Weighted inequalities in Lorentz and Orlicz spaces

= 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/

Orlicz-Moirey

$(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 = A-0+ 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 ^ - / sup ^ ^ ds J 0
>

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 HardyLittlewood-Wiener 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 (1-2.5)

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 t-i < 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; \x-y\ 0. If |x| > 1, then Mft{x)

* \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 (1-2.2) 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

fc-i 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 } , k-l

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. Vector-valued 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*
lrt-M*)«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 vector-valued 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

w e will write / £ L\oc if each component A belongs to L\oc, For such a vector / = ( A , / a , . ■ •) G L\oc we define Mf = {MfuMf2,...). THEOREM 1.3.1. Let $ 6 $, 1 < 9 < oo. Then the following are equivaient:

statements

(i) # is quasiconvex and satisfies the A2 condition; (ii) there exists a constant c\ > 0 such that the inequality

$(\)\{ xeRn;

\\Mf(x)\\9 > A }| < Cl | *(||/(*))||#) d*

holds for every A > 0 and every vector-va7uec/ function f = (Ai Ai ■ ■ ■) fromL^. D To prove the theorem we use one known result (FefFerman and Stein [1]) and an auxiliary assertion. THEOREM A. Let 1 < p < 00, 1 < 9 < 00. Then there exists a constant ci > 0 such that the inequality \{xeRn;

\\Mf(x)\\e

> A}| < cxA"" |

| | / ( * ) | | ; dz

is true for every A > 0 and every vector-valued function f = (A, Ai • • •) € L/^. In 1.1 we proved that $ £ $ is quasiconvex if and only if

(1.3.1)

^M < 'l

£») *2

1.3. Vector-valued maximal functions

17

for all 0 < t\ < <2 and some a > 1 independent of <1( t 2 . We can say more if

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;2-k-2<xi<2-k-1,

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;2-j-2

<Xi <2-k~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. Vector-valued 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«

A/2 } I < ^

- "AT"

y

ll/(x)l|e 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)

(ii).

Let

ll/MII* ^
f(x) = ,J(x) + */(*), where xf(x) and xf(x) are defined as in the proof of Theorem 1.3.1. We have 00

J #(||M/0c)||,) dx = J\{xeRn; Rn

\\Mf(x)\\e

> A }|d#(A)

0 00

<J\{x£Rn;

\\Mxf(x)\\e

> A/2}|d<2>(A)

0 00

+ J\{xeRn;\\Mx/XaOU* 0

= h + h-

> A/2 }|<*2>(A)

1.3. Vector-valued maximal functions

21

In virtue of Theorem A we have

h<ejUj\\xf{x)\\A d#(A) = cj\\

I

°

d#(A)

II/OOM*

\fll/(«)ll.>A} / II/MII<

= cij\\f(x)\\,l

J

R"

\

d#(A)

dx.

0

Making use of Theorem 1.2.1 we conclude that

h < c2 J ^{\\f{x)\\e)dx. Now we estimate I-i- Lemma 1.3.2 guarantees the existence of constants p > 1 and 6 > 1 such that #(*a) . » ( * i ) ttP22 " t{ tx whenever 0 < 11 < £2- Fix some p\ > p. By Theorem A we get \ 1

h < c /A-" / A"?

/

d#(A)

\\f{x)\\V /

Vll/(*)II«<M

= cJ\\f{x)\\V

j

d$(\) APi

da:.

Further,

[<m=m

J As $(u)/u

p

+pJm du, Pl+1

1

UP*

V?

t

J "

< b4>(l) for u > 1, we have lim - ^ u—»oo

u?1

= hm u—►oo

^ — = 0.

upUPi~P

22

Chapter 1. Integral operators

Whence oo

J

r d$(u) b
t

For Ii it implies the estimate

., *(ll/(*)ll«)

/2
*i\\f(x)\\t)dx.

R"

Putting together the estimates for Ji and i2 concludes the proof.

□

T H E O R E M 1.3.4. Let $ G $ and 1 < 9 < oo. The inequaJity (1.3.3)

sup<2>(A)|{ x e i?" ; c\\Mf(x)\\e

> A }| < oo

A>0

holds for all f = ( / i , / 2 , . . . ) . Il/Wlls G # ( £ ) , x € R", witA a constant c independent of f if and only if the function

0 such that (1.3.4)

*(2tfc)>4**(<0.

fc

= 1,2,... .

a, = J ^ S f e ) ) - 1 ' " ■

* = 1,2,....

Let fc-i

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-(j-km)

'

2m-(j-km)-l

i=

l,...,n}.

1.3. Vector-valued 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(-*+i-o^ *(**) = 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»)|/ 2t e e > k)\Ikm\ fclB | ${2h)
We arrive at >2t oo lim ^(2ijt)|{x *(2t fc )l{* € Rn j; c||M/(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 Garcia-Cuerva 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 A/2}| < ^ 1

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) -(max(c,
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 = c-i

/ * ( / ( * ) ) 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. Vector-valued 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 \x-y\< |*| + |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 Vector-valued 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 Garcia-Cuerva 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 vector-valued 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 vector-valued inequalities in Lp spaces are dealt with e.g. in the mono­ graph by Garcia-Cuerva 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 p-i

{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], Garcia-Cuerva 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, 0 such that I $(Mf{x))g(x)dx fi" for all f G

L$(Q);

< c1 I R"


34 Chapter 2. Maximal functions and potentials (ii) there is a c-i > 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)

< i7d|x£||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 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 Q-1^)/®'1 (fit), <>o

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

-

A-i/.(*)

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\ rip(r) for all r > 0, therefore \E°\ *(2*v{Q)/\Q\) |0| " a 0, conse­ quently,

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

0, and a cube Q, we have

xeRn.

M/|.)>i/,(,)%gWl Q

By duality argument, there is a /„ G £#(*$) such that | | / 0 | | L # ( « J ) = 1 and JMv)dy = = Jf J0{y)jl-Se(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\\||XQ|U,(« 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

Hxg||i#(«,)

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

l|M/||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 0 be such that the measure of E f] [0, o) is no smaller that 3a/4 for all 0 < a < a 0 . Then 4|£n[4-1a,o)| ^ 2 L > _.

(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[4-1a,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^n|0.4-m#-( 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)\V-i

— l

/A"23-14m+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 |^|>2-1|/|.

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 extra-weak 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, 0 such that for all a > 0,

(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 0 such that sup / #

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 extra-weak in­ equalities which will also justify the terminology. We claim that a weak type inequality always implies the corresponding extra-weak type one. Indeed, by homogeneity, the extra-weak 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 -J-r [ \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 extra-weak 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 / ( £ 4 ^ )

*(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.

l/k}

2.2. Two weight inequalities of weak type

49

and

x€Rn,

»M=*(I2W!OKM.

where e is a positive constant which will be fixed later. We have

J \\QW(*)J

r(x) dx ■■

£Q(Q)

\Q\

Qk Qk

= £e(Q)

r

J

\\Q\a(x)J

Qk

9k(\Q\) \Q\ '

If 9k(Q)/\Q\ is uniformly bounded with respect to k and Q by some positive constant we are done so that suppose that gk(Q)/\Q\ > c > 0 for large Jk's where c is from (2.2.5). Then using (2.2.5) and the inequality #(AS#(<)) < const. V(t), A G (0,1), t ^ 0 , we have

H

Qk Qk

,^rr9k(Q) U( c\Q\ ee(Q) \ r{x) dx < c4e / 9 ——— 9k(x)j \Q\<*)) \Q\ J \9k(Q)

a(x)dx

Q Q

(2.2.6)

+ cAeg{Q) + cAeg(Q) < c5e j V(gk(x))
i.e., for sufficiently small e > 0, substracting in (2.2.6), we get

J

Qk Qk

r(x)dx < c6Q(Q)

\\QW*)J

with some c$ independent of Qk- Thus (g,cr) € -<4|w. It remains to check the case when 9 is infinite near oo, i.e. # £ Boo- Then R$ is bounded near oo and (2.2.5) yields . '

Q(Q)
\Q\<E) \E\

for any measurable E C Q- Whence (g, a) € A\ and

(2.2.7)

H /

Q

ee(Q) \ cr(x) dx S Q\a(x)J W) QiQ) -\Q\JQ * VIQkM) Q

< sS
(jOkM)

dx

50

Chapter 2. Maximal functions and potentials

The expression on the right hand side of (2.2.7) is finite provided £ is sufficiently small. Now suppose that (g,cr) 6 J4| W . Fix a cube Q and a / > 0, / G L$(o-). Then by Young's inequality,

\Q\ ~£ Jnf{))e(Q)

+

Q

J \\QW(*)J e(Q) Q

- ^o) J0^x))^x) dx +£~lcQ

Passing to the supremum over Q centered at x, we get Mcf(x)

£-lMla{$U{x))+e-lc.

<

Thus { x € Rn ; Mcf(x) > £-\l + Cl)}c{x€Rn; Mce^(f(x)) > 1}. In virtue of this and invoking the standard approach based on the Besicovitch covering lemma, we arrive at e({ xeRn;

Mf(x)

>\})
Mcf(x)

> c2X })

^g({,E^;M-((1+c^)>(l +

Cl)A})

^^;M,(*(k^))>1}) <J0^My[x)dx where we used (2.2.4) to get the last line. This completes the proof. □ Now we turn our attention to the weak inequalities. T H E O R E M 2.2.2. Let 0 G $ and g and a be weights on Rn. (2.2.8)

$We({

x£R";

Mf{x)

> A }) < c f $(cf{x))*{x)

Then dx

2.2. Two weight inequalities of weak type 51 for all f € Z| o c and all A > 0 with c independent of f and A hoids if and only if

* (lip)eiQ) -e j *&(*))<*) **■ Q

Let K > 0 be such that the set E = { x e Rn ; ff(«) > !//<, er(a) < K } has a positive measure and fix a cube Q with IQnE 1 ! > |Q|/2. If supp / C E, we have

* ( ] £ [ / l/(*)l
f 1 #1 * | (2|QnE|

/

cK2 !/(*)! <**) < \Qf\E\

I


QnE

Let s , f > 0 , a e (0,1) and put QnE 1 = F U F ' where \F\ = a|QD.E|. Consider f(x) = s x f (x) +<x/"( a : )- ^"hen (2.2.9) becomes # ( 2 - 1 ( a * + (1 - a)t)) < cK2(a$(cs)

+ (1 - a)#(rf)),

i.e. <£ is quasiconvex. As in the preceding theorem we will assume from now that

£ £ 0 U 5 o o . Then

(2.2.10)

#(5#(0) < *(0-

Indeed,

#Wffl ..(aa)- 3 .(ffi-!a) < £ W sup{s>0;W(t)/t>&(s)/s},

t>0.

52

Chapter 2. Maximal functions and potentials

The function t H-> \P(t)/t, t > 0, is nondecreasing, so that we get from the above inequality that *(S#(t))< ^ <

= *(*)•

Let

/(*) = c-'S* ( — W ) ,

*£#">

and for a > 0, and k £ N, define Ek = {xeRn-o-(x)>l/k},

/ k ( a ) =/(*)XB k («),

*G^"-

The estimate (2.2.10) gives

\I IQI \Q\ )y -- KQ)Ey Ekk

w*)y

\a(T(x)J

T{X) K ] dx

^ c\Q\ IMQ) - «e(Q) IQI ' As IAKQ) < oo and |Q \ U ^ l = 0, we get

a

\Q\^

■

=

\Q\JS*{c«
dx\

< c.

Q

If 0 £ Bo, then for / = XE, E C Q measurable, we have c -i]£!

101

~i S

ME)

\\Q\J-

i.e. (g,
•m«**-

< e / # ( ^ ) Q

,(«)*.

2.2. Two weight inequalities of weak type 53 Putting f = XE, we obtain

W

e(Q):

i.e.

-<4 '

(Q,*)eA1,too.

Now we prove the sufficiency. If 0 6 B0 U B x , then (£,
(2.2.11)

Mf{x) > A }) < j f \f(x)\a{x) dx.

Moreover, as

V \Q\ )

<

w\h{cf{x))dx oQ

and # is increasing we have

^■'(sf)*s*(f) <sup^-

U{cf{y))dy

Q

= cM(9?(c/(z))). The last estimate together with (2.2.11) gives Q{{ xeR";

Mf(x) > X }) = g({ x G Rn ; <2>(M/(x)) >
For # i So U Boo we have

(2212) 1

'

Ifflffl < ■ / # ( « , ) « . ) *

Bi -

IOI y

lA u

*

+rai/'(=£j)"'Wfc

54

Chapter 2. Maximal functions and potentials

With no loss of generality we can assume that the function

1/1(0) .

(2.2.13)

S"

\Q\ ■

|=,/#C«.)M.) fe + D—1 ^«?)J Q

For A > 0, let

= Wl(/i(rt.)W.)*)" *»! f/*(/WM.)^'1 .. = Passing to the supremum over Q in (2.2.13), we get Mcf(x)

<\ +

cRf\cMcei0(
Whence { x G i T ; Mc/(x)

> 2A } C { x e Rn ; A ^ c 1 ^ ^ / ^ ) ) ) > fl#(A/e)} C{x G i?" ; M e %(*(/(*))) > # ( c - x A ) } ,

so that
> A ^ ^ ( c - 1 A) < c I $(f(x))
dx.

The theorem is proved. Q Let us add an observation. It is clear that g € A| w always implies g € Ax Indeed, recalling the estimate

m
e(«) g(E) e(£)

\\E\) ■(f)

<S ci e 9

XE{X),

2.2. Two weight inequalities of weak type 55 for an arbitrary cube Q and any measurable E C Q. Now, if g(E)/g(Q) then \E\/\Q\ < c/$-l(l/c6) and this is equivalent to g € Ax.

< 6,

On the other hand, if g e ^4oo, then g £ Ap for some 1 < p < oo. Assume additionally that the function Fp>(t) = ^(t1^'), t G R1, (p' = p/(p - 1)) is quasiconcave. In virtue of g G Ap we have

e(Q) V' eix) dx =- c < oo. Q\s(x)J e(Q) Q

The converse of Jensen's inequality yields

H

e(Q) \

JW\\Q\g(x)J Q\e(x)J g(Q)dX-J Q

*»' [\\Q\g(x)J

)

v

Q

7

^cFp'[CJQ \\Q\g(x)J \

Q

"

6{X)

dx g(Q)dX QiQ)

e(Q)

dx

j

< cFpi(c) < oo. Whence in this case g G A™ iff g G Aoo. Note that the assumption on the quasiconvexity of Fpi is satisfied for every 4> such that indeces of its comple­ mentary function equal 1. In the next section a detailed analysis will be carried out in the particular case i, $ 2 be nonnegative, nondecieasing functions on [0, oo), lim $ 2 ( 0 = 0, and n be a positive and 1—0+

nondecieasing function on [0, oo). Assume that
*i(A)fl({ x € Rn ; Mf(x) > A }) < c J 4>2 (^f)

<*) **

56

Chapter 2. Maximal functions and potentials

holds for all f € L*a( 0 independent off there is an e > 0 such that

SUP sup H „ ? ,K A>0 Q

if and only if

'1(X)n(X)e(Q)\ x dx ■^ OO / U Sfrt ( \ °°i^w, 1*2 f ^ T A|Qk(x) T T T T ^ ^ )J
*i(A)««) J

\

Q

where $2 is the complementary function to 2 we are getting an alternative equivalent condition for validity of (2.2.8) (the weak inequality, Theorem 2.2.2), whereas the choice 77(A) = A and ^(t) = 1 yields directly the A% condition (the extra-weak inequality in Theorem 2.2.1).

2.3 A n o t h e r characterization of A^ Recall that a weight function Q belongs to Aoo if to each e G (0,1) there is 6 G (0,1) so that if Q C Rn is a cube and E any measurable subset of Q, then \E\ < 6\Q\ implies Q(E) < Q(Q). It follows from known properties of the Ap classes that Aoo A00--

== U{JA ApP.. p>i

The following characterization of Aoo is due to Khrushchev [1]: THEOREM A. A weight Q belongs to Ax (2.3.1)

if and only if

^H^/'^-h

We will show an alternative way of proving Theorem A, getting also another equivalent condition of the type (2.3.1). First, let us reformulate Theorem 2.2.1 in a sharper form: THEOREM B. Let $ be an even function on R1, nondecreasing and quasiconvex on [0,oo), #(0+) = 0 and 9 its complementary function. Then

2.3. Another characterization of A^,

57

there is a c such that (2.3.2)

e({xeR

n

;Mf(x)>X})
f

$ (Sl&\

tt(x)dz

{|/{*)|>A/2}

for all f G Lloc if and only if there is a constant e > 0 such that

(2.3.3,

- , ^ / r

(,2^)

* ) * < . .

The proof goes in the same way as in Theorem 2.2.1; the argument yielding the smaller range of integration on the right in (2.3.2) is standard. Let us consider one particular case of Theorem A. Put i, w I 0 if 0 < t < 1, and we see that *log + - < 9{t) < f l o g + < . Theorem B implies then COROLLARY 2.3.1. Let g be a weight on Rn. Then the inequality (2.3.4)

Q({ xeRn;

Mf(x)

> A }) < c

J

exp \ ^ y ^ )

e(x) dx

{|/(*)I>V2} holds for all f € L\oc and all A > 0 with a c independent of them if and only if

(2.3.5)

sup Q

^) mh*±.J \\QW)) ■(

6

^ \ iog+

< oo.. dl<00

A,

Q Q

Indeed, the condition from Theorem B turns into ( eQ(Q) \ dx < oo

\\Q\ei')J ■yraMisi&Wsup Q

Q Q

58

Chapter 2. Maximal functions and potentials

for some e > 0. As log + ab < log + a + log + b the latter condition is equivalent to (2.3.5). We are going to prove that (2.3.5) is equivalent to g G A^. Suppose (2.3.5) holds. Observe that (2.3.2) implies g G A^ as noted toward the end of the foregoing section. Let us prove it in a more detailed way (as we actually arrived there at an equivalent condition). Let Q be a cube and F its measurable subset. For f(x) — XF(X), x G Rn, we have xeRn.

Mf(x)>^XQ(x), Setting A = | F | / | Q | in (2.3.2) we get Q(Q) = e({xeRn;

Mf(x)

> \F\/\Q\ } <

c$(c\Q\/\F\)e(F),

i.e. (2.3.6)

g(Q) < c4>(c\Q\/\F\).

Given 0 < a < 1, let c in (2.3.6) satisfy c<2>(c(l - a)) > 1. If E C Q is measurable, and \E\ < ct\Q\, then for F = Q \ E, we have \F\ > (1 - a)\Q\. Therefore (2.3.6) gives

e(Q) < c${c(\ - a))e(F), whence

^

* {' ~ ^ ( l - a ) ) ) SiQ)'

which is Q G ^4oo • On the other hand, assuming g G Ax, Thus g({xeRn;Mf(x)>X))
then g G Ap for some p 6 (0,1).

J

0 - ^ Y

{l/(x)|>A/2}

As f < cexpt we get (2.3.4) and therefore (2.3.5), too.

□

e(x)

dx.

2.3. Another characterization of A,*,

59

Now we will prove C O R O L L A R Y 2.3.2. A weight Q belongs to Ax if and only if 8{Q) sup lo d dx< a: sup oo■. log T P T T / S T77TTT < °° Q 101 J \Q\Q(X)

iol/Q

\Q\e(x)

Q

Proof: We show that (2.3.7) is equivalent to (2.3.5). The necessity of (2.3.7) is clear. The converse implication follows by Jensen's inequality. Let Q be a cube and E =

ixzQ;e(Q)/(\Q\e(x))>l}.

Then

-i- [iogJ*9L = ± [iog+J*9Ldx 101 J g \Q\e(x) \Q\ J log \Q\e(x) Q

1

=

101

\Q\Q{X)

mJ1^

-\Q\J

g

"1017Q

g

Q

=

M 7Q l

>

log

/log+ g g

- 1017 IOI 7

± \Q\ J

log

Q \E\ e{Q)E , + e(Q) dx . + \E\ log , Q(Q)

f* Q

>

+ +

Q{Q)

IOI
101


I O101 I

fiog^L \Q\e(x)

l /",l o g l iol/ ^)

dz ,

log

101 101 lol S \E\ 101

e(Q) dx + \E\ W B(Q)\E\ g \Q\e(x) + I0I Q(E)\Q\ 101 %(£)I0I |0k(*) e(Q) dx + \E\ hg \Q\+\Q\+ \Q\e(*) d£+ 101log

IQkW «"'

-

M lQl

log

m

lQT log ^)

1

e

ioie(*) « |01e(*)

aQ (we have used the fact that a; log a; > - 1 / e for 0 < a; < 1/e). Finally, if (2.3.7) is true, we get i y,log + g(0) , x . i /log \ g(0) + i

iol7 Q

Ioi^)" -Iol7 ]ol^) e Q

< — i.e. (2.3.5) holds.

