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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 678171, 21 pages
http://dx.doi.org/10.1155/2012/678171
Research Article

Weighted Hardy and Potential Operators in Morrey Spaces

Department of Mathematics, Research Center CEAF, Instituto Superior Técnico, 1049-003 Lisbon, Portugal

Received 15 November 2009; Accepted 9 June 2010

Academic Editor: V. Stepanov

Copyright © 2012 Natasha Samko. 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.

Abstract

We study the weighted 𝑝𝑞-boundedness of Hardy-type operators in Morrey spaces 𝑝,𝜆(𝑛) (or 𝑝,𝜆(1+) in the one-dimensional case) for a class of almost monotonic weights. The obtained results are applied to a similar weighted 𝑝𝑞-boundedness of the Riesz potential operator. The conditions on weights, both for the Hardy and potential operators, are necessary and sufficient in the case of power weights. In the case of more general weights, we provide separately necessary and sufficient conditions in terms of Matuszewska-Orlicz indices of weights.

1. Introduction

The well-known Morrey spaces 𝑝,𝜆 introduced in [1] in relation to the study of partial differential equations and presented in various books, see [24], were widely investigated during the last decades, including the study of classical operators of harmonic analysis—maximal, singular, and potential operators—in these spaces; we refer for instance to papers [523], where Morrey spaces on metric measure spaces may be also found. Surprisingly, weighted estimates of these classical operators, in fact, were not studied. Just recently, in [24] we proved weighted 𝑝𝑝-estimates in Morrey spaces for Hardy operators on 1+ and one-dimensional singular operators (on 1 or on Carleson curves in the complex plane).

In this paper we develop an approach which allows us both to obtain weighted 𝑝𝑞estimations of Hardy-type operators and potential operators and extend them to the multidimensional case, for the Hardy operators (related to integration over balls). Note that, in contrast with the case of Lebesgue spaces, Hardy inequalities in Morrey norms admit the value 𝑝=1 for 𝑝; see Theorem 4.3. The progress in comparison with [24] is based on the pointwise estimation of the Hardy operators we present in Sections 3.2 and 4.1. Such an estimation is helpless in the case of Lebesgue spaces (𝜆=0), but proved to be effective in the case of Morrey spaces (0<𝜆<𝑛). Roughly speaking, it is based on the simple fact that|𝑥|𝑛/𝑝𝐿𝑝(𝜆=0),but|𝑥|(𝜆𝑛)/𝑝𝐿𝑝,𝜆(𝜆>0).(1.1)

The admitted weights 𝜑(|𝑥𝑥0|) are generated by functions 𝜑(𝑟) from the Bary-Stechkin-type class; they may be characterized as weights continuous and positive for 𝑟(0,), with possible decay or growth at 𝑟=0 and 𝑟=, which become almost increasing or almost decreasing after the multiplication by some power. Such weights are oscillating between two powers at the origin and infinity (with different exponents for the origin and infinity).

We also note that the obtained estimates show that the Hardy operators (with admitted weights) act boundedly not only in local and global Morrey spaces (see definitions in Section 3.1), but also from a larger local Morrey space into a more narrow global Morrey space (see Theorems 4.3 and 4.4).

The paper is organized as follows. In Section 2 we give necessary preliminaries on some classes of weight functions. In Section 3 we prove some statements on embedding of Morrey spaces 𝑝,𝜆(Ω) into some weighted 𝐿𝑝(Ω,𝜚)-spaces. In Section 4 we prove theorems on the weighted 𝑝𝑞-boundedness of Hardy operators in Morrey spaces. Finally, in Section 5 we apply the results of Section 4 to a similar weighted boundedness of potential operators. The conditions on weights, both for the Hardy and potential operators are necessary and sufficient in the case of power weights. In the case of more general weights, we provide separately necessary and sufficient conditions in terms of Matuszewska-Orlicz indices of weights.

The main results are given in Theorems 4.3, 4.4, 4.5, and 5.3 and Corollary 5.4.

2. Preliminaries on Weight Functions

2.1. Zygmund-Bary-Stechkin (ZBS) Classes and Matuszewska-Orlicz (MO) Type Indices
2.1.1. On Classes of the Type 𝑊0and𝑊

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 𝑓(𝑥)𝐶𝑓(𝑦) for all 𝑥𝑦 (𝑥𝑦, resp.). Equivalently, a function 𝑓 is almost increasing (almost decreasing) if it is equivalent to an increasing (decreasing, resp.) function 𝑔, that is, 𝑐1𝑓(𝑥)𝑔(𝑥)𝑐2𝑓(𝑥),𝑐1>0,𝑐2>0.

Definition 2.1. Let 0<<.(1)By 𝑊0=𝑊0([0,]) one denotes the class of functions 𝜑 continuous and positive on (0,] such that there exists the finite limit lim𝑥0𝜑(𝑥), and 𝜑(𝑥) is almost increasing on(0,);(2)by 𝑊0=𝑊0([0,]) one denotes the class of functions 𝜑 on [0,] such that 𝑥𝑎𝜑(𝑥)𝑊0 for some 𝑎=𝑎(𝜑)1.

Definition 2.2. Let 0<<.(1)By 𝑊=𝑊([,]) one denotes the class of functions 𝜑 continuous and positive on [,) which have the finite limit lim𝑥𝜑(𝑥), and 𝜑(𝑥) is almost increasing on [,);(2)by 𝑊=𝑊([,)) one denotes the class of functions 𝜑𝑊 such 𝑥𝑎𝜑(𝑥)𝑊 for some 𝑎=𝑎(𝜑)1.

2.1.2. ZBS Classes and MO Indices of Weights at the Origin

In this subsection we assume that <.

Definition 2.3. One says that a function 𝜑𝑊0 belongs to the Zygmund class 𝛽, 𝛽1, if 𝑥0𝜑(𝑡)𝑡1+𝛽𝜑𝑑𝑡𝑐(𝑥)𝑥𝛽,𝑥(0,),(2.1) and to the Zygmund class 𝛾, 𝛾1, if 𝑥𝜑(𝑡)𝑡1+𝛾𝑑𝑡𝑐𝜑(𝑥)𝑥𝛾,𝑥(0,).(2.2) One also denotes Φ𝛽𝛾=𝛽𝛾,(2.3) the latter class being also known as Bary-Stechkin-Zygmund class [25].

