into (1.4.1), we obtain *(A){ x£Rn; \RjXi(x)\ > 2 } < e*(A/2) .
(X/2)xi(x),
Note that { x € Rn ; i?jX/( a; )l > 2 } is positive and independent of A, whence the last inequality turns into the A2 condition for 4>. This completes the proof. □ T H E O R E M 1.4.3. Let4> € $ . Then the following statements are equivalent: (i) 0 satisfies the A2 condition and $a is quasiconvex for some a 6 (0,1); (ii) a constant c > 0 exists such that (1.4.3)
j4>(Rjf(x))dx
J$(f(x))dx,
j =
l,...,n,
$e(L).
for allfe
Proof: First (i)=J>(ii). Let / = g + h where /(*), if/(x)>A, 9{x) = {0, if 1/(^)1 < A. It is \Rjf(x)\ < \Rjg(x)\ + \Rjh(x)\, therefore j0(RjHx))dx R
n
= J\{x
e Rn ; \Rjf(x)\
> A}d#(A)
0 00
< J \{x £ Rn ; \Rj9(x)\>
\/2}\d
0 00
+ J\{xeRn; 0
= h + h
\Rjh(x)\ > A/2}Id*(A)
28
Chapter 1. Integral operators
As the operators Rj are of weak type (1,1) we have CO
/
c
^< j\\
x dx
J
\\
\tt )\
0 " \{i{ y /(r)>A}
1 l/(*)l
c*Z>(A) j
\ da;
T h e function # a is quasiconvex for some a € (0,1) which was shown in Section 1.2 to be equivalent to
[ d$(u) djHu)
<
ci$(cit)
t>0
.
Therefore
h
l/(*)l
dx = ci
/ * ( / ( * ) ) dx .
As to 72, let us recall Lemma 1.3.2 which guarantees the existence of p > 1, b > 1 such t h a t *(
fP
2
Let pi > p. T h e operators i?j are of type ( p i . p i ) and we can write \
p
h
J \f{X)r \{/(*)I
= c3J\f(x)r
J
d${X) /
d$(\) AP»
dx
Going along the lines of the proof of Theorem 1.3.3 we arrive at h < C3 J*(\f(x)\)dx.
T h e estimates for I\ and 7 2 give (i)^(ii).
1.5. Vectorvalued Riesz transforms 29 If (ii) is valid, then fy are of weak type ( # , # ) and by Theorem 1.4.1 the function $ is quasiconvex and $ £ A 2 . Now we prove that there is a G (0,1) such that 9a is quasiconvex. Set / ( * ) = *XB(o,i)(«) where 5(0,1) is the unit ball centered at the origin. Then
/ #(pwi)<^ J_1
x>2 x /H'
Let h, k = 1,...,m,
be those indeces for which \xh  > 2; clearly \Rj, f(x)
=
J \x 
B
y\n+l
On the other hand, \xjk  yjk  > \xjk   \yjk \ > 221:xjk  and at the same time \xy\< * + y < 1 + * < c\x\. Therefore M Iffftol>rf \Rs>*/(*)! z"+l '
Further, n
X) j=\
m X
I^J/( )I
^ £
m x
\Rhf( )\
( n+1
> ct\x\~ 
k=l
'> ] T \xjk x"+1 Jfc = l
where \xjk  > 2. The absolute value of all the remaining coordinates is smaller than 1 so that
£ft/()i>eti«r<^X;i.,i^di«i. Continuing now along the lines of the proof of Theorem 1.2.1, we arrive now easily at the quasiconvexity of 9a for some a G (0,1). □
1.5 Vectorvalued Riesz transforms in 9(L)
classes
Here are the main results: T H E O R E M 1.5.1. Let9 G $, 1 < 9 < oo. Then are equivalent: (i) <2> is quasiconvex and satisfies the A2 condition;
thefoiiowimjstatements
30
Chapter 1. Integral operators
(ii) a constant c > 0 exists such that
(1.5.1)
*(A){* e Rn ; \\Rjf(x)\\e > A}
j = 1,..., n, is true for ah1 A > 0 and / = (/i, / 2 , . . . ) such that /(.)« G 4>(i) and JK//* exists for all j = 1 , . . . , n,fc= 1,2, To prove Theorem 1.5.1 we need the following LEMMA 1.5.2. Let l
\{xeRn;
Then there is a constant c\
WRjfWWe > A } < ciA"' J /(*)5 dz
for all A > 0 and all / = (fuf2,...),
\\f{.)\\, G LP(RT).
The proof of the lemma can be found e.g. in GarciaCuerva and Rubio de Francia [1, Chapt. V]. Proof of Theorem 1.5.1: The implication (i)=>(ii) can be proved in a sim ilar way as the corresponding implication in Theorem 1.3.1 for the maximal function. That (ii)=^(i) follows from the scalar case (see Theorem 1.4.1). □ THEOREM 1.5.3. Let 1 < p < oo, 1 < 6 < oo. Then the following statements are equivalent: (i) # satisfies the A 2 condition and <Pa is quasiconvex for some a G (0,1); (ii) there exists a constant c > 0 such that (1.5.3)
J *(\\Rjf{x)\\t)dx
it"
for all / = (/i, / 2 , • • •) with /(.)e G $(L) and such that Rjfk exists for all j = l , . . . , n , k = 1,2,.... The proof of Theorem 1.5.3 goes as follows: The implication (i)=>(ii) can be proved using the method for handling the maximal function in Theo rem 1.3.3. The implication (ii)=>(i) follows from Theorem 1.4.3. Q
Notes to Chapter 1 31 Notes to Chapter 1 The classical theorems on the boundedness of the maximal functions in Lp spaces (1 < p < oo) go back to Hardy and Littlewood [2] (in R1) and to Wiener [1] (in Rn). As the boundedness fails for p = 1 there was the general problem of a necessary condition when the power tp is replaced by a more general function. For a Young function in A2, this question was answered by Lorentz [2] and a different proof was given by Gallardo [1]; it turns out that the A2 condition for the complementary Young function is necessary. The exposition here is given in terms of quasiconvexity, the Young functions being replaced by more general functions from the $ class so that the concept of the complementarity used earlier need not have the usual geometric sense. Note that for Young functions the equivalence of the statements (i) and (ii) from Theorem 1.2.1 was shown by Tsereteli [1]. The weak and strong type inequalities for the vectorvalued maximal func tions in Lp spaces are subject of Fefferman and Stein [1]. M. Riesz [1] proved the boundedness of the conjugate function in Lp and the weak type inequality for the conjugate function appeared in the paper Kolmogorov [1]. These questions are discussed in details in the Zygmund monograph [4]. As to the singular integrals in Orlicz spaces with a Young function in A2 we refer to Ryan [1], Tsereteli [2], [1], Oswald [1]. The vectorvalued inequalities in Lp spaces are dealt with e.g. in the mono graph by GarciaCuerva and Rubio de Francia [1, Chapt. V, § 3]. Sections 1.2, 1.3, 1.4 and 1.5 are based on papers by Gogatishvili, Kokilashvili and Krbec [1], [2], and Gogatishvili [3], [6]. The "sufficient part" is in other terms essentially contained in Tsereteli [2] (see also his paper [1]) where arbitrary quasilinear operators of weak types (1,1) and (p,p) for sufficiently large p's are considered, the Young function in question being in A 2 . For the Hardy averaging operator see the papers by Butzer and Feher [1] and by Kokilashvili [2]; further variants can be found in Maligranda [1] and at many other authors.
Chapter 2 Maximal functions and potentials in weighted Orlicz classes 2.1 One weight inequality for the maximal functions in reflexive Orlicz spaces Starting with this chapter we will deal with weighted inequalities. Let us briefly summarize the well known concepts we will frequently use in the sequel. Let us agree that, for brevity, when talking about cubes in Rn, we will have in mind cubes with sides parallel to coordinate axes. A couple of weight functions (g, cr) is said to belong to the Ap class (the Muckenhoupt class), 1 < p < oo, if pi
{Ap) s
e{x)dx
7\W\J
x 1/{p 1)dx
)
[&\Jw »~ ~
< oo
where the sup is taken over all cubes Q C Rn (with sides parallel to the coordinate axes). For p = 1 the condition reads (Ai)
— / g{x)dx < c inf ess
If g = a, then we simply say that Q belongs to the Ap class. A weight Q belongs to Aoa if to each e £ (0,1) there corresponds 6 > 0 such that if Q is a cube in Rn, E C Q measurable, and l^l < 6\Q\, then e(E)<sg(Q). Let us recall basic properties of the Ap classes: (i) if Q € Ap for some p 6 [1, oo), then Q 6 Aq for all q £ [p, oo); (ii) if g £ Ap for some p £ (1, oo], then there is an e > 0 such that g £ A p _ e ;
32
2.1. One weight inequality 33 (iii) if g G Aoo, then the reverse Holder inequality holds: there is a positive 6 such that
5i/(*(«))1+,«fa<«
\Q\
[w\I^x)dx
Q
for all cubes Q C Rn. A detailed analysis can be found e.g. in the monographs by Garnett [1], GarciaCuerva and Rubio de Francia [1], Torchinsky [2]. Just recall that M is bounded in LP(Q), i.e. J{Mf{x)fe(x)
dx
\f(x)\"Q(x)
Rn
iff
Q
dx,
f G LP(Q),
R"
G Ap, 1 < p < oo.
If IT is another weight on Rn, then the inequality of weak type (with respect to g and cr), Q({ x£Rn;
Mf(x)
> A }) < cA"" / /(x)»V(x) dx R"
holds iff (g,
L$(Q);
< c1 I R"
34 Chapter 2. Maximal functions and potentials (ii) there is a ci > 0 such that WMfWusg) for all 6 > 0 and all f £ L$(Sg);
< c 2 /i # (j e )
for all 6 > 0 and all cubes Q, t i e following condition is satisfied:
[I
(iii) for all 6 > 0 and all cubes Q, the following condition is satisfied: d \
^/Wfc
\
for all cubes Q and all 6 > 0 with c 3 independent of Q and 6; (iv) j e % ) . To prove the theorem we will need several auxiliary assertions, each of interest itself. Recall that a sublinear operator T : £p(f>) —► Lp(g) is called of restricted weak type (p,p) with respect to g (1 < p < oo), if the weak type inequality holds on the set of all characteristic functions of all measurable sets, i.e. I
g{x)dx
EcRn,
£ measurable,
with c independent of E. The following version of the Marcinkiewicz theorem is due to Stein and Weiss [1]: If T is a sublinear operator of restricted weak types (po,po) and (Pi>Pi) wtth respect to g, 1 < p0 < px < oo, then T is bounded from Lp{g) into itself for all po < p < pi. Observe that interpolation with change of measure is possible, too. We refer to Stein and Weiss [2]. If T = M, then the restricted weak type can be characterized as follows (cf. the Aoo condition): L E M M A 2.1.2. Let g be a weight on Rn and 1 < p < oo. Then there is a C > 0 such that (2.1.1)
/ {M X E >A}
g(x)dx
2.1. One weight inequality 35 for all A > 0 and all measurable E C Rn if and only if there is a K > 0 such that for all cubes Q and all measurable sets E C Q,
m
( 2 .i.2) v
;
101 \e(Q)J
'
Proof: Let (2.1.1) hold. We have (1
iF»<*>.
xeRn,
MXE(X) >EIXQ(X)> MXE(X)>{1~^
for any e G (0,1). Therefore, putting A = (1  e)\E\/\Q\ in (2.1.1), we get
e(Q) <
(jr \Q\
CQ(E)
V
and sending e to 0, (2.1.2) follows. Conversely, let (2.1.2) hold. If
Mef(x) = sup L.
[ \f(y)\e(y)dy,
x£Rn,
Q 1,p
then MXE(X) < K(MeXE(x)) ■ At the same time, the function g satisfies the doubling condition. As known (Coifman and Fefferman [1]) the operator M is then of weak type (1,1) with respect to g. Whence the same is true for the operator XE *—► {MXE)P and (2.1.1) holds. □ Next we prove that the class A$ enjoys prominent properties analogous to those of the Ap classes. Precisely, we have L E M M A 2.1.3. Let # satisfies the assumptions of Theorem 2.1.1. Then (i) A(#) C AT for all r > i{$); (ii) if, for 7 > 0, the Young function $y is given by t
* r ( * ) = I
where fZ {r)
/ 1 r
= ((^ ) ( ))
1+7
'
tnen
A$ C A$y for all sufficiently small y.
36
Chapter 2. Mammal functions and potentials
Proof: As to (i) it suffices to show that (2.1.2) holds for all p > i($). Indeed, as M will be then of restricted weak type (r, r) for all r > i(#), it will be also of type (r, r), whence Q € AT. So let Q be a cube and E its measurable subset. By Holder's inequality and the known explicit formula for the norm of a characteristic function we have, for any 6 > 0,
\E\ 6e(y) \Q\ 101" \Q\J *e(y)
eKV)w
y
Q
(2.1.3)
< i7dx£L»(«e)X£:/(MIU*(«ff) Wl < <
*i(i/MQ))llxQlU*(* e ) c
We are going to prove that
j£l < K (?W yfm \Q\ \Q(Q)J for some K independent of E and Q. Setting A = Q(E)/Q{Q) that it is desirable to show that
(2.1.4)
 ^ M
<
in (2.1.3), we see
/al/.(*).
To start with we prove that i($) = 1/7($ _ 1 ). Indeed, i(<£) is the supremum of those a > 0 for which #(/J<) < na
t>o
2.1. One weight inequality
37
H > 0, we have according to the definition of the upper index
/ ( O = inf
]
K/i
2iiM log//
logft(l//0
— inf
0<"<1
l0gl//i
• f
logJf2(/i) logff2(A«) = oSf
Whence
Igg^j > /(^_1) = l/i(#),
0 < /i< 1,
which gives <j2(/i) > /,!/<(*). 0 < /i < 1. Especially, for our A =
Q(E)/Q(Q)
there is rA > 0 such that
*H^) . #HA*x) Sthat is
Ai/.(*)
2
HAn)
4>"Hn)
Choosing now 6 in (2.1.3) in such a way that 6Q(Q) = l/r A , we obtain the desired estimate (2.1.4). The assertion in (i) is proved. Now we prove (ii) for the particular case 6 = 1. In the course of the proof, however, it will be seen that the constant on the right in the A$ condition for 0S does not depend on S, proving thus # 7 G A*. Let Q € A$ and set v(x) = ^ ( l / ^ x ) ) , x e R", then g(x) = l/
/ 1
/
1
<£c J v?
Q
v{x
(w\I )
dx J < K.
Q
For a > 0 define E = {xeQ; V(X) > av(Q)/\Q\} and E> = Q \ E. We prove that there are a, 0 > 0 independent of Q such that \E\/\Q\ > /?. It is v(x) < av(Q)/\Q\
±f
for x <E Ec and therefore g £ A$ implies
I
<
_
\Q\J 9(av(Q)/\Q\) ~
38
Chapter 2. Maximal functions and potentials
thus \EC\ TM<*v(Q)/\Q\ \Q\
IQI which can be made arbitrarily small provided a is sufficiently small. Invoking the approach due to Coifman and Fefferman [1], we see that the reverse Holder inequality holds in the following form, namely, for small y > 0,
( Whence
\ 1/(1+7)
Q
I
Whence
*(]£
Q
UwJ cte j < Mil A" '(£)*)■
Q
As ^ 6 A2, we have C\Q\
'w/^GraH* e(QY Putting together the last two inequalities, we get g 6 A$ as C is readily seen to be independent of the choice of 6 in the very beginning of the proof. The lemma is proved. □ Proof of Theorem 2.1.1: The implication (i)^(ii) is obvious. We prove (ii)=>(iii). First observe that, for any 6 > 0, the norm is finite. In the opposite case, for all 6 > 0 and some / € L$(6g),
\\XQ/^Q\\L^(6Q)
J f(y)dy= Q
I f(y)j7nsse(y)dy Q
which is impossible as then Mf = 00 on Q.
= 00
2.1. One weight inequality 39 We claim that
xeRn.
M/.)>i/,(,)%gWl Q
By duality argument, there is a /„ G £#(*$) such that   / 0   L # ( « J ) = 1 and JMv)dy = = Jf J0{y)jlSe(y)dy fo(y)j^)6Q(y)dy jMv)dy. Q
> > cA\\XQ/{6 c4\\xQ/(6ee)\\ )\\LASe L^le))
Q Q
with some c4 > 0 depending only on # so that \\ < jgTllXQ/(fy)Hi»(*#; illxol U#(4g) << Cf>\\Mf c5\\Mf0 0 Lt(Se) In other terms,
Q Q
(*«r)
C2C5.