□

1 c+-, e

60

Chapter 2. Maximal functions and potentials

Analogous considerations as above with combination of interchanging the role of the Lebesgue measure and the measure generated by g (this is possible as both measures are doubling, see Coifman and FefFerman [1]) give an alternative proof of the theorem due to Fujii [1]: A weight g belongs to the Aoo class iff g(x) sup / log + Q J e(Q)

f2 ** oo.< e(Q)

Q

2.4 O n e - s i d e d m a x i m a l function Let g be a locally integrable and a.e. positive function on R1. The one­ sided weighted maximal function is defined by x+h M+f(x)

= sup . / h>o g(x,x + h) J

x£Rn,

\f(y)\g{y)dy,

where g{x,x + h)=

x+h j g(y)dy,

xeRn,h>0.

X

The Lp inequalities for M+ were studied by several authors in last years (see Notes to this chapter). Let us present a quintessence: THEOREM A. Let g and a be weights on R1 and p > 1. Then there is a c > 0 such that Q({ xeR1;

M+f(x)

> A }) < cA~p J | / ( x ) | V ( x ) dx fi1

for all f G Lp( 0 if and only if(g, a) £ A+(g), i.e p

(A}(9)) P

e(a,b)

suP

4 ^ (I *1

a
\g(a>c)J

p-i

1 ffg(x)Y'my-

g(x)dx

\o-(x)J

3V

'

< oo.

2.4- One-sided maximal function For p = 1 the condition reads M~(g)(x) the left version of M + .

< ci
61 is

Ortega Salvador [1] has investigated the case of the weak type inequality for M+ in a weighted Orlicz space under the restriction of reflexivity of the space involved. Here, the A2 assumption will be removed. Recall that for $ G $ the functions R$ and S$ from the foregoing section are defined by R$(t) = P(t)/t, t > 0, respectively, where !? is the function complementary to # . Observe that if 4> is convex, then (2.4.1)

*(AS#(t)) <

M(t),

for all t > 0 and 0 < A < 1. Definition 2.4.1. Let (g, a) be a couple of weight functions on R1. (i) Let<£ G <&\(-BoUBoo). Then the couple (Q,CT) is said to satisfy the A+(#, g) condition if there exists an e > 0 such that (Ai(*,9))

sup sup ^ | i ^ a>o a
(-±\g(a,c)J

[S*

\aa(x))

(jM-)9(x)dx) J

00.

(ii) Let # G 5o U Boo- Then the couple (g,(r) is said to satisfy the A+( 0 such that

M

o\ , ,


;{ \gj-0/ (*) < c-M-. »(*) »(*) 9

(iii) The couple (Q, 0 such that

(4k(*.*))

sup

/f (fee(a,b)g(x)\ eg(a,b)g(x)\ / | H ,V ; 7 ;

a(x) a(x) ,, - 7 ^ 7 dx < oo.

b

T H E O R E M 2.4.1. Let g and
62

Chapter 2. Maximal functions and potentials

(i) there is a Ci > 0 such that $(X)e({ x € R1; M + / ( z ) > A }) <

(2.4.2)

Cl

j ${cxf{x))
R

dx

1

for all f e L\oc and all A > 0; (ii) there is a c2 > 0 such that c

(2.4.3)

e (a,

6)*(/i+(/; a, 6, c)) < c2 / *(c 2 /(i))
for all f e L$(
/i+(/;a,6 l C ) = - i -

[\f(x)\g(x)dx;

(iii) the function 4> is quasiconvex and (g, a) 6 J 4 ^ ( # , y). Proof:

The implication (i)=>(ii) follows from the estimate M+f(x)>n+(f;a,b,c)x(a,h)(x),

a < b < c,

xER1.

Let us prove (ii)=>(iii). First we show that (2.4.3) implies the quasiconvexity of 4>. Fix a C > 0 such that the set E = {x E R1; C~x < g(x) < C, g(x) > C~\

a(x) < C)

has a positive measure, let a < 6 < c be such that g((a,b)nE)>2-1g{a,b),

g({b, c) H E) > 2- 1
and g{a,b) = g{b,c).

Let T\, t2 > 0, 6 G (0,1) and decompose (6, c)!"!!? into disjoint sets, say F and JS", with g(F) = 9g((b, c) n £ ) . Put /(a;) = i i * F ( z ) + hXF'(x), x € JJ1. Then ^(/;a,6,c)>4-1(^i + ( l - ^ 2 ) , 1 2 e(a,6)>2- C- <7(a,6),

2.4- One-sided maximal function and

}

63

dx < C2g(a, b)(9
/ $(c2f{x))a(x) b

Whence
+ (1

S)t2\ ^ 2 c 2 C 4 ( ^ ( c 2 < i ) + (1 _

d)Hc2t2)l

i.e. <2> is quasiconvex (Lemma 1.1.1). Now we wish to prove that (Q,(T) 6 A+($,g). We distinguish three cases. First suppose that 0 G B0. Then inserting / = XE, E C (b, c) into (2.4.3) yields

iB)^

^ ^ {gjB}) g(a'b) -

^ ^

and A+(g) follows. Let G 5 ^ . Observe that (2.4.3) implies its extra-weak counterpart (cf. Theorem 2.2.2), i.e. C

g(a,b)
Again, the choice / = XB gives e(a,b) <

c2v{E) [i^j

<

W{E)9-^

and this is A+(g).lt remains to consider the case ( P ^ B 0 U f l M . We may certainly assume that
it = 1,2,...,

and

hk(x) = C3 % f ^ , ^ ) W * ) ,

z >0

with c 3 = min(c 2 l C) and C is the constant from the inequality $(\S$(t)) CV(t), t> 0, 0 < A < 1. We get

$(hk(x)) <^(^Y^jXDk(x),

x>0.

<

64

Chapter 2. Maximal functions and potentials

The inequality (2.4.3) gives then, for any a < b < c,

Q(a,b)0\

) f S*(-^L)g{x)dz) \c3g(a,c)J \aa(x)J

<

e(a,b)
I cc

< c2

&(c2hk(x))(r(x) dx

< c2 / &(c2hk(x))a(x) < c 2 / 5
dx

G

a{X

\aa{x)J

g{x)

\a{x)J

> dx g(x)

Dividing the last inequality by a~1g(a,c)

j S* Dk

G

(£&rjg(x)dx
and letting k tend to infinity, we see that (g, a) €

A+($,g).

All we have still to prove is (iii)=^(i). By Young's inequality, for any func­ tion / € L${a), a > 0, and a < b < c, we have c

c

— ] — j\f{x)\g{x)dx<-£— 9(0., c) J b

U{f{x))a{x)dx

g(a,c) J b

+ g(a,c)J^-)[sJ^±) 9(X)dx \ao-{x)J b c

< ^ — 9(a,c) J

U{f(x))o-{x)dx

b

C59M + c^fag(a,b)

Let A > 0 and put -l

a - Xg(a,c) I /
2.4- One-sided maximal function

65

Then c ff(a,c)/

l/(x)lff(l)da; A

+

-

C5jR 1

*

6

°5

1

/ {rVffrlWfrrWr 1

^A,(a,6)7^ / ( x ) M x ) < f X J-

Let {(aj,6j)}j be a Calderon-Zygmund decomposition of the set l 1 {x {x€ReR ;M+f(x)>b\}, ; M + / ( x ) > 5A},

i.e. {{*x■eR € i 1? 1 ; ;M M++//((xx))>>5 5A} A } = U ( a j ,.*>) 6J)

= |>i j

and */ / 1 f(i\\n(t\

Jt ^ <\\

X fc (O.j,0j).

X

Let (a,6) be one of the intervals (a.j,bj). Denote xo = a and let the sequence {xk}k satisfy b

b

J f(x)g(x)

dx = = 2 J/ f(x)g(x) f(x)g(x)dx, dx,

x)t Xk

ki = 1,2,.... l,2,....

xfc+i Ik+1

Clearly, {ijb} is an increasing sequence and lim xk = b. Further, fc—*oo 6 b

i ^A <* 5A< * / f(x)g(x)dx f(x)g(x)dx ff(x*_i 9{xk-i,b),6) Jy xjt-i

4 < <— / f{x)g(x)dx f(x)g(x)dx g{xk. - l , z * + l ) / Kit Xk

( /

1

««.+» -*+»

V\

/ *(f(z))
Xfc + i Xk + l

<^(A/(4c6))^(xfc_1 ,**)<<* y ^(/(x)Hx)dx. * ( A / ( 4 c 6 ) ) e ( x t _ i , x t ) < C 6 xk/ *(/<*)M*)(fe. Taking the sum over k's and j ' s yields (i). D

66

Chapter 2. Maximal functions and potentials The extra-weak counterpart of Theorem 2.4.1 is treated next:

THEOREM 2.4.2. Let <£ be quasiconvex, and g and <x weights on R1. Then the following statements are equivalent: (i) there is a c\ > 0 such that Q({ xeR1-

M+f(x)

> A }) < cj J$

(

£ L

^ )

'(*) dx

fl1

for all f € L]oc and all A > 0; (ii) there is a c-i > 0 such that c

e{a>b)
J

*fix)

)a(x)dx

V^J(/;a,o,c)y

for all f e L\oc; (iii)

(flff)e4»(^j).

Proof: The implication (i)=^(ii) follows in the very same way as in The­ orem 2.2.1. Suppose (ii) is true and # ^ B^.

For a < b < c put

M«) = *V(<ji(o,c)o-(a;) *£^W . fc=l,2,..., y «). with Die from the proof of Theorem 2.4.1 and with e > 0 to be determined later. We have

/K^^w) ^^=e^+(,it;a'6'cMa'&)

Dk

< c2£g{a,b) + I(a,b,c)

2.4- One-sided maximal function 67 where I(a, b, c) = ep+(hk; a, b, c)g(a, b) if n+(hk; a, 6, c) > c2 and /(a, 6, c) = 0 if fi+(hk;a,b,c) < c 2 . Then (2.4.2) for A = c2/fij{hk;a,b,c) gives C

/(a,6,c)<=a^+(A»;a,fc J c) / * ( - ^ * ± — ) J \N{nk;a,b,c)J

a(x)dx

b


\g{a,c)o-(x)

)

Dk

For sufficiently small £ > 0 we get

J

Dk

,b)9(*)\ r(x) (fa; <

\g(a,c)a(x) J c)ar(z) 7 \9{a,

1 -- cc22ee

.*)

and sending k to oo concludes the proof of (iii) for $ £ Boo ■ If (ii) holds and 0 £ Boo, then easily (£,,g) condition is majorized by S${c2e). It remains to prove (iii)=>(i). Let a < b < c. Then (iii) and Young's inequality give c

c

- 1 - f\f(x)\g(x)dx
U{f{x))^Ldx Q{a,b)

J

b

b

+Czhfji^m^Ldx J

\c3g{a,c)cr(x)J

g(a,b)

b

< c3J(a,b,c)

+ c3

where c

j

{ a M

=

Jmx))4±dx. b

Given A > 0, let {(aj,bj)}j

be the following Calderon-Zygmund decomposi­

tion: {xeR1;M+f(x)>\}

=

\J(aj,bj), i

68 Chapter 2. Maximal functions and potentials and »i

; , s / f(x)g(x)dx 9{x,bj) J

> A,

xe(ajtbj).

c

As in the proof of the foregoing theorem let (a,6) = (a.j,bj) for some j and choose xjt's. It is tfk + l

A<

,

4 c

*+i)

I

f{x)g{x) dx

< 4c3J(xk--l,Xk ,xk+i) + 4c3. Put A = 8c 3 . We get /> 1

g(xk-uxk)<

or

$(f(x))ff(x)dx.

Summing over k's gives e({

xeR1;

M+f(x)

> 8c3 }) < J *(/(x))«r(e) dx.

As M+ is homogeneous the proof is complete.

□

2.5 Carleson measures Recall the notation introduced in Chapter 1: Mf(x, t) = sup ~

J |/(y)|
(x,

t)eRnxR\,

B

the supremum being taken over all balls B C Rn with x E B and rad 5 > 2 - 1 t . is the generalized maximal function. In this section we generalize the following theorem due to Carleson [1]. THEOREM 2.5.1. Let 1 < p < oo. Then the inequality J (Mf(x, 0 ) p dp < c j |/(x)|" dx,

f G 27(fl"),

2.5. Carkson measures 69 where c is a constant independent constant Cl such that

of f holds if and only if there exists a

/?S(x,r)< 0B(x,r)
x€R", xeR",

r >> 00..

We shall generalize this theorem to the case of the Orlicz-Morrey classes. T H E O R E M 2.5.2. Let 0 < A < n, $ G $ and a be quasiconvex for some a G (0,1). Then the following statements are equivalent: (i) there exists a constant ci > 0 such that

(2.5.1)

supr-* J $(Mf(y,t))dp

/eiL;

B{x,r)

(ii) there exists a constant c2 > 0 such that sup r-A<2>(5)/?({ (j/, r) G i ? ; + 1 ; M/(», r) > a } U £ ( * , r)) (2.5.2)

.«• < C 2 |C 2 /|*,A

for all f G L\oc; (iii) there exists a constant c3 > 0 such that PB(x,r)
,

x € Rn, r > 0.

Proof: The implication (i)=>(u) is trivial. We prove (ii)=>(iii). Fix a cylinder B{z0,r0). have

Mf(y,r)

=

sup jeB(i.r) r>W2

\B(x, r)!" 1

For (y,r) G B(zQ,rQ),

/

\f(t)\dt

B(z, r)

1 > i^nor isc^o^o)!-1 I/ fl(*o,ro) fl(*o,r 0)

|/(()l Jf. 1/(01*.

we

70

Chapter 2 . Maximal functions and potentials

Let /o = XB(, o,r 0 )-

T h e n

M/0(j/,r)>l,

(!/,r)eB(z 0 ,r 0 ).

Inserting / = /o into (2.5.2), we get &(2-1)pB(zo,r0)
sup e>0

Thus

a fi, pB(z 0,r0)

x

[

e

^{c2)dy.

B(x,Q)r\B(z0,ro)

w 15(2:^)0 5^0^0)1 x < c 4 sup L-s *—3—i ^-rjj ,

and PB(zQ,r0) < C4 m a x

/

'

\

sup e>r 0

|S(x l f f )nB(zo,r 0 )| A

8

r 0A , sup e
|B(x,ff)nfl(z 0l ro)| A r0 A 8

Further, . _ |B(*,e)n B(r 0 ,ro)| A \B(z0,r0)\ x sup r 0 < sup r0

£>r 0

e

e>r0

«

< csr? , and sup|5(x,,)n5(,o,,o)lr,<sup|5(x,,)|roA

ff
S

e
x€Rn

< sup e e
8

r0

2.5. Carleson measures 71 Notice that (ii)=>(iii) was proved without assuming 0 G $ and the quasiconvexity of 4>a for some a G (0,1). Let us prove the implication (iii)^-(i). Fix a ball Bo = B(xo,ro) and put h(x) = f{x)\3BB{x), g(x) = f(x)xRn\3B0(x) where 3B 0 is the ball centered at Xo and with radius three times as that of So. Clearly, Mf(x,t)

< Mh(x,t)

+

Mg(x,t).

The function &a is quasiconvex so it is $ itself. Therefore TQX [$(Mf(x,t))d/3

< c0roX [

BQ

(2

-


BQ

53)

+c 0 r 0 - A

[$(c0Mg(x,t))d!3 Bo

=h+h By Carleson's theorem and the quasiconvexity of 4>a, h = c0roX

j


J



< c10r0-A

j{u{c1oh{x,t)))1la{x,t)dx R"

x

<cur0-

j 4>(cnh(x)) dx .R"

< C12|C12/|*,A •

We have thus proved the estimate (2.5.4)

h < ci2|ci2/|
72

Chapter 2. Maximal functions and potentials

Now we find an upper bound for I2. Let (x,t) e B0 (recall that according to the definition B0 = B(xQ,r0) x [0,2r 0 )). As Mg(x,i)=

sup

\B{z,r)\~l

f

2r>t

\g(y)\dy,

B(z,r)

and g is supported in Rn \ 3 5 0 , the supremum can be taken over those balls B(x, r) for which x € B(x, r) and B(x, r) n 3 5 0 = 0. The radius of these balls is greater than ro, therefore, taking the supremum over such balls, we have by Jensen's inequality coMg(x ,t) = sup<£~■H*(Mz < sup <£" \${\B{z

< sup 4>~■\\B(z,r)

'r)l_1

'r)l_1

/

/ B(*,r)

co\g(y)\dy))

co\f(y)\dy))

r1 / *(c*f(y))

dy)

B(z,r) x n

< SUp 0' \c5r - ;

C4f{y)) dy) B(z,r)

-W-"i |C /|
< sup $~

4

Further, h < Cor 0 - A y^- 1 (c 5 r 0 A -' l |c 8 /| ( f , A (y))) f l (/? B0

=

c6ron\c8f\$tX/3(B0).

In virtue of the statement (iii), we eventually arrive at (2.5.5)

I2 < c7\c8f\$,x.

Putting together (2.5.4) and (2.5.5) concludes the proof of (iii)=>-(i). The theorem is proved.

□

2.5. Carleson measures 73

Let us make several remarks on Theorem 2.5.2. If A = 0 and 0 is a Young function, then we are in standard Orlicz spaces. Let us consider this case now. The condition (i) turns into a modular inequality, (ii) into a weak modular inequality and the condition (iii) remains. We claim that if, in addition to the assumption of Theorem 2.5.2, we suppose $ is a Young function, then any of the conditions (i)-(iii) implies (2-5.6)

l | M / | | L . ( e ) < c\\f\\Lt{Rn).

If, moreover, <£ satisfies the A2 condition, then (2.5.6) implies (i)-(iii) of The­ orem 2.5.2. The estimate (2.5.6) is obviously implied by e.g. (i). Suppose that # € A2 and that (2.5.6) holds. We prove (iii). Denote by 9 the Young function complementary to
/ My)

d

v = C||XB||L»(«»)

B

where c is independent of / and expresses the relation between the Orlicz and Luxemburg norm. Fix t > 0. As

Mf0(x,t) > YjjrA\xB\\L*{R")Xg(x,t) > jjdlXfl||jMH»)Xj?(M) we get from (2.5.6) that WXSWL^R?1,,)

$ c il 5 l HXB|lZi(«.)'

i.e.

K < ci\B\9~ #(1M5)) "

iwi)<-

#-

\\I\B\Y

which implies

iH-"U))As 0 £ A 2 , we get \B\ ~ Q{B) ' i.e. Q is a Carleson measure.

74

Chapter 2. Maximal functions and potentials

Also, another equivalent condition for the measure g can be found. As­ suming that $ satisfies the assumptions of Theorem 2.5.2, then the statements (i)-(iii) are equivalent to the following one: There exists c > 0 such that for all balls B, all measurable E C B and all e > 0, (2-5.7)

$(j^e(B)
This is easy to see. First, given e, B, E C B, we have

(*, t) G i^ + 1 •

M(eXE)(x, t) > ^ W ( M ) , Putting A = e|f?|/(2|B|) in (i), we get (2.5.7).

Conversely, inserting e — 2, and E = B into (2.5.7), the Carleson condition follows. The operator M is linked to the Poisson integral Pf(x, t) = f*Pt(x) where Pt is the Poisson kernel so that a standard argument gives the following COROLLARY 2.5.3 Let g be a positive measure on .R" +1 and 0 a Young function such that <Pa is convex for some a € (0,1). Then the Poisson integral is a bounded operator from L${Rn) into L$(Rn+l, g) if and only if g is a Carleson measure.

