LP(-)

(1)

ONE AND TWO WEIGHT ESTIMATES

FOR ONESIDED OPERATORS IN

L ^p( ^· ⁾

^SPACES

V. Kokilashvili, A. Meskhi, M. Sarwar

Communiated by R. Oinarov

Keywords and phrases: one-sided maximal funtions, one-sided potentials, one-

weightinequality,two-weight inequality, trae inequality.

Mathematis Subjet Classiation: 26A33, 42B25,46E30.

Abstrat. Various type weighted norm estimates for one-sided maximal funtions

and potentials are established in variable exponent Lebesgue spaes

L ^p( ^· ⁾

^. ^In

partiular, suient onditions (in some ases neessary and suient onditions)

governing one and two weight inequalities for these operators are derived. Among

other results generalizations of the HardyLittlewood, FeermanStein and trae

inequalitiesare given in

L ^p( ^· ⁾

^spaes.

1 Introdution

This paper deals with the boundedness of one-sided maximal funtions and

potentials in weighted Lebesgue spaes with variable exponent. In partiular, we

derive one-weight inequality for one-sided maximal funtions; suient onditions

(insomeasesneessary andsuientonditions)governingtwo-weightinequalities

for one-sided maximal and potential operators; riteria for the trae inequality

for one-sided frational maximal funtions and potentials; FeermanStein type

inequality for one-sided frational maximal funtions; generalizationof the Hardy-

Littlewood theorem for the RiemannLiouville and Weyl transforms. It is worth

mentioning that some results of this paper implies the following fat: the one-

weight inequality for one-sided maximal funtions automatially holds when both

the exponent of the spae and the weight are monotoni funtions.

The boundedness of one-sided integral operators in

L ^p( ^· ⁾

^spaes ^was ^proved ⁱⁿ

[13℄.Inthatpapertheauthorsestablishedtheboundednessofthe one-sidedHardy

Littlewood maximal funtions, potentials and singular integrals in

L ^p( ^· ⁾ (I)

^spaes

with the ondition on

p

^whih îs ^weaker ^than ^the ^log-Hôlder ôntinuity ^(weak

Lipshitz) ondition.

Solutionof theone-weight problemforone-sided operatorsinlassialLebesgue

spaeswas given in[48℄, [1℄. Trae inequalitiesfor one-sided potentialsin

L ^p

^spaes

wereharaterizedin[38℄,[40℄,[22℄.Itshouldbeemphasizedthataompletesolution

of the two-weight problem with transparent integral onditions on weights for one-

sided maximal funtions and potentials in the non-diagonal ase are given in the

(2)

monographs [16, Chapters 2 and 3℄, [9, Chapter 2℄. For Sawyer-type two-weight

riteriafor one-sided frational operators werefer to[35℄, [36℄,[34℄.

Weightedinequalitiesfor lassialintegral operatorsin

L ^p( ^· ⁾

^spaes ^were ^derived

in[6℄, [8℄, [10℄[14℄, [19℄, [23℄[32℄, [45℄, [47℄,et (see also[21℄, [44℄).

The one-weight problemfor the two-sided HardyLittlewood maximal operator

in

L ^p( ^· ⁾

^spaes ^was ^solved ⁱⁿ ^[7℄. ^Earlier, ^some generalizations of the Mukenhoupt ondition inthese spaes dened onbounded sets were disussed in[30℄ and [31℄.

Criteriafor the boundednessof two-sided frationalmaximaloperators from

L ^p _w

to

L ^q( v ^· ⁾

^were ^given ⁱⁿ ^[24℄. ^T^wo-weight ^Sawyer ^type ^riteria ^for ^two-sided ^maximal

funtions onthe real linewere announed in[23℄,[25℄.

In[2℄neessaryandsuientonditionsonaweight

v

^governing^theboundedness ompatnessofthegeneralizedRiemannLiouvilletransform

R α( · )

^from

L ^p( ^· ⁾ ( R ₊ )

^to

L ^q( v ^· ⁾ ( R ₊ )

^,

α ₋ > 1/p ₋

^,^were ^derived.

In Setion 1 we give the denition and some essential well-known properties

of the Lebesgue spae with variable exponent and formulate CarlesonHormander

typeinequalities.InSetion2westudytheone-weightproblemforone-sidedHardy

Littlewoodmaximaloperatorsin

L ^p( ^· ⁾

^spaes,^while^Setion³^is^devoted^to^the ^same

problemforone-sided frationalmaximalfuntions.InSetion4wederivesuient

(in some ases neessary and suient) onditions guaranteeing two-weight

p( · )

q( · )

^norm êstimates ^for ône-sided ^frational ^maximal ôperators. ^FeermanStein type inequalities in variable exponent spaes are disussed in Setion 5. In Setion

6 we established riteria governing the trae inequality for the RiemannLiouville

and Weyl operators in

L ^p( ^· ⁾

^spaes. ^In ^Setion ⁷ ^we ^formulate generalization of the HardyLittlewood theorem for one-sided potentials in these spaes. Setion 8 is

dediated totwo-weight inequalitiesfor one-sided operators.

Finally,wepointout that onstants(often dierent onstants inthe same series

of inequalities)willgenerally be denoted by

c

^or

C

^.

2 Preliminaries

Let

Ω

^be ^an^open ^set ⁱⁿ

R ⁿ

and let

p

^be ^a^measurable ^funtion ^on

Ω

^. ^Suppose ^that

1 ≤ p ₋ ≤ p + < ∞ , (1)

where

p ₋

^and

p +

âre^theînmumând^the^supremumrespetivelyof

p

^on

Ω

^.^Suppose

that

ρ

îs â ^weight ^funtion ôn

Ω

^, ^i.e.

ρ

îs ân âlmost êverywhere ^positive ^loally

integrable funtion on

Ω

^. ^W^e ^say ^that ^a ^measurable ^funtion

f

^on

Ω

^belongs ^to

L ^p( ρ ^· ⁾ (Ω)

^(or

L ^p(x) ρ (Ω)

⁾ ^if

S p,ρ (f ) = Z

Ω

f (x)ρ(x) ^p(x) dx < ∞ .

It is known that (see, e.g., [33℄, [26℄, [28℄, [42℄)

L ^p( ρ ^· ⁾ (Ω)

^is ^a ^Banah ^spae ^with

the norm

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k f k _L ^p(·) _ρ _(Ω) = inf

λ > 0 : S p( · ),ρ f /λ

≤ 1 .

If

ρ ≡ 1

^, ^then ^we ^use ^the ^symbol

L ^p( ^· ⁾ (Ω)

^(resp.

S p

⁾ ^instead ^of

L ^p( ρ ^· ⁾ (Ω)

^(resp.

S p( · ),ρ

^). ^It ^is ^lear ^that

k f k _L ^p(·) _ρ _(Ω) = k f ρ k L ^p(·) (Ω)

^. Ît ^should ^be âlso êmphasized ^that

when

p

^is ^onstant, ^then

L ^p( ρ ^· ⁾ (Ω)

^oinides ^with ^the ^lassial ^weighted ^Lebesgue

spae.

Further, wedenote

p ₋ (E) := inf

E p; p + (E) := sup

E

p, E ⊂ Ω, p ₋ (Ω) = p ₋ ; p + (Ω) = p + .