■>e{X)dx
Astf{£)~ ^ ( ^ ) _ 1 ( 0 for a.e. £ > 0, it follows that this can be rewritten as v
x
czllxqlk.d.) '
cr\Q\ C?IQI
Jy Q
 1i(cr\\x (°T\\XQh.V.A Qh*m\i
V S\Q\e(x) Jy v *ioie(«)
dx ix
< K. cCo

Q
Clearly there is a 5 > 0 such that ,,,,, *CSWXQWL^SB) * * M =_ x
*\Q\
"
as the term on the left is a continuous function of £ G (0, oo), taking all values from (0,oo). Indeed, (2.1.6) is equivalent to ,c cC99\\XQU,(« c9 XQ\\LA6e) e) _ C9 6=
m M
l
1 1
 ]g[#Hi/(W))) M*  d/(W)))
W Q * ( 1(MQ)) » ~ iQr^"
40
Chapter 2. Maximal functions and potentials
Whence  1
■l(i/(6e(Q)))
\Q\ and we get ,$
, —  —. e(QM\Q\/Q(Q)) Inserting this into (2.1.5), we obtain
M£)*
s
10
6
Hxgi#(«,)
e(QM\Q\/e(Q)) \QW\\Q\le{Q))Finally, this gives
m l ' U)J' 
ix Cll
S(QY
Obviously, the whole proof of (ii)=>(ia) goes in the same way with 6g instead of Q and the constant corresponding to c6 will be independent of the particular choice of Q. The proof of (iii)=>(i) follows easily by a modular interpolation (see, e.g. Krbec [1]) or by a direct computation. It is easy to check that for 1 < r < *(#) the function t i» ${tllr) is quasiconvex. Indeede if h < t2, then for 0 < e < i($)  r, we have
Wn h
^ ^
*(#'&##/') / f i V
2
<2 1 + (
W <
, .
w
_
£ ) / r )
^
/ r
)
*2
cg(j^)
According to Lemma 1.1.1, the function t i  ${tllr) is quasiconvex and we can apply the inequality of Jensen's type (the same lemma) to the maximal inequality Af/2 ((ff ) < c r /2,r (e ). We get the modular inequality (i) and the theorem is proved. □
2.1. One weight inequality 41 It is natural to ask what happens if one drops the assumption # € A2 and wants the maximal inequality from Theorem 2.1.1 to be valid. The question about a characterization of all possible weights can be answered at least partly. In the rest of this section we will suppose that <£ € $ is such that its complementary function \P is finite. We shall also restrict ourselves to the case n= 1. In Chapter 1 we saw that the A2 condition for complementary functions is indispensable for validity of the nonweighted norm inequalities. It can be shown that, at least near 00, the same is true in the case of one weight norm inequalitites. A conjecture about the global A2 condition, however, seems to be reasonable for the case of general weights. L E M M A 2.1.4. Let <5 be a Young function and Q a weight on R". Suppose there is a c > 0 such that
lM/L#(,) < c/L#(e))
/ e Md
Then the function & complementary to 0 satisfies the A2 condition near 00. Proof: It is easy to show that Q satisfies the doubling condition. Choose any K such that
E = {x e R" ; K'1
has a positive measure. With no loss of generality let us suppose that 0 is a point of density of E and let
(2.1.7) Indeed, it is
\E 0 [ 4  y a) = \E n [0, a)  \E D [0,4 x a) 3a a a
T~4
=
2
and (2.1.7) follows. As the function x t> l/x is decreasing on (0,oo), we get from (2.1.7) that (2.:1.8)
dx //
En[41a,a
aa
f dx = log2. X
X
)
Bn[4 1 a,o)
>
2~1a
J  = log2.
2~ 1 a
42
Chapter 2. Maximal functions and potentials
Set fm(x) = XEn[o,4^b)(x), for some fixed 6 G (0,a 0 ), and all m G N. Then
ll/ll^,, = ,([0,4"m6) n S ) ^ 1 (—L—}
(2.1.9) Further
Iff (A l £ n [°' 4 " m 6 )l
.
xG(4~m6,6).
X
Putting * w = *"* ( e (£n[o,6)))
x nE{x)
^
'
we obtain in virtue of (2.1.8) \\Mfm\\Lfie)
>J
Mfm(x)g(x)g(x)dx
2^n0.4m#( s s J E ^)
/ Enl4
(2.1.10)
>  £ n [ o , 4  m 6 )  ^  1 ( ——=—— j J i r
£*
"''" 1
^
n
/ £n[4»»,4»+ )
^nPl4""rl(iw)rlfflk,!l It is \E n [0,6)1 < 6 = 4 m [0,4 m 6) < 3  1 4 m + 1  £ O [0,4" m 6), and
e(£?n[0,6))<jb£n [o,6) <7
  / m   M f ) < K\En[Q,4™b)\Vi
— l
/A"2314m+1 {eiEn[Qb))
b))J
2.1. One weight inequality 43 Putting together (2.1.10), (2.1.7) and (2.1.11), we get
**
(2.1.12)
\Q(E
I1 )\ mm << £*V> ;—^ n[o,&))J ~ log 2
l m+l\
(K^4m+1)
l[0,6))J
Now choose m > 2cK2/log2 and realize that m was independent of b so that 6 can be chosen arbitrarily between 0 and a0. The estimate (2.1.12) implies 2^~1(t) < ^(ct) for all t > t0, t0 = 1/Q(E n [0,a 0 )), and a c > 0 independent of t. But this is just the A2 condition for & in terms of !P~X. D Next let us generalize the concept of the A$ condition: Definition 2.1.1. Let 4> G $ , & be its complementary function, and Q a weight on Rn. Let R$(t) = 4>(t)/t and S*(t) = !?(*)/«. * # 0. Then 0 G A$ if there is an e > 0 such that
{A ] s
* z [w\JMx)dj *• [ml5* (^y) djK °°
T H E O R E M 2.1.5. Let $ G $ and p be a weight on R*1. Then (i) If there is a constant c > 0 such that (2.1.13)
[$(Mf(x))e(x)dx
1
fSLf(e),
1
R
then 0 is quasiconvex, & satisfies the A 2 condition near 00, and Q G A$. (ii) If \P G A2 and £ G J4#, then (2.1.13) holds with some constant c > 0 independent of f. Proof: (i): The quasi convexity follows even from the weak counterpart of (2.1.13). As this is more general than what we claim here we omit the proof and refer to the very beginning of the proof of Theorem 2.2.2. Further, Lemma 2.1.4 gives !F G A 2 near 00. The membership of Q in A$ follows in a standard manner (see Theorem 2.1.1). (ii) The proof borrows from Kerman and Torchinsky [1] (Theorem 2.1.1), the technique is modified. Put R$(t) = &(t)/\t\ and S*(t) = 9(t)/\t\, t / 0.
44 Chapter 2. Maximal functions and potentials For a > 0, let v(x) = F$(l/ag(x)), x G R1. Fix a bounded interval I and put, for /? > 0, £/) = { i € / i » ( r ) > / J » ( J ) /  /  } . We will distinguish several cases: As # G A2, it is 0: (1) (2) (3) (4) (5)
S#(0,oo ) = 5#(0,oo ) = S$(0,oo ) = £#(0,00 ) = S#(0,oo I =
(0,oo); (0,a); (a, 00); [a, 00);
{a}.
If (5) holds, then #(£) = at, t > 0, and v is a constant function, whence v G Aoo If either (2), or (3), or(4) holds, then S$ is invertible on the corre sponding intervals. Let 7 G (l,i(#)). Then #(Af) < c7A7
< G iZ1,
A G (0,1).
Let e be the constant from the A$ condition. Let further /? G (0,e/2] be such that 2 7 c 7 (/?/£) 7  1 < 1/2. Fix I and suppose that j3g{I)/\I\ is in the range of S$l. Then, for some C\ > 0,
dx R${i v{I)l\I\) ~ \I\ J Si \v{x)) £1
;, 1 f
>
\I\Ep\ \I\Ep\

I
Whence
j/y^j /
CS£^(J)/ID "
J R*{ev(I)/\I\ 2cR # (2 J gt;(J)/J)

fl#(CT(/)/I
(f)' 
2'
so that ^>21/.
11
Si \pv{I)/\I\)
2.1. One weight inequality 45 If 0v(I)/\I\ does not belong to the range of 5 # ) then (3) or (4) occurs. Then, of course, Ep = I and this case is trivial. Thus g 6 Ax and the reverse Holder inequality holds for v. There is 8 > 0 such that
(2.1.14)
\
±J(v(*))l+'dx
l+<
f i , / i»(«)dx)

Define <2>{ by S#,(t) = (5*(t))1+*, * # 0. The complementary function to say !?,, is then given by 9s(t) = tS*,(t). Clearly, I(9S) = 1(9) + 6(1(9) implying i(#,) < i(#) as the case /(if) = 1 (or i(#) = oo), i.e. the case was already considered and 1(9) is finite (9 € A 2 ). Thus !(#«) > 1(9), equivalent^, i(S«) < i(9). Rewrite (2.1.14) as /
^ / « * » >
\ 1/(1+*) i/(i+«) , +
*
<^/»(x„,
If 5# is invertible on (0, oo), then
 (/(*(*))«) *(<Sr(h *) )).«■(&/«.) "■(»/**■) c—
c 4 7
which in turn implies
w «((£) £/«.)>"*) **■(&/.«*)• 5,'((^) ,+ 'i/««))^<^(^/»M^.
Whence
(2.1.15)
<* (ten/*■&*)*)
H*/*
'.;.,)  )
*4, 1), (5) or,
46
Chapter 2. Maximal Junctions and potentials If the case (3) or (4) occurs, then /
2ei ST'
[w\I \\I\
{v{x))1+tdx
\ 1/(1+*)
)
But then S${t) > a, thus Sft(t) > a1+s and Rft(t) inequality (2.1.15) holds trivially.
'
= 0 for t < a1+5.
The
Now the proof can be completed. We have g £ Ap for all p > i($s) whence (2.1.13) follows by interpolation or by applying (quasi)Jensen's inequal ity (stated in Lemma 1.1.1) and Theorem 2.1.1 to the function t t+ 4>(t1/,?) which is quasiconvex for all q, i($s) < q < i(&) □ In proofs of next theorems using the functions R$ and S$ such a detailed discussion of all the cases listed as (l)(5) in the above proof will be mostly omitted and left to the reader.
2.2 T w o weight inequalities of weak t y p e In this sections we will study the inequalities of weak type (2.2.1)
*(A)e({ x G Rn ; Mf{») > A }) < c [$(cf(x))o(x)
dx
and of extraweak type (2.2.2)
Q({ xeRn;
Mf(x)
> X }) < c f $ (^y^)
for the maximal operator. The conditions imposed on the function $ will be rather mild as in Chapter 1. We get natural analogues of the Ap condition as in the preceding section which are equivalent to the Ap condition provided Q — a and $ generates a reflexive Orlicz space. The difference is, however, that these generalized conditions work in more general spaces, too. Definition 2.2.1. Let $ € $ in the sense of Definition 1.1.1. The function <£ will be called of lower type B near 0 (near oo) and we will write 4> € Bo (f G Boo) if there is a c > 0 such that 4>(t)/t > c (#(t)/t < c) for all t > 0.
2.2. Two weight inequalities of weak type 47 It is easy to show that for a quasiconvex #, we have $ € B0 iff the comple mentary function "P vanishes on some interval [0, a), and, similarly, # € 5<x> iff If' equals to oo on some interval (a, oo). Definition 2.2.2. Let # £ $ and define V(t) = sup { st  $(s) } its formal complementary function. A couple of weights (g,cr) will be said to satisfy the A% condition (we will write (g,
(lQl/^((,)) r f a : )
Qcfl" QCfl"
V
Q
/
n
where Q denotes any cube in R and R#(t) = $(t)/t t > 0. (For 0 e BoUBoo this turns into 4 ^ r < c i n f ess a(z),
and S$(t) = !F(*)/*,
Q C Rn,
(i.e. into the Ai condition for g and
ciJ
: , /.
—■ dx < oo.
\\QHx)J g{Q)
Q
If g = o we will simply write £ 6 vl# and g € A w , respectively. Before we go on let us make a remark on the weak and extraweak in equalities which will also justify the terminology. We claim that a weak type inequality always implies the corresponding extraweak type one. Indeed, by homogeneity, the extraweak type inequality is equivalent to g({ xeRn;
Mf(x)
> 1 » < c / #(c/(*))er(x) dx
which is the weak type inequality with A = 1. Of course, the same can be directly shown for the relation of the conditions A% and A w . Putting a = \Q\/g(Q) in the A% condition, we get directly Af.
48
Chapter 2. Maximal functions and potentials Let us recall that the centered maximal operator is given by
Mcf{x) =
sup
L f \f(y)\dy,
Q centered at x v°c\ J Q
and the two weight centered maximal operator by Ml„f(x)= sup Jr [ \f(y)W(y) Q centered at x Qv°()
dy.
J Q
Standard methods show that Mcq(J is of weak type (1,1) with respect to the weights Q and er, namely, (2.2.3)
\Q{{ xeR";
MeeJ(x)
> A }) < c J \f(x)\
We will start with the extraweak type inequality in Rn: T H E O R E M 2.2.1 Let 0 £ $ be quasiconvex. Then (2.2.4)
e({
xeR";
Mf(x)
> A }) < c /
*(x)dx
Rn
for all f E L\oc and all A > 0 with a constant c independent of f and A if and onlyif{e,o)eA^. Proof: First let us realize that the function <£ can be assumed to be con vex. We will abandon all the technical details carried out in Chapter 1 as the constant characterizing the quasiconvexity do not change our qualitative estimates. Let (2.2.4) hold. Then easily,
(2.2.5)
m
We will show that this implies (g,cr) G A  W . Assume first that & is finite, i.e.
o #(t) '
/i#(A) = sup * v ' Zo
tf^At)
Then (cf. Chapter 1)
,(#) =
SUp
KA
}%2&1 = log A
inf l2f^) 0
log A
lim
A00 =
fagwW log A
lim
*!»&
A*0+
log A
2.7. Another variant of M 85 and 1 rr^r = i(q>)
. inf
O
1 I(&)
=
sup KA
\ogh$(\) \ogh$(\) ■—\± = hm ———V 1 , log A
log/i*(A)  l o— g ^ A =
A.0+
log A
logft# (A) hm Aoo
log A
Let us still introduce the function
G#(A) == sup t>0
For 6 > 0, the space L$(6) will denote the Orlicz space with the weight equal to the constant 6. We will write L$ = L#(l). T H E O R E M 2.7.1. Let 0 and & be a couple of complementary Young functions both satisfying the A2 condition and Q a weight on R". Then the following statements are equivalent: (i) there is a ci > 0 such that
/$(M I ${Meef(x))dx< f(x)) dx
Rn
for all f 6 L$; (ii) there is a C2 > 0 such that
\\Mef\\L*{i) < c3\\f\\L.(s) for all f G 1$(6) and all 6 > 0; (iii) the function Q belongs to the class B$, namely, (B*)
sup xgeU # («)
XQ/(MIU»(«)
< 00;
86
Chapter 2. Maximal functions and potentials
(iv) there is a c 3 > 0 such that for all Q C Rn and all 6 > 0,
\J9[#(1—) {se(x)J °[\Q \\Q\J \s*t*)J Q
(fg0(Sg(x))dx
J Q
where tp = &+; (v) the function g belongs to Bi{l§) D BJ(#)> i.e.,
dx) dx J
)1 '
equivalent^,
sup oo llxQelk w  HXQ/
and sup < oo. sup XQ
Proof: It will go as follows: (i)=^(ii)=>(iii)=^(iv)=>(v)=>(i). In the course of the proof, various constants independent of Q, 5, and / will generally be denoted by c and we will not keep the track of their changes from line to line. The implication (i)=*>(ii) is trivial, (v)=>(i) follows by interpolation (see the proof of Theorem 2.6.1 for the interpolation technique). It suffices to realize that the set of all p £ [1, oo) for which a weight belongs to Bp is open in [1, oo). This immediately follows from the reverse Holder inequality: If a weight, say,
{a{x)y \W\I ■*)I (»i/U)) *)/ \ Q \ Q \
Q
/
\
a? £GV Ap, there is a > 1 such As aP such that that
(
\ 1/(«P l / ( a p ))
/
Q
/
/,
1/p \. 1/p
— / (<x(x)) ^ I1TVI V' y < I ■—Qf(o{x)Y
p
~ f(o(x))aP dx J and if (ap)< is the index conjugate to op then (aPy
. ity, //
\~\Q\J V <3
\ m»p) l/(«p) t/
(
fxl
I
1 
sc
llQli IOI 7 V^)y Woo;
,
v
dx j
dx ) By Holder',3 inequalNV
I/(«P)' 1/(«P)'
dx J
I
which is cf
~< cc,'
2.7. Another variant of M 87 That (ii)=J>(iii) is easy: It is, for any cube Q and all 5 > 0, Me(x< ?/)(*) >
Q(X)
\Q\
J My)
(5dy
Q Q
so that
\\exQ/\Q\\\L.l6) Jj^6dy^
«II/XQIIM«).