2.6 Fractional order maximal functions and Riesz potentials Let 0 < 7 < n. We will deal with the fractional integral (the Riesz potential operator)

(2-6.1)

T7f(X) =

J-^dy

and the corresponding fractional order maximal function

(2.6.2)

Myf(x) = sup ^ p ^ j \f(y)\ dy Q

2.6. Fractional order maximal functions

75

where the supremum is taken over all cubes Q C Rn with sides parallel with the coordinate axes and containing the point x. Using the standard approach (inserting a suitable characteristic function into the weighted inequality) it is easy to find a necessary condition in terms of the A$ classes, nevertheless, it is possible to show a far more suitable char­ acterization using the Ap classes arriving at the result analogous to Theorem 2.1.1. First, we shall need an interpolation background. We use the interpolation theorem due to Gustavsson and Peetre [1]. Let 3>0 a n d 3>i be generalized Young functions, i. e. nonnegative, contin­ uous and increasing functions on < 0,oo), $;(0) = 0, $i(oo) = oo, i = 0,1. Assume that t(#,) > 0, /(
sa{\)

= sup ^ *>o

cr(t)

,

A > 0,

and suppose that s„(\) = o(max(l,A)) for A —> 0+ and A —> oo. Let 0 be given by

t-Ht)

= ^\t)o-{$-\t)i$-Q\t)),

t> o.

Then if T is a continuous linear mapping from L${ into L${, i = 0,1, it is also continuous from L$ into L$ and \\T\\L^L»

< c ||T|u, o ^, G a„(||r|| L#i ^ L . i /||r|| L#o ^ #o ).

Observe that modular inequalities in reflexive Orlicz spaces can be obtained from Lp inequalities by means of the modular interpolation (see Gustavsson and Peetre [1] and Krbec [1] for details). A particular case which applies here is as follows: If 0 is a Young function, 1 < i(#), /( J(#), g is a weight on Rn, and T is a sublinear operator acting continuously from LPi(o) into LPi(e), then / $(Tf(x))g(x)

dx
for some c > 0 and all / E L$(g). An alternative way to get the modular inequality in L$(g) is to start with the norm inequalities in Lpi(g) and to

76

Chapter 2. Maximal functions and potentials

smuggle e > 0 inside, namely, if T is bounded in LPi(g), then so it is the case in LPi(£g), and conversely, for any £ > 0. Whence interpolating the inequalitites in the L Pi (eg) norms we arrive at the norm inequality in L${£Q),

l|r/||L#(«,) < e||/|U#(«.).

/ e L&Q),

£ > o.

where e does not simply cancel at all. But the last inequality for T yields its modular counterpart (see Lemma 3.4.2 in the next chapter). Such an e-trick is essentially the main ingredience in the proof of Theorem 2.1.1 (Kerman and Torchinsky [1]). Our main result in this section is THEOREM 2.6.1. Let$x and$2 be Young functions such that 1 < »'(#i) = p < I($i) < oo, 1 < t(#2) = 9 < I($2) < °°- Assume that 0 < j < n and ( $ 2 ) - 1 ( 0 ~ <~ 7 / n (#i) - 1 (0> f > °- Let 6 be a weight function. Then the following statements are equivalent: (i) there is a c\ > 0 such that \\Ty(f(£Qr/rt)\\Lfiice)
fEL*x(£e),

£>0;

/ e W<^).

£ > 0;

(ii) there is a c2 > 0 such that \\M,(f(eey/n)\\L^e) (hi) ^ € ^4, where s = 1 + g/p'

< call/IU^c,). (p' = p/(p - 1)).

We split the proof of the "necessary part" of Theorem 2.6.1 into several lemmas. LEMMA 2.6.2. Let gbea weight in R71,
(2.6.4)

\\MyU^eTfn)h^t)
holds for all f € £$, (ep) and all e > 0, then (2 6 5)

--

XQ

1 r/ f(e^) d 1w H X Q I U<..,
for all cubes Q C Rn and all £ > 0.

2.6. Fractional order maximal functions 77 Proof: Let e > 0, Q C Rn be a cube and let y EQ. Then (we can suppose that / > 0 a.e. in Rn)

My(f(eey'»)(y) > ^L-j-I

nx){£Q{x)y/n

dx

Q

/f(x)(ee(*))-1+'/nee(x)dx.

= jg|i^r Q

It follows from the duality between L$1{eg) and L^^eg) that there exists a function f0 e Lfl{eg) such that supp f0 C Q, H/olU,,^) = 1, and J f0(x)(eg(x)y'n

dx =

XQ (eg)l-y/n

i* t (ee)

This means that M(/o(e«)"»)

^ j -

n

XQ (egy~y/n

XQ(V)£*j(ee)

According to (2.6.4), XQ |Q|l"7/n

(ee)

<

XQ

1_7/n

L*1(ee)

\\M{fQ{egf-^n)\\L^te)

L*i(ee)

<

c\\h\\L^ce),

so that X
(eg)1-^"

\\xQ\\L^e)
u The next lemma is essentialy contained in the proof of Theorem 2.1.1. L E M M A 2.6.3. Let 0 and \P be mutually complementary Young functions both satisfying the A2 condition. If XQ ■ ||XQIU#(*#) < 4 £6 Lq,(ee) for all cubes Q C R" and all e > 0, then g e A{($y

78

Chapter 2. Maximal functions and potentials

L E M M A 2.6.4. Let the assumptions on 7, g, $1) $2, and #1 from Lemma 2.6.3 be satisfied. Let p = z'(#i) and q = i(# 2 )- Suppose that ( # 2 ) - 1 ( 0 ~ t-iln(^i)-l[t) and let (2.6.5) hold for all cubes Q C Rn and all e > 0. Then Q € As where s = 1 + q/p' (p' — p/(p - I)). P r o o f : Set /3 = 1/(1 - j/n) = n/{n - 7 ) . We have

-pV'

•llxgll2(£e)
Let us define (<2>3)_1(0 = {{^r\\t\)f

and $3(t) = ^ ( l * ! 1 " ) ,

*eBT.

It easy to check that #3 and #3 are Young functions and that they satisfy the A2 condition. As

(^3)-1(0(^)-1(0 = (((*2)-1(0)/,((^i)-1(0)"

-(r^f*!)-1^)"1^))" ~ (tl-7/»)/» _

f)

we see that $3 and ^3 are mutually complementary Young functions. Further, HXQllL#a(««) = IIXQlU#,(£e), and from the definition of the Luxemburg norm, XQ

(eg)1-1'"

(9

inf A>0;

\

i»i(«e)

/^(A^F^)£e(x)dx£1 ■ e(x)dx < 1

= inf <{ A > 0

I

XQ CQ

L*30?)

We have proved that XQ £0

whence £ £ -*4i(tf3)

m

• HXQ||L # ,(«,) <

c|Q|

L* 3 (cff)

accordance with Lemma 2.6.3. But

i(# 3 ) = 1/Wa)-1) = 1/)9J((#2)- 1 ) = «//? = «(1 - 7/") = « and this completes the proof. D

2.6. Fractional order maximal functions

79

Proof of Theorem 2.6.1: First (iii)=>(i). If Q G A,, then from known properties of the membership of a function in the class As we have g € ASl for some sx < s and, moreover, g G As for all s > sx. It is easy to see that conditions 7 1 11 qo = 11 + - T , «s = = ' + :7> n p 9 P7 n p q p1 are equivalent to p

nn

ss \V

1 = 1(1-1).

n/ nJ

qq

ss \\

nn)/

Whence it follows that choosing Si sufficiently close to s and s 2 sufficiently large and putting, for i = 1,2, Pi

n

Si \

nJ

qi

s,- V Si \

n)

we get 1 < px < p < 7(#i) < p 2 ,

1 < gi < q < I{$2) < 32-

As the Lp theory gives

\\Ty(fg^n)\KM
i = 1,2,

we can interpolate between the Lebesgue spaces involved to get the desired estimate (i). It only remains to verify that the assumption of the interpola­ tion theorem from the beginning of this section are satisfied. We will restrict ourselves to the indeces Pi, p 2 ; the interpolation on the right hand side of the last estimate is quite similar. The function a is given by (
;

i.e. o-(t) = tPl/{p°-pi\V1)-1

(iP°Pi/(P°-Pi)) f

whence S„(X) = SUp —j-r= A P :I / ( P I

-

P°)

sup

(*l)-

1

((A*) P o P l / ( p a - P i ) )

*>0

_ . . / r ._p, - K u p (( **l i) -) ■i- 1((m(pAo p^i /^( p- o^-■Pl)f) >«) = AAP l / ( P I °) sup <>0

( * i ) ~ ■i(0

80

Chapter 2. Maximal {unctions and potentials

To justify the interpolation we have to verify that «„(A) = o(max(l, A)) for A -» 0+ and A -» oo. If A £ (0,1), then for all e > 0 there is Ce > 0 and 5(e) > 0 such that sJX) < C£APl/(pl~Po)

fAPoPl/po_Pl)>)/((*irl)+£

= C APl/(pi_Po)APoPl(/((*l)_1)+£)/(po-Pl)

= C fA P l /( p i -p a )V~ P ° / ( i ( * l ) - * ( £ ) ) n

\(Pi/(Pi-Po))-(t(#i)-Po-^(c))/(i(*i)-*(e))

The constant 6(e) can be made so small that 0 < 6(e) < i(*i) - Po- Thus lim sJX) = 0. If A G (l,oo), then one can get in a similar way the correspending estimate for A " 1 ^ ) showing that its limit at oo is 0, too. As to the the implication (i)^(ii) let us observe that the pointwise estimate M,g(x) <

nl»-*f%g{x)

is valid for a.e. nonnegative g G L\oc. Indeed, let Q be a cube in Rn, x, y € Q. Then

l*-i/|
i.e.

yeQ, i

— L _

<„(»-•>>/» |*-v|*-7 \

y€Q.

Consequently, \Q\i-y/n

B J 9(y)dy
ii J

Q

g(y) rf ——-dy, x - 2/|"- . 7 y.

Q

whence

M,g(x) 0. D

2.6. Fractional order maximal functions 81 In the end of this section we will briefly deal with the anisotropic version of Theorem 2.6.1. Let a = (ax,...,a„) be an n-tuple of real positive aj, j - 1,...,n, with the length \a\ = ai + ■ ■ ■ +an. Define the anisotropic distance 1/2

n*-»ll = ik-*=(|>-wi«°<) and the anisotropic cubes E = E(x,t)

= {yeRn;\xj-yj\<2-1ta'},

xeRn,

t > 0.

The anisotropic potential of the order s, 0 < s < \a\, is the operator

T,f(x) = J

/GO

ll*-y||s

dy

and the anisotropic Riesz potential of the order j (0 < 7 < 1) is defined as M7/(x)=sup|£(z,t)r

y

|/(y)|dy.

T H E O R E M 2.6.5. l e t $1 and $ 2 be Young functions, 1 < i(#i) = p < /(
1 < 1(^2) = 5 < ^ # 2 ) = Q < 00.

Furtner, Jet 0 < s < \a\, r= l/p - 1/q > 0, and q = \a\p(\a\ - (|a| - s)p),

Q = \a\P(\a\ - (\a\ -

s)P).

Then the following statements are equivalent: (i) there is

Cl

> 0 such that

l|T.(/(**)r)llw«> < ci||/|U#l(,f),

/ € L+(S8),

S > 0;

82

Chapter 2. Maximal functions and potentials

(ii) there is c 2 > 0 such

that

r

feLttfe),

\\Msm{f{h) )\\L»3m < call/Hi^*,), (iii) Q€A1+q/p,

6>0;

{p' = p / ( p - 1)).

T h e p r o o f is based on the same idea as the proof of T h e o r e m 2.6.1 and is omitted. Theorems on boundedness of the Riesz potentials have an immediate applications to imbedding theorems. Let us point out one simple consequence of T h e o r e m 2.6.1: Let W1L$(Q) be a weighted Orlicz-Sobolev space, consisting of / 6 L}^ with

l|/lktx#(e) = ||/e-1/-||L#(f) + £ If 1 < i(#) < / ( * ) < oo, then W1Lt{g) Riesz potential of the order n - 1, namely,

\wlL,(e)

df

o-V"

d^6

< oo . L*(e)

admits a representation via the

-£*■<*>-■

This follows in a standard manner, observing t h a t a Mikhlin type multiplier theorem holds in L${Rn) thanks to e.g. a suitable interpolation, (see Krbec [1]). If we show t h a t C^{Rn) is dense in W1L${Q), then it suffices to prove the imbedding inequality for functions from C^{Rn). Precisely: If the functions <2>i and &2 satisfy the assumptions of Theorem 2.6.1, and if / € C%°(Rn), then

0/(1/) ttf /Mlvs< v* /is/Ml


.-—1 •*

n

3= 1

it dyj

) '

therefore,

\\f\\w3u)^E\\T^(l£j < c||/||wi£,3(e) and the density argument will conclude the proof.

We)

2.6. Fractional order maximal functions

83

As to the density, the following is true: If€> is a Young function satisfying the A 2 condition and !(<£) < n, then C%°{Rn) is dense in W1L^(g). This can be proved making use of the standard mollifier procedure. Let {o be a family of mollifiers, i.e. 0, H^HLH-R") = *> an< * lim ipt * g = g in Lr(Rn) for all g € Lr(Rn), 1 < r < oo. We prove that lim (pt * / = / in W1L$(Q)

for all / 6 W1L$(g).

We can certainly assume

t—►O-i-

that / has a compact support, say K, and let fi be an open bounded set containing K. We prove that lhn j 0 ( ( / ( * ) - / t (x))(e(x))" 1 /") e(x) dx = 0. n The same will also be true for the derivatives for d(f*ipt)/dxj j=l,...,n.

=

(df/dxj)*
Choose (3 > 1 such that /? < n/(n — i()). Then for sufficiently small e > 0, we get by Holder's inequality that J

e(x)dx + [/(*) - / t ( ^ ) | / ( * ) + £ ) # ( ( ^ a : ) ) - 1 / " ) e («) dx

< | / l / W --/1(*)|«*>-')/>/(/>-i)(fa \n

i yi/(x)-/ (x)|(^^wc)j + t n

/ 1//8

1/

/

y (*((ff(*))- ")e(a)) ' dx

84

Chapter 2. Maximal functions and potentials

The last integral is finite. Indeed, let fl = { x £ fi; Q(X) < 1 } . Then / ($ ( ( ^ x ) ) - 1 / " ) g(x)Y

dx < c I'(ff(*))/»

dx

+ c / ( e (x))<—« # >-'»/>/»

dx

n\n 0 < c|fi o | + c||0Xn\n„llLH«") < oo.

2.7 Another variant of M Let £ be a weight on i£ n . For / € i ^ we define the maximal function

Mef(x) --= sup Q3T

\Q\

I

dy

9(y)

-

Q

This variant of the maximal function was used e.g. for the study of con­ vergence of trigonometric series (Y.-M. Chen [1]). It is evident that Me is bounded in Lp iff M is bounded in Lp(gp). Due to the nonhomogeneity of Young functions, an analogous observation is no more true in the case of Or­ licz spaces. Nevertheless, a characterization of weights in the reflexive Orlicz case is possible, similar to that of Theorem 2.1.1 and is due to Kazaryan [1]. First some additional notation. Throughout we will assume that

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

A-00 =

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 A-oo

-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 ||xge|U # («)

||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? £G-V 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)

(±.\\XQ/Sg\\L^

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,

JlQlf-Hi/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)dx-c'

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 < ~HxBe||L#(«)|lx<3/(*e)l|i,(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»U-g)ii) '*)icQ \ Q By the definition of the Luxemburg norm we get , 15|llxg/(« «)IU.(^(]5fllxg/(«'»)llL,(io)

I

^6>f9*(C6(x))dx' Q

whence 1 < «*-* «P- ^jje{x))dx i|lx./(^)ll W ) < j 1"■ l*'J*#(e(*))
^\\XQ/(6'Q)\\W)

1

Now we estimate Hxfipllz*^')- Let

fy6'jGMz))dz)) ' N ^- -( "(gJ55His))

B) = U-i # 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£?e|k*(«')

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))'W-6I

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(*))m-tl 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^)/(iW-6l) 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 A-W*)- 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 Stein-Weiss interpolation theorem recalled at the beginning of Section 2.1) the resulting restricted weak type inequalities, and then applying the Riesz-Thorin 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

96

Chapter 2. Maximal functions and potentials

Theorem 2.3.3 giving simultaneously an answer both to weak and extra-weak problems is due to Gogatishvili [9]. As to the Lp spaces, the first norm inequality for fractional integrals in R was proved by Hardy and Littlewood [1] who also proved a weighted inequality with power weights. The n-dimensional nonweighted problem was solved by Sobolev [1] and this was much later generalized by Stein and Weiss [1] for power weights. Let us mention further generalizations by Il'in [1], Nikolaev [1], Lizorkin [1], Walsh [1], . . . . Let us recall that M 7 : Lp —> Lq and T 7 : Lp —► Lq for 1/g = 1/p — 1/n and the corresponding norm inequality can be interpreted as an imbedding theorem for the Sobolev spaces Wp <—* Lq. The same relation between the indeces p and q hold in the weighted theorems mentioned. Details (for the nonweighted imbedding) can be found in many monographs, e.g. in Stein [4, Chapt. V]. Naturally, all these years there was the challenging problem of the characterizing all those weights Q for which T 7 and M 7 act continuously from LP{Q) into Lq(g), similarly as in the case of the maximal function. The answer was given by Muckenhoupt and Wheeden in [1], again in terms of the Ap classes. As to the reflexive Orlicz spaces, the solution to this problem presented here belongs to Kokilashvili and Krbec [1], [2]. Observe that the nonweighted inequality in a reflexive Orlicz space was proved by Torchinsky The one-sided maximal operator in Lp spaces was studied by Sawyer [6], Martin-Reyes, Ortega Salvador and de la Torre [1], Martin-Reyes [1], the reflexive Orlicz space case was treated in Ortega Salvador [1]. Our exposition follows Ortega Salvador and Pick [1]. The last section is devoted to an alternative variant of the maximal function whose study is due to Kazaryan [1]. Needless to say, we were enjoying T£}X very much while preparing the output of this section. Finally, let us observe that necessary and sufficient conditions for two weight modular inequality in reflexive Orlicz spaces is studied in Quinsheng [1]. The same paper deals also with convolution operators under more restricted as­ sumptions on the function 4> involved (

Chapter 3 Singular integrals in weighted Orlicz classes We will stick to the notation introduced earlier; if g is a weight on Rn, then L$(g) will denote the weighted Orlicz space. The function complementary to $ will be denoted by !?. We will establish one weight inequality for the Riesz transforms, two weight inequalities of weak and extra-weak types for the Hilbert transform.

3.1 Riesz transforms in Orlicz spaces In this section we are going to work with the Riesz transforms Rjf(x)

= cnJ

Xj lx _~*+if(y)

dy,

j = 1, • • •, n,

where C

" -

r((n+l)/2) x (n+l)/3

•

and with the operator n

Rf(x) =

1

£Rjf(x),

i=l T H E O R E M 3.1.1. Let 0 be a Young function, 1 < i( 0 such that f $(Rf(x))g(x)dx

< Cl f

$(f(x))g(x)dx

for all f € L$(g); (ii) there is c2 > 0 such that ||J*/IU#(«(f) < ca||/|U#(«j) 97

98

Chapter 3. Singular integrals in weighted Orlicz classes for all f G L$(6g) and all 6 > 0;

(iii) there is C3 > 0 such that #(A)

/

g(x)dx
|i?/(x)|>A


i?»

for all f e L$(g) and all A > 0; (iv)

geA^y Proof: Observe that (i)=>(ii) and (i)=>(iii) are trivial.

The implication (iv)^(i) follows by modular interpolation (Gustavsson and Peetre [1], Krbec [1], see Section 2.6 for the details) as g € -<4,'(<j) yields g € Ar for all r > i($) — e with a sufficiently small e > 0 and thus the operator R is bounded from Lr(g) into Lr(g) for these r as well known (e.g. Garcia-Cuerva and Rubio de Francia [1, Chapt. IV, Thm. 3.7]). We prove (ii)=>(iv). Let Q be a cube in Rn, f 6 L$(g), / > 0, supp f C Q and 6 > 0. Denote by Q' the cube of the same volume as Q, having a common vertex with Q and such that XJ > yj for all x £ Q' and y S Q. Then

\Rf(x)\>-^Jf(y)dyxQ'(x

)

Q

(3.1.1)

og(y) . . . \Q\J■fiy) Se(y)6eiy)dyXQ '(*), o

=

x£Rn.

Q

Choosing /o in such a way that ||/o|U*({f) = 1 and / /o(v) dy = J f0(y) j Q

^ 6g(y) dy = | | x Q / ( M I U , ( * e ) ,

Q

we get from our assumption that (3-1.2)

\\XQ/(Sg)\\L^6e)

WXQ'Wwe) < ci\Q\.

