The maximal Operator in Variable Spaces Lp(·)(Ω,ρ) with Oscillating

The maximal Operator in Variable Spaces Lp(·)(Ω, ρ) with Oscillating Weights

by Vakhtang Kokilashvili, Mathematical Institute of the Georgian Academy of Sciences, Georgia ([email protected]) Natasha Samko University of Algarve, Portugal [email protected] and Stefan Samko University of Algarve, Portugal [email protected] Georgian Mathem. J., 2006, vol.13, N1.

Abstract We study the boundedness of the maximal operator in the spaces Lp(·) (ρ, Ω) Q over a bounded open set in Rn with the weight ρ(x) = m k=1 wk (|x−xk |), n

xk ∈ Ω, where wk has the property that r p(xk ) wk (r) ∈ Φ0n , where Φ0n is a certain Zygmund-type class. The weight functions wk may oscillate between two power functions with different exponents. It is assumed that the exponent p(x) satisfies the Dini–Lipschitz condition. The final statement on the boundedness is given in terms of the index numbers of the functions wk (similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).

Key Words and Phrases: maximal functions, weighted Lebesgue spaces, variable exponent, potential operators, integral operators with fixed singularity AMS Classification 2000: 42B25, 47B38 1

1

Introduction

Nowadays there is an evident increase of interest to harmonic analysis problems and operator theory in the generalized Lebesgue spaces with variable exponent p(x) and the corresponding Sobolev spaces, we refer, in particular to surveys [10], [13], [25], [7] and to [28], [14] for the basics on the spaces Lp(·) . For the boundedness results of maximal operators we refer to L. Diening [5] for bounded domains in mathbbRn and to D.Cruz-Uribe, A.Fiorenza and C.J. Neugebauer [4] and A.Nekvinda [21], [20] for unbounded domains, and to V.Kokilashvili and S.Samko [12] for weighted boundedness on bounded domains. We refer also to L.Diening [6] and D.Cruz-Uribe, A.Fiorenza, J.M.Martell, and C.Perez [3] where there are also given new insights into the problems of boundedness of singular and maximal operators in variable exponent spaces. In [12] the power weights |x − x0 |γ were considered and one of the main points in the result obtained in [12] was that in condition on γ only the values of p(x) at the point x0 are of importance: −

n n 0 : ¯¯ dx ≤ 1 ¯   λ Ω

(1.3) Notation a.d. =almost decreasing ⇐⇒ f (x) ≥ Cf (y) for x ≤ y, C > 0; a.i. =almost increasing ⇐⇒ f (x) ≤ Cf (y) for x ≤ y, C > 0; Ω is a open bounded set in Rn ; |Ω| is the Lebesgue measure of Ω; χΩ is the characteristic function of a set Ω; f ∼ g ⇐⇒ there exist C1 > 0 and C2 > 0 such that C1 f (x) ≤ g(x) ≤ C2 f (x). Br (x) = {y ∈ Rn : |y − x| < r}; n |Br (x)| = rn |S n−1 | is the volume of Br (x); p(x) 1 1 q(x) = p(x)−1 , 1 < p(x) < ∞, p(x) + q(x) ≡1; ∗ p∗ = inf p(x), p = sup p(x); x∈Ω

q∗ = inf q(x) = x∈Ω

x∈Ω p∗ , q∗ p∗ −1

= sup q(x) = x∈Ω

p∗ ; p∗ −1

C, c may denote different positive constants.

2

Statement of the Main Results

Let

ρ(x) M f (x) = sup r>0 |Br (x)|

Z

ρ

where ρ(x) =

m Y

|f (y)| dy, ρ(y)

Br (x)∩Ω

wk (|x − xk |), xk ∈ Ω.

k=1

We write M = M0 when ρ(t) ≡ 1.

3

(2.1)

In [12] there was proved the boundedness of the operator Mρ in the case of the power weight ρ(x) = |x − x0 |β , x0 ∈ Ω under the following (necessary and sufficient) condition n n − 0 for x > 0, w(x) is a.i.}. (3.1) The numbers ³ ln lim inf mw = sup x>1

w(hx) w(h)

h→0

ln x

µ

´

ln lim sup

¶

x>1

h→0

µ

ln x

x→∞

w(hx) w(h)

ln x ¶

ln lim sup = lim

h→0

x→0

µ

w(hx) w(h)