Q
Taking sup over all / with \\fxQ IU*(«) < 1 yields S<j>. Now we prove that (iii) implies (iv); actually we will prove (i)=>(iv) and that (iii)=$>(iv). Let 6 > 0 and /£,(i) = 1 be such that
/ Q
md Jm 1 8dy> e(y) My)
\\xQ/{6e)\\Ll »(0
Q
(duality and the fact that the Luxemburg norm is majorized by the Orlicz norm). Then Me/(«) > Q Q
> XQ{^{X)
\\XQ/m\\L*w,
*£*",
whence, in virtue of (i),
(2.7.1)
J$ (^p(*)xg/(*(f)L#(o)
6dx
< c
Q
Observe that (2.7.1) is an immediate consequence of (iii) so that from now on, the proof of (iii)^(iv) starts. As g$(\)$(t)
< $(\t) for all A > 0, all* > 0, we have
g*(Q(x))0 (^Xfl/(^)HM')) ^ * ( j ^ j «(*)HX0/(^)lli.(«))
88 Chapter 2. Maximal functions and potentials and (2.7.1) gives
(2.7.2)
Jg*(fa))6dx < c. Q
For brevity, set V(S)
= 6 f g$(g(x))dx. Q
Then (2.7.2) turns into XQ/(MIU, W
< c$\ih)\Q\.
From the definition of the Luxemburg norm, 1 / •
6 dx < c,
JlQlfHi/nMx)
Q
and as V(t) ~ *v?_1(t), we get (2 7 3)

Wfnih'J1 9~ \6\Q\^^h)fa)j \6\Q\^{lh)fa)) ]QWWh)
fa fa)dxc'
Q
Now we mimick the trick from Kerman and Torchinsky [lj: Find <50 > 0 such that
60\Q\$l(Vv) =
L
First let us compute 60 up to an equivalence. With 7?0 = i?(*o) we have *1(i/i7o) =
W\Q\)
so that
9\iM
g  / g$(Q(x))dx' Q
2.7. Another variant of M 89 whence /
1
101 r—, ,L , / g$(e(x))dx fg*(g(x))dx
T~& S*00
>
j g$(e(x))dx tf w JI' g*(e(x))dz * " Q )) Q
\Q
fg${g{x))dx Si
~  g\Q\  2Q
I
101 \Q\
1 1
\\
\Q\ \Q\ dx i IMe(x)) \fg*(e(x))dx
*A (
\Q \Q
IQI^1
101
JI
)
Jg*(Q(x))dx \
By our choice of <50 we get from (2.7.3) that So
\e(x)J
Q Q
dx < c,
in other terms,
6o
l*{ix])e{x)dx^c
Q
Q
This yields
W\J* \M*))
dx
^
C
c\Q\ \
\ yi9M'))dx\' Jg$(g(x)) dx J '
i.e.
* [ W\J * \&j) \J*\e(x)) *(l£
dx dx 1 ;
c\Q\
fg*(e(x))dx ) ~ fg*(e(*))dx
Q Q \ Q I Q and (iv) follows with 6—1. In the course of the proof, however, the constant c was readily seen to be independent of 6 so that starting with an arbitrary multiple of g instead of g we arrive at the condition in (iv) for all 6 > 0.
It remains to prove (iv)=>(v). We show that restricted weak type inequal ities for M hold simultaneoously in Li^_6l(g'^~61) and Lj^+62(gI(^+S3), this yields then, by interpolation argument, the boundedness of Me in £,(#)_$'
90 Chapter 2. Maximal functions and potentials and £/(#)+«' for any 0 < 6'{ < 6{, i = 1,2, in particular, Q(W € Aif^) and g1^ € A J ( # ) , i.e. (v) holds. Let Q be a cube in Rn, E C Q measuarable, and 6 > 0. By Holder's inequality we have
(2.7.4)
j  j < ~HxBeL#(«)lx<3/(*e)li,(o •
Clearly, there is 6' > 0, namely, 6' =
1 fV(l/e(x))dx Q
such that (2.7.5)
6'\\XQ/(6'Q)\\L9{(I)
= 1.
The condition (iv) (for 6 = 1) and (2.7.5) gives
m !•«*■»"" (ml' U.)ii»Ug)ii) '*)icQ \ Q By the definition of the Luxemburg norm we get , 15llxg/(« «)IU.(^(]5fllxg/(«'»)llL,(io)
I
^6>f9*(C6(x))dx' Q
whence 1 < «** «P ^jje{x))dx ilx./(^)ll W ) < j 1"■ l*'J*#(e(*))
^\\XQ/(6'Q)\\W)
1
Now we estimate Hxfipllz*^') Let
fy6'jGMz))dz)) ' N ^ ( "(gJ55His))
B) = Ui # A(e,£?)
■
2.7. Another variant of M
91
Then —^jG${e{x))6'dx
1
E
so that
/ # ( ^ f ) * ' ' , , s / # ( = ) * w $(^(x))5'rfa; ' » * ' * "=1=1.
giving giving us us
E E
E
k*(«' > <
\\XEQ\ llX£?ek*(«')
Returning (2.7.41 we Returning to to (2.7.4) we see see that
]£[
A(e,E) C
\Q\~
a(g,Q)
It is clear that the last inequality is equivalent to \E\ \Q\~
a(Sje,Q)
with arbitrary 6 > 0, 7 > 0. For K > 0 let us denote QK = {XEQ;Q(X)>
K],
EK=EnQK,
and Q' = Q \ QK, E' = E \ EK. Choose K > 0 large enough so that (2.7.6)
(1  K)a(Sie, Q') < a(61Q, Q) < (1 + K)a(67e,
with some K close to 1 and similarly for A(6jg, E). equivalent to (277) K
;
\E\< \Q\
Q')
Then (2.7.5) will be
A{6ie,E>) a(6le,Q')
with a c' equivalent to c, this equivalence being independent of a particular choice of E and Q provided E' and Q' were chosen in such a manner that (2.7.6) and its analogue for a(Syg, Q) hold, This follows from the generalized homogeneity of $ as e.g. i ( #  1 ) = ( i ( # ) ) _ 1 and I{$~1) = ( i ( $ ) )  1 . Looking at the definition of the functions g$ and G$ we see that given 5j, 62 > 0, there is 6 > 0 such that **'<*>(0(ar))*'W < G*(^(x)) < * l(#) ~' 1 (
92
Chapter 2. Maximal functions and potentials
and *'(#)+«»(£(„.))'(*)+'» < g*(6g{x))
<
SWigWyW
for all x E E' and all x E Q', respectively. Further,
g$(y)g*(8e(x)) < g
<
G,(y)G,{6g(x)).
Applying t h e last two couples of inequalities to (2.7.7), we get
#
■
•l 6'g*(,y)6Il*)+e2
(2 7 8)
' 
\Q\ "
/'(eC*))"*'" 1 "* 2 dx
/ i
$1
I «'j#(7)« i (*> 4 i J(Q(T))'W6I
dx
Denoting f—
■
1 1 ■' » /(e(x))«'W* dx 6'gt(y)6*(*)
we have ,$'W+*2
(5')(J(*)+*2)/(''(*)*i)(G a i (7))( / (*)+^)/('(*)«i)t( J (*)+^)/( / (*)*i)' 1 V (/(*)+*,)/(!•(*)
( / ( < ? ( * ) ) ' ( * )  * ! dx") Inserting this into the argument of $
x
in the numerator in (2.7.8), we get
1 «'jf # (y)«'(*)+'» / ( g ( j ! ) ) ' ( * ) + « » d i Q' ( ( 5'<)(/(*)'(*)+<5i«2)/U(*)«i)(G !f (7))U(*)+<3)/(i(*)*i) »#(7) N (/(#)+« a )/(.(#)« 1 )
(/(e(»))«»)«Odxj
i. Q'
2.7. Another variant of M 93 Now
V(/(*) + «,)/(<(#)«!)
'
l^l"1 J(e(*))mtl dx
\ so that
B
E>
< \E'\
1
JiaixtfW*
E'
I 9*{l) / #
,
V(G$(7))( ( )+^)/( 'W^)(<5'<£')( / (*) i (*)+^+^)/«*)' 5 i) #  HSt) #"i(t)
with
/(
c= ^
/(e(x)) J (*)+«»dx'
Q'
It follows from the definition of h that there is
1^1. Cc
5# + 1/w (j( S(x)yw+ *dx\ //(«(*))'< > '»Ml/<(*)
E
f / f(e(x)yW+ /(e(x)) w+«>a dx
\Q\ I Q I"
\Q
I
which in turn gives 7(f(z))'< # >+*»ds\ (2.7.10)
IQI
< c
l/(/(#)+*a)
E
/(e(x))'(*)+*» dx
and this is nothing but the equivalent condition for the restricted weak type inequality (2.7.11) (2.1 Ml)
/ MXE>A
(Q{x))I^+^dx
94
Chapter 2. Maximal functions and potentials
So let us turn to (2.7.9). There is still a constant 7 > 0 which can be chosen arbitrarily. Denote €
1(g) + S2 ~ if*)*!
and let us look at the behaviour of g,(y)/(G$(j)f
■ K 7 > 1, then
ff*(7) g*(7) << G1_* rl« u 5: ( G # ( 7 ) ) «  ** (G#(7))« so t h a t lim * # ( 7 ) / ( G # ( 7 ) ) « = 0. If now ji is chosen from (0,<52), then for 7 7  . 00
sufficiently close to 0 we have 9*(l) (G$(j)Y
27n*)+,i 7<«(*)'i)<J(*)+*»)/<»"(*)*i)
>

>27',*2I giving
lim 0.5(7)/(G$(y))t
= 00. We have shown t h a t , indeed, 7 can be
chosen in such a way t h a t (2.7.9) turns into (2.7.10) (continuity of g
from below in terms of a power of \Q'\ and of
heix))*™1* dx Q'
and going along the lines of the tedious calculation above, we get, after a suitable choice of 7 , t h a t the numerator on (2.7.8) is majorized by ft*(l)#1(5't0) where now
S' —
ff(e(x)yw^dx\{m^)/(iW6l) f(e(i))'(*)*idi \Q'
/
Notes to Chapter 2 95 We get V (/(#) + < 2 )/(i(#)(i(*)* 1 ))
! ^
E
1 f(e(x)y(*)*> dx
)
(2.7.12) s
( f(g{x)yW > dx\ <
E
This results in (2.7.13)
/
(e(z))'(*)'i d* < c( f (£))'(*) J 1 AW*) J 1 ).
M X E >A
From (2.7.11) and (2.7.13), using Lemma 2.1.2 and interpolating with change of measures (the SteinWeiss interpolation theorem recalled at the beginning of Section 2.1) the resulting restricted weak type inequalities, and then applying the RieszThorin theorem to Me, we eventually get the boundedness of Me in both the spaces £;(
N o t e s to C h a p t e r 2 The pioneering paper on weighted inequalities in Orlicz spaces is due to Kerman and Torchinsky [1]. The crucial point is the equivalence of the A$ and Ait$\ condition for a Young function satisfying the A2 condition together with its complementary function. The generalizations of the A$ condition is the subject of Pick [2] where the A2 condition is removed. Assuming
\}
ci = sup (vB(x,2r))1/"\\xR^B(x,r)H(x'
O H I j V  ' < °°
xeR"
Then there exists a constant C2 > 0 such that (6.1.3)
u{x e Rn ; K(fj>)(x)
> A} < c 2 A«/« F .
Proof: Without loss of generality we can suppose that (vRnr1/q\\f\\LV
(6.14)
<
^
where c is the constant form Proposition 5.1.1 and c\ is from (6.1.2). In the opposite case we would have n v{xeR ;
K(M)(x)
>\}
(2c 1 c)«/« ! ..
172
Chapter 6. Potentials and Riesz transforms
Put (6.1.5)
Ex =
{x£Rn;K(W)(x)>\}.
We show that for every x £ Ex there is ro (depending generally on x) such that (61.6)
(VB(x,r0))^\\f\\LV
>^c
and M(*,2ro))1/?ll/IU>
(6.17)
> ^ 
The inequality (6.1.7) holds for some r 0 thanks to our assumption. We show that, however, it cannot be fulfilled for all r 0 > 0. Allowing for it, this would lead to /
k(x>y)Hy)f(y) dv < c\\f\\Lt \\XR\B^KX^
O/HliV'
\xy\>r0
IcCiH/IUi'O'flCx^ro)) 1 '* A <2' Letting ro tend to zero, we get K(f(xlj))(x) < A/2 which contradicts with our assumption x £ Ex This proves that a constant r > 0 exists such that (6.1.6) holds for ro = r. Now let us denote again with ro the supremum of all such constants. Then, clearly, the inequalities (6.1.6) and (6.1.7) hold for this ro. Denote b = lim vB(x, g) e>o and m = max{ifc € N; 2~ki/B(x,r0)
>b}.
If 6 = 0, then, obviously, m = oo. First let m < oo. For each it, 0 < ifc < m, let us choose r* in such a way that (6.1.8)
uB(x, rk) < 2kuB(x,
r 0 ) < vB(x,
rk+1).
6.1. Three weight estimates 173 Each rfc can be obtained from rt_i by dividing rk_\ by a sufficiently large number so that we can suppose the sequence {H,}* is nonincreasing. Using Proposition 5.1.1 and (6.1.7), we have for x G E\ and the corre sponding To, J
k(x, y)^(y)f(y)
dy < c\\f\\Lr;
\\XB'\B+K*,
O/MlxjV
\xy\>r0
k{x,y)f(y)Hy)dv
/ m 1 lit—1
(6.1.9)
.
=J2
/
H*,y)f(y)Hy)dy
i"fc+i<x —yj
/
+
Hx>y)f{y)^{y)dy
sg
k(x,y)f(y)ip(y)dy
< x\\xB(x,rk)f\\Lu\\XB(x,rk)^Kx,
O/HljV'
< cc^XRn^r^fWL'jivBix,
2r i + 1 ))~ 1 /».
r
*+i
The choice of r* (the inequality (6.1.8)) implies uB(x,2rk+1)
> 2k1i/B(x,rQ)
>
2~1vB{x,rk).
174 Chapter 6. Potentials and Riesz transforms Therefore, employing (6.1.10), we get j
Hx,y)f(y)rP(y)dy
21'*cc1\\xB(x,rk)f\\Ll(vB(x,rk))1'*
<
rk+i<\xy\
= (cc1)21'"^B(x,
rk)fl^l"{VB{x,
1/p 1/?
rk))"P\\xB{Xlrk)f\\L'w1
< c3(vB(x,r,))  (^,
r l )) /"xB(x,r t )/IU^
c3(uB(x,r0))k^"1''>Wf(x)
< where c3 = 21/*cci and Mf(x)=
\B(x,r)\l'P\\xB(x,r)f\\LU
sup r>0
The last term in (6.1.9) can be estimated as follows: /
k(x,y)f(y)ip(y)dy=
\xy\
lim
/
k(x,y)f(y)i>(y)dy
e
\im(vB(Xt2e))V*
= cc161/"Hx^,rm)/z
< CCl 2( m+1 )/H^(x,ro)) 1 /' XB( ,, rm) / i
Whence for A satisfying (6.1.4), x 6 E\, and the corresponding r0 from (6.1.6) and (6.1.7), we get A ^ < 1
m
c3J2^~Kl,P~1,9)^B(x,r0))1/p1/''Mf(x) k=0
< c4(i/5(z,r0))1/p1/?M/(i). This estimate and (6.1.6) give A < 2(2c 1 c)«( 1 /P 1 ^)/[ ( P 1 / p  l A ) A^ 1 /P 1 /«)M/(x).
6.1. Three weight estimates 175 Consequently, 1 l < ^ l C5A«/'+ i / H  H l *11/«'r (r8. A Af/( a ! )
with c 5 independent of / , x, and A. We have proved that
Mf(x) > c^'Wflfe/" for all A > 0 satisfying (6.1.6) and all x G E\ which in turn gives
EX c { x G Rn ; Mf(x) > % 1 A ^  I / I I ^ / P } Now, it suffices to use Theorem 5.2.4 to get (6.1.3).
