Weighted hardy operators in complementary morrey spaces

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Volume 2012, Article ID 283285,19pages doi:10.1155/2012/283285

Research Article

Weighted Hardy Operators in Complementary

Morrey Spaces

Dag Lukkassen,

1

_{Lars-Erik Persson,}

2

_{and Stefan Samko}

3

1_{Department of Technology, Narvik University College, P.O. Box 385, 8505 Narvik, Norway} 2_{Department of Mathematics, Lule˚a University of Technology, SE 921 87 Lule˚a, Sweden} 3_{Departamento de Matematica, Universidade do Algarve, 6005-139 Faro, Portugal} Correspondence should be addressed to Lars-Erik Persson,larserik@ltu.se

Received 23 July 2012; Accepted 20 September 2012 Academic Editor: Alois Kufner

Copyrightq 2012 Dag Lukkassen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the weighted p → q-boundedness of the multidimensional weighted Hardy-type operators Hα

w andHαwwith radial type weight w w|x|, in the generalized complementary Morrey spacesLp,ψ_{0}Rn_{defined by an almost increasing function ψ ψr. We prove a theorem} which provides conditions, in terms of some integral inequalities imposed on ψ and w, for such a boundedness. These conditions are suﬃcient in the general case, but we prove that they are also necessary when the function ψ and the weight w are power functions. We also prove that the spacesLp,ψ_{0}Ω over bounded domains Ω are embedded between weighted Lebesgue space Lp with the weight ψ and such a space with the weight ψ, perturbed by a logarithmic factor. Both the embeddings are sharp.

1. Introduction

Hardy operators and related Hardy inequalities are widely studied in various function spaces, and we refer to the books 1–4 and references therein. They continue to attract

attention of researchers both as an interesting mathematical object and a useful tool for many purposes: see for instance the recent papers5,6. Results on weighted estimations of Hardy

operators in Lebesgue spaces may be found in the abovementioned books. In the papers7–9

the weighted boundedness of the Hardy type operators was studied in Morrey spaces. In this paper we study multi-dimensional weighted Hardy operators

H_wαfx : |x|α−nw|x| |y|<|x| fydy wy , H α wfx : |x|αw|x| |y|>|x| fydy _yn wy, 1.1

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where α ≥ 0, in the so called complementary Morrey spaces. The one-dimensional case will include the versions

H_wαfx xα−1wx _x 0 ftdt wt , H α wfx xαwx _∞ x ftdt twt, x > 0 1.2

adjusted for the half-axisR1

, so that in the sequelRnwith n 1 may be read either as R1 or

R1

.

The classical Morrey spacesLp,λ_{Ω, Ω ⊆ R}n_{, defined by the norm}

_f p,λ: sup x∈Ω,r>0 1 rλ Bx,rf ypdy 1/p , 1≤ p < ∞, 0 ≤ λ ≤ n, 1.3

where Bx, r Ω ∩ Bx, r, are well known, in particular, because of their usage in the study

of regularity properties of solutions to PDE; see for instance the books10–12 and references

therein. There are also known various generalizations of the classical Morrey spacesLp,λ_,

and we refer for instance to the surveying paper13. One of the direct generalizations is

obtained by replacing rλ _in_1.3_{by a function ϕr, usually satisfying some monotonicity}

type conditions. We also denote it as Lp,ϕ_{Ω without danger of confusion. Such spaces}

appeared in14,15 and were widely studied in 16,17. The spaces Lp,ϕ_locΩ, defined by

the norm _f p,ϕ; loc: sup r>0 1 ϕr Bx0, r _f yp dy 1/p , 1.4

where x0∈ Ω, are known as generalized local Morrey spaces

Hardy type operators1.1 in the spaces Lp,ϕ_{Ω, L}p,ϕ

locΩ have been studied in 7–9.

The norm in Morrey spaces controls the smallness of the integral_B_x,r|fy|p_{dy over}

small balls Bx, r and also a possible growth of this integral for r → ∞ in the case Ω is unbounded. There are also known spaces_Lp,ψ

{x0}Ω, called complementary Morrey spaces,

with the norm controlling possible growth, as r → 0, of the integral

Ω\Bx0, r

fypdy _1.5

over exterior of balls. Such spaces have sense in the local setting only. It is introduced in 16,17 that the spaceLp,ψ_{x0}Ω is defined as the space of all functions f ∈ Lp_locΩ\{x0} with

the finite norm

f_Lp,ψ

{x0}Ω sup_r>0 ψ

1/p_r_f

Lp_Ω\Bx0_{, r}, 1.6

where admission of the multiplier ψr vanishing at r 0 controls the growth of the norm

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to18. We refer also to the paper 19 where variable exponent complementary spaces of

such type were introduced.

During the last decades various classical operators, such as maximal, singular, and potential operators were widely investigated both in classical and generalized Morrey spaces, including the complementary Morrey spaces. The mapping properties of Hardy type operators in complementary Morrey spaces were not known. In this paper we obtain conditions for the p → q-boundedness of Hardy type operators in complementary Morrey spaces. Thus we make certain contributions to the known theory of Hardy type inequalities; see, for example, the books2–4 and references given there.

