Weighted Hardy and Potential Operators in Morrey Spaces

We study the weighted p → q-boundedness of Hardy-type operators in Morrey spaces Lp,λ R or Lp,λ R1 in the one-dimensional case for a class of almost monotonic weights. The obtained results are applied to a similar weighted p → q-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.


Introduction
The well-known Morrey spaces L p,λ introduced in 1 in relation to the study of partial differential equations and presented in various books, see 2-4 , were widely investigated during the last decades, including the study of classical operators of harmonic analysismaximal, singular, and potential operators-in these spaces; we refer for instance to papers 5-23 , 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 p → p-estimates in Morrey spaces for Hardy operators on R 1 and one-dimensional singular operators on R 1 or on Carleson curves in the complex plane .
In this paper we develop an approach which allows us both to obtain weighted p → q − estimations 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 The admitted weights ϕ |x − x 0 | 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 L p,λ Ω into some weighted L p Ω, -spaces.In Section 4 we prove theorems on the weighted p → q-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.

Zygmund-Bary-Stechkin (ZBS) Classes and Matuszewska-Orlicz (MO) Type Indices
2.1.1.On Classes of the Type W 0 and W ∞ 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 x ≤ y x ≥ y, resp. .Equivalently, a function f is almost increasing almost decreasing if it is equivalent to an increasing decreasing, resp.function g, that is, 1 By W 0 W 0 0, one denotes the class of functions ϕ continuous and positive on 0, such that there exists the finite limit lim x → 0 ϕ x , and ϕ x is almost increasing on 0, ; 1 By W ∞ W ∞ , ∞ one denotes the class of functions ϕ continuous and positive on , ∞ which have the finite limit lim x → ∞ ϕ x , and ϕ x is almost increasing on , ∞ ;

ZBS Classes and MO Indices of Weights at the Origin
In this subsection we assume that < ∞.
One also denotes 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 , 29-32 for the properties of the indices of such a type.For a function ϕ ∈ W 0 , the numbers ln lim sup h → 0 ϕ hx /ϕ h ln x .

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 0 ≤ m ϕ ≤ M ϕ ≤ ∞ for ϕ ∈ W 0 and −∞ < m ϕ ≤ M ϕ ≤ ∞ for ϕ ∈ W 0 , and the formulas are valid 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 β ≥ 0, γ > 0, and ϕ ∈ W 0 .It is evidently true also for ϕ ∈ W 0 and all β, γ ∈ R 1 , in view of formulas 2.5 .

ZBS Classes and MO Indices of Weights at Infinity
Following 14, Section 4.1 and 29, Section 2.2 , we introduce the following definitions.
and Z α is the class of functions ϕ ∈ W , ∞ satisfying the condition where c c ϕ > 0 does not depend on x ∈ , ∞ .
The indices m ∞ ϕ and M ∞ ϕ responsible for the behavior of functions ϕ ∈ Ψ β α , ∞ at infinity are introduced in the way similar to 2.4 :

2.15
Properties of functions in the class Ψ β α , ∞ are easily derived from those of functions in Φ α β 0, because of the following equivalence: where ϕ * t ϕ 1/t and * 1/ • .Direct calculation shows that 2.17 Making use of 2.16 and 2.17 , one can easily reformulate properties of functions of the class Φ β α near the origin, given in Theorem 2.4 for the case of the corresponding behavior at infinity of functions of the class Ψ β α and obtain that 19 We say that a function ϕ continuous in 0, ∞ is in the class W 0,∞ R 1 if its restriction to 0, 1 belongs to W 0 0, 1 and its restriction to 1, ∞ belongs to W ∞ 1, ∞ .For functions in W 0,∞ R 1 , the notation has an obvious meaning.In the case, where the indices coincide: β 0 β ∞ : β, we will simply write Z β R 1 and similarly for Z γ R 1 .We also denote Making use of Theorem 2.4 for Φ β α 0, 1 and relations 2.17 , we easily arrive at the following statement. 2.23

On Classes V μ ±
Note that we slightly changed the notation of the class introduced in the following definition, in comparison with its notation in 32 .
Lemma 2.8.Functions ϕ ∈ V µ are almost increasing on 0, , and functions Note that the classes V μ ± being trivial for μ > 1.We also have which follows from the fact that An example of a function which is in V μ with some μ > 0, but does not belong to the total intersection μ∈ 0,1 V μ is given by where 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".
− in the case ϕ x is decreasing and there exists a number μ ≥ 0 such that x μ ϕ x is increasing.