□
T H E O R E M 6.1.2. Let 1 < s < p < q < oo. If there is c6 > 0 such that (6.1.11)
k(x,y)
for all x, x', and y satisfying \x' — y\ < Z\x — y\, then the conditions (6.1.2) and (6.1.3) are equivalent. Proof: The implication (6.1.2)=>(6.1.3) was proved in Theorem 6.1.1. We show that (6.1.3)=J>(6.1.2) provided (6.1.11) is true. Fix a ball B(a,r) C Rn. Obviously,
(6.1.12)
K{W){x)>
J
k(x,y)f{y)rP(y)dy
\ya\>r
for all measurable / > 0. Further, it is \x  y\ < 3a  y\ for all x G B(a,2r). The inequalities (6.1.12) and (6.1.3) give vB(a, 2r) < v{ x G Rn ; K(M)(x)
> ^
J
k(a, y)f(y)^(y)
dy }
j/o>r
(6.1.13)
1f
x
J
\\ya\>r
k(a, y)f{y)^{y)w{y)
\ «
dy
H/lfc}
176 Chapter 6. Potentials and Riesz transforms Let /o be such that
/OL>"
=
1 an(i
(6.1.14) c1XlI.VB(.,r)t(ol.)HI^<
/ Ka,v)f(y)~$j™(y)dyfl»\B(o,r)
Inserting / = f0 into (6.1.13) and using (6.1.14), we get {vB(a,2r))1/q
< ci]Xit«\B(«,r)^(0/Hl2£."
The theorem is proved. Q THEOREM 6.1.3. Let 1 < s < p < q < oo, let w, v, and V> be weight functions in Rn. If the kernel k satisfies the condition (6.1.11), then the following conditions are equivalent: (i) there is a constant c\ > 0 such that {vB(x,r))1/q\\xR^\B(x,r)^k(x,.)/w\\LPj.'
< ci
for all balls B(x, r); (ii) there is a constant ci > 0 such that v{ x e Rn ; K(M)(x)
> A } < C2A«/«.j.
for all nonnegative functions f € L^J. T h e proof of T h e o r e m 6.1.3 is analogous to that of Theorem 6.1.1; one has to use the continuity of vB(x,r) with respect to r. In particular, Theorem 6.1.3 gives the solution to the two weight problem in the Lorentz spaces for the Riesz potentials
I,f(x) = J ^[^dy,
0< 7
Rn
THEOREM 6.1.4. Let 1 < s < p < q < oo and let v, w, and V> be weight functions in Rn. Then the following statements are equivalent: (i) there is a constant c\ > 0 such that {vB{x,r))1l"\\xRn\B{Xir)^\x^nlw\\Lr:,
x G Rn,r
> 0;
6.1. Three weight estimates 177 (ii) there is a constant c2 such that v{x€Rn;
M/VO(*) > M < eaA'H/llij.
for ail nonnegative functions f G LJ/. Let n = 1, p = s = 2, g > 2, 1/2 < 7 4 1/g < 1. An example of a triple of weight functions which satisfy the condition (i) of the foregoing theorem is i>(x) = l,v(x) = \x\«'"l\logx\i'~lX(o,i/2)(x),w(x) = Mlog*r*'x(o,i/a)(«) (see Sawyer [2]). Now, we return to integrals with positive kernels. We will consider them on the set of characteristic functions of all measurable sets. T H E O R E M 6.1.5. Let the kernel k satisfy the conditions from Theorem 6.1.2. Ifl
;
(ii) tAere is a constant c^ > 0 such that tlv n
u{xeR ;
> A} < c2A"« I /w(y)dy j
KXE(X)
for all measurable sets E C R" ', (iii) the following condition is satisfied sup (vB(a,2r))1/9\\xR»\B(a,r)Ha,)/w\\L£°°
< °°;
a£Rn r>0
(iv) tiiere is a constant C3 > 0 such that UP
(vB(a,2r)y>i
fk(a,y)dy< E
for alla€Rn,r>0,
and E C Rn \ B(a, r).
I
Jw{y)dy \E
178
Chapter 6. Potentials and Riesz transforms
Proof: The implications (ii)=>(iv)=^(iii) are obvious. Theorem 6.1.2 guar antees (i)<=^(iii). We prove (ii)=>(iv)=^(iii). Analogously as in the proof of (6.1.3)=>(6.1.2) in Theorem 6.2.1 we get that (ii) implies N"1
/ {VB{a,2r)fl"
J
XE(y)k(a,y)dy
\XE\\L£
\iy>r
Consequently, for E C Rn \ B(a, r) we have I/P 1
{vB{a, 2r)) '« J k(a, y) dy < c 3 I f w{y) dy E
\ E
Let now (iv) be satisfied. Putting
Ea(a) = {yERn\B(a,r);]^^>a},
a > 0,
iv)
it is Ea(a) CRn\
B(a, r) and
a 1 w(t) dt< f k(a ,y)dy < c 3 E
E
Consequently, for all a > 0,
f / \i
iN
iyB{a ,2r))"■1/9.
w(y)dy I
i/p'
\
(vB(a,2r))l'*al
J \l"y>r
x £ „ (a) (y)u;(j/)dy
/
and passing to the supremum we get (»'fl(a 1 2r)) 1 /«xi i \fl(a,r)fc(a ) .)/«' Lt ' < «& which is (iii). The theorem is proved. Q
6.2. Two weight weak and strong type inequalities 179 6.2 T w o weight inequalities for potentials In this section, a characterization of couples of weight will be given which guarantees the validity of the weak type two weight inequality. The condition obtained is of the type given by Sawyer [5]. It has more difficult form than the condition from the preceding section, however, it includes the case p = q, too. Moreover, a combination of the presented results with those due to Sawyer [5] gives the possibility to characterize the weights for which the two weight strong type (p, q) inequality (1 < p < q < oo) for potentials is valid. Let us nail down the notation. Let a = ( a i , . . . , a n ) be an ntuple of positive numbers with the length n, i.e. oti + • ■ • + a„ = n. We define the quasinorm x a = max \xi\a' . l<j
Observe that 2 1 *°\x\ a  \y\a <\x + y\a < 2a°\\x\a
+ \y\a),
x,y£Rn,
with a 0 = max or,. It will be assumed throughout this section that k is an aanisotropic radi ally decreasing function (shortly anARD function): k(x) = h(\x\a), x € R", with h a positive decreasing function on [0,oo). The anisotropic ball centered at x € Rn and with diameter r > 0 is the set B — B(x,r) = {y € Rn ; \y — x\a < r } . In this section we will simply talk about balls, having in mind the anisotropic ones. The function k is said to belong to the anisotropic Muckenhoupt class A\ if there is a constant c > 0 such that — / k(y) dy < inf ess k(x) \±S\ J x£B B
for all (anisotropic) balls B C Rn. Let / i b e a Borel measure on Rn. We define the operator (6.2.4)
K(fu){x)
= Jk(x
y)f(y) du.(y).
180
Chapter 6. Potentials and Riesz transforms
Especially, K(f)(x) measure.
will be the above integral with respect to the Lebesgue
T H E O R E M 6.2.1. Let 1 < p < q < oo, 1 < s < p. Assume that v and fj. are a weight function and a nontrivial Borel measure, respectively. Let the operator K be given by (6.2.4) where the kernel k is an aARD function satisfying the Aicondition. Then the weak type inequality (6.2.5)
v{ x e Rn ; Ktftfiiz)
> A ft* <
j\\f\\L>
holds for all nonnegative measurable functions f and all A > 0 if and only if the following condition is satisfied: A constant A\ exists such that (6.2.6)
< A^vQ)1'"'
\\XBl<(XBv)nL,'.'
< oo
for all (anisotropic) balls. The constants a and A\ are independent of f, v, ip, and fj. The general idea of the proof goes back to Sawyer [4]. It is postponed after a covering lemma of the Whitney type: LEMMA 6.2.2. Given an open subset Q of Rn with a nonempty boundary dd, T > 1, Tj > ar, then there exists a sequence (finite or infinite) of (anisotropic) balls B{ = B(xt,r,), i = 1,2,..., and a constant 6 = 0(n, a, r,rj) such that (6.2.7)
n =
(jBi, t
(6.2.8)
B(xi,t)n)n(Hr\n)?t,
1 = 1,2,...
and (62.9)
J2 XB(Xl,rr,) < 0Xn(x),
x £ II.
i
Proof: XQ € R"\£l.
Let d(x) =
i n f j z  y\a, r(x) = d(x)/n,
x G ft and choose
Certainly we can assume, using the appropriate shift, if necessary,
6.2. Two weight weak and strong type inequalities 181 that x0 = 0. Put d0 = 0, dj = [a2(r) + T)/(T)  o r ) ] ' " 1 , j = 1,2,..., and define fij = {x € fi;^_i < x a < dj}, j = 1,2,.... Fix 6 € (0,1) and j G AT. Put Rj! = sup r(x) < dj/r}. As RjX < oo, there is x j X G B; such that xgrij
iyi == r(Xjl ) > « f i j l .
Xj i,... ,Xjm have been chosen and
(62.10)
VjCijBji i
with Bj, = B(xji,rj
j), we stop. If (6.2.10) does not hold, we go on, putting m
Rj m+i = sup { r(x); x Gfij\ ( J 5; i } and choosing x / m + i G fij \ U ^ i « > 6Rjm+i.
m sucn a
i
»=i
manner that r ; m + i = r ( x j m + i )
If the sequence {XJ ,}, is finite, then
(6.2.11)
Bj
c\JBit. i
Suppose now that {XJ,}, is infinite and assume that (6.2.11) does not hold. Fix i > m. We have Xj, ^ Bjm, rjm > SRjm > SRji > Srji, and if y G B(xjm, jrjm) with j  6/(a(l + <5)), then \yxji\a
> ax\xji
xjm\a
 \yxjm\a
> ( a  1 y)Srji
Whence y £ B(xji,yrji) which means that the balls B(xji,jrji) disjoint. At the same time, M a < 0.(\xj m\a + jrj m) < a ( l + j/T])dj
 jrjt
.
are pairwise
,
in other terms, 5 ( x j m , r j m ) C B(0,a(l + y/r))dj). We have allowed for the existence of some x G fij \ [jBji. Further, Xjm > 6Rjm > 6r(x) > 0 for all m. Thus
s
5(0,a(l+7//?H)>^B(xjm,rjm),
the sum on the right hand side being infinite and each term of it is bigger than (2~f6r(x))n. This contradiction proves (6.2.10). Now we estimate [CXBCiji.rry;)^) ^ or V e ^ Choose z in such a way that i
(6.2.12)
y£B{xji,Trji)\B{xji,irji).
182
Chapter 6. Potentials and Riesz transforms
Let m be the smallest i for which (6.2.12) is true. We have a(r + T))rj, > a(\y  Xji\a + dxj {) > d(y) > a  1  y — Xj, 1
> (a 7/further, for z 6 \zy\a
d(xji)
T)xjm,
B(xji,jrji), < a(\zZji\a
+ \VXji\a)
< S~1a(T
+ j)xj
m
.
Whence
(6.2.13)
B[XH
7(77  ar)
B
\
{^^V)r^)CB^^ CB(y,a6~\T
+
y)r]m)
and so (6.2.12) is satisfied by not more indeces i than ( r + 7j)(7 /a33(r ^)(r' + 7)y t)Y
(a
V \
75(77 IT) r) 7(5(77  aar)
J)
as the balls on the left hand side of (6.2.13) are pairwise disjoint. If 6 is chosen sufficiently close to 1, then
(6
"4)
g M ( „,„,„ M <2 + (gl±^I±i))", ,zn,
as y can be contained at most in one of the balls
B(xji,yrj{).
Now let {Bi} be a renumeration of {Bji}jj. The assertion (6.2.7) is obviously implied by (6.2.11) and (6.2.8) follows from the construction of Bi. It remains to check that (6.2.9) holds. Let y £ 2?(a^_i,,Trj_i,) and z G B(xj+n,rrj+n) for some j > 2. Then \y\a < a(\xjU\a
+ Ti)ld(zjU))
< a(l + T T / " 1 ) ^  !
= (a" 1  Trjl)dj < a  ^ i j + u U 
rrl1d{xj+lt)
e <4 + 2
I
77 — ar
)
■
6.2. Two weight weak and strong type inequalities 183 Now we are prepared to give Proof of Theorem 6.2.1: First the necessary part. Let (6.2.5) hold and E C Rn be such that 0 < fi(E l~l Q) < oo. Then K(XEnBipn) > inf k(x  y) / ip d/j. = 2A > 0, x,y€B J EnB
x G B,
and i>5 n {xi eE Rfln n; K(xEn ii>/j.)(z) > A vB < < v{ K(XEr\L A}} B4>n)(x)
< (A/A)« XB nL,.B i;. <(A/\y\\xEnB\\" n Bylp << OO. oo. = (A/\yn(E (A/\Yn(E nfl)«/" By Proposition 5.1.1 and (6.2.5) we have c0 sup < j / K(xBv)ipfdfi \\XBK(XBv)ip\\ K{XBv)H>fdn ;; ll/IUj\\XBK{xBv)nLt>rP<.> !
S < cCo" j ^ 1s usup p I
Jv{y&B;K(frl> Jv{y&B; K(f^)(y) > A } dX ; ;ll/lkr   /   £ j . << i1 1 ti)(y)>X}dX 0
OO OO
^KCQ C Q1 1
fmm{vB,(AXy}dX
o0 1 )!/,, = CQ c 0  q'A(vE g'A(t;5) ^, 1
so that (6.2.6) is fulfilled with Ax < c~xq'A. We prove the sufficiency. Let (6.2.6) hold, we show that (6.2.5) holds for all A > 0 and all / supported in some ball, (6.2.15)
supp / c f l ( 0 , r ) .
A density argument will conclude the proof. Observe that the function K(fi{>fi) satisfies the A\ condition. Therefore the anisotropic maximal operator Mg{x) = supi^T / \g{y)\dy B3x D J B
184 Chapter 6. Potentials and Riesz transforms is majorized in the following way: (6.2.16)
x e Rn.
M{K{ftil>n)){x)
The set fiA = {x G R" ; M{K(fipn)){x) > cA} is open. We distinguish two cases. If d£l\ ^ 0, then according to the preceding lemma there is a decomposition fiA = ti
where 5 , = B(xj,r,), with x, and r, satisfying (6.2.8) and (6.2.9), r = 2a, T) > 2a 2 . Put 5 ? = B(xi,Tn). We have (6.2.17)
k(x) < ctnk(y)
for i < 1, \y\a < t\x\a.
Indeed, for all B and almost all y' £ B,
±Jk(z)dz
Therefore, if t\ > t, then
w^mm I tn
k{z)dz
s(o,*U)
h < k(z) dz J  \B(0,h\:t\*)\ B(o,*„; B(0,*„)I
< <
y€Rn\Bf.
Then \z  y\a
K(XR"\Brfil>n)(x) =
By (6.2.7)
I k(x y)f(y)i>(y) d(i(y)
R»\B:
c"
J
]B\k(zy)dzf(yMy)My)
<~jk(f^)(z)dz ' B,
~ ClT,n \B{x],m)\
JB(*i,w)K(Mf*)(z)dz Z
6.2. Two weight weak and strong type inequalities
185
where cx = ca"(l + 2a) n . It is 5(a;,,7?r,) \ fiA ^ 0 and in virtue of (6.2.8), tf(XK»VB/^)(z)
> 7 A } n Bi C { x G Rn ; K(XB;fM(x)
> 7^/2 } D £,
provided 7 > 2cc"?7n. For i with vB* > 0 and (6.2.18)
 ^ j
K{XB:fMdv
where /? G (0,1) is a fixed constant, we have v({ x€Rn;
K(frpfi)(x) > 7A } n 5 0 < ^
(6.2.19)
/ K(XB> SM dv
< — vB*t . ~ 7
If i is such that vB* > 0, but (6.2.18) does not hold, then using (6.2.6), we get q
X"VB: < r
(vB:y" J
= /39(VB:)1"
4
K^/MA"
J KixBvWdp \B: I
< c0p\vB*)l*\\xB:K{XB:K{XB*v)n\qLr,,
XB;/HI;.
 
K(W»)(x)
> 7A } 0 Bi) < vB*{ <(^r9Ax\\XB:f\\U..