We also prove a new property for the generalized complementary Morrey spaces; see

Theorem 3.1, by showing that the spacesLp,ψ_{0}Ω over bounded domains Ω are embedded between the weighted Lebesgue space Lp_{with the weight ψ, and such a space with the weight}

ψ, perturbed by a logarithmic factor. In the case where ψ was a power function, this was

proved in19.

The paper is organized as follows. In Section 2 we give definitions and necessary preliminaries, including conditions for radial functions to belong to complementary Morrey spaces. In Section 3 we prove the abovementioned embedding of the generalized Morrey space between Lebesgue weighted spaces. In Section 4, which plays a crucial role in the preparation of the proofs of the main results, we prove pointwise estimate for the Hardy-type constructions via the norm defining the complementary Morrey space. In Section 5

we give the final theorems on the weighted p → q-boundedness of Hardy operators in complementary Morrey spaces. In the appendix we collect various properties of weights from the Bary-Stechkin class which we need when we formulate some suﬃcient conditions for the boundedness in terms of the Matuszewska-Orlicz indices of the functions ψ and w.

2. Complementary Morrey Spaces and Their Properties

2.1. Definitions

LetΩ be an open set in Rn_,_{Ω ⊆ R}n_and_{diam Ω, 0 < ≤ ∞, Bx, r {y ∈ R}n _:_{|x −y| < r}}

and Bx, r Bx, r ∩ Ω. Let also ψr be a continuous function nonnegative on 0, with ψ0 : limr→ 0ψr 0 which may arbitrarily grow as r → ∞. Let also 1 ≤ p < ∞.

Definition 2.1. Let 1≤ p < ∞. The complementary Morrey spaceLp,ψ_{x0}Ω, where x0 ∈ Ω, is

the space of functions f∈ Lp_locΩ \ {x0} such that

f_Lp, ψ {x0}Ω: sup_r>0 ψr Ω\ Bx0,r fyp dy 1/p <∞. 2.1

In the case of power factor ψr ≡ rλ_{, λ > 0, we also denote}

_Lp, λ

{x0}Ω :L

p, ψ

{x0}Ω|ψ≡rλ 2.2

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The spaceLp, ψ_{x0}Ω is nontrivial in the case of any locally bounded function ψr with an arbitrary behaviour at infinity, because bounded functions with compact support belong then to this space.

We will also use the following notation for the modular:

Mp, ψ f; x0, r : ψr Ω\ Bx0, r fypdy. 2.3

The function ψ, defining the complementary Morrey space, will be called definitive function.

Remark 2.2. The function ϕ in 1.4, defining the Morrey spaces, in some papers is called

weight function. We prefer to call both ϕ and ψ as definitive functions, keeping the word weight

for its natural use in the theory of function spaces, that is, for the cases where the function f itself is controlled by a weight.

2.2. On Belongness of Radial Functions to the Space

L

p, ψ_{x₀_}

Ω

Lemma 2.3. For a nonnegative radial type functions u u|x − x0|, x0 ∈ Ω, the condition

sup 0<r< ψr r utptn−1dt <∞ 2.4 is suﬃcient to belong to_Lp, ψ

{x0}Ω. (It is also necessary, when Ω RnorΩ is a ball centered at x0.)

The proof of Lemma 2.3 is obvious. By means of this lemma we easily obtain the following corollary.

Corollary 2.4. Let Ω be bounded. A power function |x − x0|γ _{with γ /}_{− n/p belongs to the space}

_Lp, ψ

{x0}Ω if and only if

sup

r>0

rmin{n γp, 0}ψr < ∞. _2.5

In the case n γp 0, the same holds with rmin{n γp,0}replaced by| ln r|. When Ω Rn_{, the necessary}

and suﬃcient conditions are n γp < 0 and sup_r>0rn γp_ψ_{r < ∞.}

As is known, there may be given suﬃcient numerical inequalities for the validity of the integral condition in2.4 in terms of Matuszewska-Orlicz indices mu and Mu of the

function u. We give below such suﬃcient inequalities, but in order not to interrupt the main body of the paper by notions connected with such indices and related Bary-Stechkin function classesZα,β, we put these notions in the appendix.

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Let now u be a nonnegative function in Bary-Stechkin class, namely, u∈ Z−n/p0, if < ∞, u∈ Z−n/p,−n/p R1 if ∞. 2.6

Then_rup_ttn−1 _dt _{∼ r}n_up_{r, 0 < r < , so that the condition sup}

0<r<rnψrupr < ∞

is suﬃcient for u u|x − x0| to be inL

p, ψ

{x0}Ω. Note also that 2.6 is equivalent to the

inequalities pMu n < 0, pM∞u n < 0 in terms of the indices, where the second inequality

is to be used only in the case  ∞; see A.16.

3. Complementary Morrey Spaces Are Closely Embedded between

Weighted Lebesgue Spaces

The complementary Morrey spaces are close to a certain weighted Lebesgue space as stated in the following theorem, which provides sharp embeddings of independent interest. The notation for the weighted space is taken in the form Lp_{Ω, : {f :}

Ω x|fx|pdx <∞}.

The class W00, , used in this theorem, is defined in the appendix; see DefinitionsA.1-A.2

therein. In the case where ψr is a power function,Theorem 3.1was proved in19.