It is known that the property of a function is to be almost increasing or almost decreasing after the multiplication (division) by a power function is closely related to the notion of the so called Matuszewska-Orlicz indices. We refer to [26, 27] [28, page 20], [2932] for the properties of the indices of such a type. For a function 𝑊𝜑0, the numbers𝑚(𝜑)=sup0<𝑥<1lnlimsup0(𝜑(𝑥)/𝜑())ln𝑥=lim𝑥0lnlimsup0(𝜑(𝑥)/𝜑()),ln𝑥𝑀(𝜑)=sup𝑥>1lnlimsup0(𝜑(𝑥)/𝜑())ln𝑥=lim𝑥lnlimsup0(𝜑(𝑥)/𝜑()).ln𝑥(2.4) are known as the Matuszewska-Orlicz type lower and upper indices of the function 𝜑(𝑟). Note that in this definition 𝜑(𝑥)needs not to be an 𝑁-function: only its behaviour at the origin is of importance. Observe that 0𝑚(𝜑)𝑀(𝜑)for𝜑𝑊0 and 𝑊<𝑚(𝜑)𝑀(𝜑)for𝜑0, and the formulas are valid𝑚𝑥𝑎𝑥𝜑(𝑥)=𝑎+𝑚(𝜑),𝑀𝑎𝜑(𝑥)=𝑎+𝑀(𝜑),𝑎1,[](2.5)𝑚(𝜑(𝑥)𝑎[])=𝑎𝑚(𝜑),𝑀(𝜑(𝑥)𝑎𝑚1)=𝑎𝑀(𝜑),𝑎0,(2.6)𝜑1=𝑀(𝜑),𝑀𝜑=𝑚(𝜑),(2.7)𝑚(𝑢𝑣)𝑚(𝑢)+𝑚(𝑣),𝑀(𝑢𝑣)𝑀(𝑢)+𝑀(𝑣)(2.8) for 𝑊𝜑,𝑢,𝑣0.

The following statement is known see [26, Theorems 3.1, 3.2, and 3.5]. (In the formulation of Theorem  2.4 in [26], it was supposed that 𝛽0,𝛾>0, and 𝜑𝑊0. It is evidently true also for 𝑊𝜑0 and all 𝛽,𝛾1, in view of formulas (2.5).)

Theorem 2.4. Let 𝑊𝜑0 and 𝛽,𝛾1. Then 𝜑𝛽𝑚(𝜑)>𝛽,𝜑𝛾𝑀(𝜑)<𝛾.(2.9) Besides this, 𝑚(𝜑)=sup𝛿>0𝜑(𝑥)𝑥𝛿isalmostincreasing,(2.10)𝑀(𝜑)=inf𝜆>0𝜑(𝑥)𝑥𝜆isalmostdecreasing,(2.11) and for 𝜑Φ𝛽𝛾 the inequalities 𝑐1𝑥𝑀(𝜑)+𝜀𝜑(𝑥)𝑐2𝑥𝑚(𝜑)𝜀(2.12) hold with an arbitrarily small 𝜀>0 and 𝑐1=𝑐1(𝜀),𝑐2=𝑐2(𝜀).

2.1.3. ZBS Classes and MO Indices of Weights at Infinity

Following [14, Section 4.1] and [29, Section 2.2], we introduce the following definitions.

Definition 2.5. Let <𝛼<𝛽<. One puts Ψ𝛽𝛼=𝛽𝛼, where 𝛽 is the class of functions 𝑊𝜑 satisfying the condition 𝑥𝑥𝑡𝛽𝜑(𝑡)𝑑𝑡𝑡𝑐𝜑(𝑥),𝑥(,),(2.13) and 𝛼 is the class of functions 𝜑𝑊([,)) satisfying the condition 𝑥𝑥𝑡𝛼𝜑(𝑡)𝑑𝑡𝑡𝑐𝜑(𝑥),𝑥(,),(2.14) where 𝑐=𝑐(𝜑)>0 does not depend on 𝑥[,).

The indices 𝑚(𝜑) and 𝑀(𝜑) responsible for the behavior of functions 𝜑Ψ𝛽𝛼([,)) at infinity are introduced in the way similar to (2.4):𝑚(𝜑)=sup𝑥>1lnliminf(𝜑(𝑥)/𝜑()),𝑀ln𝑥(𝜑)=inf𝑥>1lnlimsup(𝜑(𝑥)/𝜑()).ln𝑥(2.15)

Properties of functions in the class Ψ𝛽𝛼([,)) are easily derived from those of functions in Φ𝛼𝛽([0,]) because of the following equivalence:𝜑Ψ𝛽𝛼([,))𝜑Φ𝛽𝛼0,,(2.16) where 𝜑(𝑡)=𝜑(1/𝑡) and =1/. Direct calculation shows that𝑚𝜑(𝜑)=𝑀,𝑀𝜑(𝜑)=𝑚,𝜑1(𝑡)=𝜑𝑡.(2.17)

Making use of (2.16) and (2.17), one can easily reformulate properties of functions of the class Φ𝛽𝛼 near the origin, given in Theorem 2.4 for the case of the corresponding behavior at infinity of functions of the class Ψ𝛽𝛼 and obtain that𝑐1𝑡𝑚(𝜑)𝜀𝜑(𝑡)𝑐2𝑡𝑀(𝜑)+𝜀𝑊,𝑡,𝜑,𝑚(2.18)(𝜑)=sup𝜇1𝑡𝜇𝜑[)𝑀(𝑡)isalmostincreasingon,,(2.19)(𝜑)=inf𝜈1𝑡𝜈[.𝜑(𝑡)isalmostdecreasingon,)(2.20)

We say that a function 𝜑 continuous in (0,) is in the class 𝑊0,(1+) if its restriction to (0,1) belongs to 𝑊0([0,1]) and its restriction to (1,) belongs to 𝑊([1,]). For functions in 𝑊0,(1+), the notation𝛽0,𝛽1+=𝛽0([]0,1)𝛽([1,)),𝛾0,𝛾1+=𝛾0([]0,1)𝛾([1,))(2.21) has an obvious meaning. In the case, where the indices coincide: 𝛽0=𝛽=𝛽, we will simply write 𝛽(1+) and similarly for 𝛾(1+). We also denoteΦ𝛽𝛼1+=𝛽1+𝛾1+.(2.22)

Making use of Theorem 2.4 for Φ𝛽𝛼([0,1]) and relations (2.17), we easily arrive at the following statement.

Lemma 2.6. Let 𝑊𝜑0(1+). Then 𝜑𝛽0,𝛽1+𝑚(𝜑)>𝛽0,𝑚(𝜑)>𝛽,𝜑𝛾0,𝛾1+𝑀(𝜑)<𝛾0,𝑀(𝜑)<𝛾.(2.23)

2.2. On Classes 𝑉𝜇±

Note that we slightly changed the notation of the class introduced in the following definition, in comparison with its notation in [32].

Definition 2.7. Let 0<𝜇1. By 𝐕𝜇±, One denotes the classes of functions 𝜑 nonnegative on [0,],0<, defined by the following conditions: 𝐕𝝁+||||𝜑(𝑥)𝜑(𝑦)||||𝑥𝑦𝜇𝜑𝑥𝐶+𝑥𝜇+,𝐕(2.24)𝝁||||𝜑(𝑥)𝜑(𝑦)||||𝑥𝑦𝜇𝜑𝑥𝐶𝑥𝜇+,(2.25) where 𝑥,𝑦(0,] and 𝑥+=max(𝑥,𝑦),𝑥=min(𝑥,𝑦).