186
Chapter 6. Potentials and Riesz transforms
Summing (6.2.19) and (6.2.20), and using (6.2.9), we obtain v{x G Rn ; K(fi>n)(x)
2yl90v{nx)
>jX}<
+ (/Wr»Ai*'/'/«;. which by (6.2.14) yields
(7\yv{x
6 Rn ;K(Mn)(x)
> jX}
q l
(6.2.21)
xeRn;
< 26i  f}v{
K(fip»)(x)
>X}
q
+ 7'Ale'tT \\f\\l? Now let us consider the second case 0* = R" ■ Let a ball B C Rn contain the support of/. We use (6.2.16) and (6.2.6) to get vB < i f K(Mfi)
dv=lf
B
K(XBv)rl>f dn
B
C
< i\\XB{KXBv)nL>>.'
(6.2.22)
{vB)ih<
jALmt,..
The right hand side of (6.2.22) is independent of B so that B can be replaced by Rn on the left hand side. The inequalities (6.2.21) and (6.2.22) give a "good Ainequality" {cjXYv{ xeRn\
K{Wii){x)
> cyX }
< c2Pv{ xeRn;
K(Mii)(x)
>X} + c3A{\\f\\LV
where c2 = 26yq~1c^ and c 3 = ma.x{y01/p /3'1, ccoy}. Taking the supremum over 0 < X < t/(cj), we get sup X"v{ x£Rn;
K(/V>/i)(x) > A }
0
(6.2.23)
< c2/3 sup X*v{ x6Rn] 0
+ ^!ll/IUr
K{fi>n){x) > X }
6.2. Two weight weak and strong type inequalities 187 It remains to prove that the left hand side of (6.2.23) is finite for all* > 0 and to make a choise of an appropriate /? £ (OjCj 1 ). The desired inequality (6.2.6) will follow with A < AiC3{l  c 2 /?) _1/ «. The function A !► v{x e R" ; K(fipn)(x) > A } is decreasing so let us consider small A. If inf K(ftpfi)(x) = Ao > 0, then by (6.2.22) we have Wv{x e Rn ; K{f*ii){z)
> A} < cji4j/«j.
for A G (0,A 0 ). Let A0 = 0. It is K{frp/i)(x) < cK(fxpn)(z) for \x\a > a2 z (\ \a + 2r). So suppose that A = cK(fipn)(z) for large \z\a, say, \z\a > 2ar. Then { x £ Rn ; K(Mn){x) > A } C B* = 5(0, a2{\z\a + 2r)). Taking into account that f,la, + ^ \z  x\a < 5a 4 z  x\a, a \z\ar we obtain from (6.2.17) that I* ~ V\« <
a3(l
xeB,
l
c(ba4)nk(x
k(z x)<
y)<
y € B*,
c ( 5 a 4 ) " ^ j k(x  y) dv(y) B'
c(5a 4 )" ■K{fi>ix){x). < vB* Finally, c(ba4)n C \\XBK{XB' < ^\\XBl<(XB'V)n Lr'' vB* 4n 11 i A 11 < c(5a) )"
\\XBKZ
W\\Lj..
<
and \*v{ x€Rn\
K{Mn){x)
> A}< j
/' *(*  y)f(y)rl>(y) dfi(y) J vB*
D
188 Chapter 6. Potentials and Riesz transforms THEOREM 6.2.3. Let l< s
oo, 0 < 7 < n and w a. weight on
\{x€Rn;\I1f(x)\>\}\
(6.2.24)
f€l£,
holds with a constant c > 0 independent off if and only if \\XB(x,r)lw\\LTl,
(6.2.25)
x
€
#»,
r > 0,
with some ex > 0 independent ofx and r. Proof: Let (6.2.25) be true. We show that (6.2.26)
XB(,,r)J r )xj,(,,r)/H 1 y.'
Clearly,
XB(
*' r )
/7
dy *  y\ni
1 w(z)
B(x,r)
(6.2.27)
<
XB(x,r)
/
dy \zy\nr
1 w(z)
B(x,2r)
Lf
On the other hand,
n 7=
J k  j / i  fc5= l2 
B(r,2r)
fc
y
r<zy<2*+1r
dy zJ/«7
Therefore, from (6.2.27) we get
XB(x,r)
j J3(r,r)
dy
1
\zy\^M7)
< carlxBO^HIjV.'
Consequently, (6.2.25) implies B(«, r)rfp'\\XB(x,r)(I,XBM)M\L,,
)M\Li.
< c 4 r  /c
This means that we can apply Theorem 6.2.1 to prove (6.2.24).
6.2. Two weight weak and strong type inequalities 189 The converse implication also follows from Theorem 6.2.1. Just realize the obvious estimate /
\zy\ni
> const. ( 2 r ^ ,
z€B(x,r).
D We are going to give a pellucid characterization of couples of those weights for which the two weight norm inequality holds for Riesz potentials. First, let us recall a result due to Sawyer [2]: T H E O R E M A. Let 1 < p < q < oo. Then the inequality j \Iyf(x)\"v(x)dx\
\f{x)Yw(x)dx
j
holds for all f € Uw with a constant c\ independent of f if and only if the following two condition are satisfied:
ylq'
(
J(Iy(XBv)Y'(y)w1"'(y)dy
Jv(y)dy\
< co
and
tip 1
J(Ir(XBW ^')y(y)v(y)dy
n
I
l
Jw ^\y)dy
J < °°
\ B
for all (isotropic) balls B, the constant c2 and C3 being independent of B. Combining Theorem 6.2.1 in the particular case dp. = w(x)dx, ip = 1/w, with Theorem A and using duality, the following result can be obtained. Ob serve that the condition is simpler and verifiable in an easier way than that previously known (Theorem A). T H E O R E M 6.2.4. into LI if and only if
Let 1 < p < q < 00. TJien Iy is bounded from L&
W sup (vB(x, r)fl< %*0
n
j \k»l>r
\x  y^^'w1"'
< 00
(y) dy /
190
Chapter 6. Potentials and Riesz transforms
and sup (w1p'B(x,r))1'p' xefi"
f
r>0
\xy\^n^v(y)dy\
< oo.
I , ,
/
\k»l>»
/
6.3 One weight inequality for t h e Riesz potentials In this section, a characterization of weights in the case of both the weak type and norm inequalities for the Riesz potentials will be presented. We start with the weak type inequalities. THEOREM 6.3.1. Let 0 < y < n, I < s < p < n/j, Then the following statements are equivalent: (i) there is a constant c\ > 0 such that
l/q = 1/p 
j/n.
IIM/™7/n)IUt« < ciH/iUv for all f e Ll'; (ii) there is a constant ci > 0 such that ( U ;5(x,r)) 1 /«xB(x,o^ 7/ "" 1 H J : S ,'.' <
c2\B(x,r)\1/"'+1^
for all x G Rn and r > 0; (iii) it is sup (wB(x,r)Y^\\XR^B(x,r)W^nl\x.r'n\rl.l
< oo;
rgfl" r>0
(iv) w £ Ap where /? = 1 + q/p' ifp > 1, and /? = 1 ifp = 1. Proof: The equivalence of (ii) and (iv) was proved in Theorem 5.2.2. Theorem 6.4.1 gives (iii)o(iv). We prove (i)<=>(ii). First (ii)=*(i). According to Welland [1], for every e, 0 < c < min(7,117) there is ce > 0 such that for all nonnegative functions / € Lloc(Rn), (6.3.1)
Iyf(x)
< c£(M7_c/(x)M7+£/(i))1/2,
xeR
n
.
6.3. One weight inequality 191 As observed in Theorem 5.2.2, the condition (ii) guarantees the boundedness of the mapping / i+ M1{fw"iln) from Lp(w) into Lq(w) and we also have w e Ap. Whence there is an e G (0,min(7,n  7)) such that w G APl H Ap2 where & =
l
+ "771 " f I xx p ' ( l  p ( 7 + e))
and
&■ = ! + ^ 7 ; v »• p'(lp(7e))
If we put
fc =
(1
~~ \p
7 + e\ n J
l
I
and
9c 
j  c \
(\
1
\P
, n JI
l
,
then u> G ^l+j./p' n v l i + y p ; . Denote pi = 2g £ /g and p 2 = 2
7 / n \ \ l / 2n   n < \\(My+ eifw11\\(M < )My 1+c(fwi/")M «(/«> yE(fwi' )mLV
\m
1 •))!/: < 12 .My_n e{fvf>?" ")) 1 / 2 Hit* 1+1+e,{fvPl <\\(M (fw^n))^\^.M iUt* 2 " y_e(fw^ )) fXl^ n = \\MOik* )\\Ll7 .,.\\M^ .M y+c(fw^ = M 7 + £ {fvpl* Af^c/n(fwy/")\\ )\\Li. Llt5 ..•
Recalling Theorem 5.2.2 and the fact w £ Ap, we conclude that
\\Iy(f^/n)hv <4f\& ll/lll?. whence (ii)=>(i) is proved. Let (i) be fulfilled. Then the pointwise estimate M1{fvfiln){x)
< cl1(\f\w">/n)(x),
x G Rn,
implies the boundedness of the mapping / H^ My{fvPln) By Theorem 5.2.1 the condition (ii) is satisfied. The theorem is proved.
from Uw> into Ll°°.
□
COROLLARY 6.3.2. If the condition (iii) from Theorem 6.3.1 is satis fied for some s = SQ, 1 < so < p, then it holds for all s, 1 < s < 00. T H E O R E M 6.3.3. Let 0 < 7 < n, 1 < p < n/y, l/q  l/p  y/n. If w G Ap, /? = 1 + q/p', then the operator I : / —► I1{fw'yln) is bounded from
192
Chapter 6. Potentials and Riesz transforms
£,P" into LgJ for all v, \ < u < oo. Conversely, if the operator 1 is bounded from IJ™° into L^"° for some vQ, then w £ Ap. Proof: If the assumptions of the first part of the theorem are satisfied, then the continuity of 2 can be proved analogously as in the case of the fractional maximal function, making use of Theorem 6.3.1. For the remaining part it suffices to make use of the pointwise estimate of the fractional maximal function by the fractional integral and to apply Theorem 5.2.2. □ T H E O R E M 6.3.4. Let 0 < j < n, 1 < p < j/n, (6.3.1)
l/q = 1/p  j/n.
sup (u;(5(x,r)) 1 /«xi ? n j B (x 1 r)^ 7 / n  1 k  ■V~n\\L^'
If
< °°
x£Rn r>0
for some s, 1 < s < p, then the operator 1 is bounded from 1P^ into L^1 for all v, 1 < v < oo. Converseiy, if I is bounded form LJ," into L^ for some v, 1 < v < oo, then (6.3.1) holds for all s, 1 < s < oo. Proof: Both the statements follow from Theorem 6.3.1 and the preceding theorem.
6.4 Three weight inequalities with critical exponent The basic result is THEOREM 6.4.1. Let 0 < 7 < n, p — n/y, 1 < s < 00 and supppose that w, v, and ifr are weight functions on R". Then the following statements are equivalent: (i) there is a constant c\ > 0 such that (6.4.1)
supp ess t,(z)i /  / 7 / ( x )  (Iyf)B\
dx <
B
for all balls B C Rn and all f with a compact support;
C l /V> u .
6.4 Three weight inequalities with critical exponent 193 (ii) there is a constant c2 > 0 such that (6.4.2)
supp ess v{z)\B\^\\XBl^w)\\Tr,,
< c2.
Proof: Let B = B(y,r) be a ball in Rn, S = B(y, 2r), S* = Rn \ S. Put A = fxs, h = fxs<, and a = supp ess v(z). It is obvious that zeB
ili y i77/(x) (v) B i ^ < p y iv^x)  (77A)BI ^ B
B
(64.3)
+^
f\I1f2(x)(IJh)B\dx
\B\ J B
= h + hWe estimate 7i. It is >/J
r
h < , S  / \Iyfi(x)\dx <
9/7 /■ /
B
(«4.4)
r
/ 1 /
B U \ S

"
\f(t\\ 1/(01
\
ct"i '"
"'
dx
^
= \ B \ I m i ( l \ ^ )dL S
\ B
Obviously, 5(j/, 2r) C 5(*, 3r) if t e B(y, 2r). Therefore
j \xt\ni B
dx
~ J x*"" B(t,3r)
= f\{xeB(t,Zr); o 
B
\x t\in
> \}\d\
7/»l
/
5(<,3r)dA
o oo
+
J BT/»I
\{xeRn;\xt\<X^^}\dX
194 Chapter 6. Potentials and Riesz transforms oo
i
\n'^nU\
\B\yi*
= c4\B?'n. The last estimate combined with (6.4.4) yields
h<
\B\
%ft j 1/(01 dt < csalSp/1 J /(0I dt s
s
v(z)\S\l'p'\\xs/(v>1>)\\L,:'\\f1>\\Li:
< CTWMWL'J ■
As to the estimate I2, let us start with observing that h < ~ j j
IVaOO 
Iyh(z)\dxdz
B B
=  i  / / J KM \\x~t\»r
!£—}& dzdx \zt\»n)
~ \B\> J J BBS'
*jj?//(/tftoiii**r T H*r T i
\ S'
/
A s i , z £ B and t £ S', the following estimate holds (6.4.5)
 \x  t\*n  \ z  t\*~n\ < c6\B\l'n\x

t\»n1.
On the other hand, for the centre y of the ball B and all x € B, t £ 5 ' , obviously, 21*2/<x<<22/*. Whence, by (6.4.5), we obtain
1/(01
7 F  y"T
1 / 2 < cgalSI /'
S'
S'
+1
d<.
6.4 Three weight inequalities with critical exponent 195 Now put Bi = B{y, 2'r) and a, = sup ess v(z), i = 1,2,.... We have
h
, I'llL. , ,,
I
.
oo
/ /(f) dt R . Bi + 1
OO
n+7
.
•v /
\f(t)\dt. 1/(01*
Applying Holder's inequality for the Lorentz spaces (see Proposition 5.5.1), we get I2 < C 9  5  1 / " £ a i + 1  5 1 + 1  (  " + ^ 1 ) / "   x B , + 1 / M )   u ' . '   / V ' l k »=i
oo
< c9\B\lln\\fnz.>: £a*+il5.+il"1/p,llxBi+1/(^)llL'.'51/n •=1
< cn/Vz;S,~ • From the estimates for Ii and I2 we conclude that, indeed, (6.4.1) implies (6.4.2). Now we prove that (ii)=3>(i). It is easy to see that there is k > 1 such that for all balls B and all y G B,
J\xyyndy<^
j\xyVndy.
kB
B
Fix B and an integrable nonnegative function / supported in B. We have
(6.4.6)
(Iyf)B  (Iyf)kB
=
±jlj\xB
k~n
yrnf{y) dy
\ B
J(\xyrnf(y)dy\ kB
dx.
196 Chapter 6. Potentials and Riesz transforms By Fubini's theorem,
(Iyf)B ~ (Iyf)kB =    f f(y) I j \* ~ # " " *X B
\ B
k~n j \x y 7_ " dx ) dy. kB
I
Using (6.4.6), we get
(IyfB)(Iyf)kB
( j\*yV~ndx\
< ^B\Jm B
\ B
dy J
which gives (Iyf)B
< 2((Iyf)B

(J7/)*B))
for some k > 1. Applying Minkowski's inequality to the right hand side of the last inequality, we see that
(Iyf)B < 2 I ± J \{I,f)B ~ I^f(x)\ dx \ +
B
±J\I7f(x)(I^f)kB\dx
Integrating in the last term over kB instead over B, we get in virtue of (6.4.1) that (6.4.7)
( I 7 / ) B < c 12 sup ess
{v{z))l\\f^\\Lrj.
z£B
On the other hand, there is a nonnegative function g : R" —► R1 such that
\\g\\Ll. = 1
and
J &L dx =  X B/( W ) ^ . ' . B
Put / = XB9/i> Then, clearly,
Whv
= \\9hv = 1
6.5. Generalized potentials
197
It follows from (6.4.7) that (6.4.8)
sup ess v{z){I1f)B zeB
< cp.
Further,
L r (f \B\J B
m
dt) dx
\ J il>(t)\x  t\»r \ B
I
> C5x7/n jm_dt 
' '
)
J V(0 B
= a 1  7 / n llxB/W) L ,'.' ■ Inserting this into (6.4.8), we get sup ess t ; ( ^ )  S  l  T / n   x B / M ) I I L . ' . ' < c2 so that the condition (6.4.1) is fulfilled. The theorem is proved.
□
6.5 Three weight e s t i m a t e s for generalized potentials A necessary and sufficient condition for the triple of weights is presented here, guaranteeing validity of the weak type estimates for the generalized po tentials of the order 7, 0 < 7 < n, defined by
^(M) = /
ff
fr.