Theorem 3.1. Let Ω be a bounded open set, 1 ≤ p < ∞, ψ ∈ W00, , diam Ω, and let ψ be

absolutely continuous and

P : sup 0<t< tψt ψt <∞. 3.1 Then LpΩ, ψy− x0→L p, ψ {x0}Ω → ε>0 LpΩ, ψεy− x0, 3.2

where ψεt ψt/lnA/t1 ε, A > . The left-hand side embedding in3.2 is strict, when ψ

satisfies the condition

r dt tψtdt≤ C ψr 3.3

(in particular, if ψr rλ_{, λ > 0); that is, there exists a function f}

0 f0x such that f0∈L p, ψ {x0}Ω, but f0∈ L/ p Ω, ψy− x0. 3.4

The right-hand side embedding in 3.2 is strict, when ψ satisfies the doubling condition ψ2r ≤

Cψr, (in particular, if ψr rλ_{, λ}_{≥ 0), there exists a function g}

0 g0x such that g0∈ ε>0 LpΩ, ψε|x − x0| , but g0∈/L p, ψ {x0}Ω. 3.5

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Proof. Without loss of generality, we may take x0 0 for simplicity, supposing that 0 ∈ Ω.

10_{. The left-hand side embedding: since the function ψ is almost increasing, we have}

Ωψ _y_f y|p dy 1/p ≥ Ω\ B0,rψ _y_f y|pdy 1/p ≥ C ψr Ω\ B0,r fypdy 1/p 3.6 so that f_Lp_Ω,ψ|y|≥ Cf_Lp,ψ {0}Ω. 3.7

20. The right-hand side embedding: we have B0,rf ypψεydy B0,rf yp _|y| 0 d dtψεtdt dy, 3.8 where ψ_εt ψt t _tψ_t ψt 1 ε lnA/t 1 lnA/t1 ε, 3.9 so that ψ_εt ≤ Cεψt tlnA/t1 ε, Cε P 1 ε lnA/. 3.10 Therefore, Bx0,r fypψεydy≤ _t 0 ψ_εt {y∈Ω:t<|x0−y|<r} fypdy dt ≤ 0 ψ_εt · fp Lp_{Ω\ Bx0}_,tds ≤fp Lp,ψ {x0}Ω 0 _ψ εt ψt dt, 3.11

where the last integral converges when ε > 0 since|ψ_εt|/ψt ≤ Cε/tlnA/t1 ε.

30_{. The strictness of the embeddings: the corresponding counterexamples are}

f0x 1

ψ1/p_|x||x|n/p, g0x

lnlnB/|x|

|x|n/p_ψ1/p_|x|, B > e

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Calculations for the function f0, which is obviously not in LpΩ, ψ|y|, are easy. Indeed, _f₀p _Lp,ψ {0}Ω≤ Sn−1_ψr r dt tψt, 3.13

where the right-hand side is bounded by3.3.

In the case of the function g0we have

_g₀p Lp_Ω,ψε Ω lnplnB/|x| |x|n_lnA/|x|1 εdx≤Sn−1 0 lnplnB/t tlnA/t1 εdt <∞ 3.14

for every ε > 0. However, for small r∈ 0, δ/2, where δ dist0, ∂Ω, we obtain

ψr x∈Ω: |x|>r g₀p|x|dx ≥ ψr x∈Ω: r<|x|<δ lnplnB/|x|dx |x|n_ψ_|x| Sn−1_ψrδ r lnplnB/tdt tψt ≥Sn−1_ψr2r r lnplnB/tdt tψt . 3.15

Taking into account that ψt is almost increasing and satisfies the doubling condition, we get ψr _2r r lnplnB/tdt tψt ≥ C ψr ψ2rln p ln B 2r 2r r dt t ≥ C ln p lnB r −→ ∞ as r −→ 0, 3.16 which completes the proof of the theorem.

Remark 3.2. Under the assumption3.3, the upper embedding in 3.2 does not hold with ε

0. The corresponding counterexample is fx 1/|x − x0|nψ|x − x0| which is inL

p,ψ

{x0}Ω,

but f /∈ Lp_{Ω, ψ}

ε|y − x0||ε0.

4. Weighted Estimates of Functions in Complementary Morrey Spaces

In the following lemmas we give a pointwise estimate of the “Hardy-type” constructions in terms of the modularMp,ψf; x0, r with x0 0. These estimates are of independent interest

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Lemma 4.1. Let 1 ≤ p < ∞, 0 < s ≤ p, let v, ψs/p_v_{∈ W0, ∞, and v2t ≤ cvt. Then} |z|<r _f_zs v|z| dz≤ C _r 0 Vst t dt· M s/p p,ψ f; 0, r, r > 0, 4.1

for f ∈ Lp_locΩ \ {0}, where C > 0 does not depend on f and r ∈ 0, ∞ and

Vst t

n1−s/p

vtψs/p_t. 4.2

Proof. We use the dyadic decomposition as follows:

|z|<r _f_zs v|z| dz ∞ k0 Bkr _f_zs v|z| dz, 4.3

where Bkr {z : 2−k−1r < |z| < 2−kr}. Since there exists a β such that tβvt is almost

increasing, we observe that

1

v|z|≤ C

v2−k−1r 4.4

on Bkr. Applying this in 4.3 and making use of the H¨older inequality with the exponent

p/s≥ 1, we obtain |z|<r fzs v|z| dz≤ C ∞ k0 2−k−1rn1−s/p v2−k−1r Bkr _f_zp dz s/p . 4.5 Hence |z|<r fzs v|z| dz≤ C ∞ k0 2−k−1rn1−s/p M s/p p,ψ f; 0, 2−k−1r v2−k−1rψs/p₂−k−1_r. 4.6