¶

ln lim sup = lim

ln x

0 0. The following class Φ0γ was introduced and studied in [1] (with integer γ); there are also known ”two-parametrical” classes Φβγ , 0 ≤ β < γ < ∞, see [19], [18], [27] and [26], p. 253). Observe that in [30], [31] there were a(x) considered more general classes Φb(x) with limits which may ”oscillate”; the class Φ0γ corresponds to the case where a(x) = x0 = 1 and b(x) = xγ . Definition 3.2. ([1]) The Zygmund-Bary-Stechkin type class Φ0γ , 0 < γ < ∞, is defined as Φβγ := Z 0 ∩ Zγ , where Z 0 is the class of functions w ∈ W satisfying the condition Z h w(x) (Z0 ) dx ≤ cw(h) x 0 and Zγ is the class of functions w ∈ W satisfying the condition Z ` w(x) w(h) dx ≤ c , (Zγ ) 1+γ hγ h x where c = c(w) > 0 does not depend on h ∈ (0, `]. In the sequel we refer to the above conditions as (Z0 )- and (Zγ )-conditions. We note that each of the inequalities (Z0 ) and (Zγ ) is invertible if they both are satisfied. Namely, the following statement holds. Lemma 3.3. Let w(r) ∈ Φ0γ , γ > 0. Then Zh 0

w(r) dr ∼ hγ r

Z`

w(r) dr ∼ w(h) r1+γ

(3.2)

h

the latter equivalence holding on any subinterval [0, ` − δ], δ > 0. The following statement is valid, see [22],[24] for γ = 1 and [11] for an arbitrary γ > 0. 5

Theorem 3.4. A function w ∈ W belongs to Z 0 if and only if mw > 0 and it belongs to Zγ , γ > 0, if and only if Mw < γ, so that w ∈ Φ0γ ⇐⇒ 0 < mw ≤ Mw < γ.

(3.3)

Besides this, for w ∈ Φ0γ and any ε > 0 there exist constants c1 = c1 (ε) > 0 and c2 = c2 (ε) > 0 such that c1 tMw +ε ≤ w(t) ≤ c2 tmw −ε ,

0 ≤ t ≤ `.

(3.4)

mw = sup{λ ∈ (0, 1) : t−λ w(t) is a.i.},

(3.5)

Mw = inf{µ ∈ (0, 1) : t−µ w(t) is a.d.}.

(3.6)

The following properties are also valid

Corollary 3.5. Let w(t), 0 < t ≤ `, be such a function that ta w(t) ∈ Z 0 for some a ∈ R1 . Then for any ε > 0 there exist c1 > 0 such that w(t) ≤ c1 tmw −ε .

(3.7)

a

Similarly, if t w(t) ∈ Zγ , then for any ε > 0 there exist c2 > 0 such that w(t) ≥ c2 tMw +ε .

(3.8)

(The indices mw and Mw may be negative in this case). f and mω > 0, then w ∈ W . Remark 3.6. If w ∈ W 1 Indeed, let a ∈ R be such that wa (t) = ta w(t) ∈ W . Then according to (3.5) the function tmwwaa(t)−ε is a.i. for any ε > 0. But mwa = mw + a, so that tmw(t) w −ε is a.i. In particular, the function w itself is a.i., which means that it is in W . Remark 3.7. Functions w ∈ Zγ , γ > 0, satisfy the doubling condition w(2r) ≤ Cw(r), 0 ≤ r ≤ `

(3.9)

is a.d. for every µ > Mw which follows from the fact that the function w(r) rµ according to (3.6) (observe that Mw is finite since Mω < γ by Theorem 3.4). We shall need the following lemma. f and Mω < γ. Then Lemma 3.8. Let w ∈ W is,

Zr

rγ tγ−1 dt ≤ c , [w(t)]λ [w(r)]λ

tγ [w(t)]λ

∈ Z 0 if λMω < γ, that

0 < r ≤ `.

(3.10)

0

Proof. For w1 (x) = obtain

xγ , [w(x)]λ

from the definition of the lower index we easily

mw1 = γ − λMw . f , that is, there exists a number b Hence mw1 > 0. It is easily checked that w1 ∈ W b such that t w1 (t) is a.i. Then w1 ∈ W according to Remark 3.6 and consequently w1 ∈ Z 0 by Theorem 3.4. 2 6

3.3 On radial Ap -weights generated by oscillating functions w. Let ρ(x) = [w(|x − x0 |)]λ , where λ ∈ R1 , x ∈ Rn and x0 is a fixed point in Rn and w(r) is a function such that ra w(r) ∈ W for some a ∈ R1 . The following statement provides conditions in terms of the lower and upper indices mw and Mw of the function w(r), under which ρ(x) = [w(|x − x0 |)]λ is a Muckenhoupt weight of the class Ap . According to the definition of the weighted space given in (1.3), we use the following definition of the class Ap = Ap (Rn ), p = const, 1 < p < ∞,     p−1 Z Z   1 1 Ap = ρ : sup  ρ(x)dx  [ρ(x)]1−q dx 0

λ < 0.