<>o.
As earlier, /3 will denote a Borel measure on R" x [0,00). If B = B(x, r) C Rn is a ball, then B — B(x,r) will stand for the cylinder B x [0,2r]. T H E O R E M 6.5.1 Let 0 < j < n, 1
are weight functions on Rn. Then the following statements are equivalent: (i) there is a constant c\ > 0 such that (6.5.1)
/?{(*,<) G Rn x [0,oo); Ty(fi>)(x,t)
> A} < ciA«/«j.
for all nonnegative functions f £ L^ and all X > 0;
198
Chapter 6. Potentials and Riesz transforms
(ii) tiiere is a constant c2 > 0 such that (6.5.2)
^5(a,5(2r
+
0) 1 / s Xfl"\B(a,r)(a. + 0 7 _ n V ' H I ^ . ' < c 2
for alia £ Rn,r>0,t>
0.
First we prove two auxiliary assertions. The following notation will be used in the sequel: If B(x, r) is a ball and N > 0, then NB(x,r) will denote the ball concentric with B(x, r) and with radius Nr. LEMMA 6.5.2. Let E C Rn be bounded and {Ba}a€A, for some index set A, be a covering of E such that sup rad Ba < oo. Then there is a finite aeA
or countably subset {ctk}k of A such that {Bak] consists of nonoverlapping balls such that every Ba, a £ A, is contained in a ball 5Bak for some ak and rad Ba < 2 rad Baic. Proof: Let E C Bo(xo,ro) and R\ = sup rad Ba. Then a£A
(J BaC B0(x0,r0 + 2R1). a£A
Moreover, there is a ball B\ such that rad B\ > Ri/2. If Ba fl B\ = 0 , then rad Ba < 2 rad B\ and thus Ba C §B\. Let Ai be the set of those a £ A for which B1r\B2^
There is a ball B2 with rad B2 > Ri/2. Define A2 = { a £ Ax ; Ba C\B2 # 0 }. Then 5 „ C 5J02 and rad 5 a < 2rad B2 for all a £ A2. Continuing this way, we get a sequence of mutually disjoint balls {Ba} possessing the following properties: (i) lim rad Sj, = 0 provided {Bk} is an infinite sequence as the balls are k—+oo
nonoverlapping and are contained in B(x0,ro + 2i?i); (ii) every Ba, a £ A has nonempty intersection with some Bk and if fc0 is the smallest such k, then Ba C 5 5 t 0 .
6.5. Generalized potentials 199 Indeed, if a ball B does not intersect any Bk, then B must be contained in all {Ba}a£A., and, consequently, rad Ba < Rj, i = 1,2, But Ri —> 0 and this is a contradiction. The lemma is proved.
□
Let us introduce the maximal function Mf(x,t)
= sup ( / 3 ( 5 5 ) )  1 / "   X B /   L  ,
1 < s < P < oo,
where the supremum is taken over all balls containing the point x and whose diameter is not greater than t. L E M M A 6.5.3. There is a constant c > 0 such that /3{(x,t)
€ Rn x [0,oo); Mf(x,t)
> A} < c A  '   /  £ j .
for all f £ LpJ and all A > 0. Proof:
Let A > 0 and BQ be a ball in R". Define Ex =
{(x,t)eBQ;Mtf(x,t)>\}
where Mtf(x,t)
= sup(^(5S))1/PxB/U
with the supremum taken over all balls B such that x 6 B, t/2 < rad B < £/2. Further, define d(x)sup{t> 0; (x,t)£Ex}. Obviously, for all x € B0 there exists a ball Bx such that (x,d(x)/2) to Bx, rad Bx < 1/2, and (l3(5Bx)r1/p\\XBj\\L>J
belongs
> ^
The family (Bx x [0,4rad Bx))x€Bo covers £ A  According to Lemma 6.5.2 there is a sequence {Bk}k C (Bx)x€Bo of nonoverlapping balls such that ev ery Bx is contained in 5Bk0 for some k0 and rad Bx < 2rad Bk0, x e Bk0. Therefore ^ Bx x [0,4rad Bx) C 55 fco .
200
Chapter 6. Potentials and Riesz transforms
Consequently, we have
Exc\j5Bk k
and < ^£>(5B*) / ? ( 5 S t ) <
"Ell**
P < A< Appl/ll l/lll • LfJr •
Letting rad Bk and £ to tend to oo, we obtain the desired inequality.
□
Proof of Theorem 6.5.1: We prove (i)=>(ii). Let a 6 Rn, r > 0, t > 0, and (x, r) 6 B(a, 5(2r + t)), i.e. x £ B(a, 5(2r +1)) and 0 < r < 10(2r + t). The following estimate holds: T7(/V>)(* r ) >
/
1ai  yy  >> rr
f(y)Hv) rf (x 2/l + r)"". 7 y
We show that \x — y\ + T < 31(a — y\ + t) for \a — y\ > r. Indeed, I*  » + r < ar  a\ + \a ~ y\ + r < 5(2r +1) + \a  y\ + 10(2r +1) = 30r + 15* + \ay\<
31a y\ + I5t
< 31(a — y + 0 so that for such x and t,
rn7(/^)(« A ,o> y 3311„" _7 7 l
j/
ai/>r 1ay\>r
f(y)i>(y) n . 7 rf2/(a — y\+t) (l« v +n t) T
In accordance with (6.5.1) we have P$(a, 5(2r + t)) < /?{(*, r) G / T x [0, oo);
r7(/^)(x,0>3r" y 7
ay + f)»T
yj
a»l>r
I
/
J
Jay>r
/(vM»)
,
\ay\>r
(ay + t)»7ay
/
+ ,
= c(//(„to.w.,r,W*'(l"4 " "»WJ \\ay\>r
)
9
6.5. Generalized potentials 201 By Proposition 5.1.2 there is a function f0 € Lp^ such that lXfl»\B(a,r)(a ~  +
/OLS,'
= *
and
tynll>/w\\Lry
< ci f /o(y)xj»\B(«,r)(y)(a  2/ + t) 7 ""V'(!/)(^(2/))" 1 ^(3/) <*y. Consequently, (l3B(a, 5(2r + i ) ) ) 1 / j < c2\\XR.\B(a,r){\a ~ I + 0 7 " n ^ / w   ~ v and (ii) is proved. Now as to (ii)=$»(i). Suppose that (/?(*" x t C o c ) ) )  1 ^ / ! ! ^ .
\\f\\zz(PB(z,5{r
+
> A. Then
t)))lf'>^
and (6.5.4)
/ L ,.(/?5(x,5(2r + 0 ) )  1 / ? >
A 2cc 2
The estimate (6.5.4) follows from our assumptions. Observe that (6.5.4) cannot hold for all r > 0. Indeed, allowing for it, then for all positive r, then
/
(]* yj+0»7 * * "Ml** IIXHVB(..r)(l« " I +
\xy\>r
< c c 2   /   i r ^ % , 5 ( 2 r + t))) 1 /» A <2'
tynHM\Lf
202
Chapter 6. Potentials and Riesz transforms
On sending r to 0, it follows that T7(/V>)(a;,<) < A which is a contradiction. Whence (6.5.3) is true for some r > 0 and, denoting by r 0 the lowest upper bound of all such r's, we see that (6.5.3) and (6.5.4) hold for r0 and
J
,
(«  !!l + %r dy ~ C H^II W*(*,o)(l* .\+ty *i>M\I£*
\xy\>r00
7
fl(*,r0)
(xyH*)»7^>2
Put 6 = lim /?B(x,5(p + f))Let &o be the sup of all k with b<2keB{x,b(r0+t)). If 6 = 0, then obviously ko — °° For each finite k, 0 < k < ko, let us choose rjc > 0 in such a manner that pB{x, 5(rfc + 0 ) < 2kpB(x,
5(r 0 + <)) < 0B(x, 2(rk + t)).
Each rfc can be obtained from r^x by multiplying rjt_i by a sufficiently small negative power of 2 so that {rk}k can be assumed to be a decreasing sequence. We have (putting formally ko — 1 = oo if ko = oo)
f{y)i>{y) 2
j
(\xy\+t)ny
J B(x,r0) to —1
(6.5.5)
/■
<£
f(y)i>(v)
j B(x,rk)\B(x,rk
+ 1)
y)^(:
' y (X ^ B(*,r* 0 )
( l i  j / l + t)" 7
;
4
dy
6.5. Generalized potentials 203 iffco= oo, then the second term on the right hand side is missing and the summing goes from 1 to oo. For all k, 0 < k
I
f(y)Hv) (\xy\
B(x,rk)\B(x,rk
+ t)ni
dy
+ 1)
< c\\XB(X,rk)f\\L>j\\XR»\B(X,rk
+ l)(\x

 + t ) 7 " " V > / ^   L P ' . '
< cc 2  X fl( I , rfc) /Up,.(/?5(x 1 5(2r, +1 + t ) ) ) _ 1 / ' <21/qcc2\\XB{XiTk)f\\Ll.(l3B(xMrk
+
t)))l/*
= 21^cc2(/35(x,5(r4+ 0))1/p1/?(/?5(x.5(rt+0))1/pIIXB(x,rk)lk' < 2*( 1 / p  1 ^c 3 (/?.B0c, 5(r 0 + 0 ) ) 1 / p _ 1 / ? M / ( a ; , t). If ko < oo, then the second term on the right hand side of (6.5.5) can be estimated in the following way:
f(y)Hy) I „ 'Wfr>
, _= ,.lim dy
B(*,rko)
f/
f(y)Hv) (\xy\+t)nr
dy
B(x,rka)\B(x,e)
< cc2\\XB(*,rka)f\\L>J lim (/?5(x,5(2^ + 0 ) ) _ l A CC2b1/g\\xB(T,rk0)f\\Ll'
=
= 2(*°+ 1 )/' CC2 (/?5(*,5(r 0 +0))" 1/ 'IIXfl(«,r„)/Us P < 2 1 /'cc 2 (/35(x,5(r t o + 0 ) )  l A   X B ( , , r k o ) /  U < 2 fc °( 1 / p  1 /»)c 3 (/?5(i;,5(r 0 + 0)) 1 / p _ 1 / 'M/(a:,*)■ Thus for A and (z,f) satisfying
{P{R^[Q^)))^l"\\f\\Ll.
<^ 
and Ty(M)(x,t)
> A,
and for the corresponding r 0 for which (6.5.3) and (6.5.4) hold, we have k
 < <* £ 2fc(1/p1/^M/(x,0(/?5(a:, 5(r0 + <)))1/p1/? = c4M/(*, *)(/**(*> 5(r0 + 0)) 1/p_1/ '
204 Chapter 6. Potentials and Riesz transforms Recalling (6.5.3), we obtain
\
v
> A} n
< /?{(*,*) £R x
[0,oo); Mf(x,t)
> A«/"/^?)/p}
0{(x,t)
€ Rn x [0,oo); T 7 / ( z , r ) > A} < c i A ~ «   /   ^
for all nonnegative functions f 6 LJ,1; (ii) there is a constant ci > 0 such that (i{ (x,t)
eR"x
[0,00); TIXE{X,
(6.5.7) >.7) (6.1
< c2A"*
t) > X }
y/p
(f
J w(y) dy E
for all measurable sets E C Rn;
6.5. Generalized potentials 205 (iii) there is a constant c 3 > 0 such that i 1/P
1
(6.5.8) (/?£(„, 5(2r + t))) '' /
(fl_y^f)n_7
£
< c3 ( J w(y) dy' \ E
)
for all a € iff*, r > 0, < > 0, and an" measurable sets E C Rn\
B(a, r);
(iv) there is a constant C4 > 0 sucii that Xil»\B(a,r)(a.+07~7HI£j,'»(/?5(a15(2r + t)))1/' < c 4
(6.5.9)
for all a € ft", r > 0, and t > 0. Ah" the constants c,, i = 1,2,3,4, are independent of f, X, E, a, r, and t. Proof: Obviously (6.5.6)=>(6.5.7) and Theorem 6.5.1 gives (iv)^(i). We prove (ii)=>(iii)=>(iv). Let (ii) hold. Going along the lines of proving (i)=>(ii) in Theorem 6.5.1, one can show that given a measurable set E C Rn, then
(/?5(a,5(2r + t ) ) ) 1 ^ < c 5
Y1
I J
x*(y)(a  y\ + r ) ^ " dy
\ay>r
/
\\XE\\LV
On taking E C Rn \ B(a, r), (6.5.8) follows. Next suppose that (6.5.8) is valid. Let E = { y € Rn \ B(a, r); (\a y\+ Then E C Rn\B(a,r)
^  " ( ^ ( y ) ) " 1 > a }.
and
a / w(x) dx < f(\a y\ + t ) 7 ~ " dy E
E 1/P
< c6 I
/ w(x) dx)
\ E
(/?5(a,5(2r + t))) 1 /«.
■
206
Chapter 6. Potentials and Riesz transforms
This means that for any a > 0,
W llq
(/3B(a, 5(2r + t))) a
I J
XE(y)w(y)
dy J
< c6.
Passing to the supremum over all a > 0, we get (/?B(a, 5(2r + 0)) 1/? IIXfi"\B(a,r)(a  I + ^ " " / H I j V  < «* ■ The theorem is proved.
□
6.6 Riesz transforms in weighted Lorentz spaces One weight weak and strong type inequalities will be considered in this section and the corresponding weights will be characterized. THEOREM 6.6.1. Let 1 < p < oo and 1 < s < oo. If a constant cx > 0 exists such that
lia/IUs < cill/llu then w € Apa. Proof: Let Q C Rn be a cube and / a nonnegative locally integrable function supported in Q such that ( / ) Q > 0. Let Qi be a cube with the same side length, having a common vertex with Q and such that x € Qi, y E Q implies Xj > j/j for the individual coordinates, j — 1 , . . . , n. If x £ Q\, then easily,
*/(«) = *. E / ( * J  «)i*" yrn_1/(j/)dy J=lQ
1.1) (6.6.1) (6.f >
fiy)
c[J
\xy\"
Q «
dy> Q
Choose a function / such that
ll/lk, == i1 II/IIL
and
yJ /(x)dxS{x)dx>c > CXWXQ/WW^.. l\\xQlw\\Lr:.. oQ
6.6. Riesz transforms in weighted Lorentz spaces 207 where ci is the constant from Proposition 5.1.2. By our assumption we have
o
Jf(y)dy
c3\QnxQH\pp,„,
< thus
(V>QI)1,P\\XQM\:%
< CZ\Q\.
Analogously, {v>Q)llp\\XQjw\\Lr.,
(66.2)
Inserting / = \Q into (6.6.1), we obtain Rf{x) > c,
x € Qi .
Whence wQi < ciw{x£Rn <
; RXQ{X)>
C}
C
A\XQ\\PLPJ
which together with (6.6.2) gives
(«"0i)1/pllxQ1/Hli,'. < c5QiJ, i.e. the J4 P< condition.
Theorem is proved.
□
T H E O R E M 6.6.2. Let 1 < p < oo, 1 < s < oo. Then the following statements are equivalent: (i) the operator R is bounded on L£,*; (ii) t i e operator R is bounded from L?5 into L£,°°; (iii) w G Ap .
208
Chapter 6. Potentials and Riesz transforms
Proof: Clearly (i)=»(ii). By Theorem 6.5.1, (ii) implies w G Ap, which is equivalent to w G Ap for all 1 < s < oo, thus (iii) holds. It remains to prove that (iii)=>>(i). If w G Ap, then w G Api for some pi < p and w G Ap3 for all Pi > p. As the operator R is bounded in l?w (see GarciaCuerva and Rubio de Francia [1, p. 411]), we get the boundedness of R in L% by interpolation (e.g. Stein and Weiss [1, Chapt. 5, Thm. 3.15]) for all s, 1 < s < oo. Theorem is proved.
□
THEOREM 6.6.3. For 1 < p < oo the following statements are equiva lent: (i) the operator R is bounded from L p l into LJ£°; (ii) there is a constant c > 0 such that
m
\wQj
holds for all cubes Q C Rn and all measurable sets E C Q; (iii) w G Api. Proof: The implication (i)=>(iii) was proved in Theorem 6.6.1. Suppose that (iii) is true. Then w G ^oo a n d the distribution functions of Rf and Mf are linked together by the estimate (Coifman and FefFerman [1]) supA p u;{x GR"; R/(x) > A} < csup Apu>{x G Rn ; Mf(x) A>0
A>0
> A}.