On the other hand we have r 0 Vst t M s/p p,ψ f; 0, tdt ∞ k0 2−kr 2−k−1r tn1−s/p−1M s/p p,ψ f; 0, t vtψS/p_t dt. 4.7

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The function 1/ψtMp,ψf; 0, t is decreasing, and the function 1/vt is almost

decreasing after multiplication by some power functions. Therefore, _r 0 Vst t M s/p p,ψ f; 0, tdt≥ C ∞ k0 2−krn1−s/p M s/p p,ψ f; 0, 2−kr v2−krψS/p₂−k_r ≥ C∞ k1 2−krn1−s/p M s/p p,ψ f; 0, 2−kr v2−krψs/p₂−k_r C1 ∞ k0 2−k−1rn1−s/p M s/p p,ψ f; 0, 2−k−1r v2−k−1rψs/p₂−k−1_r. 4.8

Then4.1 follows from 4.6 and 4.8.

Lemma 4.2. Let 1 ≤ p < ∞, 0 ≤ s ≤ p, v ∈ WR1

and v2t ≤ Cvt, ψ2t ≤ Cψt. Let also

Vst :t n1−s/p_v_t ψs/p_t . 4.9 Then |z|>rv|z| _f_zs dz≤ C _∞ r Vs_t t M s/p p,ψ f; 0, tdt, 4.10

where C > 0 does not depend on r > 0 and f.

Proof. We use the corresponding dyadic decomposition:

|z|>rvt fzs dz ∞ k0 Bk_r vzfzsdz, _4.11

where Bk_{r {z : 2}k_{r <} _{|z| < 2}k 1_r_{}. Since there exists a β ∈ R}1 _{such that t}β_v_{t is almost}

increasing, we obtain ∞ k0 Bk_r v|z|fzsdz≤ C ∞ k0 v2k 1r Bk_r fzsdz, 4.12 where C may depend on β but does not depend on r and f. Applying the H ¨older inequality with the exponent p/s, we get

|z|>rv|z| fzsdz≤ C ∞ k0 v2k 1r2krn1−s/p Bk_r fzpdz s/p ≤ C∞ k0 v2k 1_r₂k_rn1−s/p_ψ−s/p₂k_r_Ms/p p,ψ f; 0, 2k_r_. 4.13

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On the other hand, the integral on the right-hand side of4.10 can be estimated as follows: _∞ r Vs_t t M s/p p,ψ f; 0, tdt ∞ k0 2k 1r 2k_r tn−1−ns/p Mp,ψ f; 0, t ψt s/p vtdt ≥ C∞ k0 v2krMp,ψ f; 0, 2k 1_r ψ2k 1_r s/p 2krn1−s/p ≥ C∞ k0 v2k 1rψ−1/p2kr2krn1−s/pMs/pp,ψ f; 0, 2k 1r, 4.14

which completes the proof.

Corollary 4.3. Let 1 ≤ p < ∞, let w, wψ1/p _{∈ W0, , w2t ≤ cwt, 0 < ≤ ∞ and 0 ∈ Ω.}

Let also Vr : 1 wp_rψr ∈ Z0. 4.15 Then Mp,ψ _f w; 0, r ≤ C wp_rMp,ψ f; 0, r, 0 < r < . 4.16

5. Weighted Hardy Operators in Complementary Morrey Spaces

5.1. Pointwise Estimations

The proof of our main result of this Section given in Theorems5.3and5.6is prepared by the following Theorems5.1and5.2on the pointwise estimates of the Hardy-type operators.

Theorem 5.1. Let 1 ≤ p ≤ ∞, let w, wψ1/p_{∈ W, and w2t ≤ Cwt. The condition}

_ε 0 Vt t dt <∞ with ε > 0, where V t tn/p wtψ1/p_t 5.1

is suﬃcient for the Hardy operator Hα

wto be defined on the spaceL

p,ψ

{0}Rn. Under this condition, the

pointwise estimate _Hα wfx ≤ C|x|α−nw|x| _|x| 0 Vt t dtfLp,ψ{0} 5.2 holds.

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Theorem 5.2. Let 1 ≤ p ≤ ∞ and 1/w ∈ W, or w ∈ W and w2t ≤ Cwt. The condition _∞ ε Vt t dt <∞, with ε > 0, where Vt 1 wtψ1/p_ttn/p 5.3

is suﬃcient for the Hardy operator Hα

w to be defined on the spaceL

p,ψ

{x0}Rn, and in this case the

following pointwise estimate:

Hα wfx ≤ C|x|αw|x| _∞ |x| Vt t dtfLp,ψ{x0} 5.4 holds.