The followingstatement is well-known (see, e.g., [33℄, [42℄):

PropositionA.Let

E

^be â ^meâsurable ^subsetôf

Ω

^. ^Then ^the^followinginequalities hold:

k f k ^r _L ⁺ ^r(·) ^(E) _(E) ≤ S _r( _· ₎ (f χ E ) ≤ k f k ^r _L ⁻ ^r(·) ^(E) _(E) , k f k L ^r(·) (E) ≤ 1;

k f k ^r _L ⁻ r(·) ^(E) (E) ≤ S r( · ) (f χ E ) ≤ k f k ^r _L ⁺ r(·) ^(E) (E) , k f k L ^r(·) (E) ≥ 1;

Z

E

f (x)g(x)dx ≤ 1

r ₋ (E) + 1 (r + (E)) ^′

k f k L ^r(·) (E) k g k L ^r ^′ ^(·) (E) ,

where

r ^′ (x) = _r(x) ^r(x) ₋ ₁

^and

1 < r ₋ ≤ r + < ∞ .

Let

I

^be^an ^open ^set ⁱⁿ

R

. In the sequel we shall use the notation:

I + (x, h) := [x, x + h] ∩ I, I ₋ (x, h) := [x − h, x] ∩ I;

I (x, h) := [x − h, x + h] ∩ I.

Weintrodue the following onesided maximaloperators:

M α( · ) f

(x) = sup

h>0

1 (2h) ¹ ⁻ ^α(x)

Z

I(x,h)

| f (t) | dt,

M _α( ⁻ _· ₎ f

(x) = sup

h>0

1 h ¹ ⁻ ^α(x)

Z

I − (x,h)

| f (t) | dt,

M _α( ⁺ _· ₎ f

(x) = sup

h>0

1 h ¹ ⁻ ^α(x)

Z

I + (x,h)

| f (t) | dt,

where

0 < α ₋ ≤ α + < 1

^,

I

îsânôpen ^set ⁱⁿ

R

and

x ∈ I

^.

If

α ≡ 1

^, ^then

M α( · )

^,

M _α( ⁻ _· ₎

^and

M _α( ⁺ _· ₎

^are ^the ^onesided HardyLittlewood maximaloperators whihare denoted by

M

^,

M ⁻

^and

M ⁺

respetively.

In [4℄L. Diening proved the followingstatement:

(4)

Theorem A. Let

Ω

^be ^a ^bounded ^open ^set ⁱⁿ

R ⁿ .

^Then^the ^maximal ^operator

(M Ω f

(x) = sup

r>0

1 r ⁿ

Z

B(x,r) T Ω

| f (y) | dy, x ∈ Ω,

is bounded in

L ^p( ^· ⁾ (Ω)

^if

p ∈ P (Ω)

^, ^that ^is,

a) 1 < p ₋ ≤ p(x) ≤ p ₊ < ∞ ;

b) p

^satises^theDiniLipshitz(log-Holderontinuity)ondition

(p ∈ DL(Ω))

^:

there exists a positive onstant

A

^suh ^that ^for ^all

x, y ∈ Ω

^with

0 < | x − y | ≤ ¹ ₂

^the

inequality

p(x) − p(y) ≤ A

ln _| _x ₋ ¹ _y _| (2)

holds.

The next statementwas proved in [3℄.

Theorem B.Let

Ω

^be ân ôpen ^subsetôf

R ⁿ

. Suppose that

1 < p ₋ ≤ p + < ∞

^. ^Then

the maximal operator

M _Ω

^is ^bounded ⁱⁿ

L ^p( ^· ⁾ (Ω)

^if

(i)

p ∈ P (Ω)

^;

(ii)

| p(x) − p(y) | ≤ C

ln(e + | x | ) (3)

for all

x, y ∈ Ω

^,

| y | ≥ | x | .

We shall alsoneed the followingstatements:

Proposition B ([33℄, [42℄). Let

1 ≤ p(x) ≤ q(x) ≤ q + < ∞

^. ^Suppose ^that

Ω

^is ^an

open set in

R ⁿ

with

| Ω | < ∞

^. ^Then^the ^inequality

k f k L ^p(·) (Ω) ≤ (1 + | Ω | ) k f k L ^q(·) (Ω)

holds.

PropositionC([4℄).Let

Ω

^be^an^open^setⁱⁿ

R ⁿ

andlet

p

^and

q

^be^bounded^exponents

on

Ω

^. ^Then

L ^q( ^· ⁾ (Ω) ֒ → L ^p( ^· ⁾ (Ω)

ifandonlyif

p(x) ≤ q(x)

âlmostêverywhereôn

Ω

ând^thereîsâônstant

0 < K < 1

suh that

Z

Ω

K ^p(x)q(x)/( ^| ^q(x) ⁻ ^p(x) ^| ⁾ dx < ∞ . (4)

Remark A. In the previous statement itis used the onvention

K ^1/0 := 0

^.

Denition A ([13℄). Let

P − (I)

^be ^the ^lass ^of ^all ^measurable ^positive ^funtions

p : I → R

satisfyingthe followingondition: there existapositiveonstant

C ₁

^suh

that for a.e

x ∈ I

^and ^a.e

y ∈ I

^with

0 < x − y ≤ ¹ ₂

^the ^inequality

p(x) ≤ p(y) + C 1

ln

1 x − y

(5)

holds.Further,wesaythat

p

^belongs^to

P + (I )

^if

p

^is^positive^funtion^on

I

^and^there

existsapositiveonstant

C 2

^suh^that^for^a.e

x ∈ I

^and^a.e

y ∈ I

^with

0 < y − x ≤ ¹ ₂

the inequality

p(x) ≤ p(y) + C 2

ln

1 y − x

(6)

isfullled.

DenitionB.Wesaythatameasurablepositivefuntionon

I

^belongs^to^the^lass

P ∞ (I )

⁽

p ∈ P ∞ (I)

⁾^if ⁽³⁾ ^holds ^for^all

x, y ∈ I

^with

| y | ≥ | x | .

^.

Weshall also needthe following denition:

Denition C. Let

p

^be â ^measurable ^funtion ôn ûnbounded înterval

I

ⁱⁿ

R

. We say that

p ∈ G (I)

îf ^there îs âônstant

0 < K < 1

^suh ^that

Z

I

K ^p(x)p ⁻ ^/(p(x) ⁻ ^p ⁻ ⁾ dx < ∞ .

Theorem C ([13℄). Let

I

^be ^a ^bounded ^interval ⁱⁿ

R

. Suppose that

1 < p ₋ ≤ p + <

∞

^. ^Then

(i)

^if

p ∈ P − (I)

^, ^then

M ⁻

^is ^bounded ⁱⁿ

L ^p( ^· ⁾ (I)

^;

(ii)

^if

p ∈ P ⁺ (I )

^, ^then

M ⁺

^is ^bounded ⁱⁿ

L ^p( ^· ⁾ (I)

^.