We show that ||XQ'IU*(ie) ~ H X Q I I L ^ S ) - Inserting / = \Q into (3.1.1), we get \Rf{*)\>cXQ'(x), x€Rn,

3.1. Riesz transforms in Orlicz spaces and Orlicz Spaces 99 and, consequently, C||XQ'IIL,(«<>) < CIIIXQIU,^). Interchanging the role of Q and Q', we obtain the equivalence of the norms. The inequality (3.1.2) turns into IIXQ/(MIU*(MIIXQII< C IM which is equivalent to the A$ and A^) preceding section).

condition (cf. Theorem 2.1.1 from the

We finish the proof by proving that (iii) implies g 6 A$. In virtue of Theorem 2.1.1 this is equivalent to g 6 A^y Given Q, Q' and / € L$(g) as above we have the estimate

xeRn,

\Rf(*)\>^j-Jf(y)dvxQ>(* ), thus for all A, 0 < A < (c/\Q\) J f(y) dy, we get Q

(\)e(Q')
and therefore,

#

(w\ I \

fiy) dy e{QI)

Q

)

I

C2

{fix))e{x) dx

~ /*

-

Q

Putting f = XQ here yields g(Q') < c3g(Q). Interchanging the roles of Q and Q' gives g(Q) ~ g(Q') so that

(3.1.3)

* j | | i J f(y) dy J g(Q) < c2 J #(/(*))g(*) dx. V

Q

I

Q

As
e(Q) \Q\

J/(y)dyip\W\If{y)dy) Q

-C3Jf(xMf(x))eWdx

100

Chapter 3. Singular integrals in weighted Orlicz classes

where ip is the derivative of $. Let 6 > 0 and f(x) Then from (3.1.3) we get

Mg) \Q\

=


dy c3

Ahl^(m) )- '

i.e. £ € A$, whence in virtue of Theorem 2.1.1, g G J4J(#) as # G A2. The proof is complete. Q

3.2 Multiple singular integrals in Orlicz spaces Let J be the family of all parallelepipeds in Rn with sides parallel with coordinate axes. We introduce the Ap{J) class as the set of all weight functions g which satisfy, for 1 < p < 00,

(Ap(j))

supi-y^dj^iy^x))- 1 ^- 1 )^) <<x>,

and/or, for p — 1, (Ai(J))

777 / e(x) dx
inf ess g{x)

for all J G J

x£J

J

(see Kokilashvili [14], R. Fefferman and Stein[l]). For x = (xi,...,xn) e Rn, let us write xt - (x\,..., ar,-_i,*,- +ll ... , x n ) G iZ , i = 1 , . . . , n. It is easy to show that g G Ap(J) if and only if there is a c > 0 such that n_1

(

l

— 1 g(x) jrijg(x)dx [j(Q{x))-^-i\UXi\ i d,Xi lj(g(x))-^"-^dx

\p_1


I

for all one dimensional bounded intervals J and for all x~i G .ft" -1 , i = 1 , . . . , n. Indeed, if g G Ap(J), then it suffices to use n — 1 times the Lebesgue dif­ ferentiation theorem to get the latter condition. Conversely, if this condition is satisfied, then the strong maximal function is bounded in Lp(w) and thus

3.2. Multiple singular integrals in Orlicz spaces 101 w G Ap(J). Details can be found in Kokilashvili [14], see also Garcia-Cuerva and Rubio de Francia [1, p. 454]. We will work with the multiple Hilbert transform

1

/n m *«

»„/(»)= We will work with the multiple Hilbert transform understood in the sense of the p.v. lim p.v., understood in the sense of the £-0 + /

ita

n f(y)

r.- - Vi

dy,

t=i ■

lf[JULdyi

where e = (ei,... , e n ) , £i > 0, and Ec,x - {(i/i,... ,s/n) € Rn \\XJ - yj\ > e,, j = 1,...,n }

T H E O R E M 3.2.1. Let $ be a Young function, 1 < i() < /(<*>) < oo and Q be a weight function on Rn. Then the inequality (3.2.1)

J *(Hnf(x))e{x) R

dx
n

J $(f(x))e(x)

dx

n

R

holds for all f £ L${Q) with a constant c independent of f if and only if

e e Ai(f)(J). Proof: First assume that (3.2.1) is true. Then for any S > 0 we have (3.2.2)

\\Hnf\\L,(6e) < c||/|| L # ( « e ) .

Let J = [ai - fci,Oi] x • •• x [a„ -

hn,an]

and Jx = [ai,ai + hi] x •■■ x [an,an + hn] where hi > 0, i = 1,2,..., n. supported in J, then

(3.2.3)

If / is a nonnegative measurable function

\Hnf(x)\>j±Jf(t)dtXJl(*), j

*ZRn-

102

Chapter 3. Singular integrals in weighted Orlicz classes

Let us rewrite it as

x € Rn.

\Hnf(x)\ > ±L JmSe(t)-^XJl(x), j

By duality argument, there is a function f0 € L$(6g) supported in J such that ||/o|U,(« e ) = 1 and

Jfo(t)dt

= / / o W ^ W ^ y = \\Xj/(Se)\\L,(st) •

J

J

Whence \\Hnf0\\L^Se) >

7J7||X//(<5?)||L»(^)I|XJ1|U,(^)-

As Sg € Ai($){J), we have (3.2.4) ||x//(*e)IU,(« e )||XJ l |U # (««) < h\J\. The roles of J and J± are symmetric, therefore (3.2.5) IIW(*i?)ll£.(*«>IIXj|lM* e ) ^ 6 3|J|Inserting f = XJ into (3.2.3), we get by (3.2.2) that (3.2.6)

c\\xj\\Ltise)

> WHnfhrtt)

> h\\XjA\L*(ee) •

If g is any nonnegative measurable function supported in J\, then

\Hng(x)\>-^-Jg(t)dtXj(x),

x£Rn,

and repeating the above argumentation, we arrive at (3-2.7) IIXj||L # («,)<MxJxlU # (« e ). Finally, by (3.2.6), (3.2.7) and (3.2.5) there is b5 > 0 such that (3-2.8) Hxj/(*e)llL,(«,)||Xj|U.(«ri < W\This is an inequality analogous to the condition which we met at the beginning of the proof of Theorem 2.1.1. Going through the proof of this theorem, we see that cubes can be replaced by parallelepipeds and everything works so that Q £ J4«'(#)(«7). The details can surely be left to the reader. Now, conversely, assume that Q € Ai^(J). The proof of (3.2.1) goes in an analogous way as in the one dimensional case (Theorem 3.1.1); we will apply interpolation together with the known fact that the multiple Hilbert transform is bounded in Lp(g) iff Q € AP{J) (Garcia-Cuerva and Rubio de Francia [1, Chapt. IV, Thm. 6.2]). Q

3.2. Multiple singular integrals in Orlicz spaces 103 In the sequel, we are going establish relations between the multiple frac­ tional integral

Tyf(x)= [f(y)f[-.

^Vr-

0< 7 <1,

and the strong maximal function of the fractional order, M'yf{x) = sup \J\i~1 JBx

11/(01 dt,

0 < 7 < 1.

J

For 7 = 0, My is called the strong maximal function. We prove a theorem analogous to Theorems 2.3.1 and 2.4.1 linking Ty and M 7 and giving a full characterization of weights guaranteeing their boundedness in weighted Orlicz spaces. T H E O R E M 3.2.2. Let $ x and 4>2 be Young functions, 1 < i(#i) = p < / ( * ! ) = P < n/j, 1 < i(*j) = g < I(# 2 ) = Q < oo, 0 < 7 = 1/p - 1/g = 1/P — 1/Q. Then the following conditions are equivalent: (i) it is \\Ty(f(6ey)(x)\\L^Se) for all f 6

L$(6Q)

< Cl||/|U#i(4e),

6 > 0,

with a constant c± independent of f;

(ii) it is

||M^(/(W|U^)
S>0,

+ q/pJ.

= $2, i.e. 7 = 0, then the conditions (ii) and (iii) are equivaient.

The proof goes without substantial changes along the lines of that of Theorem 2.3.1. Just observe that when proving the implication (i)=^(ii), we use the inequality Myfx < cTyf(x), x G Rn with a c independent of / as in Section 2.6. In our case this follows by using n-times the inequality Mx,f(xi)

/(y) < c(n) [ J \Xi -ViV

R

l

7

dyu

i=l,...,n,

104 Chapter 3. Singular integrals in weighted Orlicz classes where MXif(xi)

= sup I7.I 7 " 1 / \f(y)\dyu

i=l,...,n,

i,

and observing the fact that M;f(x)

<MXl ... MxJ{x),

x G Rn.

3.3 Weighted inequalities of weak type for the Hilbert transform Recall that the Hilbert transform in R1 is given by Hf(x)=

lim -

I

£ ^ 0 + 7T

-^-dy,

J

\*-y\>£

x G R1,

x - y

Analogously as in Chapter 1, L%{Q) denotes functions from L$(Q) for which Hf exists. First the inequality of extra-weak type will be dealt with. T H E O R E M 3.3.1. Let 0 E $ be quasiconvex and assume it satisfies the global A2 condition. Then the following statements are equivalent: (i) there is a c\ > 0 such that for all f € Q({ xeR1;

\Hf(x)\

L%{Q),

> A }) < cx j 0 ( J ^

e(x)

dx;

e(x)

dx;

(ii) there is a ci > 0 such that for all f € L\oc, Q({

xeR1;

Mf(x) > A }) < c2 j $ ( J ^ \

(iii) there is e > 0 such that dx ■?MlS$>)$*
1

< CO.

3.3. Weighted inequalities of weak type 105 When proving Theorem 3.3.1 we will make use of the following characterization (due to Kokilashvili [5], [14], see also Heinig and Johnson [1]): T H E O R E M A. Let 1 < p < oo, 1 < 9 < oo. Then the inequality J \\Mf(x%e(x)

dx < c J \\f(x)\fse(x)

holds for all vector-valued functions f = (f\,h,c > 0 independent of f if and only if Q 6 Ap.

dx

■■) £ L\oc with a constant

Proof of Theorem 3.3.1: Assume (i), let I be an arbitrary bounded interval and let i 7 be its shift to the right having a common endpoint with I. Put I = IU I'. Clearly, for every x € I' we have

mxs)(X)\>^!lJxi(y)dy=l. I

In virtue of the assumption (i), e(I') < ci / R

i.e. g(I') <

C2Q(I)

$(cxi(x))g(x)dx,

l

and this shows that

Q is

a doubling weight.

Let / e L%(g), f > 0, Ex = {x e R1; Mf(x) compact. Then ,

N

Kc\Jlj,

If

> A} and let K C Ex be

r

where A < y — / / ( y )\dy. .

For every Ij let l\ denote the shift of Ij to the right, having a common endpoint with Ij. Put Ij = Ij Ul'j. Ifx 6 I-, then easily

\H(fXiJ)(x)\>^I-lJf(y)dy. 3

h

106 Chapter 3. Singular integrals in weighted Orlicz classes Therefore (see, e.g. Garnett [1, Chapt. 1, Lemma 4.4]) there exists a subfamily {Ijk} C {Ij} of mutually disjoint intervals such that g ({J Ij J < 2 £3 s(Ijk)

an<

^

we can write (l'jk is the analogous shift of 7 Jt )

{K)<e\[jiA e*(#)(J;.J
k

l

1

> A/(2x)} < « * £ > { * €;\Hf R ; XIff/x/iWI A/(2»)}

<^E/*(2'i(*))«(*))«fa Ij

^x))e{x)dX. The constant C2 obviously does not depend on If so that by Fatou's lemma we get (ii). The equivalence of (ii) and (iii) has been proved in Theorem 2.2.1. Finally, assuming (iii), we prove (i). As a first step we show that g is a doubling weight. Given an interval J we have (27 is the double of J, concentric with I)

g(2I\I)

J \i\e(x) 2cJ

i

e{x)dx

2\I\g(x) e

e(X dX

>

ee(27)

+

^s/'(7)*'* 5 <4,(/) + ^(2/) = (i£ + * ) * ' > +5«<2I>-

2/

Ww*

3.3. Weighted inequalities of weak type 107 therefore g(2I) = g(2I \I) + g(I) < c4g(I) + ^g(I) = c5o(I). Consequently, g satisfies the doubling condition. As a result of (iii) we have g £ Ax 1 < p < oo.

and thus g belongs to Ap for some

Let fi = {x £ R1 ; Mf(x) > A }, F - R1 \ Q and {Ij} be a Whitney decomposition of fi, i.e. fi = (J7,, dist (F,Ij) = \Ij\ and A < \f\(Ij)/\Ij\ < 2A. 3

Let g and 6 be the good and bad part of / , respectively, g(x)-- - f(x)xi? (r) + Ef(Ij)XL ,(*)/! J/l j

b(x) == f{x) /(/;)/|/i|)x/ 3 .(x) b(x)-g(x) ■= £ ( / ( * ) --/(^)/l^l)x/ ,(*) / ( * )-- g(x) j

=E».,(*). =

!

>

(

*

)

■

i

We have g({ x£Rl;

\Hf(x)\

> 5A }) < g({ x £ R1; \Hg(x)\ > A }) + 4 A } .

As # G A 2 globally, it is 3>(A) > c\p for all A G [0,1] and some sufficiently large p. We can suppose that p is such that g £ Ap and thus (Hunt, Muckenhoupt and Wheeden [1]) g({ xER1;

\Hg(x)\ >X})
( ^ ^ J

«(*) dx


g(x) dx

l RR

(3-3-1)

\

p

*jm -j*m 1

< pJ
(^Y )

p g(x)dx cpg(Q,) g( X)dx + 22"c pg(n)

F


and as (3.3.2)

g(n) ( ^ p ) e(x) dx i? 1

by Theorem 2.2.1, the estimate for the good part follows.

108 Chapter 3. Singular integrals in weighted Orlicz classes We estimate g({x € R1 ;\Hb(x)\ > A}). If x e F, then (see Stein [4, Chapt. II, § 4.5]) (3.3.3)

\Hb(x)\ < £

\Ij | J J M I dt + cMb(x).

j

I,

It is \b(x)\ < \f{x)\ + 2A, whence by Theorem 2.2.1

(3.:5.4)

e({x c\Mb(x)\ >>A})< ■eR11;., ; c\Mb(x)\ A }) < ( ^ ) e(x) e(x)dxdx «P(2) c cf j00(^£) ++ c4>(2) e({ xeR e(n) e(fi) n


It remains to estimate the first term on the right in (3.3.3). Let tj be the center of Ij. Then

Ew/lS^s.Ei^F/Moi* 141 jj\f(t)\dt

<

X-tj\

^ < < cc V

Si

-

J,

Ujl \Wi) \h\*a 1/1(4) X-tj\

2

^[x-ttf

\lj\

'

\Ij\ ■

Denoting a,- = (I/K-J/J/Iijl) 1 ' 8 and observing that xeF, MXii{z)>r~-n \X~tj\

we get dt<

c£> a (a ixiiK*) i

and g({x eR1 (3.3.5)

j

\x-t\*dt>X})

2 < < e({x Q{{* eF; € F ; cJ2M c ^ M ^(a, I fX/>)(«) /JW > A A}) }) ++ *(«). «(«).

3.3. Weighted inequalities of weak type 109 Further, as g G Ap, Theorem A gives e({zeF;c£M2(aiX/j)(*)>A}) 3

c

\pl2

/"/

^P/(E^(W)(«))

*(*)*

p/2 -

AF72 / f e ( a i X J i ( x ) ) 2 )

g(*)•

:sr

P/2

By Young's inequality we have

e({X€F;cj:MHajXlj)(x)>X})

-i?/"!"«(3.3.6)

/

f

f zedi) ^;E unime(I)dI+j»(j4iiL\

\

dx + cg(n). Dl

Putting together the estimates (3.3.1)-(3.3.6), we get e({xeR

1

;

\Hf(x)\

> 5A}) < C l /
with a independent of / and A and we are done.

V

Q{X)dx J

□

Q(X) dx

110

Chapter 3. Singular integrals in weighted Orlicz classes

Let us formulate a particular consequence of the foregoing theorem (cf. Carbery, Chang and Garnett [1] and Krbec [2] for the equivalence of (ii) and (hi)). " C O R O L L A R Y 3.3.2. Let w be a weight function. Then the following statements are equivalent: (i) a constant c\ > 0 exists such that w({ xeR w({x eR1;1

\Hf(x)\ > A \Hf(x)\ M)}) <

Cl

R

\f(x)\ J M A

1

) w{x) dx dx ■TV

( l ++ log+ log+ ^ ('

i>(a:)

for all A > 0 and all f G Llog + L; (ii) constants e > 0, ci > 0 exists such that 11

e/xef rx vFn (

/ £ew u,(J) \ w(x) W V W dx, <, C2', r, \\I\w(x)J w(I) d x C 2

^y / H|TkwJM7) - ' (iii) a constant C3 exists such that 1 w({x M / ( x ) >> A w({ xeR }) < c3 / J ^ Gfl; 1 ; Mf(x) M)< A 1

( l ++ log log++ l ^ H u(a:) u>(a:)da; da; (>

A

for all f G L ^ . Now we prove inequalities of weak and extra-weak type linking M to the maximal Hilbert transform

#*/(*) == sup H*f{x) 00 00

/

J

M-dy x-y

.

\x-y\>t k-y|>£

T H E O R E M 3.3.3. Let 0 G $ be quasiconvex and satisfies the A 2 con­ dition near t i e origin, let Q be a weight on Rn. Then the following statements are equivalent:

3.3. Weighted inequalities of weak type 111 (i) there is Cx > 0 such that

Q({ xeR1;

H*f(x) > A }) < Cl j ( ^ ^ ) A


1

for all f G i j ( g ) and all A > 0; (ii) there is c2 > 0 such that e({x£R>;

Mf(x) > A } < c2 / ( £ i y ^ )
for aii / e L j ^ and a/J A > 0; (iii)

g£A%<.

Proof: The proof of (i)=^(ii) and (iii)=>(i) goes similarly as in Theorem 3.3.1, observe that (3.3.3) holds for H*, too. The implication (ii)=>(iii) was proved in Theorem 2.2.1 (when showing that (2.2.5) implies -4| w ). Q T H E O R E M 3.3.4. Let ^ e $ and g be a weight on Rn. following statements are equivalent:

Then the

(i) there is a c\ > 0 such that (A)
> A }) < cx I $(cif(x))g(x) R

dx

l

for all f E L%{g) and all A > 0; (ii) $ £ A 2 and there is a c2 > 0 such that
R1 ■ Mf(x)

> A}) < c2 f R

for all f £ Ljoc and all A > 0;

1

$(c2f(x))g(x)dx

112 Chapter 3. Singular integrals in weighted Orlicz classes (iii) $ £ A 2 and there is a c 3 > 0 such that * (^W)

< ^ f y J0(c3f(x))e(x)

dx,

f e Li(Q);

I

(iv) 0 G A2, 0 is quasiconvex, and g G A$. Proof: The equivalence of (ii), (iii) and (iv) follows from Theorem 2.2.2. Further, (ii)=>(i) can be proved using an appropriate good-A inequality (Coifman and FefFerman [1]). The remaining implication (i)=>(ii) can be proved in the following manner. First observe that the inequality in (ii) is implied by (i) (see Theorem 3.3.1). It suffices to show that (i) gives 0 G A 2 . But this can be seen on taking f(x) — C~1XXE{X) with 0 < g(E) < 00. Then the inequality in (i) turns into C

0(X) < k ;

Theorem is proved.

~ □

4^

1

#(A/2). K

Q({xeR ;HXE(x)>2c})

'

There is a more general theorem which include both the one weight in­ equalities of weak and extra-weak type for the Riesz transforms as particular cases. It can be proved using the approach from preceding theorems. T H E O R E M 3.3.5. Let Q be a weight on R1, 0X and 02 nonnegative, nondecreasing functions on [0,00), lim 02(t) = 0, and n be a positive and nont—>0.|.

decreasing function on [0,oo). Assume that both 02 and 77^2 are quasiconvex and satisfy the A2 condition. Then the inequality

# i ( % ( { * G Rn ; |2I,/(*)| >\}
(^f)

e(x)dx, j = l t ...,n,

holds for all f G L%(g) with a constant c independent of f if and only if there is an e > 0 such that

l sup 1*2 (ci±mM9i) < «, eix)dx T>o #x(A) (Q) J * \ \\Q\g{x) ) > < °°" 2

fi

Q

Q

where #2 is the complementary function to 02.

e{x dx

3.4- Weighted inequalities of strong type 113 One can obtain the particular cases by a proper choice of $ j and t] exactly as in Theorem 2.2.3.