Proof. ApplyLemma 4.2with s 1.

5.2. Weighted

p

→ q Boundedness of Hardy Operators in Complementary

Morrey Spaces

5.2.1. The Case of the Operator H

α

w

The Lp _{→ L}q_{-boundedness of the multidimensional Hardy operators within the frameworks}

of Lebesgue spacesthe case ϕ ≡ 1 in 1.4 with 1 < p < ∞ and 0 < q < ∞ is well known: see

for instance,4, page 54. For Morrey spaces, both local and global, the boundedness of Hardy

operators was studied in7,8. We call attention of the reader to the fact that, in contrast to the

case of Lebesgue spaces, Hardy-type inequalities in both usual and complementary Morrey spaces diﬀerent from Lebesgue spaces i.e., in the case ϕ0 0 or ψ0 0 admit the value

p 1.

Theorem 5.3. Let 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ and let

w, ψ1/pw∈ WR1, w2t ≤ cwt. 5.5 The operator Hα wis bounded fromL p,ψ {0}Rn toL q,ψ {0}Rn if supr>0Wr < ∞, where Wr : ψr _∞ r n−1 α−n/p ψ1/p_V 0 Vt t dt q d , 5.6

and Vt is the same as in 5.1. Under this condition,

H_wαf_Lq,ψ

{0}≤ C sup_r>0 W

1/q_r_f

_Lp,ψ

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Proof. From the estimate5.2 ofTheorem 5.1we have H_wαfx ≤ C|x| α−n/p ψq/p_|x| 1 V|x| _|x| 0 Vt t dt f_Lq,ψ {0}. 5.8 Hence, calculating Hα wf _Lp,ψ

{0}, passing to polar coordinates, we obtain5.6.

Remark 5.4. Note that

1 V 0 Vt t dt 1 5.9 in5.6 if we suppose that V ∈ Z0_.

The following corollary gives suﬃcient conditions for the boundedness of the operator

Hα

win terms of the Matuszewska-Orlicz indices of the function ψ and the weight wwe refer

to the appendix for these notions.

Corollary 5.5. Let 1 ≤ p ≤ q < ∞ (with q p admitted in the case α 0) and the conditions 5.5

be satisfied. Suppose also that

ψr ≥ Crαpq/q−p−n 5.10

in the case α / 0. The operator Hα

wis bounded fromL p,ψ {0}Rn toL q,ψ {0}Rn if min{mV , m∞V } > 0, minmψ, m∞ψ> 0. 5.11

The condition min{mV , m∞V } > 0, is guaranteed by the inequalities

Mw < n

p− M

ψ, M∞w < _pn− M∞ψ. 5.12

In the power case ψr rλ_{, λ > 0, and w}_{r r}μ_{, the conditions}_5.10_-_5.11_{reduce to}

1≤ p < n λ α , 1 q 1 p − α n λ, μ < n p− λ p; 5.13

conditions5.13 are also necessary for the operator Hα

wto be bounded fromL

p,ψ

{0}Rn toL

q,ψ

{0}Rn.

Proof. We have to check that the condition sup_r>0Wr < ∞ ofTheorem 5.3holds under the

assumptions5.10-5.11. From the inequality min{mV , m∞V } > 0 it follows that

Wr ≤ Cψr _∞ r n−1 qα−n/p ψq/p d . 5.14

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We represent ψq/p_{as ψ}q/p_ψq/p−1_{ψ and by}_5.10_obtain Wr ≤ Cψr _∞ r d ψ , 5.15

which is bounded in view of the second assumption in5.11, by the property A.4.

The suﬃciency of the conditions 5.13 in the case of power functions is then obvious

since mψ m∞ψ λ and mw m∞w μ in this case.

Let us show the necessity of these conditions. From the boundedness Hα w _Lq,ψ

{0}Rn ≤

C f _Lp,ψ

{0}Rn with ψr r

λ_{, λ > 0, and w}_{r r}μ_{, by standard homogeneity arguments}

with the use of the dilation operatorΠδfx : fδx it is easily derived that the conditions

1≤ p < n λ/α, 1/q 1/p − α/n λ necessarily hold, via the relation Πδf_Lp,ψ {0}Rn δ −n λ/p_f _Lp,ψ {0}Rn, H α wΠδf δ−αΠδHwαf. 5.16

We take into account that we excluded q ∞ in this theorem.

The necessity of the remaining condition μ <n/p − λ/p in 5.13 follows from the

fact that|y|−n λ/p ∈ Lp,ψ_{0}Rn_by_{Corollary 2.4}_{, so that this condition is necessary for the}

operator Hα

wwith w rμto be defined on the spaceL p,ψ

{0}Rn.

5.2.2. The Case of the Operator

H

αw

Let Wr : ψr _∞ r wq qα n−1 _∞ Vt t dt q d , 5.17

whereV is the same as in 5.3.