Lemma 2.8. Functions 𝜑𝐕𝝁+ are almost increasing on [0,], and functions 𝜑𝐕𝝁 are almost decreasing on [0,].

Proof. Let 𝜑𝐕𝜇+ and 𝑦𝑥. By (2.24) we have |𝜑(𝑥)𝜑(𝑦)|𝐶𝜑(𝑥)(1𝑦/𝑥)𝜇𝐶𝜑(𝑥). Then 𝜑(𝑦)|𝜑(𝑥)𝜑(𝑦)|+𝜑(𝑥)(𝐶+1)𝜑(𝑥). The case 𝜑𝐕𝜇 is similarly treated.

Corollary 2.9. Functions 𝜑𝐕𝜇+ have non-negative indices 0𝑚(𝜑)𝑀(𝜑), and functions 𝜑𝐕𝜇 have non-positive indices 𝑚(𝜑)𝑀(𝜑)0, the same being also valid with respect to the indices 𝑚(𝜑),𝑀(𝜑) in the case =.

Note that𝑉𝜇+𝑉𝜈+,𝑉𝜇𝑉𝜈,0<𝜈<𝜇1,(2.26) the classes 𝑉𝜇± being trivial for 𝜇>1. We also have𝑥𝛾𝜇[0,1]𝐕𝜇+𝛾0,𝑥𝛾𝜇[0,1]𝐕𝜇𝛾0,(2.27) which follows from the fact that 𝑥𝛾𝐕1+𝛾0 and 𝑥𝛾𝐕1𝛾0 (see [24, Subsection 2.3, Remark 2.8]) and property (2.26).

An example of a function which is in 𝐕𝜇+ with some 𝜇>0, but does not belong to the total intersection 𝜇[0,1]𝐕𝜇+ is given by𝜑(𝑥)=𝑎𝑥𝛾||+𝑏𝑥𝑥0||𝛽𝜇[0,𝛽]𝐕𝜇+,(2.28) where 𝑥0>0 and 𝛾0,0<𝛽<1,𝑎>0,𝑏>0.

The following lemmas (see [24], Lemmas 2.10 and 2.11) show that conditions (2.24) and (2.25) are fulfilled with 𝜇=1 not only for power functions but also for an essentially larger class of functions (which in particular may oscillate between two power functions with different exponents). Note that the information about this class in Lemmas 2.10 and 2.11 is given in terms of increasing or decreasing functions, without the word “almost”.

Lemma 2.10. Let 𝜑𝑊. Then(i)𝜑𝐕1+ in the case 𝜑 is increasing and the function 𝜑(𝑥)/𝑥𝜈 is decreasing for some 𝜈0;(ii)𝜑𝐕1 in the case 𝜑(𝑥) is decreasing and there exists a number 𝜇0 such that 𝑥𝜇𝜑(𝑥) is increasing.

Lemma 2.11. Let 𝜑𝑊𝐶1((0,]). If there exist 𝜀>0 and 𝜈0 such that 0𝜑(𝑥)/𝜑(𝑥)𝜈/𝑥 for 0<𝑥𝜀, then 𝜑𝐕1+. If there exist 𝜀>0 and 𝜇0 such that 𝜇/𝑥𝜑(𝑥)/𝜑(𝑥)0 for 0<𝑥𝜀, then 𝜑𝐕1.

3. On Weighted Integrability of Functions in Morrey Spaces

3.1. Definitions and Belongness of Some Functions to Morrey Spaces

Let Ω be an open set in 𝑛.

Definition 3.1. The Morrey spaces 𝑝,𝜆(Ω),𝑝,𝜆loc(Ω)1𝑝<,0𝜆<𝑛, are defined as the space of functions 𝑓𝐿𝑝loc(Ω) such that 𝑓𝑝,𝜆=sup𝑥Ω,𝑟>01𝑟𝜆𝐵(𝑥,𝑟)|𝑓(𝑦)|𝑝𝑑𝑦1/𝑝,𝑓𝑝,𝜆;loc=sup𝑟>01𝑟𝜆𝐵(0,𝑟)|𝑓(𝑦)|𝑝𝑑𝑦1/𝑝,(3.1) respectively, where 𝐵(𝑥,𝑟)=𝐵(𝑥,𝑟)Ω.

Obviously,𝑝,𝜆(Ω)0𝑝,𝜆loc(Ω).(3.2)

The spaces 𝑝,𝜆(Ω),𝑝,𝜆loc(Ω) are known under the names of global and local Morrey spaces; see for instance, [9, 10].

The weighted Morrey space is defined as𝐿𝑝,𝛾(Ω,𝜔)={𝑓𝜔𝑓𝐿𝑝,𝛾(Ω)}.(3.3)

Remark 3.2. As is well known, the space 𝑝,𝜆(Ω) as defined above is not necessarily embedded into 𝐿𝑝(Ω), in the case when Ω is unbounded. A typical counterexample in the case Ω=𝑛 is 𝑓(𝑥)=|𝑥|(𝜆𝑛)/𝑝𝑝,𝜆(𝑛).(3.4) Indeed, we have 𝑓𝑝,𝜆=sup𝑥,𝑟1𝑟𝜆𝐵(𝑥,𝑟)|𝑦|𝜆𝑛𝑑𝑦1/𝑝(3.5) which is bounded (when |𝑥|2𝑟, take into account that |𝑦|𝑟, and when |𝑥|2𝑟, make use of the inclusion 𝐵(𝑥,𝑟)𝐵(0,3𝑟)).

Lemma 3.3. Let =diamΩ<, 𝑊𝑢0(0,), and 𝑥0Ω. The condition 𝑚(𝑢)>𝜆𝑛𝑝(3.6) is sufficient for the function 𝑓(𝑥)=𝑢(|𝑥𝑥0|) to belong to 𝑝,𝜆(Ω),0𝜆<𝑛. In the case 𝑢(𝑡)=𝑡𝛾, the inclusion |𝑥𝑥0|𝛾𝑝,𝜆(Ω) with 0<𝜆<𝑛 holds if 𝛾(𝜆𝑛)/𝑝, the latter condition being necessary, when 𝑥0 is an inner point of Ω or 𝑛=1 and Ω=(𝑎,𝑏),𝑎<𝑏.