Definitions and Belongness of Some Functions to Morrey Spaces
Let Ω be an open set in R n .
respectively, where B x, r B x, r ∩ Ω.

Journal of Function Spaces and Applications
Obviously, The spaces L p,λ Ω , L p,λ loc Ω are known under the names of global and local Morrey spaces; see for instance, 9, 10 .
The weighted Morrey space is defined as As is well known, the space L p,λ Ω as defined above is not necessarily embedded into L p Ω , in the case when Ω is unbounded.A typical counterexample in the case Ω R n is Indeed, we have 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 .
Proof.We have which is bounded under the choice ε < m u n − λ /p.In the case 2 , we observe that B x, r ⊂ B 0, 3r and then the same estimate f p,λ ≤ Csup r>0 r m u n−λ /p−ε follows.
In the case u t t γ , 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

3.10
Corollary 3.4.If u ∈ W 0 0, and there exists an a < n−λ /p such that t a u t is almost increasing, then To derive this corollary from Lemma 3.3, it suffices to refer to formula 2.10 .

Some Weighted Estimates of Functions in Morrey Spaces
Lemma 3.5.
where C > 0 does not depend on y and f and under the assumption that the last integral converges.
Proof.We have Making use of the fact that there exists a β such that t β v t is almost increasing, we observe that Applying this in 3.13 and making use of the H ölder inequality with the exponent p/s ≥ 1, we obtain 3.17 We have Making use of the fact that t β v t is almost increasing with some β, we easily obtain that which proves 3.17 .
≤ cB y f p,λ; loc , y / 0, 3.21 where C > 0 does not depend on y and f and Proof.The proof is similar to that of Lemma 3.5.We have where Since there exists a β ∈ R 1 such that t β v t is almost increasing, we obtain where C may depend on β, but does not depend on y and f.Applying the H ölder inequality with the exponent p/s, we get

3.25
It remains to prove that We have which completes the proof.
Remark 3.8.The analysis of the proof shows that estimate 3.21 remains in force, if the assumption v ∈ W 0 R 1 is replaced by the condition that 1/v ∈ W 0 R 1 and v satisfies the doubling condition v 2t ≤ cv t .≤ cy b n/s− n−λ /p f p,λ;loc , y / 0. 3.28

Pointwise Estimations
We consider the generalized Hardy operators In the sequel R n with n 1 may be read either as R 1 or R 1 with the operators interpreted as In the case ϕ t is a power function, we also use the notation and their one-dimensional versions adjusted for the half-axis R 1 .
I Let ϕ ∈ W 0 .Then the Hardy operator H α ϕ is defined on the space L p,λ R n or on the space L p,λ loc R n , if and only if and in this case II Let 1/ϕ ∈ W 0 or ϕ ∈ W 0 and ϕ 2t ≤ Cϕ t .Then the Hardy operator H α ϕ is defined on the space L p,λ R n or on the space L p,λ loc R n , if and only if for every ε > 0 and in this case 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| λ−n /p in a neighborhood of the origin and zero beyond this neighborhood.Then f ∈ L p,λ by Lemma 3.3.For this function f, the existence of the integral H α ϕ f 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/t n ϕ t .The "Only If" Part.We choose a function f x equal to x λ−n /p in a neighborhood of infinity and zero beyond this neighborhood.Then f ∈ L p,λ by Remark 3.2.For this function f, the existence of the integral H α ϕ f is nothing else but condition 4.7 .

if and only if γ < n/p λ/p, and in this case
(II) The Hardy operator H α γ is defined on the space L p,λ R n or on the space L p,λ loc R n , if and only if γ > λ − n/p, and in this case

Weighted p → q-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.

The Case of Power Weights
Proof.The "only if" part follows from Corollary 4.2 and the "if part" from 4.9 and 4.10 , since |x| α− n−λ /p |x| n−λ /p ∈ L q,λ R n ; see Remark 3.2.