In the ase of unbounded set wehave

TheoremD([13℄).Let

I

^beânârbitraryôpen^setⁱⁿ

R

.Supposethat

1 < p ₋ ≤ p + <

∞

^. ^If

p ∈ P ⁺ (I ) ∩ P ∞ (I)

^, ^then ^the ^operator

M ⁺

^is ^bounded ⁱⁿ

L ^p( ^· ⁾ (I)

^. ^F^urther, ^if

p ∈ P − (I ) ∩ P ∞ (I)

^. ^Then^the ^operator

M ⁻

^is ^bounded ⁱⁿ

L ^p( ^· ⁾ (I)

In partiular, the previous statementyields

Theorem E ([13℄).Let

I = R ₊

and let

1 < p ₋ ≤ p + < ∞

^. ^Suppose ^that

p ∈ P ⁺ (I)

and there is a positive number

a

^suh ^that

p ∈ P ∞ ((a, ∞ ))

^. ^Then

M ⁺

^is ^bounded

in

L ^p( ^· ⁾ (I)

^. ^F^urther, ^if

p ∈ P − (I)

ând ^there îs â ^positive ^number

a

^suh ^that

p ∈ P ∞ ((a, ∞ ))

^, ^then

M ⁻

^is ^bounded ⁱⁿ

L ^p( ^· ⁾ (I)

^.

Thenext statementgivesoneweightriteriaforone-sidedmaximaloperatorsin

lassialLebesgue spaes (see [48℄, [1℄).

Theorem F ([1℄). Let

I ⊆ R

be an interval. Assume that

0 ≤ α < 1

^and

1 < p <

1/α

^, ^where

p

^and

α

^are ^onstants

(1/α = ∞

^if

α = 0)

^. ^We ^set

1/q = 1/p − α

^.

(i)

^Let

T := M _α ⁻

^. ^Then^the ^inequality

Z

I | T f (x) | ^q v(x)dx 1/q

≤ C Z

I | f (x) | ^p v ^p/q (x)dx 1/p

(7)

holds if and only if

sup

h>0

1 h

Z

I + (x,x+h)

v (t)dt ¹ _q

1 h

Z

I − (x − h,x)

v ⁻ ^p ^′ ^/q (t)dt _p ¹ ′

< ∞ . (8)

(6)

(ii)

^Let

T := M _α ⁺

^. ^Then

(7)

^holds îf ând ônly îf

sup

h>0

1 h

Z

I − (x − h,x)

v(t)dt ¹ _q

1 h

Z

I + (x,x+h)

v ⁻ ^p ^′ ^/q (t)dt _p ¹ _′

< ∞ . (9)

Denition D. Let

I ⊆ R ₊

be an interval. Suppose that

1 < p < q < ∞

^, ^where

p

and

q

^are ^onstants.^We^say ^that ^the ^weight

v ∈ A ⁻ _p,q (I) (

^resp.

v ∈ A ⁺ _p,q (I) )

^if

(8) (

resp.

(9))

^holds.

If

p = q

^,^then^we^denote^the^lass

A ⁺ _p,q (I ) (

^resp.

A ⁻ _p,q (I) )

^by

A ⁺ _p (I ) (

^resp.

A ⁻ _p (I)

^).

Notie that

v ∈ A ⁺ _p,q (I)

^(resp.

v ∈ A ⁻ _p,q (I)

⁾ îs êquivalent ^to ^the ôndition

v ∈ A ⁺ _1+q/p ′ (I)

^(resp.

v ∈ A ⁻ _1+q/p ′ (I )

^).

Further, we denoteby

D( R )

^(resp.

D( R ₊ )

⁾ ^a ^dyadi ^lattie ⁱⁿ

R

(resp. in

R ₊

).

Denition E. We say that a measure

µ

^belongs ^to ^the ^lass

RD ^(d) ( R ⁿ )

^(dyadi

reverse doublingondition) if there exists a onstant

δ > 1

^, ^suh ^that ^for ^all^dyadi

ubes

Q

^and

Q ^′

^,

Q ⊂ Q ^′

^,

| Q | = ^| ^Q ₂ n ^′ ^| ,

^the ^inequality

µ(Q ^′ ) ≥ δµ(Q)

holds.

Denition F. We say that measure

µ

^on

R ⁿ

satises the doublingondition (

µ ∈ DC( R ⁿ )

⁾ îf ^there îsâ ^positive ^number

b

^suh ^that

µB(x, 2r) ≤ bµB (x, r)

for all

x ∈ R ⁿ

and

r > 0.

It is known (see [51℄, p. 11) that if

µ ∈ DC( R ⁿ )

^, ^then

µ ∈ RD( R ⁿ )

^, ^i.e., ^there

are positiveonstants

η 1

^and

η 2

^,

0 < η 1 , η 2 < 1

^, ^suh ^that

µB(x, η 1 r) ≤ η 2 µB(x, r),

for all

x ∈ R ⁿ

and

r > 0.

It is easy tohek that if

µ ∈ DC( R ⁿ )

^, ^then

µ ∈ RD ^(d) ( R )

^.

We shall need some lemmasgivingCarleson-Hormandar type inequalities.

Lemma 2.1 ([52℄). Let

1 < p ≤ r < ∞

^and ^let

ρ ⁻ ^p ^′ ∈ RD ^(d) ( R ⁿ ),

^where

ρ

^is ^a

weight funtion on

R ⁿ

. Then there is a positive onstant

C

^suh ^that ^for ^all ^non-

negative

f

^the ^inequality

X

Q ∈ D( R n )

Z

Q

ρ ⁻ ^p ^′ (x)dx

₋ _p ^r _′ Z

Q

f (y)dy r

≤ C Z

R n

(f(x)ρ(x)) ^p dx ¹ _p

holds.

(7)

Lemma 2.2 ([50℄, [53℄). Let

u(x) ≥ 0

^on

R ⁿ ; { Q _i } i ∈ A

îs â ôuntable ôlle^tion ôf

dyadi ubes in

R ⁿ

and

{ a i } ⁱ ∈ A , { b i } ⁱ ∈ A

^be ^positive ^numbers ^satisfying

(i)

Z

Q i

u(x)dx ≤ Ca i

^for ^all

i ∈ A;

(ii) X

j: Q j ⊂ Q i

b j ≤ Ca i

^for ^all

i ∈ A.

Then

X

i ∈ I

b i

1 a i

Z

Q i

g(x)u(x)dx p ¹ _p

≤ C p

Z

R n

g ^p (x)u(x)dx _p ¹

for all

g ≥ 0

^on

R ⁿ

and

1 < p < ∞

^.

3 HardyLittlewood one-sided maximal funtions. One-

weight problem

Inthissetionwedisusstheone-weightproblemfortheone-sidedHardyLittlewood

maximaloperators.

Webegin with the following statement:

Lemma 3.1 ([13℄). Let

I

^be â ^boundedînterval ând ^let

(1)

^hold^on

I

^. ^If

p ∈ P + (I)

^,

thenthereisa positiveonstantdependingonlyon

p

^suh^that^for^all

f

^,

k f k L ^p(·) (I) ≤ 1

^, ^the ^inequality

M ⁺ f (x) p(x)

≤ C 1 + M ⁺ | f | ^p( ^· ⁾ (x)

holds.

Now we formulate the main results of this setion.

Theorem 3.1. Let

I

^be ^a ^bounded ^interval ⁱⁿ

R

and let

1 < p ₋ ≤ p + < ∞

^.

(i)

^If

p ∈ P + (I)

^and ^a^weight^funtion

w

^satises^the^ondition

w( · ) ^p( ^· ⁾ ∈ A ⁺ _p ₋ (I)

^,

then for all

f ∈ L ^p( w ^· ⁾ (I)

^the ^inequality

k (Nf)w k L ^p(·) (I) ≤ C k wf k L ^p(·) (I) (12)

holds, where

N = M ⁺

^.