Proof. We have 𝑓𝑝,𝜆=sup𝑥Ω,𝑟>01𝑟𝜆𝐵(𝑥,𝑟)𝑢𝑝||𝑦𝑥0||𝑑𝑦1/𝑝sup𝑥Ω𝑥0,𝑟>01𝑟𝜆𝐵(𝑥,𝑟)𝑢𝑝||𝑦||𝑑𝑦1/𝑝,(3.7) where Ω𝑥0={𝑥𝑥+𝑥0Ω}. Then 𝑓𝑝,𝜆𝐶sup𝑥Ω𝑥0,𝑟>01𝑟𝜆𝐵(𝑥,𝑟)||𝑦||𝑝𝑚(𝑢)𝑝𝜀𝑑𝑦1/𝑝,(3.8) by (2.12). If 𝑚(𝑢)>0, we choose 0<𝜀<𝑚(𝑢) and then the right-hand side of the last inequality is bounded. So let 𝑚(𝑢)0. We distinguish the cases (1) |𝑥|2𝑟 and (2) |𝑥|2𝑟. In the case (1), |𝑦||𝑥||𝑥𝑦|𝑟. Therefore, 𝑓𝑝,𝜆𝐶sup𝑥Ω,𝑟>0𝑟𝑝𝑚(𝑢)𝑝𝜀𝑟𝜆𝐵(𝑥,𝑟)𝑑𝑦1/𝑝=𝐶sup𝑟𝑟>0𝑚(𝑢)+(𝑛𝜆)/𝑝𝜀(3.9) which is bounded under the choice 𝜀<𝑚(𝑢)+(𝑛𝜆)/𝑝. In the case (2), we observe that 𝐵(𝑥,𝑟)𝐵(0,3𝑟) and then the same estimate 𝑓𝑝,𝜆𝐶sup𝑟>0𝑟𝑚(𝑢)+(𝑛𝜆)/𝑝𝜀 follows.
In the case 𝑢(𝑡)=𝑡𝛾, the proof of the “if” part follows the same lines as above with 𝜀=0. To prove the “only if” part, it suffices to observe that 𝑓𝑝,𝜆sup0<𝑟<𝛿1𝑟𝜆𝐵(𝑥0,𝑟)|𝑦𝑥0|𝑝𝛾𝑑𝑦1/𝑝𝐶sup0<𝑟<𝛿𝑟𝛾+(𝑛𝜆)/𝑝.(3.10)

Corollary 3.4. If 𝑊𝑢0(0,) and there exists an 𝑎<(𝑛𝜆)/𝑝 such that 𝑡𝑎𝑢(𝑡) is almost increasing, then 𝑢(|𝑥𝑥0|)𝑝,𝜆,𝑥0Ω.
To derive this corollary from Lemma 3.3, it suffices to refer to formula (2.10).

3.2. Some Weighted Estimates of Functions in Morrey Spaces

Lemma 3.5. Let 1𝑝<,0<𝑠𝑝,0𝜆<𝑛, and 𝑊𝑣0([0,]),0<. Then ||||𝑧|<|𝑦||||𝑓(𝑧)𝑠𝑣(|𝑧|)𝑑𝑧1/𝑠||𝑦||𝑐𝒜𝑓𝑝,𝜆;loc||𝑦||,0<,(3.11) where 𝐶>0 does not depend on 𝑦 and 𝑓 and 𝒜(𝑟)=𝑟0𝑡𝑛1((𝑛𝜆)/𝑝)𝑠𝑑𝑡𝑣(𝑡)1/𝑠(3.12) under the assumption that the last integral converges.

Proof. We have |𝑧|<|𝑦|||||𝑓(𝑧)𝑠𝑣(|𝑧|)𝑑𝑧=𝑘=0𝐵𝑘(𝑦)||||𝑓(𝑧)𝑠𝑣(|𝑧|)𝑑𝑧,(3.13) where 𝐵𝑘(𝑦)={𝑧2𝑘1|𝑦|<|𝑧|<2𝑘|𝑦|}. Making use of the fact that there exists a 𝛽 such that 𝑡𝛽𝑣(𝑡) is almost increasing, we observe that 1𝑣𝐶(|𝑧|)𝑣2𝑘1||𝑦||.(3.14) Applying this in (3.13) and making use of the Hölder inequality with the exponent 𝑝/𝑠1, we obtain |𝑧|<|𝑦|||𝑓||(𝑧)𝑠𝑣(|𝑧|)𝑑𝑧𝐶𝑘=02𝑘1||𝑦||𝑛(1s/𝑝)𝑣2𝑘1||𝑦||𝐵𝑘(𝑦)|𝑓(𝑧)|𝑝𝑑𝑧s/𝑝.(3.15) Hence, |𝑧|<|𝑦|||𝑓||(𝑧)𝑠𝑣(|z|)𝑑𝑧𝐶𝑘=02𝑘1||𝑦||𝑛(𝑛𝜆)𝑠/𝑝𝑣2𝑘1||𝑦||𝑓𝑠𝑝,𝜆;loc.(3.16) It remains to prove that 𝑘=02𝑘1||𝑦||𝑛(𝑛𝜆)𝑠/𝑝𝑣2𝑘1||𝑦||𝒜||𝑦||𝐶𝑠.(3.17) We have 0|𝑦|𝑡𝑛1((𝑛𝜆)/𝑝)𝑠𝑑𝑡=𝑣(𝑡)𝑘=02𝑘2|𝑦|𝑘1|𝑦|𝑡𝑛1((𝑛𝜆)/𝑝)𝑠𝑑𝑡.𝑣(𝑡)(3.18) Making use of the fact that 𝑡𝛽𝑣(𝑡) is almost increasing with some 𝛽, we easily obtain that 0|𝑦|𝑡𝑛1((𝑛𝜆)/𝑝)𝑠𝑑𝑡𝑣(𝑡)𝐶𝑘=02𝑘||𝑦||𝑛((𝑛𝜆)/𝑝)𝑠𝑣2𝑘||𝑦||𝐶𝑘=02𝑘1||𝑦||𝑛((𝑛𝜆)/𝑝)𝑠𝑣2𝑘1||𝑦||,(3.19) which proves (3.17).

Corollary 3.6. Let 1𝑝<,0<𝑠𝑝,0𝜆<𝑛, and 𝑎<𝑛/𝑠(𝑛𝜆)/𝑝. Then |𝑡|<|𝑦||𝑓(𝑡)||𝑡|𝑎𝑠𝑑𝑡1/𝑠||𝑦||𝑐𝑛/𝑠(𝑛𝜆)/𝑝𝑎𝑓𝑝,𝜆;loc||𝑦||,0<.(3.20)

Lemma 3.7. Let 1𝑝<,0𝑠𝑝,0𝜆<𝑛, and 𝑊𝑣0(1+). Then ||𝑦|||𝑧|>||||𝑣(|𝑧|)𝑓(𝑧)𝑠𝑑𝑧1/𝑠||𝑦||𝑐𝑓𝑝,𝜆;loc,𝑦0,(3.21) where 𝐶>0 does not depend on 𝑦 and 𝑓 and (𝑦)=||𝑦||𝑡𝑛1((𝑛𝜆)/𝑝)𝑠𝑣(𝑡)𝑑𝑡1/𝑠.(3.22)