3.4 Weighted inequalities of strong type for the Hilbert transform We are going to characterize those weights for which the one weight (strong) type inequality holds. Observe that the two weight inequality remains to be a challenge. T H E O R E M 3.4.1. Let g be a weight on Rn and # £ $ . inequality I
Then the

R*

holds for all f 6 L%{Q) with a constant c > 0 independent of f if and only if $ is quasiconvex, satisfies the A 2 condition together with its complementary function \P, and Q 6 A$. We postpone the proof of Theorem 3.4.1 after the following two lemmas. L E M M A 3.4.2. Let 0 6 $, Q and a be weights on Rn, and T a positive homogeneous operator on L$(Q). Then I ${Tf{x))e{x)

dx
R"

$(Clf(x))cr(x)

dx,

cj

e L#(
fi"

with a C! > 0 independent off if and only if there is a c2 > 0 sucA that for all 6>0,

||27|U#(,f) < c 2 | | / | | M M ,

/GI#(H-

Proof : The "only if" part is obvious. Assume that the norm inequality holds. Then

H

R

n

Tf(x)

6Q(X)

,C2||/||L*(*
Copyrighted Material

dx < 1

114 Chapter 3. Singular integrals in weighted Orlicz classes for all / G L$(6Q)

and all 6 > 0. Fix 0 ^ / G L$(
< oo

and put

*0-X = J ${f{X))
Then f$(f(x))60
= l

n

and so ||/||£,(4 0ff ) = 1- From this we have j^(c-1Tf(x))6oe(x)dx,

1> R"

whence /$(czlTf{x))e{x)dx<

[${f{x))a(x)dx n

R"

R

and the modular inequality follows.

□

Observe that this lemma is a personal communication of D. Gallardo. LEMMA 3.4.3. Let $ G $, Q be a weight on Rn. Define Sef(*)

=

-^H(fe)(x).

Then (3.4.1)

f 0(Hf(x))e(x) it

dx
1

R

dx,

/ G L%(e),

1

with a Cj > 0 independent of f if and only if j V(Sef(x))g(x) R

l

dx
V(cf(x))Q(x) 1

R

where c2 > 0 is a constant independent of f.

dx,

f G L%(e),

3.4- Weighted inequalities of strong type 115 Proof : By Lemma 3.4.2, the inequality (3.4.1) is equivalent to ll#/IU.(««)
f€L*(6Q),

6>0,

where Cl

>

sup \\Hf\\L*(*«) ll/IU*(«<.) < i

=

sup sup / ll/IU,(««)
Hf(x)g(x)6g(x)dx

~ fl

= =

sup

sup f
sup \\SM\L. IUIU*(f«) < i

~

f{x)-±-H{gg){x)8e{x) dx e{x)

R1

(*«)

which in turn is equivalent to / V(Seg(x))g(x)dx

< c2 f V(c2g(x))g(x)

1

R

R

dx.

1

D Now we are ready to prove Theorem 3.4.1. Proof of Theorem 3.4.1: First the sufficiency. We have

sup

i?<j>

(»i/*U))*)

< Cl < 00

where R$(t) = &(t)/t, S$(t) = V(t)/t, t G Rn. Similarly as in the proof of Theorem 2.2.1 one can find a function ^ 0 such that i(3>o) < «'(#) and g G A$0. Denote p = i(# 0 ) and define F p (i) = #(< 1 / p ). It is z'(Fp) = i( 1, whence the centered weighted maximal operator

Mcef(x)=

sup

-L-[\f(t)\dt

centre of
116

Chapter 3. Singular integrals in weighted Orlicz classes

is of strong type (Fp,Fp) with respect to g, j Fp(McJ(x))

dx
Fp(cf(x))e(x)

J?i

R

dx.

1

By Theorem 2.2.1 we have g 6 Ap and thus {Mcf{x)f

cMce(f(x)r.

<

Whence f$(Mf(x))g(x) R

=

1

JFp((Mf(x)Y)g(x)dx 1

R

<jFp(c3(McfY)e(x)dx R1

JFp(c4Mce\f\"(x))g(x)dx

< R1

=

JFp(MP(c4\f\P)e(x)dx R

1


= c6 / $(c6f(x))g(x) R

dx.

1

Using the good-A inequality (Coifman [1]) and our assumption $ £ A2, we arrive at the desired inequality for the Hilbert transform. As to the necessity, the quasiconvexity of <£ and 0 € A2 was proved in Theorem 3.3.3. Even the weak type inequality guarantees that. It remains to check that \P £ A2. This follows from Lemma 3.4.3; we have g({ xeRn;

\Sef(x)\

> A })*(A) < c2 J#{cf{x))e{x) R 1

dx.

1

1

Take K > 0 such that the set E = { x G R ; K' < g(x) < K } has a positive measure and, for A > 0 and any bounded EQ C E, define f(x)

= 7TTXE0{x), 2c2

x£Rl.

3.5. The Hilbert transform of odd functions 117 Then »(A) <

-CeiEo)

e({xeR};\Sef(x)\

>2c2}/\2)

and we are done. The reader will easily notice that results of Sections 3.2, 3.3 and 3.4 remain to be true for the Riesz transforms.

3.5 T h e Hilbert transform of odd functions If / is an odd function on R1 and its Hilbert transform Hf is denned, then Hf(x) = HQf(\x\), xeR1, where oo

(3-5.1)

Hof(x)^l-f4^ldy. 2 i * J x o

-y

The operator H is bounded in Lp(g) for 1 < p < oo if and only if g G Ap, further, H maps L\{g) continuously into L\(g) if and only if g £ A\ (Hunt, Muckenhoupt and Wheeden [1]). Andersen [1] characterized the class of weights g for which HQ is bounded in Lp(g), 1 < p < oo. This class contains Ap as a proper subset and consists of weight g for which / g{x) dx j j /(e(a:)) 1 - p V

dx \

< c(b2 - a2)"

for all bounded intervals (a, 6), with c independent of a, 6. An extension to L$(g) classes will be given in this section. Throughout we will assume that $ is an even function on R1, <£(0) = 0, # increasing on (0,oo). As introduced earlier (Sections 2.1 and 2.2), R$(t) = $(t)/\t\ and S
118 Chapter 3. Singular integrals in weighted Orlicz classes oo

1

(3.5.2) e({*eR l\HQf{x)\>\})
feL%(g),

0

holds with a constant c\ > 0 independent of f and A if and only if an e > 0 exists such that

<->

T/*GS£))IH~

where the supremum is taken over all bounded intervals I C R ■ Proof: First we prove that g is a doubling weight provided (3.5.2) holds. Let I = (a, b), a < 6 < oo, I' = (fe, 2b- a). For all i 6 / ' we have #oX/(*) >

s(Z) w6(6

_

-

Q)

2s(J) 7rs(/ u 7/)

>

J_ 2ir

so that (3.5.2) yields B(I>)

< Cl€>(2nCl) J e(x) dx =

C2Q(I)

I

and the doubling condition follows. We show that (3.5.2) implies (3.5.3). Let I and I ' be as above and put

»<*>= s »(if$)) *<<*>• *>»■ with some e > 0 which will be chosen properly later. Let

x=

^(i\yT)J9^ds^-

Then I' C { x G R1; \H0g(x)\ > A } as for x £ I' and y G I, it is x2 - y2 < 2S(IUI'), whence

H 9

°^ =

i

>

ljx^ds^

MTUnI9iy)ds^-

3.5. The Hilbert transform of odd junctions We have

119

(W <„/*(!f) lW -,

and the proof of (3.5.3) now follows along the lines of that of Theorem 2.2.1. Just observe that

I«(M)«>-

■

i(x)ds(x).

«

Conversely, let (3.5.3) hold Let a(x) = e ( v / R / ( 2 v / R ) . * ± 0. Then (7 G A# if and only if g satisfies (3.5.4). Thus (T{{X G R1; |// A}) < c U R

for all A > 0 and all g G L$(g). Hg(x) = H0f(y/F) and

(^Y^j


l

Given / , put g(x) = f(,/x),

x > 0. Then

0; |ff 0 /(*)| > A }) < A})


Uf^l\e{x)dx. l

a Now let us turn our attention to the weak inequalities. T H E O R E M 3.5.2. (3.5.4)

There is c> 0 such that

°? *(A)e({ i G fl ; |Jf«/(*)| > A }) < c I #(c/(x))e(i) dx o 1

for ai/ / G L%(g) and all A > 0 if and on/y if <2> is quasiconvex, # G A2> and g satisfies the condition (3.5.5)

sup ( ~

j

^ds(x)

J ii* (^jjjs*

( ^ y )

d»(«) ] < °°

where the supremum is taien over a/i a > 0 and ah1 bounded intervals I.

120

Chapter 3. Singular integrals in weighted Orlicz classes

Proof: We restrict ourselves to the "only if" p a r t . T h e rest of the proof is similar to proof of the foregoing theorem and we leave it to the reader. Let a > 0, I = (a, 6), I' = (6,26 - a), a < 6 < oo, assume (3.5.4) holds. Put

™=°~is* GSSJ) *'w-

x

> °'

and

x

= ^hjT)j^ds^

As (3.5.4) implies its extra-weak counterpart (3.5.2) the weight g is doubling and g(I) ~ Q{I') (see the proof of Theorem 3.5.1). Whence

) j Q(X) dx

e(l) i

C

< ^Jf(x)ds(x)l i.e. ag(I)

.to

•(dpj/*^)**^

^ "/M^W

(3.5.6)

^ _c£_ 5(7)

Let A = ff(x)ds{x).

If 0 < A < oo, then divide (3.5.6) by A/s(i"). If

A = 0, then (3.5.5) holds trivially. If A = oo, then

K

£X

^ag(x)

j £>(*) dx — OO

I

and by duality there is h € L$(Q)

such that

ex h(x)g(x)dx / ; ag(x)

== - / h(x)dx =■■ o o . I

Notes to Chapter 3

Whence H0(eh)(x)

121

= oo on V for all e > 0 and

g(I')$(\)


A>0,

e > 0.

But h G L$(g) and there is e > 0 such that the last integral is finite and thus tending to zero when e —► 0. We get g(I') = 0 and, consequently, g(I) = 0 and (3.5.5) holds. The A 2 condition for <£ follows like this: Let us fix A > 0, a bounded interval I C (0, oo), and put f(x) = (2c)) _ 1 Ax/(x), x > 0, with c from (3.5.4). Then #(A)e({* > 0; \HoXi(x)

>2c})<

cg(I)$

%

■

□ The next theorem deals with the strong inequality. T H E O R E M 3.5.3. There is a constant c > 0 such that OO

[
OO


$(cf(x))g(x)dx

0

for all f G L^{g) if and only if$ is quasiconvex, 4> G A 2 , 9 G A 2 and p satisfies (3.5.5). Proof : It follows from the preceding theorem. Note that for the suffi­ ciency one can use an interpolation argument and Theorem 3.5.2 in analogy to the considerations in Section 2.1 when the strong inequality for M was dealt with. The interpolation is justified as the variant of the Marcinkiewicz theo­ rems we need can actually be proved by handling inequalities for individual functions and truncated odd functions remain to be odd. The link between (3.5.5) and the A$ condition was given in the proof of the foregoing theorem and this yields the "openess" of the condition (3.5.5). □

Notes to Chapter 3 Theorems for singular integrals in Lp with power weights appeared in Hardy-Littlewood [2], Babenko [1], Khvedelidze [1], Stein [1], Gurielashvili

122

Chapter

3. Singular

integrals

in weighted

Orlicz

classes

[1], [2]. Generalizations to more general weights with singularities having zero Lebesgue measure can be found in Gaposhkin [1], Forelli [1], Simonenko [1], Danilyuk [1], Kokilashvili [1]. The characterization of weights in L2 is due to Helson and Szego [1] and the complete answer to the one weight problem in Lp was given by Hunt, Muckenhoupt and Wheeden [1]. The Lp version of the Helson and Szego result is due to Jones [1]. For the singular integrals with the Calderon-Zygmund kernels see Coifman and Fefferman [1]. A detailed analysis is given in the monograph by Garcia-Cuerva and Rubio de Francia [1]. Observe that the two weight problem for conjugate functions remains to be an open problem. This chapter follows papers by Kokilashvili and Krbec [1], [2], another paper under preparation by the same authors, and Kokilashvili [3] (Sections 3.1, 3.2), Pick [3], Gogatishvili and Pick [1] (Sections 3.3, 3.4, 3.5). Theorem 3.3.5 is due to Gogatishvili [9].

Chapter 4 Integral operators in weighted Zygmund classes 4.1 Maximal function in weighted XlogZ We start with a weighted version of Zygmund's inequality. Let Qo be a closed dyadic cube in Rn and {K:) = {I<:;r

= 0,l...,

S

=

l,...,2k}

a dyadic decomposition ofQo, i.e. {K$}^=1, r ~ 0 , 1 , . . . , are the cubes of the Ar-th generation, and K° = QQ. For / G Lloc(Q0), let Mdf(x) be the ioca7 dyadic maximal

= sup —— /

\f(y)\dy

function.

A Calderon-Zygmund decomposition ofQo will be here a system of dyadic cubes {Qj } where the letter k stand for the number of a generation and the 0-th generation consists of Qo, which has the following properties: (J (J* = Qo, any two different cubes have disjoint interiors, and every cube of the k + 1-st generation is contained in some cube of the fc-th generation (k — 1, 2 , . . . ) . For k = 1,2,..., let Ek denote those Q'j which resulted from the cutting in the ifc-th step and which are not divided into smaller cubes in any of the next steps. T H E O R E M 4.1.1. Let Q be a weight on Rn and Q0 a dyadic cube in Rn. Then the inequality (4.1.1)

J Mdf(x)g{x)

dx
Qo

J(\f(x)

log+ \f(x)\)e(x)

dx + Q(Q0)

V?„

/

+

holds for all f 6 £ l o g L with a constant c > 0 independent of f and Q0 if and only if there is an e > 0 such that (4.1.2)

sup J Qo

M

p I.£ \

^

i

p

>3

,

- W N g i * < oo I

123

124

Chapter 4. Integral operators in weighted Zygmund classes

where the supremum is taken over all dyadic Calderon-Zygmund tions of Q0.

decomposi-

Proof: Let (4.1.1) hold and E = {x e Qo ; M*d{XQQ)(x) > Q{X) } where M* is the formal adjoint of Md in (nonweighted) L2. Put , ( . ) = exp {e{Q{x))-'M*d{XQe){x))

*G

jlfr^X*)'

Q

°'

with e > 0 to be chosen later. We have / exp {e(e{x))-1M*d(xQ

e)(x))g(x)dx

g(x)(g(x))-1Md(xQg)(x)g(x)dx

= /

E

E

— I Mdg(x)g(x) dx Qo 'a +

< c• / (g(x) log g(x))g(x) dx + cg(Q) Qo

< c / exp (e(g(a;))_1MJ(xQe)(:c))

x

E

g(x) M*(xog)(x)'

eM*d(XQg(x)) g(x) 6\x)ax

+ cg{QQ). For e < 1/c we get f

(zMUXQQ){x)\

Q{X) ,

E

and as the result of integration over Q0 \ E is bounded by expe we arrive at

(tl.3)

UKfef^f^. J

V

e(*)

/ e(Q)

"

Set T a;

/( ) = Z l 17PT / f(v)dvx
x€Qo-

4-1- Maximal function in weighted LlogL 125 Then

T*(XQe){x)

= £ ^ % S r ^ X q * ( * ) < 2Md*(XQ
x G Q0,

and (4.1.2) holds due to (4.1.3). Conversely, let (1.4.2) be satisfied and {Q* } be Calderon-Zygmund decom­ positions corresponding to the sets { x G QQ ; Mdf(x) > 2 lt (" +1 ) }, it = 0 , 1 , . . . , i.e. 2fc(n+1)<

]^|/l/(2/)|rfy-2i(n+1)+"' * Q)

By Young's inequality and (4.1.2) we get f Mdf(x)e(x)

2Hn+1)S(Ek)

dx
J

+

CQ(Q0)

k

Qo

1

IQ)

-J\f(zMQJ)-t■ CQ{QO) Q)

^ ^ < cj cj |/(x)| \f(x)\ ((J2 ^ p XQ*(*)) XQ}(*))
ij

'

J

^/(W'W 1 " 1 )*)* Qo

+ + ccJexp(eY y exp f/ £ 2 Qo Qo

h i

^'

Lk, igti

a ! + c cg(Q ) XQ^X)) *QJ (*) ) ddx + *(0o)0

3J

< Ccj(\f(x)\ < y(1/(1)1 log log++ \f(x)\)g(x) \f(x)\)S(x)

dx dx + + CQ(Qo). CQ(QO).

Qo

The theorem is proved.

□

A standard arguments show that Theorem 4.1.1 holds for the Hardy-Littlewood-Wiener maximal function, too. In complete analogy with the above Calderon-Zygmund decomposition of a dyadic cube one can introduce a Cal-

126

Chapter 4- Integral operators in weighted Zygmund classes

deron-Zygmund decomposition of an arbitrary cube in Rn. corresponding results:

Let us state the

T H E O R E M 4.1.2. Let g be a weight on Rn. Then the following state­ ments are equivalent: (i) a constant c\ > 0 exists such that

J Mf(x)e(x) dx 0 and c^ > 0 exist such that

<-•)

Q

w

c

/^w^ )^H '

/exp

\e^2 "•*

\

for every cube Q and every Calderon-Zygmund

decomposition {(?*} ofQ.

It is of interest to compare the condition in (ii) with the following condition, usually denoted by A*: there is an e > 0 such that

(A*)

s

f

exp

f eB{Q) \ Q(X)

yJQ {\Q\e(*))e(Q)

J

dx<0

^

°-

Q

It is clear that A\ =>■ A** => A*. In Carbery, Chang and Garnett [1] some examples are given which show that the converse implications do not hold.

4.2 Hilbert transform in Zygmund classes We are going to characterize the weights for which the weighted inequality of the Zygmund type for the Hilbert transform are true. Let us recall that the Hilbert transform in R1 is defined by

Hf(x)=

lim £-.0+ 7T

/ J

\x-t\>£

IQ-dt. X

- t

4-2. Hilbert transform in Zygmund classes

127

As well known (Hunt, Muckenhoupt and Wheeden [1]) t h e one weight in­ equality of weak type (1,1) for the Hilbert transform holds if and only if g G Ax. It follows from t h e "if part" of this theorem t h a t the A\ condition is sufficient for t h e weighted analogue of Zygmund's inequality, on t h e other hand, it turns out t h a t Ai is not necessary. A necessary and sufficient condition is established in t h e next theorem. on R1. Then the

T H E O R E M 4 . 2 . 1 . Let w be a weight function statements are equivalent: (i) a constant c\ exists such that f \Hf(x)\w(x)

dx <

Cl

(w(I) + J(\f{x)\log+

i

\f(x)\)w(x)

following

dx

i

for all intervals I and f £ L\oc supported in I; (ii) constants £\ > 0 and c 2 > 0 exist such that 1 f ~ T n / w(I) w(I) J

J°

ex

)(«)IVw•JJ(X) x dx dx

4P ■("

fei\H{XEw)(x)\\ .\H{XEW

V

)

t \ w(x) u>(x)

()

< 2 - °C2

i

for all intervals I and all Borel subsets E of I; iii) constants £2 > 0 and C3 > 0 exist such that \H(XE\H(XEW)(X)\\ w)(x)\\ 1 f 1 ( dx < C3exp w(x) J (-

m jI

1

for all intervals I and all Borel subsets E of I. Proof: First we verify (i)=>(ii). To start with we show that (i) implies the doubling condition for w. Let I be any bounded interval in R\ and divide it in two intervals of equal length, say h and I 2 . Given a function / € Lloc, / > 0, supp / C I\, then for x € h w e have

(4.2.1)

W(l)]>_l.jM-^>^L.Jmst.

(4.2.1)

i*«*>i*i/i,-,i***niii/'w*

Setting f(x)

= 4irciXh, it follows from (4.2.1) that

2ci / w(x) dx < ci / w(x) dx + 4TTC\ log 47rci / w(x) dx,

128

Chapter A. Integral operators in weighted Zygmund classes

whence Jw{x)dx
and the doubling condition follows. Keep the notation 7 for an arbitrary (bounded) interval and let 7' be the interval sup 7 + 7. Set

( ew(I) \ w(x)I
XE

,^ W

where e is some positive number (to be specified later) and

£? == {x e l ;

>!}•

\I\w{x) |/|ti/(x)

Then

Mii)^=m/^™/'™'>-

E

Our assumption (i) gives then

/exp(^iIL)mdx<

J

v

\\I\w(x)J

w(I)

°

f\Hg(x)\w(x)dx

- w{I')J 1 '

a w l

K

>

I

E

-^F)

\H9^)H^^
j IUI1

E

^ c + ^^y e x H^wJlm^)J wW)w{x)dx
ew(I) \ w(x)

£

/ e x p ( ^ ' ) ^

\I\w(x)J w{I)

d

E

If now e < c - 1 , then £W(I) \ w(X) /

G X P

( |/|u»(*)y w(i)

dx < c.