(ii)

^Let

p ∈ P − (I)

^and ^let

w( · ) ^p( ^· ⁾ ∈ A ⁻ _p ₋ (I)

^. ^Then ^inequality

(12)

^holds ^for ^all

f ∈ L ^p( w ^· ⁾ (I)

^, ^where

N = M ⁻

^.

The result similar to Theorem 3.1 has been derived in [30℄, [31℄ for

M _Ω

^, ^where

Ω ⊂ R ⁿ

isa bounded domain.

In the ase of unbounded intervalswe have the next statement:

Theorem 3.2. Let

I = R ₊

and let

1 < p ₋ ≤ p + < ∞

^. ^Suppose ^that ^there ^is ^a

positive number

a

^suh ^that

p(x) ≡ p _c ≡ const

^outside

(0, a)

^.

(i)

^If

p ∈ P ⁺ (I)

^and

w( · ) ^p( ^· ⁾ ∈ A ⁺ _p ₋ (I)

^, ^then

(12)

^holds ^for

N = M ⁺

^.

(ii)

^If

p ∈ P − (I)

^and

w( · ) ^p( ^· ⁾ ∈ A ⁻ _p ₋ (I)

^, ^then

(12)

^holds ^for

N = M ⁻

^.

(8)

Theorem 3.1 yieldsthe followingorollaries:

Corollary 3.1. Let

p

^be înreasing ^funtion ôn ân înterval

I = (a, b)

^suh ^that

1 < p(a) ≤ p(b) < ∞

^. ^Suppose ^that

w

îsînreasing ^positive ^funtionôn

I

^. ^Then^the

oneweight inequality

k w ^1/p( ^· ⁾ (M ⁺ f )( · ) k L ^p(·) (I) ≤ c k w ^1/p( ^· ⁾ f ( · ) k L ^p(·) (I)

holds.

Corollary 3.2. Let

p

^be ^dereasing ^funtion ôn ân înterval

I = (a, b)

^suh ^that

1 < p(b) ≤ p(a) < ∞

^. ^Suppose ^that

w

^is ^dereasing^positive ^funtion ^on

I

^. ^Then^the

oneweight inequality

k w ^1/p( ^· ⁾ (M ⁻ f )( · ) k L ^p(·) (I) ≤ c k w ^1/p( ^· ⁾ f ( · ) k L ^p(·) (I)

holds.

Nowwe proveTheorems 3.1and 3.2.

Proof of Theorem 3.1. Sine the proof of the seond part is similar to the rst

one, we prove only

(i)

^. Ît îs ênough^to^show ^that

S p wM ⁺ (f /w)

≤ C

for

f

^satisfying^the ^ondition

k f k L ^p(·) (I) ≤ 1

^.

First we prove that

S p ^∗ f w

< ∞

^,^where

p ^∗ (x) = ^p(x) _p ₋ ·

By using Holder's inequality we nd that

S p ^∗

f w

= Z

I

[f /w] ^p ^∗ ^(x) (x)dx ≤ Z

I | f(x) | ^p(x) dx _p ¹

− ·

Z

I

w(x) ^p(x)(1 ⁻ ^(p ⁻ ⁾ ^′ ⁾ dx ¹ _´

(p − ) ′

< ∞ ,

beause

w ^p( ^· ⁾ ( · ) ∈ A ⁺ _p ₋ (I)

^.

Thus Lemma3.1 might be appliedfor

p ^∗

^. Consequently,

S p w(M ⁺ f /w)

= Z

I

M ⁺

f w

(x)

p(x)

w ^p(x) (x)dx

= Z

I

M ⁺ (f /w) (x) p ^∗ (x) p −

w ^p(x) (x)dx

≤ C Z

I

1 + M ⁺ f w ^p

∗ ( · ) (x)

p −

(w(x)) ^p(x) dx

≤ C Z

I

(w(x)) ^p(x) dx + C Z

I

M ⁺ f w ^p

∗ ( · ) (x)

p −

w ^p(x) (x)dx

(9)

≤ C + C Z

I

f /w ^p(x) w ^p(x) (x)dx ≤ C.

Proof of Theorem 3.2. First we prove

(i)

^. ^Without ^loss ^of ^generality ^we ^an

assume that

M ⁺ f(a) < ∞

^. ^Sine

M ⁺

îs ^sub-linear ôperator ît îs ênough ^to ^prove

that

S p,w (M ⁺ f ) < ∞

^, ^whenever

S p,w (f) < ∞ .

^We^have

Z

R ₊

M ⁺ f p(x)

(x)w(x) ^p(x) dx ≤ c Z a

0 M ⁺ f χ _[0,a] p(x)

(x)w(x) ^p(x) dx

+ Z a

0 M ⁺ (f χ [a, ∞ ) ) p(x)

(x)w(x) ^p(x) dx + Z _∞

a

M ⁺ (f χ [0,a] ) p(x)

(x)w(x) ^p(x) dx +

Z _∞

a

M ⁺ f χ [a, ∞ )

p(x)

(x)w(x) ^p(x) dx

= c[I 1 + I 2 + I 3 + I 4 ].

Sine

M ⁺ f (x) = M ⁺ (f χ [0,a] )(x)

^for

x ∈ [0, a]

^, ^using ^the assumptions

w( · ) ^p( ^· ⁾ ∈ A ⁺ _p ₋ ([0, a])

^,

p + ∈ P + ((0, a))

^and ^Theorem ^3.1^we^nd ^that

I 1 < ∞

^.

Further, the ondition

w( · ) ^p( ^· ⁾ ∈ A ⁺ _p ₋ (I)

^implies ^that

w( · ) ^p( ^· ⁾ ∈ A ⁺ _p ₋ ((a, ∞ ))

^.

Consequently, sine

p ≡ p c ≡

^onst ^on

(a, ∞ )

^,^by ^Theorem ^F^we ^have

I 4 < ∞ .

Now observe that

M ⁺ (f χ _[0,a] )(x) = 0

^when

x ∈ (a, ∞ )

^. ^Therefore

I 3 = 0

^.

It remainsto estimate

I 2

^. ^F^or^this ^notie ^that ^if

x ∈ (0, a)

^,^then

M ⁺ f · χ [a, ∞ )

(x) = sup

h>0

1 h

Z x+h

x | f(y) | χ [a, ∞ ) (y)dy

= sup

h>a − x

1 h

Z x+h

a | f (y) | χ _[a, _∞ ₎ (y)dy

≤ sup

h>a − x

1 x + h − a

Z a+(x+h − a)

a | f(y) | χ [a, ∞ ) (y)dy ≤ M ⁺ f (a) < ∞ .

Hene,

I ₂ ≤ c Z a

0 w(x) ^p(x) dx < ∞

beause

w( · ) ^p( ^· ⁾

îs^loallyîntegrableôn

R ₊

.

To prove

(ii)

^we ûse ^the ^notationôf ^the ^proof ôf

(i)

substituting

M ⁺

^by

M ⁺

^.^In

fat,the proof is similar tothat of

(i)

^.^The ônly ^diereneîs ⁱⁿ ^the êstimates ôf

I 2 :=

Z a 0

M ⁻ (f χ [a, ∞ ) ) p(x)

(x)w(x) ^p(x) dx

and

I 3 :=

Z _∞

a

M ⁻ (f · χ [0,a] )(x) p(x)

(x)w(x) ^p(x) dx.