Now applying Theorem 5.2.1 with \j) = 1 and v = w, we get (i). As to the equivalence of (ii) and (iii), see Proposition 5.1.6. The theorem is proved.
□
Notes to Chapter 6 The weak type inequality with two weights for potentials was investigated by Sawyer [2] and the analogous two weight problem for both isotropic and
Notes to Chapter 6 209 anisotropic potentials was solved by Gabidzashvili [1] and [4] who has found a more easily verifiable necessary and sufficient condition. Theorems 6.1.2, 6.1.3, and 6.1.5 can be found in Kokilashvili [6], Theorems 6.1.4 and 6.2.2 are from [1] by Kokilashvili and Gabidzashvili, Theorem 6.2.1 will appear in the forthcoming paper Kokilashvili and Rakosnik [1]. Sawyer [5] has found necessary and sufficient conditions for validity of two weight inequalities of strong type in Lebesgue spaces for Riesz poten tials and the Poisson integral. Theorem 6.2.4 can be found in Gabidzashvili, Genebashvili and Kokilashvili [1], see also Sawyer and Wheeden [1]. The contents of Section 6.3 is essentially an amalgam of Kokilashvili's pa pers [5], [7], [8] and results in Section 6.4 are due to Muckenhoupt and Wheeden [1]. For the special case s — p, a generalization of these results to anisotropic potentials was given by Gabidzashvili [5]. Two weight weak type and one weight strong type inequality for generalized potentials on spaces of the homo geneous type were obtained by Gabidzashvili [2]. As to the critical exponent we refer to Gogatishvili [1]. The exposition in Section 6.5 follows Gabidzashvili, Genebashvili and Kokilashvili [1]. We also refer to Genebashvili [2], Genebashvili and Koki lashvili [1], Sawyer and Wheeden [1] for generalizations to spaces of the homo geneous type. For other versions of weighted Lorentz spaces see Sawyer [3], [4] and Stepanov [2].
Problems Here we list some open problems whose solution seem to resist up to now.
(1) To characterize couples of weights (v,w) for which *(A)w{x € Rn ; \Hf(x)\
> A} < c R
/&(f(x))w(x)dx. 1
The same question in the case of the extraweak inequality v{xeRn;
\Hf(x)\>X}
l$(J^p\w{x)dx. R1
This has not been solved yet even in the case <£(£) = fp, p > 1.
(2) To characterize couples of weights (v,w) for which I ${Tf{x))v(x)dx
$(f(x))w(x)dx
1
Ri
R
where T is either the maximal operator or the Hilbert transform. Also, this problem for H remains to be open even if $(t) = \t\p, p > 1.
(3) Let M' be the strong maximal function and k > n—1. It would be desirable to describe all the couples of weights (v,w) such that v{ x e Rn ; M'f(x)
>\}
i ^ i R
(1 + log+ ^ ^ \
w(x) dx.
n
For v = w and k = n — 1, the solution was given in Chapter 4.
(4) Let Hn be the multiple Hilbert transform,
Hnf(x)= ff(y)f[±dy. 210
Problems 211 The question is about a characterization of weights for which nl
w{ x e Rn ; \Hnf(x)\ >\}
(l + log+ ^ A
M
w{x) dx.
Another problem is, of course, is the two weight norm inequality for the strong maximal function.
(5) To give a characterization of weights w for which \\Hf\\L*j < c\\f\\LC,
1 < s < oo,
1 < p < oo.
Recall that in Section 6.6, a necessary and sufficient condition on ID is found, guaranteeing validity of \Wf\\L'j
< CII/IUM ,
K
« < oo,
1 < p < oo.
(6) A problem arises of a characterization of those w for which the fractional maximal function M1 satisfies
\\My(fwy'"l)\\Ll.
0<7
p
q
7
For 7 = 0, this has been solved in Section 5.3.
(7) To solve one and two weight problems for the multiple Hilbert transform in Lorentz spaces.
(8) To find an easily verifiable condition on couples (v,w) so that \\Myfhy
< Il/lln»'.
K
P < 9 < oo, 1 < s, r < oo.
An analogous problems for potentials.
(9)
Let p = ( p i , . . . , P n ) , 1 < Pi < oo, i = l,...,n. One asks about a characterization of weights w(x) = w(xi,... ,xn) for which the multiple
212 Problems Hilbert transform is bounded in weighted mixed norm spaces Lp(w) with the norm
Il/llV) = / • ■ ( / l/(*)lP"™Wd*n)
\
The answer seems to be known only in the case w(x) = w{x\)...
dxA
.
w(xn).
(10) To solve one and two weight problems for singular integrals in the ideal spaces (see. e.g. Krein, Petunin and Semenov [1] for the definition).
References D . R. Adams: [1] Trace inequality for generalized potentials. 99105.
Studia Math. 48(1973),
K. F. Andersen: [1] Weighted norm inequalities for Hilbert transforms and conjugate func tions of even and odd functions. Proc. Amer. Math. Soc 56(1976), 99107. K. F. Andersen and W.—S. Young: [1] On the reverse weak type inequality for the Hardy maximal function and the weighted classes L(logL)k. Pacific J. Math. 112(1984), 257264. K. I. Babenko: [1] Oil conjugate functions (Russian). Dokl. Akad. Nauk SSSR 62(1948), 157160. R. J. Bagby: [1] Weak bounds for the maximal function in weighted Orlicz spaces. Studia Math. 95(1990), 195204. [2] Maxima/ functions and rearrangements: Some new proofs. Indiana Univ. Math. J. 32(1983), 879891. R. J. Bagby and D. S. Kurtz: [1] L(log L) spaces and weights for the strong maximal function. J. Analyse Math. 44(1984/85), 2131. C. Bennett and R. Sharpley: [1] Interpolation of operators. Academic Press 1988. F. J. Ruiz Blasco and J. L. T o r r e a H e r n a n d e z : [1] Weighted and vectorvalued inequalities for potential operators. Trans. Amer. Math. Soc. 295(1986), 213232. S. Bloom and R. Kernian: [1] Weighted norm inequalities for generalized Hardy operators, (personal communication).
213
214 References D. W . Boyd: [1] Indices of function spaces and their relationship to interpolation. J. Math. 21(1969), 12451254.
Canad.
P. L. Butzer and F. Feher: [1] Generalized Hardy and HardyLittlewood inequalities in rearrangementinvariant spaces. Comment. Math. Special Issue (Tomus specialis in honorem Ladislai Orlicz). PWN, Polish Acad. Sci., Warsaw 1978, pp. 4164. C. P. Calderon: [1] On the existence of singular integrals near L1. J. 32(1983), 615633.
Indiana Univ. Math.
A. Carbery, S.—Y. Chang, and J. Garnett: [1] Ap weights and LlogL. Pacific J. Math. 120(1985), 3345. L. Carleson: [1] Interpolation by bounded analytic functions and the corona problem. Annals of Math. 76(1962), 547559. Y .  M . Chen: [1] Theorems on asymptotic approximations. 360407. F. Chiarenza and M. Frasca: [1] Morrey spaces and HardyLitlewood 7(1987), 273279.
Math. Annalen 140(1960),
maximal function.
Rend. Mat.
H. M. Chung, R. A. Hunt, and D. S. Kurtz: [1] TJie HardyLitlewood maximal function on L(p, q) spaces with weights. Indiana Univ. Math. J. 31(1982), 109120. R. R. Coilman: [1] Distribution function inequalities for singular integrals. Acad. Sci. USA 69(1972), 28382839.
Proc. Nat.
References 215 R. R. Coifman a n d C. Fefferman: [1] Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51(1974), 241250. A.Cordoba and R. Fefferman: [1] A geometric proof of the strong maximal theorem. (1975), 95100.
Ann. Math. 102
I. I. Danilyuk: [1] On boundedness of singular integral operators in Lp spaces with weight (Russian). Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 33(1967), 3244. E. M. Dynkin a n d B. P. Osilenker: [1] Weighted estimates for singular integrals and their applications (Russian). Itogi Nauki i Tekhniki, Matematicseskii analiz Vol. 21, Moscow 1983, pp. 42129. N . A. Fava, E. A. G a t t o a n d C. G u t i e r r e z : [1] On the strong maximal function and Zygmund class L(log+ L)n. Studia Math. 69(1980), 155158. C. Fefferman a n d E. M . Stein: [1] Some maximal inequalities. Amer. J. Math. 93(1971), 107115. R. Fefferman a n d E. M . Stein: [1] Singular integrals on product spaces. Adv. Math. 45(1982), 117143. F . Forelli: [1] The Marcel Riesz theorem on conjugate functions. Trans. Amer. Math. Soc. 106(1963), 369390. Fujii: [1] Weighted bounded mean oscillation and singular integrals. Math. Jap. 22,5(1978), 529534.
216 References M. Gabidzashvili: [1] Weighted inequalities for anisotropic potentials (Russian). Naucsnye Trudy Gruzinskogo Politekhnicseskogo Instituta 3(1984), 4856. [2] Weighted norm inequalities for potential type integrals and fractional maximal functions on homogeneous type spaces (Russian). In: Trudy Konf. Molod. Ucsennykh MGU, Moscow 1985, pp. 1518. [3] Potential type integrals and maximal functions in weighted spaces (Russian). Thesis, University of Tbilisi 1986. [4] Weighted inequalities for anisotropic potentials (Russian). Trudy Tbiliss. 82(1986), 2536. [5] Weighted inequalities for anisotropic potentials (Russian). Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 89(1988), 311. M. Gabidzashvili, J. Genebashvili, and V. Kokilashvili: [1] Two weight inequalities for generalized potentials (Russian). To appear in Trudy Mat. Inst. Steklov in 1991. D. Gallardo: [1] Orlicz spaces for which the HardyLittlewood bounded. Publ. Matematiques 32(1988), 261266.
maximal operator is
V. F. Gaposhkin: [1] One generalization of M. Riesz theorem on conjugate functions. Sb. 46,3(1958), 359372.
Mat.
J. Garcia—Cuerva and J. Rubio de Francia: [1] Weighted Norm Inequalities and Related Topics. North Holland, Ams terdam 1985. J. Garnett: [1] Bounded analytic functions. Academic Press, New YorkLondonTorontoSydneySan Francisco 1981. F. W. Gehring: [1] The Lp integrability of the partial derivatives of a quasiconformal map ping. ActaMath. 130(1973), 265277.
References
217
I. Genebashvili: [1] Two weight problem for fractional order maximal function defined on spaces of the homogeneous type (Russian). Soobshch. Akad. Nauk Gruzin. SSR 135, 2(1989), Part II, 5356. [2] Carleson measures and potentials defined on spaces of the homogeneous type (Russian). Soobshch. Akad. Nauk Gruzin. SSR 135,3(1989), 505508. [3] Two weight problem for fractional order maximal functions in Lorentz spaces (Russian). Soobshch. Akad. Nauk Gruzin. SSR 136,1(1989), 2124. [4] Two weight estimates for generalized fractional order maximal functions and potentials (Russian). Thesis. Tbilisi State University 1990. I. Genebashvili and V. Kokilashvili: [1] Weighted norm inequalities for fractional maximal functions and in tegrals defined on homogeneous spaces. Preprint, A. Razmadze Math. Inst. Georg. Acad. Sci., Tbilisi 1991, 18 pp. A. Gogatishvili: [1] Potentials in weighted Lorentz spaces for critical exponent (Russian). In: Reports of Enlarged Sessions of the I. N. Vekua Institute of Applied Mathematics Vol. 3, No. 2, Tbilisi 1988, pp. 2528. [2] Weighted norm inequalities for potentials in spaces of the homogeneous type (Russian). Soobshch. Akad. Nauk Gruzin. SSR 129,3(1988), 493495. [3] Riesz transform and maximal functions in $(L) classes (Russian). Soob shch. Akad. Nauk Gruzin. SSR 137(1990), 489492. [4] Riesz transforms in weighted LlogL classes (Russian). In: Union School on Operator Theory in Function Spaces, Abstracts of Lectures, Ulyanovsk 5.12.9.1990, p. 68. [5] Riesz transforms and maximal functions in MorreyCampanato type spaces and in weighted Zygmund spaces (Russian). To appear in Reports of Enlarged Session of the Seminar of I. N. Vekua Institute of Applied Math ematics, Tbilisi State University 1990. [6] Riesz transforms and maximal functions in
218 References [8] Weak type inequality for strong maximal functions (Russian). In: Ab stracts of Symposium on Continuum Mechanics and Related Problems of Anal ysis, Tbilisi 5.12.6.1991. [9] Two weight inequalities of weak type for maximal functions and Riesz transforms in Orlicz classes. To appear in Soobshch. Akad. Nauk Gruzii. [10] Weak type weighted inequalities for maximal functions with respect to a general basis. To appear in Soobshch. Akad. Nauk Gruzii. A. Gogatishvili, M. Kokilashvili, and M. Krbec: [1] Maximal functions in $(L) classes (Russian). Dokl. Akad. Nauk SSSR 314,3(1990), 534536. [2] Maxima] functions in $(L) classes (Russian). To appear in Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzii . A. Gogatishvili and L. Pick: [1] Weighted inequalities of weak type and extraweak type for the maximal operator and the Hilbert transform. Preprint, Math. Inst. Czech. Acad. Sci., Prague 1991. M . de G u z m a n : [1] Differentiation of Integrals in Rn. Lecture Notes in Math., Vol. 481, SpringerVerlag, BerlinHeidelbergNew York 1975. (Russian transl.: Mir, Moskva 1978.) [2] Real Variable Method in Fourier Analysis. NorthHolland, AmsterdamNew YorkOxford 1981. R. I. Gurielashvili: [1] On the Hilbert transform. Analysis Mathematica 13(1987), 121137. J . Gustavsson and J . P e e t r e : [1] Interpolation of Orlicz spaces. Studia Math. 60(1977), 3359. G. H. Hardy and J. E. Littlewood: [1] Some properties of fractional integrals. Math. Z. 27(1927), 565606. [2] A maximal theorem with functiontheoretic applications. Acta Math. 54(1930), 81116.
References 219 [3] Some theorems on Fourier series and Fourier power series. Duke Math. J. 2(1936), 354381. H. P. Heinig and R. J o h n s o n : [1] Weighted norm inequalities for Lrvalued integral operators and appli cations. Math. Nachr. 107(1982), 161174. H. Helson and D. Szego: [1] A problem in prediction 107138.
theory.
Ann. Mat. Pura Appl. 51(1960),
R. A. Hunt: [1] On L(p,q) spaces. L'Ens. Mathematique 12(1966), 249276. R. A. Hunt and D . S. Kurtz: [1] The HardyLittlewood Math. J. 32(1983), 155158.
maximal function on L(p,l).
Indiana Univ.
R. A. Hunt, B. Muckenhoupt, and R. L. Wheeden: [1] Weighted norm inequalities for the conjugate function and Hilbert trans form. Ttans. Amer. Math. Soc. 176(1973), 227251. V. P. Il'in: [1] Some integral inequalities and their applications in the theory of differ entiate functions of several variables (Russian). Mat. Sb. 54,96,3(1961), No. 3, 331380. T. Iwaniec: [1] On IPintegrability in PDE's and quasiregular mappings for large ex ponents. Ann. Acad. Sci. Fenn. Ser. A I Math. 7(1982), 301322. B. Jessen, J. Marcinkiewicz, and A. Zygmund: [1] JVote on differentiability of multiple integrals. Fund. Math. 25(1935), 217234. P. Jones: [1] Factorization of Ap weights. Annals of Math. 111(1980), 511530.