4.13
The conditions are necessary for the boundedness of the operators H α ϕ and H α ϕ , respectively.
Proof.By 2.10 and 2.11 , the function ϕ t /t m ϕ −ε is almost increasing, while ϕ t /t M ϕ ε is almost decreasing for every ε > 0. Consequently, for 0 < t ≤ r and then supposing that f y ≥ 0. From the right-hand side inequality in 4.16 and Theorem 4.3, we obtain that the operator H α ϕ 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 ϕ ∈ Z λ/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 H 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 < n − λ, and 1 ≤ p < 1 − λ /α and ϕ ∈ W 0,∞ R 1 .Then the weighted Hardy operators H α ϕ and

4.18
The conditions are necessary for the boundedness of the operators H α ϕ and H α ϕ , respectively.
Proof.The restriction of H α ϕ f x to B 0, 1 is covered by Theorem 4.4, so that it suffices to estimate H α ϕ f L p,λ R n \B 0,1 .For |x| > 1 we have where C f B 0,1 f y /ϕ |y| dy.By Lemma 3.5 we have |C f |≤ c f p,λ;loc 1 0 t n−1− 1−λ /p dt/ ϕ t , where the integral converges since ϕ t ≥ Ct M ϕ ε with an arbitrarily small ε > 0 and n − λ /p M ϕ < 1.Then 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 H α ϕ is considered in a similar way.

Application to Potential Operators
We consider the potential operator 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 R n and Y Y R 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 wI α 1/w from X to Y , it is necessary that the Hardy operators H α w and H α w α are bounded from X to Y , where w α x |x| −α w x .
The proof of the sufficiency of the obtained conditions is based on the pointwise estimate of the following lemma.Then the following pointwise estimate holds:

5.2
Proof.We have We first consider the case n − α ≤ 1.For w ∈ V μ − ∪ V μ with μ n − α in this case, by the definition of the classes V n−α ± , we have The procedure is similar to the previous case; we can first manage with the fractional part {n − α}, treating w as a function in V {n−α} − ∪ V {n−α} like in the previous case, and then repeat a similar procedure m times treating w as a function in V 1 − ∪ V 1 .For definiteness we consider the case where w ∈ V 1 ; the case of w ∈ V 1 − is similarly treated.By the definition of the class V {n−α} , we have We are ready for the following statement, where notation 2.22 is used.is sufficient for the boundedness of the potential operator 5.1 from the weighted space L p,λ R 1 , ϕ to the space L q,λ R 1 , ϕ , where 1/q 1/p − α/ n − λ .
ii Let ϕ ∈ W 0,∞ R1 .Then the condition is necessary for the boundedness of the potential operator 5.1 from L p,λ R 1 , ϕ to L q,λ R 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 ϕI α 1/ϕ from L p,λ R 1 to L q,λ R 1 .Since the non-weighted L p,λ R 1 → L q,λ R 1 -boundedness of the potential operator I α is known 5 , it suffices to show the boundedness of the operator ϕI α 1/ϕ − I α .For that it remains to make use of Theorem 4.5.This completes the proof.

5.16
Remark 5.5.As can be seen from the proof of Theorem 5.3, its statement remains valid under the condition ϕ ∈ Z α λ−n /p , if ϕ ∈ V μ , μ min 1, n − α , 5.17 more general than 5.13 .Correspondingly, condition 5.14 may be written in a more general form:
| w≡|x| −α and prove 5.2 .Let now n − α > 1.We denote n − α m {n − α}, where m n − α and {n − α} stands for the fractional part of n − α.Now be omitted when n − α is an integer , that is,K x, y ≤ K x, y K − x, y , χ R 1 t .We only have to take care about the kernel K x, y .We haveK x, y C w |x| − w y w y • θ |x| − y |x| {n−α} x − y m C θ |x| − y |x| {n−α} x − y m .5.10We make use of the fact that w ∈ V 1 and obtainK x,y ≤ C w |x| w y • θ |x| − y |x| {n−α} 1 x − y the first term must be studied.We repeat the same procedure m − 1 times more and finally arrive at the kernel w |x| w y • θ |x| − y |x| {n−α} m |x| α−n w |x| w y • θ |x| − y , 5.12 which is the kernel of the Hardy operator H α w .