Obviously, we have that

I ₂ = 0

^.^F^urther, ^we ^represent

I ₃

^as ^follows:

I 3 = Z _∞

a

M ⁻ (f · χ [0,a] )(x) p c

(x)w(x) ^p ^c dx

(10)

= Z 2a

a

M ⁻ (f · χ [0,a] )(x) p c

(x)w(x) ^p ^c dx +

Z _∞

2a

M ⁻ (f · χ [0,a] )(x) p c

(x)w(x) ^p ^c dx := I ₃ ⁽¹⁾ + I ₃ ⁽²⁾ .

Observe that for

x ∈ (a, 2a]

^,

M ⁻ (f · χ _[0,a] )(x) ≤ sup

x − a<h<x

1 a − x + h

Z a

a − (a − x+h) | f (y) | dy ≤ M ⁻ f (a) < ∞ .

Hene,

I ₃ ⁽¹⁾ ≤ (M ⁻ f) ^p ^c (a) Z 2a

a

(w(x)) ^p ^c dx < ∞ .

If

x > 2a

^, ^then

M ⁻ f

(x) ≤ 1 a − x

Z a

0 | f (y) | dy.

Therefore by using Holder's inequality with respet to the exponent

p( · )

^(see

propositionA) we nd that

I ₃ ⁽²⁾ ≤ Z _∞

2a

(w(x)) ^p ^c (a − x) ⁻ ^p ^c dx

Z a

0 | f (x) | dx p c

≤ c Z _∞

2a

(w(x)) ^p ^c (a − x) ⁻ ^p ^c dx

k f w k ^p _L ^c p(·) ([0,a])

k w ⁻ ¹ k ^p ^c

L ^p _([0,a]) ^′ ^(·)

:= cJ 1 · J 2 · J 3 .

It is lear that

J 2 < ∞

^. ^F^urther, ^sine

w( · ) ^p( ^· ⁾ ∈ A ⁻ _p ₋ (a, ∞ )

, by Holder's

inequalitywe have that

w( · ) ^p( ^· ⁾ ∈ A ⁻ _p _c (a, ∞ )

beause

p c ≥ p ₋

^. ^Hene, ^by ^applying

Theorem F(for

α = 0

⁾^we^have ^that^the ^operator

M ⁻ f := M ⁻ (f χ (a, ∞ ) )

^is^bounded

in

L ^p _w ^c ((a, ∞ ))

^.Consequently, the Hardyoperator

H a f(x) = 1 x − a

Z x

a | f(t) | dt, x ∈ (a, ∞ ),

is bounded in

L ^p _w ^c ((a, ∞ ))

^. ^This ^implies^(see, ^e.g., ^[20℄, ^[37℄) ^that

J 1 < ∞

^.

It remains tosee that

J ₃ < ∞

^. ^Indeed, Proposition B yields

k w ⁻ ¹ k _L ^p ^′ ^(·)

([0,a]) ≤ (1 + a) k w ⁻ ¹ k _L ^(p − )′· ([0,a])

≤ c k χ _{ w ⁻¹ ≥ 1 } ( · )w ⁻ ¹ ( · ) k _L ^(p − ) ′ (·) ([0,a]) + k χ _{ w ⁻¹ <1 } ( · )w ⁻ ¹ ( · ) k _L ^(p − ) ′

([0,a])

≤ c χ _{ _w ⁻¹ _≥ ₁ _} ( · )w ⁻

p(·) p − (x)

L ^(p ⁻ ⁾ ^′ ([0,a]) + c

≤ c Z a

0 w ^p(x)(1 ⁻ ^(p ⁻ ⁾ ^′ ⁾ (x)dx

1/(p − ) ^′

+ c.

Thus

I ₃ ⁽²⁾ < ∞

^.

(11)

4 Frational maximal operators. One-weight problem

In this setionwe derivethe one-weight inequality forone-sided frational maximal

operators.Our main results are the following statements:

Theorem 4.1. Let

I

^be â ^bounded înterval ând^let

1 < p ₋ ≤ p + < ∞ .

^Suppose ^that

α

^is ^onstant ^satisfying

0 < α < 1/p ₊

^. ^Let

q(x) = ₁ ₋ ^p(x) _αp(x)

^.

(i)

^If

p ∈ P ⁺ (I )

^and ^a ^weight

w

^satises ^the ^ondition

w( · ) ^q( ^· ⁾ ∈ A ⁺ _p ₋ _,q ₋ (I)

^, ^then

the inequality

k (N _α f )w k L ^q(·) (I) ≤ C k wf k L ^p(·) (I) , f ∈ L ^p( _w ^· ⁾ (I) (10)

holds for

N α = M _α ⁺

^.

(ii)

^Let

p ∈ P − (I)

^and ^let

w( · ) ^q( ^· ⁾ ∈ A ⁻ _p ₋ _,q ₋ (I)

^. ^Then ^inequality

(13)

^holds ^for

N α = M _α ⁻

^.

Theorem 4.2. Let

I = R ₊

,

1 < p ₋ ≤ p ₊ < ∞

^and ^let

p(x) ≡ p _c ≡

^onst ^outside

some interval

(0, a)

^. ^Suppose ^that

q(x) = ₁ ₋ ^p(x) _αp(x)

^, ^where

α

^is ^onstant ^satisfying

0 < α < 1/p +

^.

(i)

^If

p ∈ P + (I)

^and

w( · ) ^q( ^· ⁾ ∈ A ⁺ _p ₋ _,q ₋ (I)

^, ^then

(10)

^holds ^for

N _α = M _α ⁺

^.

(ii)

^If

p ∈ P − (I)

^and

w( · ) ^q( ^· ⁾ ∈ A ⁻ _p ₋ _,q ₋ (I)

^, ^then

(10)

^holds ^for

N α = M _α ⁻

^.

Proof of Theorem 4.1. We prove

(i)

^. ^The ^proof ^of

(ii)

^is^the ^same. ^First^we ^show

that the inequality

M _α ⁺ (f /w)(x) ≤ M ⁺ f ^p( ^· ^)/s( ^· ⁾ w ⁻ ^q( ^· ^)/s( ^· ⁾

(x) s(x)/q(x) Z

I

f ^p(y) (y)dy α

,

holds,where

s(x) = 1 + q(x)/p ^′ (x)

^.

Indeed, denoting

g( · ) := (f ( · )) ^p( ^· ^)/s( ^· ⁾ (w( · )) ⁻ ^q( ^· ^)/s( ^· ⁾

^we ^see ^that

f ( · )/w( · ) = (g( · )) ^s( ^· ^)/p( ^· ⁾ w ^q( ^· ^)/p( ^· ⁾ ⁻ ¹ = (g( · )) ¹ ⁻ ^α g ^s( ^· ^)/p( ^· ^)+α ⁻ ¹ w ^αq( ^· ⁾

^. ^By ûsing ^Hôlder's înequality

with respet to the exponent

(1 − α) ⁻ ¹

^and ^the ^fats ^that

s( · )/q( · ) = 1 − α

^,

(s(y)/p(y) + α − 1)/α = s( · )

^we ^have

1 h ¹ ⁻ ^α

Z

I + (x,x+h)

f (y) w(y) dy

≤ 1

h Z

I + (x,x+h)

g(y)dy

1 − α Z

I + (x,x+h)

g (s(y)/p(y)+α − 1)/α (y)w ^q(y) (y)dy α

≤ M ⁺ g(x) s(x)/q(x) Z

I + (x,x+h)

g ^s(y) (y)w ^q(y) (y) α

≤ M ⁺ g (x) s(x)/q(x) Z

I

f ^p(y) (y)dy α

.