(3.13)

(3.14)

Proof. For radial weights the Ap -condition (3.11) takes the form Zr

 [ρ(t)]p tn−1 dt 

0

p−1

Zr

[ρ(t)]−q tn−1 dt

≤ Crnp ,

0 < r ≤ ` = diam Ω,

0

(3.15) where C > 0 does not depend on r > 0, see [8], [9]. We rewrite this for ρ(t) = [w(t)]λ as Zr 0

 w1 (t)  dt t

Zr

p−1 w2 (t)  dt t

≤ Crnp

(3.16)

0

where w1 (t) = [w(t)]λp tn , w2 (t) = [w(t)]−λq tn . The feasibility of condition (3.16) is obviously connected with validity of Z0 - condition of Subsection 3.2 7

for functions w1 (t) and w2 (t). By Theorem 3.4, the functions w1 (t) and w2 (t) satisfy this condition if and only if their lower indices mw1 and mw2 are strictly positive. To calculate these indices, suppose that λ > 0. We have mw1 = n + λpmw ,

mw2 = n − λpMw

and the positivity of these numbers leads to condition (3.13). Similarly the case λ < 0 is considered. Under condition (3.13) we then have  r p−1 Zr Z w1 (t)  w2 (t)  dt dt ≤ cw1 (r)[w2 (r)]p−1 = crnp . t t 0

0

Thus, (3.16) and consequently (3.15) are satisfied.

4

2

Preliminaries related to the variable exponent space

4.1 Some basics We recall some basic facts for the variable exponent spaces Lp(·) (Ω) and refer e.g. to [14] for details. The H¨older inequality holds in the form Z ° ° ° ° |f (x)g(x)| dx ≤ k °f °p(·) · °g °q(·) (4.1) Ω

with k =

1 p∗

+ q10 . The modular Ip (f ) =

R

|f (x)|p(x) dx and the norm kf kp(·) are

Ω

simultaneously greater than one and simultaneously less than 1: ° °p∗ ° ° ° ° °f ° ≤ Ip (f ) ≤ °f °p∗ if °f ° ≤ 1 p(·) p(·) p(·) and

° °p∗ ° °∗ ° ° °f ° ≤ Ip (f ) ≤ °f °p if °f ° ≥ 1. p(·) p(·) p(·)

(4.2)

(4.3)

From (4.2) and (4.3) it follows that c1 ≤ kf kp ≤ c2

=⇒ c3 ≤ IΩp (f ) ≤ c4

and

(4.4)

(4.5) C1 ≤ IΩp (f ) ≤ C2 =⇒ C3 ≤ kf kp ≤ C4 ³ ´ ³ ´ ³ ´ ∗ ∗ 1/p 1/p∗ with c3 = min cp1∗ , cp1 , c4 = max cp2∗ , cp2 , C3 = min C1 ∗ , C1 and ³ ´ 1/p 1/p∗ C4 = max C2 ∗ , C2 . The imbedding Lp(x) ⊆ Lr(x) , 1 ≤ r(x) ≤ p(x) ≤ p∗ < ∞ 8

is valid if |Ω| < ∞. In that case ° ° ° ° °f ° ≤ m°f ° , m = a2 + (1 − a1 )|Ω|, r(·) p(·) r(x) x∈Ω p(x)

where a1 = inf

(4.6)

r(x) and a1 = sup p(x) . x∈Ω

Lemma 4.1. Let Ω be a bounded set in Rn , the exponent p satisfy conditions (1.1), (1.2) and let w be any function such that there exist exponents a, b ∈ R1 and the constants c1 > 0 and c2 > 0 such that c1 ra ≤ w(r) ≤ c2 r−b , 0 ≤ r ≤ ` = diam (Ω). Then 1 [w(|x − x0 |)]p(x0 ) ≤ [w(|x − x0 |)]p(x) ≤ C[w(|x − x0 |)]p(x0 ) , C

(4.7)

where C > 1 does not depend on x, x0 ∈ Ω. Proof. Let g(x, x0 ) = [w(|x − x0 |)]p(x)−p(x0 ) . To show that C1 ≤ g(x, x0 ) ≤ C, that is, |ln g(x, x0 )| ≤ C1 , C1 = ln C, we observe that |ln g(x, x0 )| = |p(x) − p(x0 )| · |ln w(|x − x0 |)| Therefore, |ln g(x, x0 )| = |p(x) − p(x0 )| · |ln w(|x − x0 |)| ≤ A`

|ln w(|x − x0 |)| 2` ln |x−x 0|

which is bounded by the condition on w.