220 References M. Kaneko and S. Jano: [1] Weighted norm inequalities for singular integrals. J. Math. Soc. Japan 27,4(1975), 570588. S. S. Kazaryan: [1] Inequalities in Orlicz spaces for one maximal function (Russian). Izv. Akad. Nauk Arm. SSR 22,4(1987), 358377. [2] Integral inequalities in weighted reflexive Orlicz spaces for the conjugate function (Russian). Izv. Akad. Nauk Arm. SSR 25,3(1990), 261273. R. Kerman and A. Torchinsky: [1] Integral inequalities with weights for the Hardy maximal function. Studia Math. 71(1982), 277284. S. V. Khrushchev: [1] A description of weights satisfying the Aoo condition of Muckenhoupt. Proc. Amer. Math. Soc. 90(1984), 253257. B . V. Khvedelidze: [1] The method of Cauchy type integrals in discontinuous boundary value problems of the theory of holomorphic functions of a complex variable (Rus sian). Itogi Nauki i Tekhniki, Seria Sovremennye problemi matematiki, Vol. 7, Moscow 1975, 5162. (English transl.: J. Soviet Math. 7,3(1977), 309415. V. Kokilashvili: [1] On boundedness of singular integral operators in Lp spaces with weight (Russian). In: Proceedings of Symposium on Continuum Mechanics and Re lated Problems of Analysis, Tbilisi 23.29.9.1971. Metsniereba, Tbilisi 1973, pp. 125141. [2] On traces of functions with partial derivatives from Orlicz classes. Com ment. Math. Special issue (Tomus specialis in honorem Ladislai Orlicz). PWN, Polish Acad. Sci., Warsaw 1978, pp. 183189. [3] Singular integrals in weighted Orlicz classes (Russian). Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 89(1988), 4250. [4] Weighted estimates for classical integral operators. In: Proceedings of the International Spring School "Nonlinear Analysis, Function Spaces and
References
221
Applications IV", Roudnice nad Labem (Czechoslovakia), May 2125, 1990. TeubnerTexte zur Mathematik, TeubnerVerlag, Leipzig 1990, pp. 86103. [5] Maximal Functions and Singular Integrals in Weighted Function Spaces (Russian). Metsniereba, Tbilisi 1985, 114 pp. [6] Three weight problem for integrals with positive kernels. Preprint, Math. Inst. Georg. Acad. Sci., Tbilisi 1991. To appear in Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzii . [7] Weighted inequalities for maximal functions and fractional integrals in Lorentz spaces. Math. Nachr. 133(1987), 3342. [8] Anisotropic maximal functions and potentials in weighted Lorentz spaces (Russian). Trudy Mat. Inst. Steklov 180(1987), 136138. (English transl.: Proc. Steklov Inst. Mat. 3(1989), 159161.) [9] Fractional integrals on curves. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 95(1990), 5670. [10] On a weight problem for integrals with positive kernels. Bull. Georg. Acad. Sci. 140,3(1990), 145148. [11] Maxima] functions and integrals of potential type in weighted Lebesgue and Lorentz spaces (Russian). Trudy Mat. Inst. Steklov 172(1985), 192201. (English transl.: Proc. Steklov Inst. Mat. 172,3(1987).) [12] Maxima] inequalities and multipliers in weighted LizorkinTriebel spaces (Russian). Dokl. Akad. Nauk SSSR 239(1978), 4245. (English transl.: Soviet Math. Dokl. 19(1978), 272276. [13] On weighted LizorkinTriebel spaces. Singular integrals, multipliers, imbedding theorems (Russian). Trudy Mat. Inst. Steklov 161(1983), 125149. (English transl.: Proc. Steklov Inst. Mat. 3(1984), 135162. [14] Maximal functions in weighted spaces (Russian). Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 65(1980), 110121. V. Kokilashvili and M. Gabidzashvili: [1] Two weight weak type inequalities for fractional type integrals. Preprint # 4 5 , Math. Inst. Czech. Acad. Sci. Prague 1989. [2] Weighted inequalities for anisotropic potentials and maximal functions (Russian). Dokl. Akad. Nauk SSSR 282,6(1985), 583585.
222 References V. Kokilashvili and M. Krbec: [1] Weighted inequalities for Riesz potentials and fractional maximal func tions in Orlicz spaces (Russian). Dokl. Akad. Nauk SSSR 283(1985), 280283. (English transl.: Soviet Math. Dokl. 32(1985), 7073.) [2] On boundedness of anisotropic fractional order maximal functions and potentials in weighted Orlicz spaces (Russian). Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 82(1986), 106115. [3] Maximal functions in 4>(L) classes and Carleson measures. Soobshch. Akad. Nauk Gruzin. SSR 137(1990), 269271. V. Kokilashvili and A. Kufner: [1] Fractional integrals on spaces of homogeneous type. Comment. Math. Univ. Carolinae 30,3(1989), 511523. V. Kokilashvili and J. Rakosnik: [1] A two weight weak type inequality for potential type operators. appear in Comment. Math. Univ. Carolinae 32(2) (1991), 251263.
To
A. Kolmogorov: [1] Sur les fonctions harmoniques conjugees et les series de Fourier. Fund. Math. 7(1925), 2328. A. S. Krantsberg: [1] On the basis of a Haar system in weighted spaces (Russian). Trudy Mosk. Inst. Elektroniki i Mashinostroeniya 24(1972), 1426. M. A. Krasnoselskii and J. B. Rutitskii: [1] Convex functions and Orlicz spaces. Noordhof, Groningen 1961. (Origi nal Russian edition: Gos. Izd. Fiz. Mat. Lit., Moskva 1958.) M. Krbec: [1] Modular interpolation spaces I. Z. Anal. Anwendungen 1(1982), 2540. [2] Two weight weak type inequalities for the maximal function in the Zygmund class. In: Function Spaces and Applications. Proceedings of the U S Swedish Seminar held in Lund, June 1986. Lecture Notes in Mathematics., Vol. 1302, SpringerVerlag BerlinHeidelbergNew York 1988, pp. 317320.
References 223 [3] Weighted norm inequalities in Orlicz spaces. In: Function spaces, differ ential operators and nonlinear analysis. Research Notes in Mathematics Series, Longman Sci. &; Tech., Harlow 1989, pp. 7788. S. G. Krein, J. I. Petunin, and E. M. Semenov: [1] Interpolation of Linear Operators (Russian). Nauka, Moskva 1978. Eng lish transl.: Translations of Mathematical Monographs, Amer. Math. Soc, Providence, R. I. 1982. L. D . Kudryavcev and S. M. Nikolskii: [1] Spaces of differentiate functions of several variables and imbedding theorems (Russian). Itogi Nauki i Tekhniki. Seriya "Sovremennye Problemi Matematiki. Fundamentalnye Napravleniya". Vol. 26(1988), 5148. P. I. Lizorkin: [1] Multipliers of Fourier integrals and estimates for convolutions in spaces with mixed norm. Applications (Russian). Izv. Akad. Nauk SSSR, Ser. Mat. 34(1970), 218247. G. G. Lorentz: [1] On the theory of spaces A. Pacific J. Math. 1(1951), 411429. W . A. J. Luxemburg: [1] Banach function spaces. Thesis, Delft 1955. L. Maligranda: [1] Generalized Hardy inequalities in rearrangement invariant spaces. J. Math. Pures et Appl. 59,3(1980), 405415. [2] Indeces and interpolation. Dissertationes Mathematicae #234, 154. Polish Sci. Publ., Warsaw 1985. F. J. Martin—Reyes: [1] New proofs of weighted inequalities for the onesided functions. Preprint.
HardyLittlewood
F. J. MartinReyes, P. Ortega Salvador and A. dela Torre: [1] Weighted inequalities for the onesided maximal functions. Amer. Math. Soc. 319(1990), 517534.
Trans.
224 References W . Matuszewska and W. Orlicz: [1] On certain properties of$ functions. Bull. Acad. Polon. Sci. 8(1960), 439443. B. Muckenhoupt: [1] Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165(1972), 207226. [2] Weighted norm inequalities for classical operators. Proc. Symp. Pure Math. 35(1979), 6983. B. Muckenhoupt and R. L. Wheeden: [1] Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc. 192(1974), 261274. J. Musielak: [1] Orlicz Spaces and Modular Spaces. SpringerVerlag, Lecture Notes in Math., Vol. 1034 BerlinHeidelbergNew YorkTokyo 1983. K. T. Mynbaev and M. O. Otelbaev: [1] Weighted Functional Spaces and Spectrum of Differential (Russian). Nauka, Moscow 1988.
Operators
V . P. Nikolaev: [1] An estimate for integrals of potential type in weighted norms and their applications to the imbedding theorey (Russian). Thesis, AlmaAta 1974. S. M. Nikolskii, P. I. Lizorkin and N. V. Miroshin: [1] Weighted function spaces and their applications to the study of bound ary value problems for elliptic equations in divergent form (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 315,8(1988), 455. P. Ortega Salvador: [1] Weighted inequalities for the onesided maximal function in Orlicz spaces. Preprint. P. Ortega Salvador and L Pick: [1] Two weight weak and extraweak type inequalities for the onesided maximal operator. Preprint, 1991.
References 225 P. Oswald: [1] Fourier series and conjugate function in ${L). Analysis Math. 8(1982), 287303. G. Palmieri: [1] Un approccio alia teoria degli spazi di traccia relativi agli spazi di OrliczSobolev. Bolletino U. M. I. (5) 16B(1979), 100119. L. Pick: [1] Two weights weak type inequality for the maximal function in L{\ + log L)K. In: Constructive Theory of Functions '87. Proceedings of the International Conference held in Varna, May '87. Publ. House Bulg. Acad. Sci., Sofia 1988. [2] Two weight weak type maximal inequalities in Orlicz classes. To appear in Studia Math. [3] Weighted estimates for the Hilbert transform of odd functions. Preprint, Math. Inst. Czech. Acad. Sci., Prague 1991. L. Quinsheng: [1] Two weight ^inequalities for the Hardy operator, HardyLittlewood maximal operator and fractional integrals. Preprint, University of Wales, College of Cardiff. M . M . Rao and Z. D. Ren: [1] Theory of Orlicz spaces. M. Dekker, Inc., New York 1991. M. Riesz: [1] Sur les fonctions conjugees. Math. Z. 27,2(1927), 218244. M. Rosenblum: [1] Summability 165(1962), 3242
of Fourier series in Lp(d{i).
Trans. Amer. Math. Soc.
F. J. Ruiz: [1] A unified approach to Carleson measures and Ap weights. Pacific J. Math. 117(1985), 397404.
226 References F. J. Ruiz and J. L. Torrea: [1] A unified approach to Carleson measures and Ap weights II. Pacific J. Math. 120(1985), 189197. [2] Vectorvalued CalderonZygmund theory and Carleson measures on spaces of homogeneous nature. Studia Math. 88(1987), 221243. R. Ryan: [1] Conjugate functions in Orlicz spaces. 13711377.
Pacific J. Math. 13(1963),
C. Sadosky: [1] Interpolation of Operators and Singular Integrals. M. Dekker Inc., New YorkBasel 1979. E. T. Sawyer: [1] A characterization of a twoweight norm inequality for maximal opera tors. Studia Math. 75(1982), 111. [2] A two weight weak type inequality for fractional integrals. Trans. Amer. Math. Soc. 281(1984), 339345. [3] Boundedness of classical operators on Lorentz spaces. Studia Math. 96 (1990), 144158. [4] Weighted Lebesgue and Lorentz norm inequalities for the Hardy opera tor. Trans. Amer. Math. Soc. 281(1984), 329337. [5] A characterization of two weight norm inequalities for fractional and Poisson integrals. Trans. Amer. Math. Soc. 308(1988) 533545. [6] Weighted inequalities for the onesided HardyLittlewood maximal function. Trans. Amer. Math. Soc. 297(1986), 5361. E. T . Sawyer a n d R. L. W h e e d e n : [1] Weighted inequalities for fractional integrals on Euclidean and homoge neous spaces. Preprint, 1989. J. S err in: [1] Local behaviour of quasilinear equations. Acta Math. 111(1964), 247302.
References
227
I. B. Simonenko: [1] The Riemann boundary value problem (Russian). Izv. Akad. Nauk SSSR. Ser. Mat., 28,2 (1964), 277306. S. L. Sobolev: [1] On a theorem of functional analysis (Russian). Mat. Sb. 4,3(1938), 471497. [2] Some applications of functional (Russian). Nauka, Moscow 1988.
analysis to mathematical
physics
E. M. Stein: [1] Fractional integrals in ndimensional Euclidean spaces. J. Math. Mech. 7(1958), 507514. [2] Note on singular integrals. Proc. Amer. Math. Soc. 8(1957), 250254. [3] Note on the class L(logL). Studia Math. 32(1969), 309310. [4] Singular Integrals and Differentiability Properties of Functions. Prince ton Univ. Press, Princeton 1970. E . M . Stein and G. Weiss: [1] Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton, New Jersey 1971. [2] Interpolation of operators with change of measures. Trans. Amer. Math. Soc. 87(1958), 159172. V. D. Stepanov: [1] Two weight estimates for the RiemannLiouville integrals (Russian). Preprint, Dal'nevostochnoe Otdelenie Akad. Nauk SSSR, Vladivostok 1988, 33 pp. [2] Weighted inequalities for Riesz potentials on classical Lorentz spaces. In: Abstracts of Symposium on Continuum Mechanics and Related Problems of Analysis, Tbilisi, June 611, 1991, p.125. E. W . Stredulinsky: [1] Weighted Norm Inequalities and Degenerate Elliptic Partial Differen tial Equations. Lecture Notes in Math., Vol. 1074, SpringerVerlag, BerlinHeidelbergNew YorkTokyo 1984.
228 References J.—O. Stromberg: [1] Bounded mean oscillation and duality of Hardy spaces. Indiana Univ. Math. J. 28,3(1977), 511544. J.—O. Stromberg and A. Torchinsky: [1] Weighted Hardy Spaces. Lecture Notes in Math., Vol. 1381, SpringerVerlag, BerlinHeidelbergNew York 1989. G. E. Tkebuchava: [1] Integral inequalities in weighted reflexive Orlicz spaces (Russian). Soobshch. Akad. Nauk Gruzin. SSR 121,3(1986), 477479. A. Torchinsky: [1] Interpolation of operators and Orlicz classes. Studia Math. 59(1976), 177207. [2] Real Variable Methods in Harmonic Analysis. Academic Press, New York 1985. H. Triebel: [1] Interpolation Theory, Function Spaces, Differential Operators. Deutscher Verlag der Wissenschaften, Berlin 1977.
VEB
O. D . Tsereteli: [1] On interpolation of operators by the truncation method (Russian). Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 26(1969), 111122. [2] On the integrability of conjugate functions (Russian). Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 43(1973), 149168. T. Walsh: [1] On weighted norm inequalities for fractional and singular integrals. Canad. J. Math. 23(1971), 907928. G. Welland: [1] Weighted norm inequalities for fractional integrals. Proc. Amer. Math. Soc. 51(1975), 143148.
References
229
N. Wiener: [1] The ergodic theorem. Duke Math. J. 5(1939), 118. A. C. Zaanen: [1] Linear Analysis. North Holland, Amsterdam 1953. L. V. Zhizhiashvili: [1] Conjugate Functions and Trigonometric Series (Russian). Tbilisi State University 1969, 271 pp. A. Zygmund: [1] Trigonometric Series, Vol. 1. Cambridge Univ. Press, Cambridge 1959.
This page is intentionally left balnk
Index A* condition, 126 Ap condition, 139 A*p condition, 141 Ay condition, 140 AOQ condition, 32 Ap condition, 32 Ap(B) condition, 140 AP(J) condition, 100 Ap, condition, 145 A+(g) condition, 60 A  w condition, 47 A# condition, 47 A i*,($i9) condition, 61 A+(4>,g) condition, 61 ascattered sequence, 134 anisotropic ball, 179 anisotropic cubes, 81 anisotropic distance, 81 anisotropic Muckenhoupt class, 179 anisotropic radially decreasing function (ARD function), 179 anisotropic Riesz potential, 81 Bp condition, 86 B$ condition, 85 CalderonZygmund decomposition of a cube, 124 centered maximal function, 48 centered weighted maximal function, 115 complementary function, 1 A2 condition (global), 2 A2 condition near 0, 2 A2 condition near 00, 2 doubling condition, 41, 107 extraweak type inequality, 46, 104 fractional integral, 74 fractional order maximal function, 74, 163
231
232 Index function of type BQ , 46 function of type Boo, 46 generalized potentials, 197 Hilbert transform, 104, 126 local dyadic maximal function, 123 lower index of a Young function, 3, 84 Luxemburg norm, 1 maximal Hilbert transform, 110 modular inequality, 6 modular interpolation, 6 Muckenhoupt's class, 32 multiple Hilbert transform, 101 multiple fractional integral, 103 onesided weighted maximal function, 60 OrliczMorrey class, 2 Orlicz norm, 1 Orlicz space, 1 $ class, 1 $(L) class, 1 restricted weak type, 34 restricted weighted inequality, 166 reversed Holder's inequality, 33, 38 Riesz potential, 176 Riesz transforms, 24, 97 strong maximal function, 103, 133 strong maximal function of the fractional order, 103 quasiconvex function, 3 two weight centered maximal function, 48 upper index of a Young function, 3, 84
Index 233 vectorvalued maximal function, 16 vectorvalued Riesz transform, 29 weak type (p,p), 8 weak type ($,4>), 8 weight function, 2 weighted Lorentz space, 143 weighted Orlicz class, 2 weighted Orlicz space, 2 Young's function, 1 Young's inequality, 1