(12)

Now we prove that

S _q wM _α ⁺ (f /w)

≤ C

^when

S _p (f) ≤ 1

^. ^By ^applying ^the

abovederived inequalitywend that

S _q wM _α ⁺ (f /w)

≤ c Z

I

M _α ⁺ (f ^p( ^· ^)/s( ^· ⁾ w ⁻ ^q( ^· ^)/s( ^· ⁾ ) s(x)

(x)w ^q(x) (x)dx

= cS s M ⁺ (f ^p( ^· ^)/s( ^· ⁾ w ⁻ ^q( ^· ^)/s( ^· ⁾ )w ^q( ^· ^)/s( ^· ⁾ .

Observenowthat theondition onthe weight

w

îsêquivalent^to^the âssumption

w ^q( ^· ⁾ ( · ) ∈ A ⁺ _s ₋ (I)

^. ^On ^the ^other ^hand,

k f ^p( ^· ^)/s( ^· ⁾ k L ^s(·) (I) ≤ 1

^. ^Therefore ^taking

Theorem 3.1intoaount we have the desired result.

Proof of Theorem 4.2. (i) Let

f ≥ 0

^and ^let

S p,w (f ) < ∞

^. ^We^have

S q,w (M _α ⁺ f ) =

Z

I

M _α ⁺ f q(x)

(x)w(x) ^q(x) dx

≤ c Z a

0 M _α ⁺ f χ [0,a] (x) q(x)

(x)w(x) ^q(x) dx +

Z a 0

M _α ⁺ (f · χ _[a, _∞ ₎ )(x) q(x)

(x)w(x) ^q(x) dx +

Z _∞

a

M _α ⁺ (f · χ [0,a] )(x) q(x)

(x)w(x) ^q(x) dx +

Z _∞

a

M _α ⁺ (f χ _[a, _∞ ₎ )(x) q(x)

(x)w(x) ^q(x) dx

:= c[I ₁ + I ₂ + I ₃ + I ₄ ].

Itiseasytoseethat

I 1 < ∞

^beauseôf^Theorem^4.1ând^the ôndition

w ^q( ^· ⁾ ( · ) ∈ A ⁺ _p ₋ _,q ₋ ([0, a])

^. ^Fûrther, ît îs ôbvious ^that

I 3 < ∞

^beause

M _α ⁺ (f χ [0,a] )(x) = 0

^for

x > a

^.^Further,^observe^that

I ₂ ≤ c Z a

0 w(x) ^q(x) dx < ∞ ,

where the positive onstant depends on

α

^,

f

^,

p

^,

a

^.

It is easy tohek that by Holder's inequality with respet to the power

(p c ) ^′ /q c

/ (p ₋ ) ^′ /q ₋

theondition

w( · ) ^q ^c ∈ A ⁺ _p ₋ _,q ₋ ([a, ∞ ))

^implies

w( · ) ^q ^c ∈ A ⁺ _p _c _,q _c ([a, ∞ ))

^.^Hene,^by^using

Theorem Fwe nd that

I 4 < ∞

^.

(ii)

^Wê^keep^the ^notationôf ^the^proofôf ⁽ⁱ⁾^but ^substitute

M _α ⁺

^by

M _α ⁻

^.^The^only

dierenebetween the proofs of

(i)

^and

(ii)

îs ⁱⁿ^the êstimates ôf

I 2

^and

I 3

^.

It is obviousthat

I 2 = 0

^,^while ^for

I 3

^we ^have

I 3 = Z 2a

a

M _α ⁻ (f · χ [0,a] )(x) q(x)

(x)w(x) ^q(x) dx +

Z _∞

2a

M _α ⁻ (f · χ [0,a] )(x) q c

(x)w(x) ^q ^c dx := I ₃ ⁽¹⁾ + I ₃ ⁽²⁾ .

(13)

If

x > a

^, ^then

M _α ⁻ f (x) ≤ sup

x − a<h<x

h ^α ⁻ ¹ Z a

x − h | f(y) | dy ≤ cM _α ⁻ f (a).

Consequently,

I ₃ ⁽¹⁾ ≤ c M _α ⁻ f (a) q c

Z 2a a

w(x) q c

dx < ∞ .

Now observe that when

x > a

^we ^have^the ^following ^pointwise ^estimates:

M _α ⁻ (f χ _[0,a]) )(x) ≤ (x − a) ^α ⁻ ¹ Z a

0 | f (y) | dy ≤ (x − a) ^α ⁻ ¹ k f w k L ^p(·) ([0,a]) k w ⁻ ¹ k L ^p ^′ ^(·) ([0,a])

:= (x − a) ^α ⁻ ¹ J 1 · J 2 .

Hene,

I ₃ ⁽²⁾ ≤

Z _∞

2a

(x − a) ^(α ⁻ ^1)q ^c (w(x)) ^q ^c dx

(J 1 · J 2 ) ^q ^c .

It is obviousthat

J 1 < ∞

^.^F^urther,

J ₂ ≤ k w ⁻ ¹ ( · )χ _w ⁻¹ _>1 ( · ) k L ^p ^′ ^(·) ([0,a]) + k w ⁻ ¹ ( · )χ _w ⁻¹ _≤ ₁ ( · ) k L ^p ^′ ^(·) ([0,a])

:= J ₂ ⁽¹⁾ + J ₂ ⁽²⁾ .

Itislearthat

J ₂ ⁽²⁾ < ∞

^.^T^o^estimate

J ₂ ⁽¹⁾

^observe^that^byPropositionBwehave

J ₂ ⁽¹⁾ ≤ (1 + a) k w ⁻ ¹ χ w ⁻¹ >1 k L ^p ⁻ ([0,a]) ≤ (1 + a) k w ⁻ ^q( ^· ^)/q ⁻ χ w ⁻¹ >1 k L ^p ⁻ ([0,a])

≤ (1 + a) k w ⁻ ^q( ^· ^)/q ⁻ k ^L ^p ⁻ ^([0,a]) < ∞ .

Sine

M _α ⁻

^is ^bounded ^from

L ^p _w ^c ([a, ∞ ))

^to

L ^q _w ^c ([a, ∞ ))

^we ^have ^the ^Hardy

inequality

Z _∞

a

(x − a) ^(α ⁻ ^1)q ^c w ^q ^c (x) Z x

a | f (t) | dt q c

dx 1/q c

≤ c Z _∞

a | f(x) | ^p ^c w ^p ^c (x)dx 1/p c

.

From this inequality itfollows that (see, e.g., [20℄, [37℄)

Z _∞

2a

(x − a) ^(α ⁻ ^1)q ^c (w(x)) ^q ^c dx < ∞ .

5 Generalized frational maximal operators. Two-weight

problem

Let

I = [a, b]

^be â ^bounded înterval ând ^let

I ⁺ := [b, 2b − a); I ⁻ := [2a − b, a).

Let

Q = I 1 × I 2 × · · · × I n

^be ^a ^ube ⁱⁿ

R ⁿ .

^W^e ^denote:

Q ⁺ := I ₁ ⁺ × I ₂ ⁺ × · · · × I _n ⁺ , Q ⁻ := I ₁ ⁻ × I ₂ ⁻ × · · · × I _n ⁻ .

(14)

Let

α

^be ^a ^measurable ^funtion ^on

R ⁿ , 0 < α ₋ ≤ α(x) ≤ α ₊ < n.