2

4.2 Auxiliary lemma for averages Let λ Mw r f (x)

[w(|x − x0 |)]λ = |Br (x)|

Z Br (x)

|f (y)| dy, [w(|y − x0 |)]λ

x0 ∈ Ω,

(4.8)

denote the weighted means related to the weighted maximal operator (2.1). ¯ In ¯ wλ (4.8) we assume that f (y) = 0 for y 6∈ Ω. We write Mr f (x) := Mr f (x)¯ . λ=0 In general, it will be admitted that λ may depend on the point x ∈ Ω. Observe that the function [w(t)]λ(x) is also of the type of the function w(t), that is, it also oscillates between two power functions. Lemma 4.1 . Let w(r) ∈ Zn and λ(x) ≥ 0. Then the inequality Z [w(|x − x0 |)]λ(x) dy wλ Mr (1) = ≤c |Br (x)| [w(|y − x0 |)]λ(x)

(4.9)

Br (x)

holds with c > 0 not depending on r > 0, x0 ∈ Rn and on x in any set D ⊆ Rn on which sup λ(x) < Mnw . x∈D

9

Proof. We distinguish the cases |x − x0 | ≥ 2r and |x − x0 | ≤ 2r. In the case |x − x0 | ≥ 2r we have 1 |y − x0 | ≥ |x − x0 | − |y − x| ≥ |x − x0 | − r ≥ |x − x0 |. 2 ¡ ¢ Since the function w ∈ Zn ⊂ W is a.i., we have w(|y − x0 |) ≥ cw 12 |x − x0 | . Taking also into account the doubling property (3.9), we obtain w(|y − x0 |) ≥ cw(|x − x0 |) and then estimate (4.9) becomes evident since λ(x) ≥ 0. Let |x − x0 | ≤ 2r. Observe that in this case B(x, r) ⊂ B(x0 , 3r) since |y − x| < r

=⇒

λ Mw r (1)

|y − x0 | ≤ |y − x| + |x − x0 | < 3r. Hence

[w(|x − x0 |)]λ(x) ≤ |Br (x)|

[w(|x − x0 |)]λ(x) = |Br (x)|

Z

Z B3r (x0 )

dy [w(|y − x0 |)]λ(x)

dy [w(|x − x0 |)]λ(x) = c [w(|y|)]λ(x) rn

Z3r

ρn−1 dρ . [w(ρ)]λ(x)

0

B3r (0)

Then by Lemma 3.8 we get µ Mw r (1)

≤c

w(|x − x0 |) w(3r)

¶λ(x)

µ ≤c

w(2r) w(3r)

¶λ(x) ≤ c. 2

5

Proof of Theorem A

5.1 Reduction to the case of a single weight Remark 5.1. It suffices to prove Theorem A for a single weight w(|x−x0 |), x0 ∈ n Ω, t p(x0 ) w(t) ∈ Φ0n . n S Indeed, let Ω = Ωk where Ωk contains the point xk in its interior and does k=1

not contain xj , j 6= k in its closure. Then kf k

µ ¶ n Q Lp(·) Ω, wk (|t−tk |)

∼

X k=1

k=1

10

kf kLp(·) (Ωk ,wk (|t−tk |))

(5.1)

whenever 1 ≤ p− ≤ p+ < ∞. This equivalence follows from the easily checked modular equivalence à ! n Y X p IΩp f (x) wk (|x − xk |) ∼ IΩ (f (x)wk (|x − xk |)) , k=1

k=1

if we take (4.4) and (4.5) into account. Then, because of (5.1), the statement of Remark 5.1 is obtained via introducn P ak (x), where ak (x) are smooth tion of the standard partition of unity 1 = k=1

functions equal to 1 in a neighborhood B(xk , ε) of the point xk and equal to 0 outside its neighborhood B(xk , 2ε), so that ak (x) [wk (|x − xj |)]±1 ≡ 0 in a neighborhood of the point xk , if k 6= j. In what follows, Ω is an open bounded set in Rn and x0 ∈ Ω.

5.2 A pointwise estimate for the weighted means Theorem 5.2. Let p(x) satisfy conditions (1.1) and (1.2) and let w ∈ W . If 0 ≤ mw ≤ Mw < where

1 q(x)

=1−

  w(|x − x0 |)  |Br (x)|

1 , p(x)

Z Br (x)

(p(·)

n , q(x0 )

(5.2)

then p(x)

|f (y)|dy   w(|y − x0 |)

for all f ∈ L (Ω) such that kf kp(·) depending on x, r and x0 .