Theorem 5.6. Let 1 ≤ p < ∞, 1 ≤ q < ∞ and

wψ1/p∈ WR1 or w∈ WR1 , w2t ≤ Cwt. 5.18 The operatorHα wis bounded fromL p,ψ {0}Rn toL q,ψ {0}Rn if sup r>0 Wr < ∞, _5.19 and then Hα wf_Lq,ψ {0}≤ C sup_r>0 W 1/q_r_f _Lp,ψ {0}. 5.20

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Proof. From the estimate5.4 we obtain Mq,ψ Hα wf; 0, r ≤ Cψr _∞ r n−1 qα−n/pψ−q/p 1 V _∞ Vt t dt q d fq Lp,ψ {0}, 5.21

from which5.20 follows.

As above, we provide also suﬃcient conditions for the boundedness of the operator Hβ

win terms of the Matuszewska-Orlicz indices.

Corollary 5.7. Let 1 ≤ p ≤ q < ∞ (with q p admitted in the case α 0 and w and ψ satisfy the

conditions5.18. The operator Hα

wis bounded fromL

p,ψ

{0}Rn toLq,ψ{0}Rn if

max{MV, M∞V} < 0, minmψ, m∞ψ> 0, 5.22

and the condition 5.10 holds; the assumption max{MV, M∞V} < 0 is guaranteed by the

conditions mw > −m ψ n p , m∞w > − m_∞ψ n p . 5.23

In the case ψr rλ_{, λ > 0, and w}_{r r}μ_{, the conditions}_5.22_and_5.10_{reduce to}

1≤ p < n λ α , 1 q 1 p − α n λ, μ > n λ p ; 5.24

conditions5.24 are also necessary for the operator Hα_wto be bounded fromLp,ψ_{0}Rn_to_Lq,ψ

{0}Rn.

Proof. We have to find, in terms of the Matuszewska-Orlicz indices, conditions suﬃcient for

the validity of5.19. For the latter we have

Wr ≤ Cψr _∞ r n−1 qα−n/pψ−q/p 1 V _∞ Vt t dt q dρ ≤ Cψr _∞ r n−1 qα−n/pψ−q/p dρ 5.25

by the assumption max{MV, M∞V} < 0. With ψq/pρ ψq/p−1ρψρ and the

condition5.10 we arrive at Wr ≤ Cψr_r∞d / ψ , where the boundedness of the

right-hand side is guaranteed by the condition min{mψ, m∞ψ} > 0.

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Appendix

A. Zygmund-Bary-Stechkin (ZBS) Classes and

Matuszewska-Orlicz (MO) Type Indices

In the sequel, a nonnegative function f on 0, , 0 < ≤ ∞, is called almost increasing almost decreasing if there exists a constant C ≥ 1 such that fx ≤ Cfy for all x ≤ y x ≥ y, resp.. Equivalently, a function f is almost increasing almost decreasing if it is equivalent to an increasingdecreasing, resp. function g, that is, c1fx ≤ gx ≤ c2fx,

c1> 0, c2> 0.

Definition A.1. Let 0 < <∞:

1 by W W0, we denote the class of continuous and positive functions ϕ on 0, such that there exists finite or infinite limit limx→ 0ϕx;

2 by W0 W00, we denote the class of almost increasing functions ϕ ∈ W on

0, ;

3 by W W0, we denote the class of functions ϕ ∈ W such that xa_ϕ_{x ∈ W}

0

for some a aϕ ∈ R1_;

4 by W W0, we denote the class of functions ϕ ∈ W such that ϕt/tb_{is almost}

decreasing for some b∈ R1_.

Definition A.2. Let 0 < <∞:

1 by W∞ W∞, ∞ we denote the class of functions ϕ which are continuous and

positive and almost increasing on, ∞ and which have the finite or infinite limit limx→ ∞ϕx,

2 by W∞ W∞, ∞ we denote the class of functions ϕ ∈ W∞such xaϕx ∈ W∞

for some a aϕ ∈ R1_.

By WR1

we denote the set of functions on R1 whose restrictions onto0, 1 are in

W0, 1 and restrictions onto 1, ∞ are in W∞1, ∞. Similarly, the set WR1 is defined.

A.1. ZBS Classes and MO Indices of Weights at the Origin

In this subsection we assume that <∞.

We say that a function ϕ belongs to a Zygmund classZβ_{, β} _{∈ R}1_{, if ϕ}_{∈ W0, and}

_x

0ϕt/t1 βdt ≤ cϕx/xβ, x ∈ 0, , and to a Zygmund class Zγ, γ ∈ R1, if ϕ∈ W0,

and_xϕt/t1 γdt ≤ cϕx/xγ, x ∈ 0, . We also denote

Φβ γ: Zβ

Zγ, A.1

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For a function ϕ∈ W, the numbers mϕ sup 0<x<1 lnlim sup_h_{→ 0}ϕhx/ϕh ln x limx→ 0 lnlim sup_h_{→ 0}ϕhx/ϕh ln x , Mϕ sup x>1 lnlim sup_h_{→ 0}ϕhx/ϕh ln x limx→ ∞ lnlim sup_h_{→ 0}ϕhx/ϕh ln x A.2

are known as the Matuszewska-Orlicz type lower and upper indices of the function ϕr. The property of functions to be almost increasing or almost decreasing after the multiplication division by a power function is closely connected with these indices. We refer to 22–28 for

such a property and these indices.