^Let^us ^dene

dyadi frational maximal funtionson

R ⁿ

:

M _α( ^+,(d) _· ₎ f

(x) = sup

Q∋x Q∈D( Rn )

1 | Q | ¹ ⁻ ^α(x) ⁿ Z

Q ⁺

| f (y) | dy;

M _α( ⁻ ^,(d) _· ₎ f

(x) = sup

Q∋x Q∈D( Rn )

1 | Q | ¹ ⁻ ^α(x) ⁿ Z

Q ⁻

| f (y) | dy.

If

α(x) ≡ 0

^, ^then ^we ^have Hardy-Littlewood dyadi maximal funtions

M ^+,(d)

^,

M ⁻ ^,(d)

^.

Inthe paper[39℄the two-weightweak-typeinequalitywasproved inthelassial

Lebesgue spaes for one-sided dyadi Hardy-Littlewood maximal funtions dened

on

R ⁿ

.

Theorem 5.1. Let

p

^be ^onstant ^and ^let

1 < p < q ₋ ≤ q + < ∞

^,

0 < α ₋ ≤ α + < n

where

q

^and

α

âre ^meâsurable ^funtions ôn

R ⁿ

. Suppose that

w ⁻ ^p ^′ ∈ RD ^(d) ( R ⁿ ).

Then

M _α( ^+,(d) _· ₎

^is ^bounded ^from

L ^p _w ( R ⁿ )

^to

L ^q( v ^· ⁾ ( R ⁿ )

îf ând ônly îf

A := sup

Q,Q ∈ D( R n )

χ _Q ( · ) | Q | ^α(·) ⁿ ⁻ ¹ v( · )

L ^q(·) ( R n )

χ _Q ⁺ w ⁻ ¹

L ^p ^′ ( R n ) < ∞ . (11)

Proof.Neessity.Assuming

f = χ Q ⁺ w ⁻ ^p ^′

⁽

Q ∈ D( R ⁿ )

⁾ ⁱⁿ^the ^inequality

M _α( ^+,(d)

· ) f

L ^q(·) v ( R n ) ≤ C k f k L ^p _w ( R n ) (12)

we have that

χ _Q ( · )

1 | Q | ¹ ⁻ ^α(x) ⁿ Z

Q ⁺

f

L ^q(·) v ( R n )

= χ _Q ( · ) | Q | ^α(·) ⁿ ⁻ ¹

L ^q(·) v ( R n )

Z

Q ⁺

w ⁻ ^p ^′ (y)dy

≤ M _α( ^+,(d) _· ₎ f

L ^q(·) v ( R n ) ≤ C Z

Q ⁺

w ⁻ ^p ^′ (y)dy ¹ _p

.

Thus, to show that (11) holds it remains to prove that for all dyadi ubes Q,

S Q = R

Q

w ⁻ ^p ^′ (y)dy < ∞ .

^Indeed, ^suppose ^the ^ontrary ^that

S Q = ∞

^for ^some ^ube

Q.Then

S Q = k w ⁻ ¹ k L ^p ^′ (Q) = ∞ .

^This împlies ^that ^there îs ânon-negative funtion

g

^suh ^that

g ∈ L ^p (Q)

^and

R

Q

g (y)w ⁻ ¹ (y)dy = ∞ .

^F^urther, ^let

Q = ¯ Q ⁺ ,

^where

Q ¯ ∈ D( R ⁿ ).

^Then^taking

f = χ Q gw ⁻ ¹

^we ^have

k f k L ^p w ( R n ) = Z

Q ¯ ⁺

g ^p (x)dx ¹ _p

< ∞ ;

(15)

M _α( ^+,(d)

· ) f

L ^q(·) v ( R n ) ≥ χ Q ¯ ( · ) | Q ¯ | ^α(·) ⁿ ⁻ ¹

L ^q(·) v ( R n )

Z

Q ¯ ⁺

f (y)dy

= χ Q ¯ ( · ) | Q ¯ | ^α(·) ⁿ ⁻ ¹

L ^q(·) v ( R n )

Z

Q ¯ ⁺

g(y)w(y) ⁻ ¹ dy = ∞ .

This ontradits (12).

Suieny. Forevery

x ∈ R ⁿ

we take

Q x ∈ D( R ⁿ )

⁽

Q x ∋ x

⁾ ^so^that

| Q x | ^α(x) ⁿ ⁻ ¹ Z

Q ⁺ x

| f (y) | dy > 1

2 M _α( ^+,(d) _· ₎ f

(x). (13)

Assume that

f

^benon-negativebounded with ompat support. Then itis easy to see that we an take maximal ube

Q _x

^ontaining

x

^for ^whih ⁽¹³⁾ ^holds. ^Let

Q ∈ D( R ⁿ )

ând ^letûs întrodue ^the ^set

F Q :=

x ∈ Q : Q

^is^maximal ^for ^whih

| Q | ^α(x) ⁿ ⁻ ¹ Z

Q ⁺

f(y)dy > 1

2 M _α( ^+,(d) _· ₎ f(x) .

Dyadi ubeshave the followingproperty:if

Q ₁ , Q ₂ ∈ D( R ⁿ ),

^and

Q ^o ₁ T ^o Q ₂ 6 = ∅ ,

then

Q 1 ⊂ Q 2

^or

Q 2 ⊂ Q 1

^, ^where

Q o

^denotes ^the înner ^part ôf âûbe

Q

^.

Now observe that

F Q 1

T F Q 2 6 = ∅

^if

Q 1 6 = Q 2

^. ^Indeed, ^if

Q o 1 T ^o

Q 2 = ∅ ,

^then ^it ^is

lear. If

Q o 1 T ^o

Q 2 6 = ∅ ,

^then

Q 1 ⊂ Q 2

^or

Q 2 ⊂ Q 1 .

^Let^us ^take

x ∈ F Q 1

T F Q 2 .

^Then

x ∈ Q 1 , x ∈ Q 2

^and

1 | Q 1 | ¹ ⁻ ^α(x) ⁿ Z

Q ⁺ ₁

f (y)dy > 1 2

M _α( ^+,(d) _· ₎ f

(x); 1

| Q 2 | ¹ ⁻ ^α(x) ⁿ Z

Q ⁺ ₂

f (y)dy > 1 2

M _α( ^+,(d) _· ₎ f

(x).

If

Q 1 ⊂ Q 2

^, ^then

Q 2

^would ^be ^the ^maximal ^ube ^for ^whih ⁽¹³⁾ ^holds.

Consequently

x 6∈ F Q 1

^and

x ∈ F Q 2 .

Analogously we have that if

Q 2 ⊂ Q 1 ,

^then

x ∈ F Q 1

^and

x 6∈ F Q 2 .

Further, it is lear that

F Q ⊂ Q

^and

S

Q ∈ D m ( R n )

F Q = R ⁿ ,

^where

D m ( R ⁿ ) = Q : Q ∈ D( R ⁿ ), F Q 6 = ∅ .

Suppose that

k f k L ^p w ( R n ) ≤ 1

^and ^that

r

îs â ^number ^satisfying ^the ôndition

p < r < q ₋ .

^We ^have

M _α( ^+,(d) _· ₎ f ^r

L ^q(·) v ( R n ) = v ^r M _α( ^+,(d) _· ₎ f r

L ^q(·) ^r ( R n ) = sup Z

R n

v ^r (x)

M _α( ^+,(d) _· ₎ f r

(x)h(x)dx,

wherethe supremum is taken overallfuntions

h

^,

k h k

L

q(·) r

^′

( R n )

≤ 1.