  ≤ c 1 +

1 |Br (x)|

Z

  |f (y)|p(y) dy 

Br (x)

(5.3) ≤ 1, where c = c(p, w) is a constant not

Proof. From (5.2) and the continuity of p(x) we conclude that there exists a d > 0 such that (5.4) Mw q(x) < n for all |x − x0 | ≤ d. Without loss of generality we assume that d ≤ 1. Let pr (x) = min p(y) |y−x|≤r

and

1 qr (x)

=1−

1 pr (x)

. From (5.2) it is easily seen that

Mw qr (x) < n if |x − x0 | ≤

d d and 0 < r ≤ . 2 2 4

(5.5) 2

11

10 The case |x − x0 | ≤

d 2

d 4

and 0 < r ≤

(the main case)

In this case, applying the H¨older inequality with the exponents pr (x) and qr (x) to the integral on the right-hand side of the equality ¯ ³ ¯ ¯Mr ¯

 ¯p(x) Z ´ ¯ f (y) c  ¯ = np(x)  w(|y − x0 |) ¯ r

p(x) |f (y)|  dy  w(|y − x0 |)

Br (x)

,

we get ¯ µ ¯ ¯Mr ¯

 ≤

c rnp(x)

 

Z

Br (x)

¶¯p(x) ¯ f (y) ¯ ≤ w(|y − x0 |) ¯  pp(x)   qp(x) r (x) r (x) Z dy    |f (y)|pr (x) dy  · (5.6)  [w(|y − x0 |)]qr (x) Br (x)

where the last integral converges, since for small |y − x0 | one has [w(|y − x0 |)]qr (x) ≥ c|y − x0 |(Mw +ε)qr (x) where one may choose ε sufficiently small so that according to (5.4), |y − x0 |(Mw +ε)qr (x) ≥ |y − x0 |n−δ for some δ > 0. We may make use of estimate (4.9) in (5.6), since w ∈ W and w ∈ Zn under the condition Mw < q(xn0 ) < n according to Theorem 3.4. We obtain ¯ µ ¯ ¯Mr ¯

  pp(x) r (x) ¶¯p(x) Z −p(x) ¯ [w(|x − x0 |)] f (y)   ¯ ≤c |f (y)|pr (x) dy  .  np(x) ¯ w(|y − x0 |) r pr (x) B (x) r

Here Z

Z |f (y)|

Br (x)

pr (x)

Z

dy ≤

|f (y)|p(y) dy,

dy + Br (x)

Br (x) {y : |f (y)| ≥ 1}

since pr (x) ≤ p(y) for y ∈ Br (x). Since p(x) is bounded, we see that ¯ µ ¯ ¯Mr ¯

 pp(x)  r (x) ¶¯p(x) Z −p(x) ¯ f (y)) [w(|x − x0 |)]   n 1 ¯ ≤ c1 |f (y)|p(y) dy  . r + np(x) ¯ w(|y − x0 |) 2 r pr (x) B (x) r

12

Since r ≤ d2 ≤ 12 and the second term in the brackets is also less than or equal to 1 , we arrive at the estimate 2   Z c   p(x) |Mw ≤ np(x) rn + |f (y)|p(y) dy  ≤ r f| r pr (x) Br (x)   Z pr (x)−p(x) 1   ≤ c rn pr (x) 1 + n |f (y)|p(y) dy  . r Br (x)

From here (4.2) follows, since rn Indeed, rn where

pr (x)−p(x) pr (x)

pr (x)−p(x) pr (x)

≤ c.

n

1

= e pr [p(x)−pr (x)] ln r ,

¯n £ ¯ ¯ 1 ¤ 1 ¯¯ ¯ p(x) − pr (x) ln ¯ ≤ n¯p(x) − p(ξr )¯ ln ¯ pr r r

with ξr ∈ Br (x), and then by (2.2), ¯n £ ¤ 1 ¯¯ ln 1r ¯ ≤ nA, p(x) − pr (x) ln ¯ ≤ nA ¯ 1 pr r ln |x−ξ | r since |x − ξr | ≤ r.

20 The case |x − x0 | ≥ d2 , 0 < r ≤ d4 . This case is trivial, because |y − x0 | ≥ |x − x0 | − |y − x| ≥

d d d − = . 2 4 4

w(t) Taking ¡ d ¢ into account that t2 is almost decreasing, we then have w(|y − x0 |) ≥ Cw 4 = const. Since w(|x − x0 |) ≤ Cw( diam Ω), it follows that

Mw r f (x) ≤ cMr f (x), and one may proceed as above for the case β = 0 (the condition |x − x0 | ≤ not needed in this case).

13

d 2

is

30 The case r ≥ d4 . This case is also easy. It suffices to show that Mw r f (x) is bounded. We have   Z Z cw( diam Ω)  |f (y)| |f (y)|  ¡ d ¢n dy + dy  . Mw  r f (x) ≤ w(|y − x0 |) w(|y − x0 |) 4 |y−x0 |≥ d8

|y−x0 |≤ d8

Here the first integral is estimated via the H¨older inequality with the exponents pd = 8

min p(y) and q d = p0d

|y−x0 |≤ d8

8

8

which is possible since αq d < n. The estimate of the second integral is trivial 8 since |y − x0 | ≥ d8 . Corollary 5.3. Let 0 ≤ mw ≤ Mw < q(xn0 ) . If conditions (1.1), (1.2) are satisfied, then ¡ £ ¤ ¢ |Mw f (x)|p(x) ≤ c 1 + M |f (·)|p(·) (x) (5.7) for all f ∈ Lp(·) (Ω) such that kf kp(·) ≤ 1 . Remark 5.4. In the non-weighted case ω(x) ≡ 1 the estimate (5.7) is known to be valid if 1 ≤ p(x) ≤ p∗ < ∞ instead of condition (1.1), see [5].