Proof. The proof is similar to that of Lemma 3.5. We have ||𝑦|||𝑧|>||||𝑣(𝑧)𝑓(𝑧)𝑠𝑑𝑧=𝑘=0𝐵𝑘(𝑦)||||𝑣(𝑧)𝑓(𝑧)𝑠𝑑𝑧,(3.23) where 𝐵𝑘(𝑦)={𝑧2𝑘|𝑦|<|𝑧|<2𝑘+1|𝑦|}. Since there exists a 𝛽1 such that 𝑡𝛽𝑣(𝑡) is almost increasing, we obtain 𝐵𝑘(𝑦)||||𝑣(|𝑧|)𝑓(𝑧)𝑠𝑑𝑧𝐶𝑘=0𝑣2𝑘+1||𝑦||𝐵𝑘(𝑦)||||𝑓(𝑧)𝑠𝑑𝑧,(3.24) where 𝐶 may depend on 𝛽, but does not depend on 𝑦 and 𝑓. Applying the Hölder inequality with the exponent 𝑝/𝑠, we get ||𝑦|||𝑧|>||||𝑣(|𝑧|)𝑓(𝑧)𝑠𝑑𝑧𝐶𝑘=0𝑣2𝑘+1||𝑦||2𝑘||𝑦||𝑛(1𝑠/𝑝)𝐵𝑘(𝑦)||||𝑓(𝑧)𝑝𝑑𝑧𝑠/𝑝𝐶𝑘=0𝑣2𝑘+1||𝑦||2𝑘||𝑦||𝑛((𝑛𝜆)/𝑝)𝑠𝑓𝑝,𝜆;loc.(3.25) It remains to prove that 𝑘=0𝑣2𝑘+1||𝑦||2𝑘|𝑦|𝑛((𝑛𝜆)/𝑝)𝑠𝐶|𝑦|𝑡𝑛1((𝑛𝜆)/𝑝)𝑠𝑣(𝑡)𝑑𝑡.(3.26) We have ||𝑦||𝑡𝑛1((𝑛𝜆)/𝑝)𝑠𝑣(𝑡)𝑑𝑡=𝑘=02𝑘+12|𝑦|𝑘||𝑦||𝑡𝑛1((𝑛𝜆)/𝑝)𝑠𝑣(𝑡)𝑑𝑡𝐶𝑘=0𝑣2𝑘||𝑦||2𝑘+12|𝑦|𝑘||𝑦||𝑡𝑛1((𝑛𝜆)/𝑝)𝑠𝑑𝑡=𝐶𝑘=0𝑣2𝑘||𝑦||2𝑘||𝑦||𝑛((𝑛𝜆)/𝑝)𝑠𝐶𝑘=0𝑣2𝑘+1||𝑦||2𝑘||𝑦||𝑛((𝑛𝜆)/𝑝)𝑠,(3.27) which completes the proof.

Remark 3.8. The analysis of the proof shows that estimate (3.21) remains in force, if the assumption 𝑊𝑣0(1+) is replaced by the condition that 𝑊1/𝑣0(1+) and 𝑣 satisfies the doubling condition 𝑣(2𝑡)𝑐𝑣(𝑡).

Corollary 3.9. Let 1𝑝<,0<𝑠𝑝, and 𝑏<(𝑛𝜆)/𝑝𝑛/𝑠. Then |𝑧|>|𝑦||𝑧|𝑏|𝑓(𝑧)|𝑠𝑑𝑧1/𝑠𝑐𝑦𝑏+𝑛/𝑠(𝑛𝜆)/𝑝𝑓𝑝,𝜆;loc,𝑦0.(3.28)

4. On Weighted Hardy Operators in Morrey Spaces

4.1. Pointwise Estimations

We consider the generalized Hardy operators𝐻𝛼𝜑𝑓(𝑥)=|𝑥|𝛼𝑛𝜑(|𝑥|)||𝑦||<|𝑥|𝑓(𝑡)𝑑𝑡𝜑(|𝑡|),𝛼𝜑𝑓(𝑥)=|𝑥|𝛼𝜑(|𝑥|)𝑛𝑓(𝑡)𝑑𝑡|𝑡|𝑛.𝜑(|𝑡|)(4.1)

In the sequel 𝑛 with 𝑛=1 may be read either as 1 or 1+ with the operators interpreted as𝐻𝛼𝜑𝑓(𝑥)=𝑥𝛼1𝜑(𝑥)𝑥0𝑓(𝑦)𝜑(𝑦)𝑑𝑦,𝛼𝜑𝑓(𝑥)=𝑥𝛼𝜑(𝑥)𝑥𝑓(𝑦)𝜑(𝑦)𝑦𝑑𝑦,𝑥>0.(4.2)

In the case 𝜑(𝑡) is a power function, we also use the notation𝐻𝛼(𝛾)𝑓(𝑥)=|𝑥|𝛾+𝛼𝑛||𝑦||<|𝑥|𝑓(𝑦)||𝑦||𝛾𝑑𝑦,𝜈(𝛾)𝑓(𝑥)=|𝑥|𝛾+𝛼||𝑦||>|𝑥|𝑓(𝑦)||𝑦||𝛾+𝑛𝑑𝑦(4.3) and their one-dimensional versions𝐻𝛼(𝛾)𝑓(𝑥)=𝑥𝛾+𝛼1𝑥0𝑓(𝑦)𝑦𝛾𝑑𝑦,𝛼(𝛾)𝑓(𝑥)=𝑥𝛾+𝛼𝑥𝑓(𝑦)𝑦𝛾+1𝑑𝑦,𝑥>0(4.4) adjusted for the half-axis 1+.

Lemma 4.1. Let 1𝑝< and 0<𝜆<𝑛.(I)Let 𝑊𝜑0. Then the Hardy operator 𝐻𝛼𝜑 is defined on the space 𝑝,𝜆(𝑛) or on the space 𝑝,𝜆loc(𝑛), if and only if0+𝑡𝑛1(𝑛𝜆)/𝑝𝜑(𝑡)𝑑𝑡<,(4.5) and in this case||𝐻𝛼𝜑||(𝑥)𝐶|𝑥|𝛼𝑛𝜑(|𝑥|)0|𝑥|𝑡𝑛1(𝑛𝜆)/𝑝𝜑(𝑡)𝑑𝑡𝑓𝑝,𝜆;loc.(4.6)(II)Let 𝑊1/𝜑0 or 𝑊𝜑0 and 𝜑(2𝑡)𝐶𝜑(𝑡). Then the Hardy operator 𝛼𝜑 is defined on the space 𝑝,𝜆(𝑛) or on the space 𝑝,𝜆loc(𝑛), if and only if𝜀𝑡𝑛1(𝑛𝜆)/𝑝𝜑(𝑡)𝑑𝑡<(4.7) for every 𝜀>0 and in this case||𝛼𝜑||(𝑥)𝐶|𝑥|𝛼𝜑(|𝑥|)|𝑥|𝑡𝑛1(𝑛𝜆)/𝑝𝜑(𝑡)𝑑𝑡𝑓𝑝,𝜆;loc.(4.8)