4-2. Hilbert transform in Zygmund classes 129 Consequently, /exp J

(4.2.2)

\ w(x)

( ew{I)

\\I\w(x)J w(I)

dx <

exp

/ ew(I)

\\JHx)J

E

\ w(x) dx + exp i w(I)

< C

with c independent of I. Taking into account that tp < cexpt, estimate (4.2.2) gives

Km

t > 0, the

wix)

; c / -j-± dx < < c,
(4-2.3)

i i

i.e. w £ Api, p' = p/(p — 1). Set now Ek-{xeI;l
k = l,%,...,

w(x) w(x)

gk{x)=e^f^m^m)^MM. V

«>(*)

/

,etf,

\H{XEW){X)\

t=1>2

where t\ is again a positive constant to be chosen in a proper way later. It readily follows that

j E (4.:2.4)

v

w x

()

J H1)

Ekk

^vfj j

9k(x)\H(xEw)(x)\dx

1

Rfli s x w)(x) dx, ~ w(l) / Kx)9k(x)H(xE ( )9k(x)H(xEw)(x)dx,

" w{I) j1 " R R1

where s is defined by fl S(X)

1 -1

UH(XEW)(X)>0,

iiH(XEw)(x)<0.

We have already proved that w G v4p/, thus by the Holder reversed inequality we have XEW £ L 1 + a for some 8 > 0 (see, e.g., Garcia-Cuerva and Rubio de Francia [1, p. 261]) and g G i ( i + 1 ) / * . Therefore we can apply the Riesz

130

Chapter 4- Integral operators in weighted Zygmund classes

equality to (4.2.4) to get

j e x p {—sw—J W)dx=W)i Bi'M')*"*')** Ek

Ri

< j \H(sg)(x)\^dx


Cl

i
"

J(\g(x)\log+

\9(x)\)^dx

i +r 1+

fr^f^\H(XEw)(x)\\ P

V

V

W{x)

»(*)

£ l

| H(XEw)(x)

>/|ff(x^)(*)l

\ W(X)

«;(*) " MI)

*

fey\H(XEw)(x)\\ = c i +, c i r£ i f e x p £I\H{XEW)(X)\\

7

(—M?)—J

Ek

w(x)

^

w(I)

Thus choosing ei < cj" 1 , the above chain of inequalities provides the desired conclusion. We do (ii)=>(iii) now. Let I, Ii, I2 be as in the proof of (i)=>(ii); by (4.2.1) we have w (I2)
Making use of the elementary inequality st < exp s + t log + t,

s, t > 0,

we see that w(I2) <

exp 1

2c2J '

\H(xi2w)(x)\ w(x)

u;(x) t/x + clog + C2Ciu(/i)

h

<

\Hh -u/2) + c3w(Ii).

At once it follows that w(I2) < cw(h) Let I' be the interval 7 + s u p I . Again, by (4.2.1), we can write w

h)Iexp (ii§)) w(x)dx - /exp (|7>M) w^dx I

1

}

^M^*'' ^**— {

-Xfp^)Jw{x)dx<^ v w(x) > ~

W(I)

4.2. Hilbert transform in Zygmund classes

131

the constant e being chosen in such a manner that ec<sv Whence (4.2.2) is true and, consequently, (4.2.3) holds, too. If e 2 is such that £2p/(p- 1) < «i. we have

*M

fi |#(X£™ (*))h dx

== im/en jty^,("'g(g;^»i).-(, <) r'(iWiw.))<<',1

,

.(w-Gx-wH (y--^*r1 «(A/-(.*w)-*r(// ,


( i7i/ p ( Ti ^ /

e x p


ex

AP-I p

| tf(X£ «,(*)) | \ w(x) w(x) J

|ff(

\

" (i)rfx

V/

\

1/p

p

/

I.P-1J/P

dxj\ ( p - l ) / p

*4r)(3:))l) $ } d x )

The implication (ii)=*(iu) is proved, next we will prove (iii)=>(ii). If I, h, h have the same meaning as above, from (4.2.1) we get

ra/-

(i/ 2 |iw) c (a;

(4.2.5)

<

j ^ / e x p /'ce|F(x/ 2 ut )(*)l\ w(x) ,

i,

2 < ' -|Au/

2

|

u>(:c)

)

yl , ( 1 ^ M ) ^

dx

r.ur,

0 , we have

f (( w(I2) V 11 1 dx < c. I2\w(x)J

ray \h\lK\ l i r a d°^By Holder's inequality, w(h) tv{h)
Z

11

/ f

\ vJ \ ■- (x)dx) Bp /

-I/P

<

Cttf(/i) cw(h) \h\

132

Chapter 4- Integral operators in weighted Zygmund classes

thus w

(h)

< cw(Ii).

This and (4.2.5) imply (4.2.2), so that w G Ap> and as well known, the reversed Holder inequality holds, namely, there are 6 > 0 and c > 0 such that \ 1/(1+*)

( (

1

1+s

— j w (x)dx [-jw'+\x)dx\

Z- jI dx dx < ■£-

j

(x)dx. ww(x)dx.

1

We are now able to prove (ii). If t\ is such that t\6/{\ + 6) < €2, it is

1 w(I) T n / <

I\H(XEW ex

P' (

£

w(x) 7~\

1

«,(/)

exp

)(*)iyiy(a;) cfa; *"(*) d x

( £iS \H(XEw)(x)\\

{i

(/

+

w{x)

s

//

"1

)

*

\ 1/(1+*) 1/(1+*) 1+s 1+6

x I fw (x)dx (x)dx\ /


dx"

J

j \ */(!+*)

\H(XEw)(x)\\

(|71 1

w(x)

A

J 'j

< < c.c.

It remains to check that (ii)=>-(i). As already shown, (ii)=^(iii), in particular, it is w G Apt. Whence x™ € L1+s for some 6 > 0. Let / G l / ^ + ^ n L l o g " 1 " L(iu), supp / C I, and E Ei1 =

{xel; {xeI;Hf(x)<0}, Hf(x) < 0 } ,

Ei Ei =

l\E\. I\EL

Then 2

J\Hf{x)\x / | f f / ( * ) Mo{x) x ) dx
i=l

*

4-3. One weight inequality 133 Employing the Riesz equality, we get 2 i+1 f(x)H(XE w(x)) dx J \Hf(x)\w(x) dx = £ ( - l )l), + 1 Jj f(x)H( M*)) XEi

R1 2

/■

;U;(a;))|(fx \f(*)H(X(E <E/|/(*)ir(xB i»(«))|
-t

1

g , a; / ' l * / - M £ l H{x ( X g i Mx) ( )l-/-N.I-

X

f^ L(x)dx. w(x) - z ^ y i/(*)i—^j—"(*)<<*• -V"

I

Whence, by Young's inequality, 1

( | / ( x ) | l olog g | / (\f(x)\)w(x) a : ) | H x ) r fdx x sE I| iIy *y"(|/(x)| ++

J \Hf(x)\w(x) dx < £

«=i

als M /-( ^r )-w*) +j-r<%;™)-MA + < ^ ++ ~f /(l/(x)|log /(|/(x)|log++ !/(«)!)«;(«)&, < ^2j2^\f(x)\)w(x)dx, which completes the proof of the theorem.

□

Notice that an analogous result is true for the Riesz transforms.

4.3 One weight inequality for the strong m a x i m a l function For / 6 L\oc, the strong maximal function M'f is defined by

M'f{x)= sup jj-J\f{y)\dy where the supremum is taken over all parallelepipeds containing the point x and whose edges are parallel with the coordinate axes. Here we will present a characterization of the class of weights which guar­ antee validity of the one weight analogue of the well known inequality due to

134

Chapter 4- Integral operators in weighted Zygmund classes

Jessen, Marcinkiewicz and Zygmund [1]. The condition w G Ai(J) is sufficient but not necessary. To get a necessary and sufficient condition we will need the following concept: Definition 4.3.1. a-scattered if

A finite sequence {Jk}k=1 of parallelepipeds is called

WuV)
1 < k < N.

THEOREM 4.3.1. Let wbea weight function on EP. Then the following conditions are equivalent: (i) there is c\ — c\(n, w) > 0 such that w{x€Rn-M'f(x)>\}
j 1 ^ 1 (l + log + ^Y^)

for allf £ L ( l + log + L)n~l; (ii) tiiere are positive constants e and ci such that for every sequence { / t } ^ ! of parallelepipeds,

N

J

exp \

fc=i

wJk \Jk\w(x)XJ> .(*)

l / ( n - -1)

u;(x) dx <

IK

C2U;

w(x)dx

l/2-scattered

(54

k= l

Proof: First we prove (ii)=>(i). Let A > 0 and {x€Rn\M'f(x)>\}.

Ex =

Let K\ be a compact subset of E\. Then there is a finite sequence of paral­ lelepipeds {Ji}fL1 such that M

Kx C [J Ji t=i

and \Ji\

\f{y)\dy>\,

i=

l,...,M.

J,.3. One weight inequality 135 From {J^f^i we choose a 1/2-scattered sequence in the following way: Let Ji be Jj_. If J\,J2,--,Jk have been selected, then Jk+i will be the first parallelepiped after those already selected from the original sequence {Ji}fLi and such that the sequence {Jj}jt\ is a 1/2-scattered. Clearly, M

| J ^ c {x e Rn ; M'{Xyjk)(x) > 1/2}.

(4.3.1)

Any system consisting of one parallelepiped, say, {
_ w_y <"-

J

Making use of the elementary inequality tp < cp exp (4.3.2)

(~jX

we have

J(w(x)y-"' dx < cwJ, j

i.e. w belongs to the strong Ap class. Therefore (see Section 2.2), w{x(ERn;

M'f(x)

> A } < cX'" J

\f(y)\pw(y)dy.

Combining it with (4.3.1), we get w{jJi

<w{x£Rn;

M ' ( X l J ~ ) ( z ) > 1/2}

(4-3.3) < cw [J Jk . Further, ^[jJk < c^wJk < j^wlk^r k k k

=

J

[\f(y)\dy k£

J_ f 2cc1\mj(ey 2cc2 J

\J7k

e

y*

A I VlAk(y) ; \Jk\w(v) ^

xr{y)))w{y)dy.

7

136

Chapter 4- Integral operators in weighted Zygmund classes

By Young's inequality in the form st < •t < c f((1 ( l + log+ s ) " " 1 + exp t 1 / ^ " 1 ) ) ,,

s,t>0, a, t > 0,

we obtain the estimate u>\}Jk k

<J-/ -

2c 2Ryn +

[W??

2cc2 l/(y)l £ A

I 2cc2

y

1

I 7fc

/ exp 1

V^

wJ

r

, v

k

1

iu(y) dy

/(n - 1 ) \ lVC"-*)

w(y)dy

^E| J i H y ) ^(^


1/(2/), ) " A

-1

w(y)dy

k

From this we conclude that -i

1/(2/) 1 i/(»)iy w{Jj <2c J\!f\(l u>(y) dy. f l ++ log+ lo g+^y~\(y)dy. A u ; | J j f ck < 2 k

R"

Using (4.3.3) we get

1 < j \m ^ ^ m y wKx cl w{y)dy. -1

w(y)
R"

As K\ was an arbitrary compact subset of E\, (i) holds. We prove (i)^-(ii). Let {Jk)i^=i be a 1/2-scattered sequence of paral­ lelepipeds. It is v I / O- i )

/ le

expf e

WJk £ \Jk\w(y) > «(y)J XJ

w(y)dy

4.3. One weight inequality 137 00

-i)

m = ll m =

, |J

U k

iC'EiAwa^w)""" >

Jb

'

'

'

'

w{y)dy

'

k fc

-m/(n-l)

n-1

LI- 7 ' OO

n —1

:?^-«)

J

ml

m=l

/•

m/(n-l) Ji;(j/)


k

oo

E - mE= l ••■+ m=n E ••• zz

=l zz. m /l+

+

m=n

/

= h + h2 ■ By Holder's inequality,

(4.;1.4)

n " -- 11

/l

sE

m= 0

! m/(r»-l)

m / \ l --(». ( » --ll))//m

/

-^j—

(w{Jjk) j.

\

m=0

/

\

\ (n-( n -ll))//m m j.

/

From (4.3.2) and by a repeated use of Holder's inequality, we get that for any parallelepiped / and an arbitrary measurable EcJ,

,.

(43 5)

-

ffl_.

wj

f\J\Y

^E ~ [W\J '

If we set

Ek = Jk\\J

Jj ,

j
then | / i | < 2\Ek\. Whence (4.3.5) yields u»Jt < T>cwEk. Therefore, < c^wEk

*It

k*

*

and by (4.3.4) we arrive at

(4.3.6)

h
£ ) m=l

— = T - <™{JJ*'

k

Now we need an analogous estimate for I2. Let functions gm : Rn -» E 1 satisfy || < 1,

m =

n,n+l,...,

138

Chapter 4- Integral operators in weighted Zygmund classes

and (f

f

-1) \ m/(n —1)

wJ j

*XJk{y) \\mfin~

( f(sr

J\^\hRy) \l\^\hHy)

) XJt{y)

)

\\ (*" - l ) "/ m

u;(j/) (fy 1 w{y)dy

j

-/(?I^»)-«-«*

Then .x

, wJk

_1)

x XJ> (/(?iSMri ') "

x

((nn--ll))//m m

1

)wWds j w(y)dy )

= ?"/'15F/pm(y)rfy J

h

s « X > ^ ■JiM* J*

*

(4.3.7)

*Ek-±-J gm(y)dy l J f c Jk l/

Vk
/

J M M'g ( 2 ) w ; ( z ) ddz 2 f lm m(z)w(z)

k

= m(z)w(z)dz m (z)u;(z)
/(/m( m -„-„ < c||M c||Mssyymm|L || Jm. m ,fuw | J j t ) + i)<x + i)=f It

-l)/m

*)

(see Proposition 5.1.1.). Making use of the inequality 11

p ((1++. loo6g++, ,) r"--1<<(_± , «(l ( ' , T))" t„,

«>!,„>!, t>\,p>h

4-3. One weight inequality 139 and putting p = m/(m - n + 1), we get from (i) that

5

| | ^ < / m | | L m / ( m - n + l)oo

<

( t

\ ((m-n+l)(n-l)/m m—n+l)(n —l)/m

c mm

1

1 _ ji )1

C

\ m-n +1 m-n+1

1

II/TII |||fl|| | 0 | | ££ -./(—»+i) m A m - » + l ) »,

/

consequently, m 1 /(m-n -"+ )("--11)/m )/ m m _ n 4+ 1 \ (l\(m -n+1)(n m ] | ^ g mm\\| |LZ£H~ + M » +. l+) o1o) „ <<|( m / ( m- i_ \\M°g ^" \I n n--11 )/

From the last estimate and (4.3.7) we have oo -

h =E

W(n-l) ££m/(n-

ii»

^ y~ . ££m/(n-l) W(n-l) m=n ~ "

^E m=n m=n

\ m / ( n - -1) NW(n-l)

wJk wJk XJ

m!

m=n

, /

\Jk\w(y) '

*

//

m-n +11 \V m -n + 1 (m

r

l/i-li m/(n-l) ccm/(n-l)

ral

/

° i '[n-1 v X

^

n 111

w(y)
m _ n + 1

"+1J

i.il

1 T

W(J7A;

\m '7

1 m mm7ulw 1Jk7, U jfc it

Observe that if e < (2ec/(n - l ) ) " ^ " 1 ) , then > ^ ^

m/(n-l) ££ m/(n-l)

/s

\xm m

m 1 m m m
so that

/a < cw{Jjk h
(4.3.8)

k

which is the desired estimate. Putting together (4.3.6) and (4.3.8) concludes the proof. □ Theorem 4.3.1 can be generalized to the case of the maximal functions with respect to a general basis: If B is a basis in Rn, i.e. a family of bounded open sets in Rn, then one puts *f tl M Bf(x)

\

=

( 8uP4r/|/(y)|d»

if * x€ e U B, 5,

I 0

otherwise.

J) B 3 i

B

BB£B €B

140 Chapter 4- Integral operators in weighted Zygmund classes In analogy to the concept of Ap(J) (see Section 3.2 in the preceding chapter) we define the Ap(B) class. The result for Mg is as follows: THEOREM 4.3.2. Let B be a basis in Rn and w a weight, 0 < 0 < oo. Then the following statements are equivalent: (i) there is c\ > 0 such that w({ xeR";

MBf(x)

> A }) < f J M i ( l + log + i f f l i )

«;(*) d*

for aJJ / G X,^ and aJi A > 0; (ii) there is c2 > 0 and a function si : (0,oo) —» (0, oo) such that for every ct-scattered finite sequence {5*}* C B and for every e > 0, v\ 1/0 1//8 / w(Bk) w(x)dx exp i .{*), / \Bk\w(x)XB'

J cxp^^Ej^^y^wJ 1 »(«)<& twE \

•(B))

^y

s

B J t <(l + eMU *(U
w({ x£Rn;

MBXE(X)

> A }) < c2u>(£)

for some 0 < A < 1 and all measurable E C Rn; (iii) there is C3 > 0 and a function s 2 : (0,00) —> (0,00) such that to every £ > 0 and every finite sequence {fli}t C 6 there corresponds a subsequence {Bkj}} C {5t} f c such that "MU5*)


and

I U

exp (.,(«) £ | ^ ^ y X B k j (*)) fc

i

M«) dx < (1 + Oil f U 5 *,

4-3. One weight inequality 141 Consider the condition: There is an e > 0 such that (MB))

sup/exp(^)^
BZBJ

\\B\W(X)J

W(B)

B

i.e. a generalization of A^1 for the exponential function. Then A;*(B)cA*p(B)cf)Ap(B). Observe that A*AB) is not a sufficient condition in the case of a general basis, nevertheless, if B is the basis consisting of all cubes in Rn, then it works as w is then a doubling weight and one gets Theorem 4.1.1. (Carbery, Chung and Garnett [1]). Considering now J, the basis of parallelepipeds (see Section 3.2), then Mj is the strong maximal function Ms and if w belongs to some Ap(J), then also the second condition in (ii) is satisfied (Bagby and Kurtz [1]) whence we get Theorem 4.3.1. Let us point out the particular case of the above theorem for the strong maximal function: T H E O R E M 4.3.3. Let J be a basis of parallelepipeds in Rn and w a weight, 0 < /? < oo. Then the following statements are equivaient: (i) there is c\ > 0 such that

<-l

i/(*)iy

0

n n s w({ xeR ; ;M 'f{x) > A }) < J J M 1 (l ++ log log++ i f f l ! ) w(a;)
for all f € (ii) there is c2 > 0 and a function si : (0, oo) —> (0, oo) such that for every ascattered finite sequence {Jk}k C 3 and for every e > 0,

I / ( ^

(J))

// exp |^ «iWE (e) £| ^ ) X \

\ J t

fc

< *); <(i+e)«»[U (i+o^tU-J/ *);

l l P \ 1//9

(x)J /

1 «(«) dx
142

Chapter 4- Integral operators in weighted Zygmund classes

(iii) there is c 3 > 0 and a function s2 : (0,oo) -> (0,oo) such that to every e > 0 and every finite sequence {Jjjfc C J there corresponds a subsequence {Jkj}j C {/*}* such that W w

j (IK) \i /

((i>) U J * ) <
/

and

/

IK J

MA,) exp |(■»(«)E \

t

1//9

1

,<*>) /

ui(x) da: < (1 + e)w

H

\}

/

For /3 = n — 1 and u> = 1 we get the theorem due to Cordoba and Fefferman [1], see also Bagby [2]. Observe that the condition A^l_1(J) is not suffi­ cient. On the other hand, A\(J) is a proper subset of v4**_1(,7), the condition Ai(J) is only sufficient.