^Now^for ^suh

an

h

^, ^using ^Lemma^2.1, ^we^have ^that

Z

R n

v ^r (x)

M _α( ^+,(d) _· ₎ f r

(x)h(x)dx = X

Q ∈ D m ( R n )

Z

F Q

v ^r (x)

M _α(x) ^+,(d) f r

(x)h(x)dx

(16)

≤ C X

Q ∈ D m ( R n )

Z

F Q

v ^r (x) | Q | ⁽ ^α(x) ⁿ ⁻ ^1)r h(x)dx Z

Q ⁺

f(y)dy r

≤ C X

Q ∈ D m ( R n )

v ^r ( · ) | Q | ⁽ ^α(·) ⁿ ⁻ ^1)r χ Q ( · )

L ^q(·) ^r ( R n )

h

L

q(·) r

^′

( R n )

Z

Q ⁺

f(y)dy r

= C X

Q ∈ D m ( R n )

v( · ) | Q | ^α(·) ⁿ ⁻ ¹ χ Q ( · ) ^r

L ^q(·) ( R n )

h

L

q(·) r

^′

( R n )

Z

Q ⁺

f(y)dy r

≤ C A ^r X

Q ∈ D m ( R n )

Z

Q ⁺

w ⁻ ^p ^′ (y)dy

₋ ^r _p Z

Q ⁺

f(y)dy r

≤ CA ^r k f k ^r L ^p w ( R n ) .

In the last inequality we used also the fat that

Q ⁺ ∈ D( R ⁿ )

îf ând ônly îf

Q ∈ D( R ⁿ ).

Letuspassnowtoanarbitrary

f

^,^where

f ∈ L ^p _w ( R ⁿ ).

^F^or^suh^an

f

^we^take^the

sequene

f m = f χ Q(0,k m ) χ _{ f <j m }

^,^where

Q(0, k m ) := { (x 1 , · · · , x n ) : | x i | < k m , i = 1, · · · , n } .

LP(-)

L p( · )

L p( · )

L p( · )

L p( · )

L p( · ) (I)

p

L p

L p( · )

L p( · )

L p w

L q( v · )

v

R α( · )

L p( · ) ( R + )

L q( v · ) ( R + )

α − > 1/p −

L p( · )

p( · )

q( · )

L p( · )

c

C

Ω

R n

p

Ω

1 ≤ p − ≤ p + < ∞ , (1)

p −

p +

p

Ω

ρ

Ω

ρ

Ω

f

Ω

L p( ρ · ) (Ω)

L p(x) ρ (Ω)

S p,ρ (f ) = Z

Ω

f (x)ρ(x) p(x) dx < ∞ .

L p( ρ · ) (Ω)

k f k L p(·) ρ (Ω) = inf

λ > 0 : S p( · ),ρ f /λ

≤ 1 .

ρ ≡ 1

L p( · ) (Ω)

S p

L p( ρ · ) (Ω)

S p( · ),ρ

k f k L p(·) ρ (Ω) = k f ρ k L p(·) (Ω)

p

L p( ρ · ) (Ω)

p − (E) := inf

E p; p + (E) := sup

E

p, E ⊂ Ω, p − (Ω) = p − ; p + (Ω) = p + .

E

Ω

k f k r L + r(·) (E) (E) ≤ S r( · ) (f χ E ) ≤ k f k r L − r(·) (E) (E) , k f k L r(·) (E) ≤ 1;

k f k r L − r(·) (E) (E) ≤ S r( · ) (f χ E ) ≤ k f k r L + r(·) (E) (E) , k f k L r(·) (E) ≥ 1;

Z

E

f (x)g(x)dx ≤ 1

r − (E) + 1 (r + (E)) ′

k f k L r(·) (E) k g k L r ′ (·) (E) ,

r ′ (x) = r(x) r(x) − 1

1 < r − ≤ r + < ∞ .

I

R

I + (x, h) := [x, x + h] ∩ I, I − (x, h) := [x − h, x] ∩ I;

I (x, h) := [x − h, x + h] ∩ I.

M α( · ) f

(x) = sup

h>0

1 (2h) 1 − α(x)

Z

I(x,h)

L ^p( ^· ⁾

L ^p( ^· ⁾

L ^p( ^· ⁾

L ^p( ^· ⁾

L ^p( ^· ⁾ (I)

L ^p

L ^p( ^· ⁾

L ^p( ^· ⁾

L ^p _w

L ^q( v ^· ⁾

L ^p( ^· ⁾ ( R ₊ )

L ^q( v ^· ⁾ ( R ₊ )

α ₋ > 1/p ₋

L ^p( ^· ⁾

L ^p( ^· ⁾

R ⁿ

1 ≤ p ₋ ≤ p + < ∞ , (1)

p ₋

L ^p( ρ ^· ⁾ (Ω)

L ^p(x) ρ (Ω)

f (x)ρ(x) ^p(x) dx < ∞ .

L ^p( ρ ^· ⁾ (Ω)

k f k _L ^p(·) _ρ _(Ω) = inf

L ^p( ^· ⁾ (Ω)

L ^p( ρ ^· ⁾ (Ω)

k f k _L ^p(·) _ρ _(Ω) = k f ρ k L ^p(·) (Ω)

L ^p( ρ ^· ⁾ (Ω)

p ₋ (E) := inf

p, E ⊂ Ω, p ₋ (Ω) = p ₋ ; p + (Ω) = p + .

k f k ^r _L ⁺ ^r(·) ^(E) _(E) ≤ S _r( _· ₎ (f χ E ) ≤ k f k ^r _L ⁻ ^r(·) ^(E) _(E) , k f k L ^r(·) (E) ≤ 1;

k f k ^r _L ⁻ r(·) ^(E) (E) ≤ S r( · ) (f χ E ) ≤ k f k ^r _L ⁺ r(·) ^(E) (E) , k f k L ^r(·) (E) ≥ 1;

r ₋ (E) + 1 (r + (E)) ^′

k f k L ^r(·) (E) k g k L ^r ^′ ^(·) (E) ,

r ^′ (x) = _r(x) ^r(x) ₋ ₁

1 < r ₋ ≤ r + < ∞ .

I + (x, h) := [x, x + h] ∩ I, I ₋ (x, h) := [x − h, x] ∩ I;

1 (2h) ¹ ⁻ ^α(x)

M _α( ⁻ _· ₎ f

1 h ¹ ⁻ ^α(x)

M _α( ⁺ _· ₎ f

1 h ¹ ⁻ ^α(x)

0 < α ₋ ≤ α + < 1

M _α( ⁻ _· ₎

M _α( ⁺ _· ₎

M ⁻

M ⁺

R ⁿ .

1 r ⁿ

L ^p( ^· ⁾ (Ω)

a) 1 < p ₋ ≤ p(x) ≤ p ₊ < ∞ ;

0 < | x − y | ≤ ¹ ₂

ln _| _x ₋ ¹ _y _| (2)

R ⁿ

1 < p ₋ ≤ p + < ∞

M _Ω

L ^p( ^· ⁾ (Ω)

R ⁿ

k f k L ^p(·) (Ω) ≤ (1 + | Ω | ) k f k L ^q(·) (Ω)

R ⁿ

L ^q( ^· ⁾ (Ω) ֒ → L ^p( ^· ⁾ (Ω)