Proof. (I) The “If” Part. The sufficiency of condition (4.5) and estimate (4.6) follow from (3.12) under the choice 𝑠=1 and 𝑣(𝑡)=𝜑(𝑡).
The “Only If” Part. We choose a function 𝑓(𝑥) equal to |𝑥|(𝜆𝑛)/𝑝 in a neighborhood of the origin and zero beyond this neighborhood. Then 𝑓𝑝,𝜆 by Lemma 3.3. For this function 𝑓, the existence of the integral 𝐻𝛼𝜑𝑓 is equivalent to condition (4.5).
(II) The “If” part. The sufficiency of condition (4.7) and estimate (4.8) follow from (3.21) under the choice 𝑠=1 and 𝑣(𝑡)=1/𝑡𝑛𝜑(𝑡).
The “Only If” Part. We choose a function 𝑓(𝑥) equal to 𝑥(𝜆𝑛)/𝑝 in a neighborhood of infinity and zero beyond this neighborhood. Then 𝑓𝑝,𝜆 by Remark 3.2. For this function 𝑓, the existence of the integral 𝛼𝜑𝑓 is nothing else but condition (4.7).

Corollary 4.2. s
(I) The Hardy operator 𝐻𝛼(𝛾) is defined on the space 𝑝,𝜆(𝑛) or on the space 𝑝,𝜆loc(𝑛), if and only if 𝛾<𝑛/𝑝+𝜆/𝑝, and in this case ||𝐻𝛼(𝛾)𝑓||(𝑥)𝐶|𝑥|𝛼(𝑛𝜆)/𝑝𝑓𝑝,𝜆;loc.(4.9) (II) The Hardy operator 𝛼(𝛾) is defined on the space 𝑝,𝜆(𝑛) or on the space 𝑝,𝜆loc(𝑛), if and only if 𝛾>𝜆𝑛/𝑝, and in this case ||𝛼(𝛾)||𝑓(𝑥)𝐶|𝑥|𝛼(𝑛𝜆)/𝑝𝑓𝑝,𝜆;loc.(4.10)

4.2. Weighted 𝑝𝑞-Estimates for Hardy Operators in Morrey Spaces

The statements of Theorem 4.3 are well known in the case of Lebesgue space 𝜆=0 when 1<𝑝<𝑛/𝛼; see, for instance, [33, p. 6, 54]. As can be seen from the results below, inequalities for the Hardy operators in Morrey spaces admit the case 𝑝=1 when 𝜆>0.

4.2.1. The Case of Power Weights

Theorem 4.3. Let 0<𝜆<𝑛,0<𝛼<𝑛𝜆, and 1𝑝<(𝑛𝜆)/𝛼. The operator 𝐻𝛼(𝛾) (𝛼(𝛾), resp.) is bounded from 𝑝,𝜆(𝑛) or 𝑝,𝜆loc(𝑛) to 𝑞,𝜆(𝑛), where 1/𝑞=1/𝑝𝛼/(𝑛𝜆), if and only if 𝛾<𝑛/𝑝+𝜆/𝑝(𝛾>(𝜆𝑛)/𝑝, resp.).

Proof. The “only if” part follows from Corollary 4.2 and the “if part” from (4.9) and (4.10), since |𝑥|𝛼(𝑛𝜆)/𝑝=|𝑥|(𝑛𝜆)/𝑝𝑞,𝜆(𝑛); see Remark 3.2.

4.2.2. The Case of General Weights

We first deal with Hardy operators on a ball 𝐵(0,),0<< of a finite radius .

Theorem 4.4. Let 0<𝜆<𝑛,0<𝛼<𝑛𝜆, and 1𝑝<(1𝜆)/𝛼 and 𝑊𝜑0. Then the weighted Hardy operators 𝐻𝛼𝜑 and 𝛼𝜑 are bounded from 𝑝,𝜆(𝐵(0,)) or 𝑝,𝜆loc(𝐵(0,)) to 𝑞,𝜆(𝐵(0,)), where 1/𝑞=1/𝑝𝛼/(𝑛𝜆), if 𝜑𝜆/𝑝+𝑛/𝑝,𝜑(𝜆𝑛)/𝑝,(4.11) respectively, or, equivalently, 𝜆𝑀(𝜑)<𝑝+𝑛𝑝fortheoperator𝐻𝛼𝜑,(4.12)𝑚(𝜑)>𝜆𝑛𝑝fortheoperator𝛼𝜑.(4.13) The conditions 𝜆𝑚(𝜑)𝑝+𝑛𝑝𝜆,𝑀(𝜑)𝑝𝑛𝑝(4.14) are necessary for the boundedness of the operators 𝐻𝛼𝜑 and 𝛼𝜑, respectively.

Proof. By (2.10) and (2.11), the function 𝜑(𝑡)/𝑡𝑚(𝜑)𝜀 is almost increasing, while 𝜑(𝑡)/𝑡𝑀(𝜑)+𝜀 is almost decreasing for every 𝜀>0. Consequently, 𝐶1𝑟𝑚(𝜑)𝜀𝑡𝑚(𝜑)𝜀𝜑(𝑟)𝜑(𝑡)𝐶2𝑟𝑀(𝜑)+𝜀𝑡𝑀(𝜑)+𝜀(4.15) for 0<𝑡𝑟 and then 𝐶1|𝑥|𝑚(𝜑)𝜀+𝛼𝑛𝐵(0,|𝑥|)𝑓(𝑦)𝑑𝑡||𝑦||𝑚(𝜑)𝜀𝐻𝛼𝜑𝑓(𝑥)𝐶2|𝑥|𝑀(𝜑)+𝜀+𝛼𝑛𝐵(0,|𝑥|)𝑓(𝑦)𝑑𝑦||𝑦||𝑀(𝜑)+𝜀(4.16) supposing that 𝑓(𝑦)0. From the right-hand side inequality in (4.16) and Theorem 4.3, we obtain that the operator 𝐻𝛼𝜑 is bounded if 𝑀(𝜑)+𝜀<𝜆/𝑝+1/𝑝, which is satisfied under the choice of 𝜀>0 sufficiently small, the latter being possible by (4.12). It remains to recall that condition (4.12) is equivalent to the assumption 𝜑𝜆/𝑝+1/𝑝 by Theorem 2.4. The necessity of the condition 𝑚(𝜑)𝜆/𝑝+𝑛/𝑝 follows from the left-hand side inequality in (4.16). The case of the operator 𝜑 is similarly treated.

In the case of the whole space (=), we admit that the weight 𝜑(|𝑥|) may have an “oscillation between power functions” different at the origin and infinity. Correspondingly, the behavior at the origin and infinity is characterized by different indices 𝑚(𝜑),𝑀(𝜑) and 𝑚(𝜑),𝑀(𝜑), as described in Section 2.1.3.