N o t e s to C h a p t e r 4 The classical theorems on the maximal operators in the Zygmund class are due to Hardy and Littlewood (n = 1) [2] and Jessen, Marcinkiewicz and Zygmund [1]. This is discussed e.g. in the well known monographs by Zygmund [1], Sadosky [1], etc. Section 4.1 follows Carbery, Chang and Garnett [1]. The weighted version of the Zygmund inequality appeared in the last cited paper, two weight weak type inequality in Krbec [2] and further generalization to L(l + log + L)K in Pick [1]. The study of a behaviour of the Hilbert transform presented here is due to Gogatishvili [7]. As to the strong maximal function, see Bagby and Kurtz [1] for the sufficient condition. A necessary and sufficient condition was found by Gogatishvili [8], [10] and is presented in Section 4.3.

Chapter 5

Fractional maximal functions in Lorentz spaces 5.1 Weighted Lorentz spaces Let (X, v) be a space with a positive, «r-additive measure v. For 1 < p < oo and 1 < s < oo, the Lorentz space Lpa = LP'(X, dv) is the space of all im­ measurable functions / for which ||/||L».(.Y > ( J„) < oo, where

||/||i> W )

= I * J(y{ * e X ; !/(*)! > r

})'/PT'-1

dr\

, if 1 < p <" CO, and 1 < s < oo,

and | | / I M x , d , ) = sup r({ z 6 X ; | / ( x ) | > r J ) 1 ^ , T>0

if 1 < p < oo, and s = co.

If 1 < p < oo and l < s < c o o r p = s = l o r p = s = co, then IJ"(X, dv) is a Banach space with a norm equivalent to || . ||iP«(A',di/)In the sequel, we will assume that X = Rn and that the measure v is absolutely continuous with respect to the Lebesgue measure, i.e. dv — w(x) dx. The space LPJ = Lp'(Rn,w dx) will be called the Lorentz space with the weight w. Observe that \\XE\\Ll.={wEflp. Further, if p = s, then ££,' is the weighted space

Lp(w).

For a fixed p, and s2 < «i, we have Lg,*2 C ££,*l» in particular,

(5.1.1)

ll/lli^ < l l / l k 143

144 Chapter 5. Fractional maximal functions in Lorentz spaces Let us recall some other important facts from the theory of the Lorentz spaces (see Hunt [1], Lorentz [1], Chung, Hunt, Kurtz [1]). PROPOSITION 5.1.1. Let 1/p = l/pi + l/p 2 , V* = l/si + l/s2. there exists a constant c > 0 such that (5.1.2)

Then

H/i/alk'<e||/illx!HI/allL«'» 1

for every / i £ L$' ,

and every / 2 £ LPj"2.

The inequality (5.1.2) is an appropriate variant of Holder's inequality in Lorentz spaces. PROPOSITION 5.1.2. TAere exists a constant c > 0 suci that (5.1.3)

C-l\\f\\Lf

< SUP

/ f(y)h(y)™(y) dy < cll/IUf

for all f £ L^J where the supremum is taken over the closed unit ball in the space Uw' . If s < oo, then the inequality (5.1.3) follows from the fact that Uw' is the dual of Z/Ps. If s = oo, then the first inequality in (5.1.3) can be obtained by a suitable choice of the function h as an appropriate multiple of a characteristic function. The second inequality follows from (5.1.2). P R O P O S I T I O N 5.1.3. Let 1 < s < p < oo and {Ej} be a sequence of measurable subsets of Rn such that oo oo

J2XE ,{*)

<S

J'=l

where £ is a positive constant. (5.1.4)

forallfeLl*.

Then

5>*i/IEt-<«ll/llk

5.1. Weighted Lorentz spaces 145 Proof: Let 0 = p/s > 1. By Minkowski's inequality we get (^2\\XEj\\pLr.)

(w{xeEj)\f(x)\>r})1^T'~1dT

= s "y

o oo

<sJ\\(w{ oo X€Er,

x !/(«)!> r ) 1 ' " ! ^ r*- iT

0

0

(y£w{xeEj;\f(x)\>T})

= s

7

< sj(Zw{x o

r'-Ur

€ Q E , ; |/(x)| > r})'/"^-1

=efp\\f\\hm-

dr

J'=1

D In Chung, Hunt and Kurtz [1], the following class of functions was introduced: Definition 5.1.1. Let either 1 < p < oo and 1 < s < oo, or p = s = 1. Then a weight function w : Rn — R1 belongs to the Aps class if supllxQllx-IIXQ/HljV.'IQr 1 < oo

(5.1.5)

where the supremum is taken over all cubes Q C Rn with sides parallel with coordinate axes. It is easy to see that

jw(x)dx Q

(J -M-i) w

Q

{x)dx)

\ p-i

= \\xQ\rL„\\xQMrL„,

I

i.e. App coincides with the Muckenhoupt Ap class.

146

Chapter 5. Fractional maximal functions in Lorentz spaces

P R O P O S I T I O N 5.1.4. Let w E Ap>. Then w G APlll and 1 < si < s or p < pi and 1 < si < oo.

if either p = pl

Proof: Let p = pi and Si < s. Then s[ > s' and therefore

HxoMljV.; This proves Aps C ApSl

<\\XQ/™\\L£''-

for p = Pi and si < s.

Now let p < pi, 1 < si < oo, and w £ Aps. Define p 0 by 1/p'j = l / V + l / p o Using (5.1.2), we have < CCIIXQILL I U ||LP'.' ■||XQ/W IIXQIL^-IIXQIIL^ HXQIILSHIXQ/HIJVII < ~HXQIILS,"'IIXQ/ 'IILL'-' < ^clQlllxQtl^-llxolko-llxollZr. CIQIIIXQILCSHIXQIIxHlxdllv 1

1/p = c|Q|(™Q)/Pi1 / P+l + 1l //Pp o°-- 1/p = clQK^Q) = c\Q\. = c|Q|.

Consequently, w G APl0O and in virtue of the preceding part of the proof, we get w £ ApiSl for all 1 < sj < oo. Q P R O P O S I T I O N 5.1.5. A weight w belongs to Api if and only if a constant Ci > 0 exists such that for all cubes Q and all measurable sets E C Q E \1^1 \ <

/*,**

101-

l,p

C 1fwE\(wE\llp

UJ •

Proof : Let p > 1, w G A p l , £ C Q.. Then by (5.1.2) \E\:

-I

w(x) dx < C IIXQH lit'' -HXBIIL'I w(x)

E E

<

■"rafe UQWL'J

•«(=§)* •MET=

5.1. Weighted Lorentz spaces 147 Now let (5.1.6) be fulfilled and set E = {x £ Q; w(x) < \/y }. Then ywE = / yw(x) dx <

^ f

~ J w(x)

J

dx = \E\

*«m£fIf p > 1, then wE'y->>'(wQ)-r'''>, i.e. tu €

Apl.

If p = 1, then for all y <

\\XQ/W\\L™°°

y <

c

> WE > 0, we get

3—77,

whence w G A n , in other words w £ A\.

D

Observe that Proposition 5.1.4 and 5.1.5 imply the doubling condition for each w G Ap,: There is a constant c > 0 such that for all cubes Q C Rn, w(2Q) < cwQ, where 1Q is the cube concentric with Q and with side length twice of that of Q. P R O P O S I T I O N 5.1.6.

The class Ap is a proper subset of

Apl.

This is easy to see: In virtue of Proposition 5.1.5, we have Ap C Ap\. On the other hand, the function w(x) = |x|"( p _ 1 ) belongs to Api, but not to Ap (see Section 5.5). P R O P O S I T I O N 5.1.7. Let w G Ap„ 1 < p < oo, 1 < s < oo. Then there exists pi and si such that 1 < f\ < p, s\ > 1, and w £ J 4 P I J I . P R O P O S I T I O N 5.1.8. Let 1 < p < oo and 1 < s < oo. A function w belongs to Aps if and only ifw 6 Ap. When proving Proposition 5.1.7 and 5.1.8 we will follow the paper Chung, Hunt and Kurtz [1].

148

Chapter 5. Fractional maximal junctions in Lorentz spaces

First notice that Proposition 5.1.8 readily follows from Propositions 5.1.7 and 5.1.4. Indeed, let w G Ap3, 1 < p < oo, 1 < s < oo. Then according to Proposition 5.1.7 there are numbers pi and s\, p\ < p, 1 < «i < oo such that w G APlSl. In virtue of Proposition 5.1.4, the function w belongs to ApSo for all s 0 G [l,oo), whence also to App = Ap. If w G Ap, we get in a similar way that w G Apt, 1 < s < oo. It remains to prove Proposition 5.1.7. This will be based on several follow­ ing lemmas. PROPOSITION 5.1.9. Let w G Ap\. Then there is a constant c > 0 such that for all cubes Q and all (3, 0 < /? < 1,

c y w{ X e Q ; w{xeQ;l/w(x)>0\Q\/(wQ)}. \/w(x) >P\Q\/(wQ)}.

wQ wQ<(j^p) <-(-, -PI

Proof: Let E = { x G Q ; \/w{x) > f3\Q\/(wQ)} and E = Q \ E. Then \E\ P\Q\

^Qlm\dx-^Qlw{x)dx

=

E

E

^ < 1 . wQ ~

=

Therefore l^l = |Q| - \E\ > (1 - /3)\Q\. According to Proposition 5.1.5, w

*m*
U

PROPOSITION 5.1.10. Let 1 < p < oo, 1 < s < p and w G Api. Then there are constants c0 > 0 and /?, 0 < /? < 1 such that for all cubes Q and any A > \Q\/(wQ), \\XEJW\\LPJ.. < coXiwEpx)1'"' where E\ = {x G Q; l/w(x)

> A }.

Proof: Observe that A

^

=

^/^)_1^^

5.1.

Weighted Lorentz spaces 149

The measure w(x) dx satisfies the doubling condition, therefore we can use the Calderon-Zygmund decomposition and as a result we get a sequence of nonoverlapping cubes {Qk}k>i contained in Q and such that A

and \/w{x)

< 7 7 T / ( w O O r M * ) dx = i % i < cA, wQk J wQk < A for a. a. x G Q \ (J Qk.

Let / G ££? satisfy we have / XE\(x)(w(x)) «»

jfc = 1,2,...,

||/||LJ;

< 1. In virtue of Propositions 5.1.1. and 5.1.3,

1

x w x f(x)w(x)dx f( ) { )dx<2_]< 2_] fc>i fc>i

w x ((w(x)) ( ))

1lf(x)w(x)dx

f{x)w(x)dx

Qk

M\ M\ l& l& Jb>l

i fc>l

ifp
fzi
QJ\Ll-X(wQk)^'

fc>l \1/p /

/

Q

\1/p'

S'MDI/XQXSTJ (|> *J
/

xl/p'

<^ii/ik-(E^O /

< cA 1 £

V 1/P'

wQk\


t

V /p '

< cX I £ wQk\

150

Chapter 5. Fractional maximal functions in Lorentz spaces

According to Proposition 5.1.4, w 6 Aps implies w G Api, whence Proposi­ tion 5.1.9 gives XEAX)(W(X))

1

f{x)w{x)di

< c\ ( J ^ C(P)W{ *£Qk; i M * ) > P\Qk \/(v>Qk)} \Js>l

< cX(w{xeQ;

l/w(x)

j3X}ylp'.

>

Passing to the supremum over all / from the unit ball of L^, we get the desired inequality. □ PROPOSITION 5.1.11. Let 1 < s < p < oo and w G Apa. Then there are constants c > 0 and 6 > 0 such that for all cubes Q, l | X 9 /

^<-

C

(^)

i | I X Q /

<'-'

where p'x — p' + p'6/s and s'x = s' + 6.

Proof: Let Q be an arbitrary cube and put Ex =

{xEQ;l/w{x)>\}.

Clearly, for all 8 > 0, oo

IQI/(»Q)

6 1

\\XBJw\\'^.,\s-ld\

J\\XEjw\\^,,X - dX= j (5-1.7)

°

"

°

+

M

/

|| WHI^.'A'- 1 dA.

IQI/C^Q) We estimate the right hand side of (5.1.7). First, IQI/(<"Q)

/ 0o

1

IQI/C^Q) lQl/(»0)

||x^Ht,'.'^- dA<|| XQ / u ,||'' F .. .

f

Xs'1 d\

o0

6 \wQJ \\xQM\°i,.,

5.1. Weighted Lorentz spaces 151 Further, CO

/

WXEjwW'^yX^dX

IQI/(»Q) oo


(w{x£Q;l/w(x)>f3\}y'!p'\s+,'-1d\

f \Q\/(*Q) oo

< c/3 f(w{xeQ;

l/w(x)

> A»'i/p''A5i-1dA

0

c

< 7\\xQH\yA as s'/p' = s^/pi. Applying Fubini's theorem to the left hand side of (5.1.7), we get oo

j\\XEjw\\^,,\S-U\ 0 oo

oo

= f s' f(w{ xeEx; 0

l/w(x)

> T }y'lp'T''-1

dr A*"1 d\

0

oo

oo

= f s' f(w{ x£Q;

\/w{x) > A, l/w(x)

> r } ) J ' / p ' r ' ' - 1 dr A*"1 dX

0

0

oo A

= s' f f(w{ xeQ; 0

l/w(x)

> A, l/w(x)

> T })»7P V ' - i

dr A *-i

dX

l/w{x)

> A, l/w(x)

> r }y'/p'T''-1

dr Xs'1 dX

\/w{x)

> X, })>'/<>'T''-1 drX6-1 dX

0

oo oo

+ / f(w{ xeQ; 0 A

oo A

= s' f f(w{xeQ0

0 oo oo

+ s' 0 f Af(w{xeQ;

l/w(x)

> T}Y',P'T''-1

drX6'1 dX

152

Chapter 5. Fractional maximal functions in Lorentz spaces oo

= f(w{xeQ;

l/w(x) > A, } ) ' ' / / v ' + * - i d X

o OO T

l/w(x) > T } ) S ' / P V ^ A * - 1 dXdr

+ s' [ f(w{ x£Q; oo oo

= f(w{xeQ; i/w(x)>\}y'>'p>yi-1d\ o oo

+ J f(w{xeQ;

l/w(x) > r }y'/p'r*'+t-i

o

oo oo

1

= ^TWXQHI^ IXQ/H

dr

+ jjJ(w{xeQ;l/w(x) 1(w{x eQ; l/u;(x)

&

l r}Y'^T^-- 1dr dT >>r } ) * 1/PVJ-

0 0

s'

= jr\\x ^ I I Xq QM%., /HI^, + + ^ l-|IXQ/HI'1.< lxg/HI^ = jiix,/»ii;v,-6\\xQM\%,1. Whence (5.1.7) yields

< JllxgHI^ J l l x o H l ^ + J (*g)' Hxg/u-lliV.' llxg/Hfe..

^HXQ/W JIIXQ/HI^.! <

^tu

1

and we have

WXQM\%>


T i l

Now it suffices to choose 5 > 0 sufficiently small so that 1/6 — c/(s' + 6) > 0 and we are done. D P r o o f of Proposition 5.1.7: By Proposition 5.1.4 we can assume that 1 < s < p. Take p\ and s[ from Proposition 5.1.11. Then p[ > p', and

5.2. Three weight weak type inequalities

consequently,

Pl

153

< p and

,
1

rPi'l

(\Q\Y' "' (c (J^Y

< c(wQy"» ^ M j '

\\XQH\'^.)

\\XQM\-L(.\Q\>'J

l/»i

= cmwQ^p^'/^-'-npO = c|Q|, using the fact that 1 Pi WegetweAPlJl.

s' 1+1 s[

s* 1- = s[p

1 s' 7 + -TT = 0 P'I 4K

D

5.2 T h r e e weight weak t y p e inequalities A complete characterization of the weights for which the inequalities in question are valid will be given in this section. Besides that the one weight problem in Lorentz spaces will be solved for the strong type inequalities. Recall that for a / £ Ljoc(Rn), of the order 7 is defined by

0 < 7 < n, the fractional maximal function

M 7 / ( z ) = sup I Q p / " - 1 Q3x

[ff(y)\dy, J

the supremum being taken over all cubes Q C R" whose edges are parallel with the coordinate axes. Observe that if fXq G L& and \\xQM\LrJ.> Qas follows from (5.1.2).

< oo, tnen / *s integrable over

T H E O R E M 5.2.1. Let 0 < 7 < n, 1 < s < p < q < 00, V, v, and w be weight functions. Then the following two statemenss are equivalent:

154 Chapter 5. Fractional maximal functions in Lorentz spaces (i) a constant ct > 0 exists such that

(5-2.1)

l|MT(/V»)|Ur < dll/lUsr

forallftli'; (ii) a constant c2 > 0 exists such that

(5.2.2)

< c->\Q\l-lln

(vQy'WxqtML,"

for ail cubes Q C fl". Proof: First we prove (5.2.1)=>(5.2.2) assuming that either 1 < p < q < oo and 1 < s < oo, or p = s = 1 and p < q, or p = q = s = oo. Let Q be a cube in Rn and / a nonnegative function supported in Q such that \\f\\t~ = 1 and y /(*)^(r) c*x > C H X Q ^ / H I J V . ' • If x e Q, then

Af,(/*)(*) > |Qr / n _ 1 / f(y)i>(y)dy. Q

In virtue of (5.2.1) we have

vQ<

dlQI^/")* | ff(yWy)dy)

i

I

\\f\\i~

and the choice of / implies

vQ
which proves (5.2.2). Now we prove the converse implication (5.2.2)=>(5.2.1). Put HX = {x e Rn ; My(frl>)(x) > A } n Q 0 , where A > 0 and Q 0 is an arbitrary (fixed) cube. For every x £ Hx let Qx be the cube centered at x and such that

l«.l*-/l/(»M»)*>».

5.2. Three weight weak type inequalities 155 Invoking the Besicovitch covering lemma (e.g. de Guzman [1, Thm. 1.1]), we can find a sequence of cubes (Qj)j C (Qx)x€Hx such that HxC\jQj

and

£>Qi(aO<£

for some f only depending on n. Combining this with Proposition 5.1.1 and 5.1.3, and with (5.2.2), we get

V

(r p/

{vHxy'' < 2>Q i ) »A-'|Q/| p/

p

< C 4^(^ J ) ^- |Q J l

(7/n 1)p

-

(7/n 1)p

-

i

/ \f{y)My)dy

\i

|l/XQ J ll

p

)

LS ,.||xQ^/^ir i| ,-.-


< cell/Ill,. • As Qo was arbitrary, (5.2.1) is proved.

□

T H E O R E M 5.2.2. Let 0 < 7 < n, 1 < p < n/y, 1/q = 1/p - -y/n, 1 < v < 00, 1 < s < 00. Tien tie following statements are equiva7ent: (i) a constant c\ > 0 exists such that (5-2.3)

||M7(/^")||Lr <

Cl||/||Lr

Aoidsforatf/elf/; (ii) a constant c-i > 0 exists such that (5.2.4)

(WQ^IIXQ"^"-1!!^'.'


for ah* cubes Q C i2 n ; (iii) it is (5.2.5)

weAp,

where 0 = l + q/p'.

156

Chapter 5. Fractional maximal junctions in Lorentz spaces

For the proof we will need the following LEMMA 5.2.3. Let 1 < s < p < q < oo, f/n = 1/p - 1/q. Then the condition (5.2.4) is equivalent to w G Apt for f3 = 1 + q/p' and £' = (I/P' + I / S K . Proof: The Apt condition follows by a straightforward computation. We have i/,'

/n

\\xQ^ -\^

n

1

= / ( M x e Q ; wil -\x) > A})'>'A*'" dX l/s'

= j f{w{x£Q;

l/w(x)

7

> A"/("- >

/P

s 1

}Y' 'X '- dX i/,'

= I l(w{ x G Q ; 1/ W (x) > t } ) » 7 P V ' ( « - 7 ) / » - i (l/p' + lfq)/? 1

= ( /(™{ * € Q; l/w(«) > * }/'/<»V'- dt = IIXQ/<<;> 1 / ? • D Proof of Theorem 5.2.2: Let (5.2.4) hold. In virtue of (5.1.1.), the inequality r/B IIXQ^"- 1 !! .... < ||xg«/ -V-' is true for all So < s. Whence from the very beginning we can assume that s /?• Choose p\ and gi in such a manner that 1 < pi < p < qi < q, /?i = 1 + 9 I / P 1 , and j/n = 1/pi - l/qi. In virtue of Lemma 5.2.3, the condition w G Agx is equivalent to (u;Q) 1/ "||XQ«' 1,/n - 1 |LPI < csWI 1 - 7 '",

5.2. Three weight weak type inequalities 157 and, therefore, using Theorem 5.2.1 (for ip = w"1^), we obtain

(5.2.6)

WM.ifw^W^Kc.WfW^.