Note that in this definition ϕx need not to be an N-function: only its behaviour at the origin is of importance. Observe that 0≤ mϕ ≤ Mϕ ≤ ∞ for ϕ ∈ W0, and−∞ < mϕ ≤

Mϕ ≤ ∞ for ϕ ∈ W, and the following formulas are valid:

mxaϕx a mϕ, Mxaϕx a Mϕ, a∈ R1, A.3 mϕxa amϕ, Mϕxa aMϕ, a≥ 0, A.4 m 1 ϕ −Mϕ, M 1 ϕ −mϕ, A.5

muv ≥ mu mv, Muv ≤ Mu Mv A.6

for ϕ, u, v∈ W.

The proof of the following statement may be found in21, Theorems 3.1, 3.2, and 3.5.

In the formulation of Theorems in 21 it was supposed that β ≥ 0, γ > 0 and ϕ ∈ W0. It is

evidently true also for ϕ∈ W and all β, γ ∈ R1_{, in view of formulas}_A.3_.

Theorem A.3. Let ϕ ∈ W and β, γ ∈ R1_{. Then ϕ} _{∈ Z}β _{⇔ mϕ > β and ϕ ∈ Z}

γ ⇔ Mϕ < γ.

Besides this, mϕ sup{μ > 0 : ϕx/xμ _{is almost increasing}_{}, and Mϕ inf{ν > 0 :}

ϕx/xν _{is almost decreasing}_{}, and for ϕ ∈ Φ}β

γthe inequalities

c1xMϕ ε≤ ϕx ≤ c2xmϕ−ε A.7

hold with an arbitrarily small ε > 0 and c1 c1ε, c2 c2ε.

We define the following subclass in W0:

W0,b ϕ∈ W0: ϕt tb , is almost increasing , b∈ R1. A.8

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A.2. ZBS Classes and MO Indices of Weights at Infinity

Following Section 2.2 in29, we use the following notation.

Let−∞ < α < β < ∞. We put Ψβα : Zβ∩ Zα, where Zβis the class of functions ϕ ∈

W∞ satisfying the conditionx∞x/t

β_{ϕtdt/t ≤ cϕx, x ∈ , ∞, and Z}

αis the class of

functions ϕ ∈ W, ∞ satisfying the condition xx/tαϕtdt/t ≤ cϕx, x ∈ , ∞, where c cϕ > 0 does not depend on x ∈ , ∞.

The indices m∞ϕ and M∞ϕ responsible for the behavior of functions ϕ ∈

Ψβ

α, ∞ at infinity are introduced in the way similar to A.2:

m∞ϕ sup x>1 lnlim infh→ ∞ϕxh/ϕh ln x , M∞ϕ inf x>1 lnlim sup_h_{→ ∞}ϕxh/ϕh ln x . A.9

Properties of functions in the class Ψβα, ∞ are easily derived from those of

functions inΦα_β0, because of the following equivalence

ϕ∈ Ψβα, ∞ ⇐⇒ ϕ∗∈ Φ−β−α0, ∗, A.10

where ϕ∗t ϕ1/t and ∗ 1/. Direct calculation shows that

m∞ϕ −Mϕ∗, M∞ϕ −mϕ∗, ϕ∗t : ϕ 1 t . A.11

ByA.10 and A.11, one can easily reformulate properties of functions of the class Φβγ

near the origin, given inTheorem A.3for the case of the corresponding behavior at infinity of functions of the classΨβαand obtain that

c1tm∞ϕ−ε≤ ϕt ≤ c2tM∞ϕ ε, t≥ , ϕ ∈ W∞, m∞ϕ sup μ∈ R1: t−μϕt is almost increasing on , ∞, M∞ϕ inf ν∈ R1 : t−νϕt is almost decreasing on , ∞. A.12

We say that a continuous function ϕ in0, ∞ is in the class W0,∞R1 if its restriction

to0, 1 belongs to W0, 1 and its restriction to 1, ∞ belongs to W_∞1, ∞. For functions in W0,∞R1 the notation Zβ0,β∞R1 Zβ00, 1 ∩ Zβ∞1, ∞, Z γ0,γ∞ R1 Zγ00, 1 ∩ Zγ∞1, ∞ A.13

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has an obvious meaning note that in A.13 we use Zβ∞1, ∞ and Z_γ

∞1, ∞, not

Zβ∞1, ∞ and Z_γ

∞1, ∞. In the case where the indices coincide, that is, β0 β∞ : β,

we will simply writeZβR1 and similarly for ZγR1 . We also denote

Φβ γ R1 : Zβ_R1 ∩ Zγ R1 . A.14

Making use ofTheorem A.3forΦα

β0, 1 and relations A.11, one easily arrives at

the following statement.

Lemma A.4. Let ϕ ∈ WR1

. Then ϕ∈ Zβ0,β∞R1 ⇐⇒ mϕ> β0, m∞ϕ> β∞, A.15 ϕ∈ Zγ0,γ∞ R1 ⇐⇒ Mϕ< γ0, M∞ϕ< γ∞. A.16

Acknowledgment

The third author thanks Lule˚a University of Technology for financial support for research visits in June of 2012.