Theorem 4.5. Let 0<𝜆<𝑛,0<𝛼<𝑛𝜆, and 1𝑝<(1𝜆)/𝛼 and 𝑊𝜑0,(1+). Then the weighted Hardy operators 𝐻𝛼𝜑 and 𝛼𝜑 are bounded from 𝑝,𝜆(𝑛) or 𝑝,𝜆loc(𝑛) to 𝑞,𝜆(𝑛),1/𝑞=1/𝑝𝛼/(𝑛𝜆), if 𝜑𝜆/𝑝+𝑛/𝑝1+𝜑(𝜆𝑛)/𝑝1+,(4.17) respectively, or, equivalently,   max𝑀(𝜑),𝑀<𝜆(𝜑)𝑝+𝑛𝑝fortheoperator𝐻𝛼𝜑,𝑚min(𝜑),𝑚>(𝜑)𝜆𝑛𝑝fortheoperator𝛼𝜑.(4.18) The conditions max𝑚(𝜑),𝑚𝜆(𝜑)𝑝+𝑛𝑝,min𝑀(𝜑),𝑀𝜆(𝜑)𝑝𝑛𝑝(4.19) are necessary for the boundedness of the operators 𝐻𝛼𝜑 and 𝛼𝜑, respectively.

Proof. The restriction of 𝐻𝛼𝜑𝑓(𝑥) to 𝐵(0,1) is covered by Theorem 4.4, so that it suffices to estimate 𝐻𝛼𝜑𝑓𝑝,𝜆(𝑛𝐵(0,1)). For |𝑥|>1 we have 𝐻𝛼𝜑𝑓(𝑥)=𝐶(𝑓)|𝑥|𝛼𝑛𝜑(|𝑥|)+|𝑥|𝛼𝑛||𝑦||1<<|𝑥|𝜑(|𝑥|)𝜑||𝑦||𝑓(𝑦)𝑑𝑦,(4.20) where 𝐶(𝑓)=𝐵(0,1)𝑓(𝑦)/𝜑(|𝑦|)𝑑𝑦. By Lemma 3.5 we have |𝐶(𝑓)|𝑐𝑓𝑝,𝜆;loc10𝑡𝑛1(1𝜆)/𝑝𝑑𝑡/𝜑(𝑡), where the integral converges since 𝜑(𝑡)𝐶𝑡𝑀(𝜑)+𝜀 with an arbitrarily small 𝜀>0 and (𝑛𝜆)/𝑝+𝑀(𝜑)<1. Then ||𝐶(𝑓)𝑥𝛼𝑛||𝜑(𝑥)𝑐𝑥𝛼𝑛+𝑀(𝜑)+𝜀𝑓𝑝,𝜆;loc(4.21) by (2.18). Here 𝑥𝛼𝑛+𝑀(𝜑)+𝜀𝑞,𝜆(1,), since 𝛼1+𝑀(𝜑)+𝜀<(𝜆𝑛)/𝑞 for sufficiently small 𝜀; see Remark 3.2.
To deal with the second term in (4.20), it suffices to observe that for 1|𝑦||𝑥|< we have inequality (4.15) with 𝑚(𝜑),𝑀(𝜑) replaced by 𝑚(𝜑),𝑀(𝜑) and then the proof follows the same lines as in Theorem (4.4) after formula (4.16).
The operator 𝛼𝜑 is considered in a similar way.

5. Application to Potential Operators

We consider the potential operator𝐼𝛼𝑓(𝑥)=𝑛𝑓(𝑦)𝑑𝑦||||𝑥𝑦𝑛𝛼,0<𝛼<𝑛,(5.1) and in Theorem 5.3 show that its weighted boundedness in Morrey spaces—in the case of weights 𝜑𝑉𝜇+𝑉𝜇 with 𝜇=min{1,𝑛𝛼}—is a consequence of the nonweighted boundedness due to Adams [5] and the weighted boundedness of Hardy operators provided by Theorem 4.5.

The necessity of the boundedness of the Hardy operators for that of potential operators is a consequence of the following simple fact, where 𝑋=𝑋(𝑛) and 𝑌=𝑌(𝑛) are arbitrary Banach function spaces in the sense of Luxemburg (cf., e.g., [34]).

Lemma 5.1. Let 𝑤=𝑤(𝑥) be any weight function. For the boundedness of the weighted potential operator 𝑤𝐼𝛼(1/𝑤) from 𝑋 to 𝑌, it is necessary that the Hardy operators 𝐻𝛼𝑤 and 𝛼𝑤𝛼 are bounded from 𝑋 to 𝑌, where 𝑤𝛼(𝑥)=|𝑥|𝛼𝑤(𝑥).

The proof of the sufficiency of the obtained conditions is based on the pointwise estimate of the following lemma.

Lemma 5.2. Let 𝑤𝐕𝜇𝐕𝜇+ with 𝜇=min{1,𝑛𝛼} be a weight and 𝑓 a non-negative function. Then the following pointwise estimate holds: 𝑤𝐼𝛼1𝑤𝑓(𝑥)𝐼𝛼𝐻𝑓(𝑥)+𝑐𝛼𝑤𝑓(𝑥)+𝛼𝛼𝑓(𝑥),if𝑤𝐕𝜇+,𝐻𝛼𝑓(𝑥)+𝛼𝑤𝛼𝑓(𝑥),if𝑤𝐕𝜇.(5.2)