5.3 Proof of Theorem A itself To prove Theorem A, we have to show that ° w ° °M f ° ≤ c p(·)

(5.8)

in some ball kf kp(·) ≤ R, which is equivalent to the inequality ° ° Ip (Mw f ) ≤ c for °f °p(·) ≤ R. According to (4.7) we obtain Z w

Ip (M f ) ≤ c

w(|x − x0 |)

p(x0 )

¯ µ ¯ ¯M ¯

Ω

f (y) w(|y − x0 |)

¶

¯p(x) ¯ dx. (x)¯¯

(5.8) first for We prove (5.8) first for −

n n < mw ≤ Mw < , p(x0 ) q0 14

(5.9)

1 q0

=

p∗ −1 . p(x0 )

1 q0

Observe that

≤

1 q(x0 )

so that the interval (5.9) for mw , Mw ³ ´ is somewhat narrower than the whole interval − p(xn0 ) , qn0 . After that we treat the remaining case. where

10 The case − p(xn0 ) < mw ≤ Mw < qn0 . Following the idea in [5], we represent this as ¯ µ ¶ ¯p1 (x) !p∗ Z à ¯ ¯ f (y) dx, Ip (Mw f ) ≤ c (x)¯¯ [w(|x − x0 |)]p1 (x0 ) ¯¯M w(|y − x0 |) Ω

(5.10) where p1 (x) =

p(x) . p∗

Estimate (5.7) with w ≡ 1 says that ³ £ ¤ ´ |Mψ(x)|p1 (x) ≤ c 1 + M ψ p1 (·) (x)

(5.11)

(see Remark 5.4) for all ψ ∈ Lp1 (·) (Ω) with kψkp1 (·) ≤ 1 (or equivalently, for all ψ with kψkp1 ≤ C with fixed C < ∞). f (x) with f ∈ Lp(·) in (5.11). Let us show We intend to choose ψ(x) = w(|x−x 0 |) that in this case ° ° ° ° f (x) ° kψkp1 = ° (5.12) ° w(|x − x0 |) ° ≤ C p1 n

for all f ∈ Lp(·) with kf kp ≤ c. Since r p(x0 ) w(r) ∈ Φ0n , by Corollary 3.5 we have w(|x − x0 |) ≥ c|x − x0 |Mw +ε , ε > 0 and then Z

Z

p(x)

|f (x)| p∗ dx. |x − x0 |(Mw +ε)p1 (x0 )

|ψ(x)|p1 (x) dx ≤ c Ω

Ω

(5.13)

To obtain (5.12), it remains to apply the H¨older inequality in (5.13) with the ∗ exponents p∗ and q∗ = p∗p−1 and take into account that µ Iq∗

¶

1 |x − x0 |(Mw +ε)p1 (x0 )

Z = Ω

dx |x − x0 |(Mw +ε)q0

where (Mw + ε)q0 < n under the choice of small ε < qn0 − Mw . In view of (5.12), we may apply estimate (5.11). Then (5.10) implies " ï ¯p1 (y) !#!p∗ Z à ¯ ¯ f (y) ¯ dx. Ip (Mw f ) ≤ c [w(|x − x0 |)]p1 (x0 ) 1 + M ¯¯ w(|y − x0 |) ¯ Ω

15

By property (4.7), this yields Ip (Mw f ) ≤ · µ Z ½ p(x0 ) p1 (x0 ) ≤c [w(|x − x0 |)] + [w(|x − x0 |)] M Ω

|f (y)|p1 (y) [w(|y − x0 |)]p1 (x0 )

¶¸p∗ ¾ dx.

Here the integral of the first term is finite since w(|x − x0 |) ≤ |x − x0 |mω −ε according to (3.4) and mw p(x0 ) > −n. Hence Z h ip∗ p (x ) w Mw 1 0 (|f (·)|p1 (·) )(x) dx Ip (M f ) ≤ c + c (5.14) Ω

in notation (2.1). p As is known [29], p. 201, the weighted maximal operator Mw 1 is bounded in Lp∗ with a constant p∗ > 1 if the weight [w(|x−x0 |)]p1 (x0 ) is in Ap∗ . According to condition (3.13) of Lemma 3.9 this is the case if − p(xn0 ) < mw ≤ Mw < qn0 which is satisfied in the case under consideration. p (x ) Therefore, by the boundedness of the weighted operator Mw 1 0 in Lp∗ , from (5.14) we get Z Z w p1 (y)·p∗ Ip (M f ) ≤ c + c |f (y)| dy = c + c |f (y)|p(y) dy < ∞. (5.15) Ω