References

1 D. E. Edmunds, V. Kokilashvili, and A. Meskhi, Bounded and Compact Integral Operators, vol. 543, Kluwer Academic, Dordrecht, The Netherlands, 2002.

2 V. Kokilashvili, A. Meskhi, and L.-E. Persson, Weighted Norm Inequalities for Integral Transforms with

Product Kernels, Nova Science, New York, NY, USA, 2010.

3 A. Kufner, L. Maligranda, and L.-E. Persson, The Hardy Inequality—About Its History and Some Related

Results, Pilsen, Czech, 2007.

4 A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type, World Scientific, River Edge, NJ, USA, 2003.

5 D. Lukkassen, A. Medell, L.-E. Persson, and N. Samko, “Hardy and singular operators in weighted generalized Morrey spaces with applications to singular integral equations,” Mathematical Methods in

the Applied Sciences, vol. 35, no. 11, pp. 1300–1311, 2012.

6 H. Triebel, “Entropy and approximation numbers of limiting embeddings—an approach via Hardy inequalities and quadratic forms,” Journal of Approximation Theory, vol. 164, no. 1, pp. 31–46, 2012. 7 L.-E. Persson and N. Samko, “Weighted Hardy and potential operators in the generalized Morrey

spaces,” Journal of Mathematical Analysis and Applications, vol. 377, no. 2, pp. 792–806, 2011.

8 N. Samko, “Weighted Hardy and singular operators in Morrey spaces,” Journal of Mathematical

Analysis and Applications, vol. 350, no. 1, pp. 56–72, 2009.

9 N. Samko, “Weighted Hardy and potential operators in Morrey spaces,” Journal of Function Spaces and

Applications, vol. 2012, Article ID 678171, 21 pages, 2012.

10 M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, NJ, USA, 1983.

11 A. Kufner, O. John, and S. Fuˇc´ık, Function Spaces, Noordhoff International, Leyden, Mass, USA, 1977. 12 M. E. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, vol. 81 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2000.

13 H. Rafeiro, N. Samko, and S. Samko, “Morrey-Campanato spaces: an overview,” in Operator Theory,

(19)

14 G. T. Dzhumakaeva and K. Zh. Nauryzbaev, “Lebesgue-Morrey spaces,” Izvestiya Akademii Nauk

Kazakhsko˘ı SSR. Seriya Fiziko-Matematicheskaya, no. 5, pp. 7–12, 1982 Russian.

15 C. T. Zorko, “Morrey space,” Proceedings of the American Mathematical Society, vol. 98, no. 4, pp. 586– 592, 1986.

16 V. Guliyev, Integral operators on function spaces on homogeneous groups and on domains inRn_{[Ph.D. thesis],} Steklov Mathematical Institute, Moscow, Russia, 1994.

17 V. Guliyev, Function Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups, Some applications, Baku, Azerbaijan, 1999.

18 V. I. Burenkov, H. V. Guliyev, and V. S. Guliyev, “On boundedness of the fractional maximal operator from complementary Morrey-type spaces to Morrey-type spaces,” in The Interaction of Analysis

and Geometry, vol. 424 of Contemporary Mathematics, pp. 17–32, American Mathematical Society,

Providence, RI, USA, 2007.

19 V. Guliyev, J. Hasanov, and S. Samko, “Maximal, potential and singular operators in the local “complementary” variable exponent Morrey type spaces,” Journal of Mathematical Analysis and

Applications. In press.

20 N. K. Bari and S. B. Steˇckin, “Best approximations and diﬀerential properties of two conjugate functions,” Proceedings of the Moscow Mathematical Society, vol. 5, pp. 483–522, 1956Russian. 21 N. K. Karapetyants and N. Samko, “Weighted theorems on fractional integrals in the generalized

H ¨older spaces Hω

0ρ via indices mωand Mω,” Fractional Calculus & Applied Analysis, vol. 7, no. 4, pp. 437–458, 2004.

22 S. G. Kre˘ın, Ju. I. Petunin, and E. M. Sem¨enov, Interpolation of Linear Operators, Nauka, Moscow, Russia, 1978.

23 L. Maligranda, “Indices and interpolation,” Dissertationes Mathematicae, vol. 234, p. 49, 1985.

24 W. Matuszewska and W. Orlicz, “On some classes of functions with regard to their orders of growth,”

Studia Mathematica, vol. 26, pp. 11–24, 1965.

25 L.-E. Persson, N. Samko, and P. Wall, “Quasi-monotone weight functions and their characteristics and applications,” Mathematical Inequalities and Applications, vol. 12, no. 3, pp. 685–705, 2012.

26 N. Samko, “Singular integral operators in weighted spaces with generalized H¨older condition,”

Proceedings of A. Razmadze Mathematical Institute, vol. 120, pp. 107–134, 1999.

27 N. Samko, “On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class,”

Real Analysis Exchange, vol. 30, no. 2, pp. 727–745, 2004/2005.

28 L. Maligranda, Orlicz Spaces and Interpolation, Departamento de Matem´atica, Universidade Estadual de Campinas, Campinas, Brazil, 1989.

29 N. G. Samko, S. G. Samko, and B. G. Vakulov, “Weighted Sobolev theorem in Lebesgue spaces with variable exponent,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 560–583, 2007.

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