Proof. We have 𝑤𝐼𝛼1𝑤𝐼𝛼𝑓(𝑥)=𝑛𝑤||𝑦||𝒦(𝑥,𝑦)𝑓(𝑦)𝑑𝑦,𝒦(𝑥,𝑦)=(|𝑥|)𝑤𝑤||𝑦||||||𝑥𝑦𝑛𝛼.(5.3) We first consider the case 𝑛𝛼1. For 𝑤𝐕𝜇𝐕𝜇+ with 𝜇=𝑛𝛼 in this case, by the definition of the classes 𝐕±𝑛𝛼, we have 𝒦(𝑥,𝑦)𝑐|𝑥|𝛼𝑛𝑤(|𝑥|)𝑤||𝑦||,||𝑦||||𝑦||<|𝑥|,𝛼𝑛,||𝑦||>|𝑥|,𝑤𝑒𝑛𝑤𝐕+𝑛𝛼,𝒦(𝑥,𝑦)𝑐|𝑥|𝛼𝑛,||𝑦||||𝑦||<|𝑥|,𝛼𝑛𝑤(|𝑥|)𝑤||𝑦||,||𝑦||>|𝑥|,𝑤𝑒𝑛𝑤𝐕𝑛𝛼,(5.4) which yield |||𝑤𝐼𝛼1𝑤𝐼𝛼|||𝐻𝑓(𝑥)𝑐𝛼𝑤𝑓(𝑥)+𝛼𝛼𝑓(𝑥),if𝑤𝐕+𝑛𝛼,𝐻𝛼𝑓(𝑥)+𝛼𝑤𝛼𝑓(𝑥),if𝑤𝐕𝑛𝛼,(5.5) with 𝛼𝛼=𝛼𝑤|𝑤|𝑥|𝛼 and prove (5.2).
Let now 𝑛𝛼>1. We denote 𝑛𝛼=𝑚+{𝑛𝛼}, where 𝑚=[𝑛𝛼] and {𝑛𝛼} stands for the fractional part of 𝑛𝛼. Now 𝑤𝐕1𝐕1+𝐕{𝑛𝛼}𝐕+{𝑛𝛼}.(5.6)
The procedure is similar to the previous case; we can first manage with the fractional part {𝑛𝛼}, treating 𝑤 as a function in 𝐕{𝑛𝛼}𝐕+{𝑛𝛼} like in the previous case, and then repeat a similar procedure 𝑚 times treating 𝑤 as a function in 𝐕1𝐕1+.
For definiteness we consider the case where 𝑤𝐕1+; the case of 𝑤𝐕1 is similarly treated. By the definition of the class 𝐕+{𝑛𝛼}, we have ||||𝐶𝐾(𝑥,𝑦)||||𝑥𝑦𝑚𝑤(|𝑥|)𝑤||𝑦|||𝑥|{𝑛𝛼},||𝑦||||𝑦||<|𝑥|,{𝑛𝛼},||𝑦||>|𝑥|(5.7) (this step should be omitted when 𝑛𝛼 is an integer), that is, ||||𝐾(𝑥,𝑦)𝐾+(𝑥,𝑦)+𝐾(𝑥,𝑦),(5.8) where 𝐾+(𝑥,𝑦)=𝐶𝑤(|𝑥|)𝑤||𝑦||𝜃||𝑦|||𝑥||𝑥|{𝑛𝛼}||||𝑥𝑦𝑚,𝐾𝜃||𝑦||(𝑥,𝑦)=𝐶|𝑥|||𝑦||{𝑛𝛼}||||𝑥𝑦𝑚𝐶2{𝑛𝛼}||||𝑥𝑦𝑛𝛼(5.9) and 𝜃(𝑡)=𝜒1+(𝑡). We only have to take care about the kernel 𝐾+(𝑥,𝑦). We have 𝐾+𝑤||𝑦||(𝑥,𝑦)=𝐶(|𝑥|)𝑤𝑤||𝑦||𝜃||𝑦|||𝑥||𝑥|{𝑛𝛼}||||𝑥𝑦𝑚𝜃||𝑦||+𝐶|𝑥||𝑥|{𝑛𝛼}||||𝑥𝑦𝑚.(5.10) We make use of the fact that 𝑤𝑉1+ and obtain 𝐾+(𝑥,𝑦)𝐶𝑤(|𝑥|)𝑤||𝑦||𝜃||𝑦|||𝑥||𝑥|{𝑛𝛼}+1||||𝑥𝑦𝑚1+𝐶||||𝑥𝑦𝑛𝛼,(5.11) where again only the first term must be studied. We repeat the same procedure 𝑚1 times more and finally arrive at the kernel 𝑤(|𝑥|)𝑤||𝑦||𝜃||𝑦|||𝑥||𝑥|{𝑛𝛼}+𝑚=|𝑥|𝛼𝑛𝑤(|𝑥|)𝑤||𝑦||||𝑦||𝜃|𝑥|,(5.12) which is the kernel of the Hardy operator 𝐻𝛼𝑤.

We are ready for the following statement, where notation (2.22) is used.

Theorem 5.3. Let 0<𝛼<𝑛, 0𝜆<𝑛, and 1<𝑝<(𝑛𝜆)/𝛼. (i)Let 𝑊𝜑0,(1+)(𝐕𝜇𝐕𝜇+) with 𝜇=min(1,𝑛𝛼). Then the condition𝜑Φ𝛼+(𝜆𝑛)/𝑝𝑛/𝑝+𝜆/𝑝1+,(5.13)or equivalently𝛼𝑛𝜆𝑝<min𝑚(𝜑),𝑚(𝜑),max𝑀(𝜑),𝑀<𝑛(𝜑)𝑝+𝜆𝑝,(5.14)is sufficient for the boundedness of the potential operator (5.1) from the weighted space 𝑝,𝜆(1+,𝜑) to the space 𝑞,𝜆(1+,𝜑), where 1/𝑞=1/𝑝𝛼/(𝑛𝜆).(ii)Let 𝑊𝜑0,(1+). Then the condition𝛼𝑛𝜆𝑝min𝑀(𝜑),𝑀(𝜑),max𝑚(𝜑),𝑚(𝜑)𝑛/𝑝+𝜆/𝑝(5.15)is necessary for the boundedness of the potential operator (5.1) from 𝑝,𝜆(1+,𝜑) to 𝑞,𝜆(1+,𝜑).

Proof. The necessity part (ii) follows from Lemma 5.1 and Theorem 4.5.
Part (i). We have to prove the boundedness of the operator 𝜑𝐼𝛼1/𝜑 from 𝑝,𝜆(1+) to 𝑞,𝜆(1+). Since the non-weighted 𝑝,𝜆(1+)𝑞,𝜆(1+)-boundedness of the potential operator 𝐼𝛼 is known [5], it suffices to show the boundedness of the operator 𝜑𝐼𝛼1/𝜑𝐼𝛼. For that it remains to make use of Theorem 4.5. This completes the proof.

Corollary 5.4. Let 0𝜆<𝑛,0<𝛼<𝑛𝜆, 1<𝑝<(𝑛𝜆)/𝛼, and 𝜚(𝑥)=|𝑥𝑥0|𝛾,𝑥01. Then the potential operator (5.1) is bounded from 𝑝,𝜆(𝑛,𝜚) into 𝑞,𝜆(𝑛,𝜚), 1/𝑞=1/𝑝𝛼/(𝑛𝜆), if and only if 𝑛𝛼𝑝𝜆<𝛾𝑝<𝑛𝑝.(5.16)

Remark 5.5. As can be seen from the proof of Theorem 5.3, its statement remains valid under the condition 𝜑𝛼+(𝜆𝑛)/𝑝,if𝜑𝐕𝜇+,𝑛/𝑝+𝜆/𝑝1+,if𝜑𝐕𝜇,𝜇=min(1,𝑛𝛼),(5.17) more general than (5.13). Correspondingly, condition (5.14) may be written in a more general form: max𝑀(𝜑),𝑀<𝑛(𝜑)𝑝+𝜆𝑝,if𝜑𝐕𝜇+,𝛼𝑛𝜆𝑝<min𝑚(𝜑),𝑚(𝜑),if𝜑𝐕𝜇.(5.18) (Recall that min(𝑀(𝜑),𝑀(𝜑))0 in the case 𝜑𝐕𝜇+ and max(𝑚(𝜑),𝑚(𝜑))0 in the case 𝜑𝐕𝜇; see Corollary 2.9.)

Acknowledgment

This work was supported by Research Grant SFRH/BPD/34258/2006, FCT, Portugal.

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