Ω

20 The remaining case To get rid of the right-hand side bound in (5.9), we may split integration over Ω into two parts, one over a small neighborhood Bδ = Bδ (x0 ) of the point x0 , and another over its exterior Ω\Bδ , and to choose δ sufficiently small so that the (Bδ )−1 0 )−1 number p∗p(x is arbitrarily close to p(x = q(x10 ) . To this end we put p(x0 ) 0) Mw = χBδ Mw χBδ +χBδ Mw χΩ\Bδ +χΩ\Bδ Mw χBδ +χΩ\Bδ Mw χΩ\Bδ (5.16) w w w = : Mw 1 + M2 + M3 + M4 .

Since the weight is strictly positive beyond any neighborhood of the point x0 , we have (5.17) Mw 4 f (x) ≤ CMf (x). For Mω3 we have Mw 3 f (x)

χΩ\Bδ (x0 ) (x) = sup |Br (x)| r>0

Z Br (x)∩Bδ (x0 )∩Ω

w(|x − x0 |) |f (y)| dy. w(|y − x0 |)

Here |x − x0 | > r > |y − x0 |. Observe that the function wε (t) = for any ε > 0 according to (3.6). Therefore

w(t) tMw +ε

is a.d.

wε (|x − x0 |) |x − x0 |Mw +ε |x − x0 |Mw +ε w(|x − x0 |) = · ≤C . w(|y − x0 |) wε (|y − x0 |) |y − x0 |Mw + ε) |y − x0 |Mw + ε 16

Hence Mw +ε Mw f (x) 3 f (x) ≤ CM

(5.18)

where MMw +ε f (x) is the weighted maximal function with the power weight |x − x0 |Mw +ε . Similarly we conclude that mw −ε Mw f (x). 2 f (x) ≤ CM

(5.19)

Thus from (5.16) according to (5.17), (5.18) and (5.19) we have Mw f (x) ≤ χBδ Mw χBδ f (x) + Mf (x) + MMw +ε f (x) + Mmw −ε f (x). (5.20) Here the operators M, MMw +ε and Mmw −ε are bounded in the space Lp(·) (Ω), because the boundedness condition (2.2 is satisfied for β = Mw + ε and β = mω − ε under a choice of ε sufficiently small. It remains to prove the boundedness of the first term on the right-hand side of (5.20). This is nothing else but the boundedness of the same operator Mw over a small set Ωδ = Bδ (x0 ) ∩ Ω. According to the previous case, this boundedness holds if n n − < mw ≤ Mw < (5.21) p(x0 ) qδ p∗ (Ωδ )−1 and p∗ (Ωδ ) = min p(x). p(x0 ) x∈Ωδ condition − p(xn0 ) < mw ≤ Mw < q(xn0 ) , one can small such that (5.21) holds. Given Mw < q(xn0 ) , Mw < qnδ ≤ q(xn0 ) . We have

where qδ =

n n = − a(δ), qδ q(x0 )

where

a(δ) =

Let us show that, given the always choose δ sufficiently we have to choose δ so that

n [p(x0 ) − p∗ (Ωδ )] . p(x0 )

By the continuity of p(x) we can choose δ so that a(δ) < q(xn0 ) − Mw . Then n > Mw and condition (5.21) is fulfilled. Then the operator Mw is bounded in qδ the space Lp(·) (Bδ ) which completes the proof.

References [1] N.K. Bari and S.B. Stechkin. Best approximations and differential properties of two conjugate functions (in Russian). Proceedings of Moscow Math. Soc., 5:483–522, 1956. [2] C. Bennett and R. Sharpley. Interpolation of operators., volume 129 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1988. [3] D. Cruz-Uribe, A. Fiorenza, J.M. Martell, and C Perez. The boundedness of classical operators on variable Lp spaces. Preprint, Universidad Autonoma de Madrid, Departamento de Matematicas, http : //www.uam.es/personalp di/ciencias/martell/Investigacion/research.html, 2004. 26 pages. 17