On the other hand, choose exponents pi and 92 i n s u c n a w a v t h a t 1 < P < P2, 1 < 9 < 32, l/n = l/p2 - I/92 and put /?2 = 1 + ftM- Clearly /32 > /?• Thus w G A ^ and Theorem 5.2.1 implies

HAf^C/^/-)!!^-. < CBII/IIX^. According to the Marcinkiewicz interpolation theorem we get (5.2.3) for all v, 1 < v < 00. We have proved (5.2.4)=$>(5.2.5)=>(5.2.1). Also, we see that if (5.2.4) holds for some s, 1 < s < 00, then (5.2.3) is valid for all v, 1 < v < 00. Besides that, we have also proved the implication (5.2.4)^(5.2.5). We show that (5.2.3)=S>(5.2.4) in the following situation: If (5.2.3) holds for some 1/, 1 < v < 00, then (5.2.4) holds, too, for all s, 1 < s < 00. So let (5.2.3) be valid with some v 6 (0,oo]. Then \\M^vPln)\\L*~

< | | M 7 ( / ^ / " ) | j x r < ci||/||i!r.

This and Theorem 5.2.1 yield

(wQ)1/q\\xQw1/n-%,>»> < c 2 |gr- 7/n . As shown above this gives w G A3 whence w £ Ap which means that we have proved (5.2.3)=$>(5.2.5). It remains to show that w £ Ag implies (5.2.4) for all s, 1 < s < 00. We have already seen that w £ Ag gives the validity of (5.2.3) for all v, 1 < v < 00, in particular, for u = s. Applying Theorem 5.2.1, we see that (5.2.4) is fulfilled. □ Now we introduce another maximal function which under our assumptions majorizes the function M 7 / . Let v and w be weights in Rn. Then we define

Mf(x) := sup Q3*

1 (vQ)1/'- fllxg/IUf-

As no confusion can arise in what follows, the dependence of M on p, q, s will not be explicitly stated.

158

Chapter 5. Fractional maximal functions in Lorentz spaces If the couple (v,w) satisfy the condition SUPO^IIXQ/HI^.'^CIQI1-^", Q

then, clearly, Myf(x)

<

cMf{x).

We prove a theorem which will find its applications in the next chapter, but is also of independent interest. THEOREM 5.2.4. Let 1 < s < p 0 the inequality v{x<ERn;

Mf{x)

> A} < CA-»||/H£M

holds with a constant c > 0 independent of f and A. Proof: We will make use of the standard approach. For A > 0 let Ex =

{x€Rn;Mf(x)>\}

For all x € E\ there exists a cube Qx such that {vQ*rllq\\xQj\\L>j

> A.

The family (Qx)x£Ex covers E\ and the Besicovitch covering lemma guarantees the existence of a sequence (Qj)j C (Qx)xeEx with the properties

ExcljQj, ;'

$ > Q j (*)<£, j

where £ only depends on n. Therefore

(vExy"> < Yl(vQj)pfq <

<€*-'ll/lifco

^-'T.WXQMLV

5.3. Maximal functions in spaces L'J

159

5.3 M a x i m a l functions in spaces ZJ,1 The results of the preceding sections immediately give the following two theorems. T H E O R E M A. Let 1 < p < oo, 1 < s < s < oo. Then there is a constant c > 0 such that (5-3-1)

\\Mf\\LrJ < c\\f\\Ll.

for all f e ISJ if and only if w G Ap. The proof of Theorem A is included in that of Theorem 5.2.2. T H E O R E M B. Let 1 < p < oo. Then there is a constant c > 0 such that

WMfhir < c\\f\\LV 1

for all f G L^ if and only if w € Ap\. Theorem B is a particular case of Theorem 5.2.1 for rp = 1, p = q, y = 0, and s = 1. Both the above theorems were prove earlier by Chung, Hunt and Kurtz [1]. Observe that Theorem B does not give a complete characterization of the class of weights w for which the operator M is bounded on IJQ-. The solution was given by Hunt and Kurtz [1] and the following section is devoted to it. T H E O R E M 5.3.1.

Let 1 < p < oo and 1 < s < oo. Then there is

constant c > 0 such that (5-3.2)

\\Mf\\LV < c\\f\\L» ,

fGLPj,

if and only if w € Ap. As to the sufficiency we refer to the foregoing section. It remains to prove the necessity. Our Theorem 5.3.1 will actually follow from the next lemma whose proof is included in that of Theorem 5.3.1 in the end of this section. We state the lemma explicitly to make the things more transparent. L E M M A 5.3.2. l e t 1 < p < oo and 1 < s < oo. If

\\Mf\\LU < C l | | / | U ,

/62#,

160 Chapter 5. Fractional maximal functions in Lorentz spaces for some Ci > 0 independent of f, then there is 1 < q < p and r such that

||Ar/lUv < c3|I/Hi«x,

fELtf,

for some ci independent of f. From Lemma 5.3.2 we get by interpolation (see, e.g. Stein and Weiss [1, Chapt. V, Thm. 3.15]) that M is bounded in Lpo(w) for q < Po < PBut then w G Apo and thus w G Ap. We will still need an auxiliary assertion concerning distribution functions. Let us introduce the following notation: If E C Rn is measurable, then we put Ex = {xeRn;MXE(x)> A}. PROPOSITION 5.3.3. There exists a constant a > 1 exists such that for all X G (0,1), t G (0,1/a), and for any measurable set E C Rn, w{xERn;

> atX] <w{xeRn;

MXE(X)

Proof: Clearly, MXEX{X) \Ex\{x

MXE(X)

> t}.

= 1 > t for X £ E\. Therefore E RT ; MXB>(x)

>*}\ = Q

for all 0 < t < 1, It remains to prove that w{x G Rn \Ex ;

>at\}<w{xeRn;

MXE(X)

MXE{X)

> t}

for t G (0,1/a) with a suitable a > 1. We show that MXE{X)

<

aXMxEx(x)

for all x G Rn \ Ex where a only depends on n. The set E\ is open. Let {Q,},>i be a Whitney decomposition of E\ into cubes, i.e. £A

= (J Qi

and dist (Qi,Rn \EX)<

diam Q,- < cdist (Q{,Rn

\ Ex).

5.3. Maximal functions in spaces Lj,1 161 We have

(5-3.3)

mJ

2" XEiy)dy ^jXE(v)dy<^J <2"A. XE(y)dy XE{y)dy<2^.

-

Qi Qi

|2»g.-i

2Qj 2Qi

Let x G Rn \E\ and Q be an arbitrary cube containing x. If Q D Q, ^ 0 for some i, then Qt C iVQ where AT = iV(n). As | £ \ £ \ | = 0 we get by (5.3.3.) that

W\IXEiy)dy

=

Q

Wig

< |0| <

E

/

E

Nn \NQ\

XE(y)dy XE(y)dy

2 A

E

" W.-I

QiCNQ

< (2JV)' NQ

< aXMxEx( )< a\MxEx{x). x

Passing to the supremum over all Q containing x, we finish the proof.

□

P r o o f of Theorem 5.3.1: It is sufficient to show that there are q < p and r > 1 such that WMXEWL'J


n

for all measurable E C R ■ In other terms, we need to prove that l

f(w{ xeRn; o

MXE(X)

> t })r/nr-1

dt <

(as M X £ ( x ) < 1). Our assumptions imply l

(5.3.4)

f{wEtyln>-1 o

dt < c(wE)>'r.

c{wE)rlq

162

Chapter 5. Fractional maximal functions in Lorentz spaces

We prove that for k = 0 , 1 , . . . , n and for all measurable E C Rn, l

(5.3.5)

k

J(wEty'"t'o

1

(log 1 )

^ <

c(ca')k(wE)'f"

where c is the constant from (5.3.4) and a is from Proposition 5.3.3. If k — 0, then (5.3.5) and (5.3.4) coincide. Let (5.3.5) hold for some k, we prove that it holds for k + 1. Let us start with the equality

Jew"--' (.os I)' £ = <^/<>*r'v- (* ?)' f, 0

0

which is true for all 0 < A < 1. Therefore A

k

~ JiwErY/Pr'o

1

log (?)

^

<

(ca'^X'-HwExY'r.

Integrating this with respect to A and taking into account (5.3.4), we get l

A

J \ J(wETyf"r'0

1

k

1

(log ?)

^dX<

(ad>)

0

k+1

j \'-\wEx)'lp

dX

0

c(cas)k+l(wEy/p.

< By Fubini's theorem, 1

f(wETTY"r-' y/pTs- [ / { (log £ ) ' £)'/>■ >> J(wB

■'(/}('- ; ) '

0

» ) $

d

I

= M>''''--'(/H)' K))£ F))S j{wETylpr'-

=

0

' ( / ( -

1

f(wETy/pT'- - 1 (log 0

i.e. (5.3.5) holds for ifc + 1.

r

;)'

d(log

dr

(* + !)!

5.4- Three weight inequalities 163 Now we choose an arbitrary a, 0 < a < 1. Multiplying (5.3.5) by Sk — (a/(cd'))k, we obtain I

J(wEt)'/n-\\og

I)* ^ <

cak(wEy/",

0

i.e. l

J(wEt)''n'-l(6\og

i) f c ^ <

ca?{wE)'lr

0

Summing up over k gives l

({wEtylns-lt-6 dt < — — (wE)'lp. J 1— a o Put r = s — 6, 5 = pr/s. The constant a can obviously be chosen in such a way that q > 1. Then (5.3.6) turns into (5.3.6)

l

I{wEtftif-1 o and we are done.

dt


Q

5.4 Three weight inequalities for generalized maximal functions This section is devoted to the solution of this problem in the weak type setting. Recall the definition of the fractional order maximal function

Myf(x,t)=sup\Qp/n-1J\f(y)\dy Q

where the supremum is taken over all cubes Q C Rn containing the point x and with sidelength l(Q) > t. Denote Q = Q X [0,£(Q)) and let v, w, rp be weights in Rn and 0 be a Borel measure on Rn x [0, oo). T H E O R E M 5.4.1. Let 0 < j < n, 1 < s < p < q < oo. Then the following statements are equivalent:

164

Chapter 5. Fractional maximal functions in Lorentz spaces

(i) there is a constant ci > 0 such that (5.4.1)

/3{(x,t)

£R"x

[0,oo); M 7 ( / V 0 ( M ) > A} < ciA-«||/||«j.

for all f e LPf and all A > 0; (ii) there is a constant c-i > 0 such that (5.4.2)

ifiQ)xl9\\xQ+M\Lv

^

c

*\Q\l~lln

for an arbitrary cube Q. Proof: First we prove (i)=>(ii). Let Q C Rn be a cube, / > 0, (x,t) G Q, then

M7(/V<)0M) > \QV'n~l j ' f{yWv)dy. Q

In virtue of (5.4.1) we have

PQ < /?{(*.*) e fl" x [o,oo);Af7(/iM*,t) > \\Q?ln~l j f{y)i>{y)dy) Q

(5.4.3) -9

n 1


ll

* l l

Il/H

On the other hand, for the function g = XQ^M there is a nonnegative function /o such that ||/o|| rp» = 1 and WXQIP/W\\LVJ.'

< / fo{y)iP{y)xQ(y)—rr™(y)dy-

Inserting f — fo into (5.4.3) we get

(0Q)1,q\\xQ4>M\L,>.'-(i). Fix a cube Q0 and consider

ax =

{(x,t))eQ0;My(M)(x,t)>\}.

5.4- Three weight inequalities 165 The set Qx is bounded. Denote d(x) = sup{t > Q;(x,t) G n A } . Clearly, d(x) < £(Q0) and it is (x,t) G ft A for all t < d(x). For every (x,d(x)/2) 6 fiA there is a cube Qx such that *(Q.) >

^

and \QxVln~l

(5-4.4)

j \f(y)My)dy

> A.

The family ( Q r x [0,2^(Q j; ))) xe Q o covers Q\ and (Qx)x€Qo covers Q0. Now if sup ^(Qx) = oo, then there is a cube Q' G (Qx)xeQ0 such that Q0 C 5Q'. If xtQo

sup ^(Qx) = di < oo, then there is Qx G (QiJxeQo such that i(Q\) > rfi/2. xeQo Then for any cube Qx which has a nonempty intersection with Q\ we have (■{Qx) < 2^(Qi) a n d Q * C 5Qi- Let d2 - s u p { £ ( Q s ) ; x G Qo,Qs nQi =«>}■ Let Q-i be a cube from our family (Qx)xeQa,QxnQl=t satisfying ((Q^) > d2/2. If there is a cube Qx from (Qx)X£Q0iQxnQl=t having a nonempty intersection with Qi, then £(QX) < 1l(Q-i) and Qx C 5Q2- Continuing this way and invoking the fact that (J Qx is a bounded set, we obtain a sequence (finite x€Qo

or infinite) of nonoverlapping cubes {Qk}k such that for all Qx G (Qx)x£Qa there is fc0 such that Qx C 5Qfc0 and £(QX) < 2t{Q]Ca). As readily seen, Qx x [0,2*(Q,)) C 5 0 ^ , whence (5.4.5)

fiA

C ( J 5Qi.

Therefore by (5.4.5), (5.4.4) and (5.4.2) we have

/? ((w W / s < X>5Q*) X>5Qi),'//'« p

ifc>l

< \->Y,Wk)plq\QA-p{1-lln) [j \f(y)My)dy)
166

Chapter 5. Fractional maximal functions in Lorentz spaces

< c3X-" S(/ 3 50*) p/, |50*l" ,,(1 " 7/n) IIXQ lk /|IU.||X5Q^/«'^ F ../ fc>l

l p


□

5.5 Restricted weighted inequalities of weak type for generalized fractional order maximal functions It will be shown that the class of those weight couples {w,f5) for which the two weight inequality holds for the operator M 7 on the set of all characteristic functions of all measurable sets (which will be called a restricted weighted inequality) is larger than the corresponding class when the whole space l%£ is considered. THEOREM 5.5.1. Let 0 < y < n, 1 < p < q < oo. Tien the following statements are equivalent: (i) there exists a positive constant ci such that (5.5.1) n

/3{(x,t)€R

x [0,oo); MyXE(x,t)

> A} < ciA"

g

/ 1 / w(x) dx YE

for all measurable sets E C Rn and all A > 0; (ii) there exists a positive constant C2 > 0 such that (5.5.2)

p{{x,t)

eRnx

[0,oo); M7f(x,t)

> A} < c a A-«||/||»

for all f e Ltf and all A > 0; (iii) there exists a positive constant C3 > 0 such that

(5.5.3)

IQP/"- 1 ^) 1 /* <

c3\E\-\wE)^

for all cube Q C Rn and all measurable sets E C Q;

IIP

5.5. Restricted weighted inequalities of weak type 167 (iv) there exists a constant c4 > 0 such that

(5.5.4)

< c 4 |Qr- 7/n

WQfl'WxQMy-

for all cubes Q C iJ". Proof: Clearly (ii)=j-(i). In virtue of Theorem 5.4.1 we have (ii)'O'(iv). Now we prove (i)=»(iii)=>(iv). Let (i) be true. Let Q be a cube and E C Q measurable. For (x,t) 6 Q we have

MyXE(x,t) > \\E\\QV>n-\ Therefore

PQ < H («,<) G ^" x [O.oo); M 7 X B ( * , 0 > j | £ | IQP 7 "- 1 }


.

Thus (5.5.3) holds. Now (iii)=>(iv). Fix a cube Q and set Ex =

{xeQ;w(x)
By (5.5.3) we have A / w{y) dy <

^r-rU)(y) dy = \EX |

< c 5 |Q| 1 -' r/ "(/?Q)- 1/ «(u;£? A ) 1 /''. It means that /

{PQflq\\

\

,

J Qn{w(x)
>

-1/p' -1/p'

< cslQI1"77"-

w{y)dy j

Passing to the supremum over A > 0 on the left hand side we get (5.5.4).

□

5.5. Restricted weighted inequalities of weak type

169

If a = 0, t h e n we would have (6P)i/p

b

(\E\py/p

<

^'SC

i

\X\P~1

If - 6 < - a < 0, t h e n one can use the fact t h a t x ^

is an even function.

T h e last case is / = [-a, b], 0 < a
/i*r^> E £

\E\/2 l-EI/2 /

i»r-> d,

-|B|/2 -|B|/2

-1 (£)'

ol-p

= —

l^|P,

and

f\xr1dx=-(aP J P

+ bP)!

|I| = o + l.

I

It suffices to show that

( aP aP ++ bP VY'rV / P : + b)Pj ~

1/p f2^\E\P\ {21-P\E\Py V Pl^lp /

( 2 ^ ' \ ?

for all a, b > 0 and some c > 0, but this is obvious. Thus the class of couples (w,0) for which the two weight inequality for the operator M1 on the set of characteristic functions of all measurable subsets is larger then the class of those couples for which this inequality holds on the whole space £&'. A question arises whether the validity of the weak type inequality for M 7 on the set of characteristic functions implies its boundedness on this set. The answer is negative as the following example shows: Let us consider the function x ,_► \X\P-I on R1 for which an interval I C R1 exists such that j{MXI{X))P\X\P-* R

dx = oo

1

and

J \X\P~' dx < oo. I

Let 1 = [1,2], then for m > 2, MXli,2](*) > ~ n > 1

5.5. Restricted weighted inequalities of weak type 169 If a = 0, then we would have (&P)1/P

bb

(|£|P)1/P

< ~

<$■!< 1 S C c.

\E\ \E\

^

-

If — b < — a < 0, then one can use the fact that x \-► | x | p _ 1 is an even function. The last case is J = [-a, 6], 0 < a < 6. Then clearly |£|/2

i

p

l

f i*p-d* = - ( ^ j

J\xr dx>

-\E\/2 2l-p

\E\P,

P

and /

\x\p-1dx=-(ap

I

+ bp),

P

|/| = a + 6.

It suffices to show that / gP + bP \

1 / p

(21-P\E\P\1/P C

\p(a + b)p)

- {

P\E\P

(21-P\1,P

)

C

- {~p~)

for all a, b > 0 and some c > 0, but this is obvious. Thus the class of couples (w,0) for which the two weight inequality for the operator M1 on the set of characteristic functions of all measurable subsets is larger then the class of those couples for which this inequality holds on the whole space U£. A question arises whether the validity of the weak type inequality for M 7 on the set of characteristic functions implies its boundedness on this set. The answer is negative as the following example shows: Let us consider the function x i-> | z | p - 1 on R1 for which an interval I C R1 exists such that f(Mxi(x))p\x\p-1dx

= oo

and

Rl

Let / = [1,2], then for x > 2, Mx[i,2](x) ^ Z—T Z X — 1 > X

j \x\P~1 dx < oo.

170

Chapter 5. Fractional maximal functions in Lorentz spaces

and oo

J(MXlil2](x)y\x\''-

1

dx > J * = oo. 1

R>

On the other hand, 2

Jx>-Ux
N o t e s to C h a p t e r 5 The one weight problem for the Hardy- Littlewood maximal function in Lorentz spaces was solved by Chung, Hunt and Kurtz [1]. The material of Section 5.2 comes from Kokilashvili [5], [7]. In particular, Theorem 5.2.4 gen­ eralizes Chung, Hunt and Kurtz [1, Theorem 6]. The exposition in Section 5.3 follows Hunt and Kurtz [1]. As to Sections 5.4 and 5.5, they contain gen­ eralizations of Ruiz and Torrea [2] and belong to Genebashvili [2], [3], [4]. A further generalization of results from Sections 5.4 and 5.5 to spaces of the homogeneous type can be found in Genebashvili [3]. Let us also observe that another Lorentz quasinorm, namely, •■if

(

f{f*{x)Yw(x)dx

where /* is the nondecreasing rearrangement of / with respect to the Lebesgue measure was used by Sawyer [4] in connection with the study of the behaviour of the Hardy averaging operator I H I " 1 f* f(t) dt. Other classical operators in this setting are studied in [2] by the same author.

Chapter 6 Potentials and Riesz transforms 6.1 T h r e e weight e s t i m a t e s for integrals with positive kernels In this chapter we present solutions to one and more weight problems for the Riesz potentials in weighted Lorentz spaces. A more general setting is considered, namely, integrals with positive kernels. In the end of the chapter we solve the one weight problem for the Riesz transforms in weighted Lorentz spaces. Let k : R" x Rn —► Rl be a measurable, a.e. positive function and

(6.1.1)

Kf(x) = J k(x,y)f(y)dy Rn

the associated integral operator. We will investigate the three weight problem of weak type for the operator K. T H E O R E M 6.1.1. Let 1 < s