[4] D. Cruz-Uribe, A. Fiorenza, and C.J. Neugebauer. The maximal function on variable Lp -spaces. Ann. Acad. Scient. Fennicae, Math., 28:223–238, 2003. [5] L. Diening. Maximal function on generalized Lebesgue spaces Lp(·) . Math. Inequal. Appl., 7(2):245–253, 2004. [6] L. Diening. Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math., 129(8):657–700, 2005. [7] L. Diening, P. H¨ast¨o, and A. Nekvinda. Open problems in variable exponent Lebesgue and Sobolev spaces. Preprint http://www.math.helsinki.fi/analysis/varsobgroup/pdf/opFinal.pdf, pages 1–17, 2004. [8] E.M. Dyn’kin and B.B. Osilenker. Weighted estimates for singular integrals and their applications. Math. Anal., vol. 21 Itogi nauki i techniki VINITI Akad. Nauk SSSR, (Russian). Moscow: Nauka, 1983. 542 pages. [9] E.M. Dyn’kin and B.B. Osilenker. Weighted norm estimates for singular integrals and their applications. J. Sov. Math., 30:2094–2154, 1985. [10] P. Harjulehto and P. Hasto. An overview of variable exponent Lebesgue and Sobolev spaces. Future Trends in Geometric Function Theory (D. Herron (ed.), RNC Workshop, Jyvaskyla, 85-94, 2003. [11] N.K. Karapetiants and N.G. Samko. Weighted theorems on fractional integrals in the generalized H¨older spaces H0ω (ρ) via the indices mω and Mω . Fract. Calc. Appl. Anal., 7(4), 2004. [12] V. Kokilashvili and S. Samko. Maximal and fractional operators in weighted Lp(x) spaces. Revista Matematica Iberoamericana, 20(2):495– 517, 2004. [13] V.M. Kokilashvili. On a progress in the theory of integral operators in weighted Banach function spaces. In ”Function Spaces, Differential Operators and Nonlinear Analysis”, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28 - June 2, 2004. Math. Inst. Acad. Sci. Czech Republick, Praha. [14] O. Kov´acˇik and J. R´akosnˇik. On spaces Lp(x) and W k,p(x) . Czechoslovak Math. J., 41(116):592–618, 1991. [15] S.G. Krein, Yu.I. Petunin, and E.M. Semenov. Interpolation of linear operators. Moscow: Nauka, 1978. 499 pages. [16] S.G. Krein, Yu.I. Petunin, and E.M. Semenov. Interpolation of linear operators, volume 54 of Translations of Mathematical Monographs. American Mathematical Society, Providence, R.I., 1982. 18

[17] L. Maligranda. Indices and interpolation. Dissertationes Math. (Rozprawy Mat.), 234:49, 1985. [18] Kh. M. Murdaev and S.G. Samko. Fractional integro-differentiation in the weighted generalized H¨older spaces H0ω (ρ) with the weight ρ(x) = (x − a)µ (b − x)ν ) (Russian) given continuity modulus of continuity (Russian). Deposited in VINITI, Moscow. [19] Kh. M. Murdaev and S.G. Samko. Weighted estimates of the modulus of continuity for fractional integrals of function with a given continuity modulus of continuity (Russian). Deposited in VINITI, Moscow. [20] A. Nekvinda. Hardy-Littlewood maximal operator on Lp(x) (Rn ). Math. Inequal. and Appl. (submitted). [21] A. Nekvinda. Hardy-Littlewood maximal operator on Lp(x) (Rn ). Mathematical Preprints: Faculty of Civil Engineering, CTU, Prague, (02/ May 2002), 2002. [22] N.G. Samko. Singular integral operators in weighted spaces with generalized H¨older condition. Proc. A. Razmadze Math. Inst, 120:107–134, 1999. [23] N.G. Samko. On compactness of Integral Operators with a Generalized Weak Singularity in Weighted Spaces of Continuous Functions with a Given Continuity Modulus. Proc. A. Razmadze Math. Inst, 136:91, 2004. [24] N.G. Samko. On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class. Real Anal. Exch., 30(2), 2005. [25] S.G. Samko. On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integr. Transf. and Spec. Funct, 16(5). [26] S.G. Samko, A.A. Kilbas, and O.I. Marichev. Fractional Integrals and Derivatives. Theory and Applications. London-New-York: Gordon & Breach. Sci. Publ., (Russian edition - Fractional Integrals and Derivatives and some of their Applications, Minsk: Nauka i Tekhnika, 1987.), 1993. 1012 pages. [27] S.G. Samko and Kh.M. Murdaev. Weighted Zygmund estimates for fractional differentiation and integration, and their applications. Trudy Matem. Ibst. Steklov, 180:197–198, 1987. translated in in Proc. Steklov Inst. Math., AMS, 1989, issue 3, 233-235. [28] I.I. Sharapudinov. The topology of the space Lp(t) ([0, 1]) (Russian). Mat. Zametki, 26(4):613–632, 1979. [29] E.M. Stein. Harmonic Analysis: real-variable methods, orthogonality and oscillatory integrals. Princeton Univ. Press, Princeton, NJ, 1993. 19

[30] A.Ya. Yakubov. Classes of pseudo-concave type functions and their applications. Proc. A. Razmadze Math. Inst., 129:113–128, 2002. [31] A.Ya. Yakubov. Fractional type integration operators in weighted generalized H¨older spaces. Fract. Calc. Appl. Anal., 5(3):275–294, 2002.

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