Weighted Hardy Operators in Complementary Morrey Spaces

We study the weighted p → q-boundedness of the multidimensional weighted Hardy-type operators H w and Hw with radial type weight w w |x| , in the generalized complementary Morrey spaces L {0} R defined by an almost increasing function ψ ψ r . 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 spaces L {0} Ω over bounded domains Ω are embedded between weighted Lebesgue space L with the weight ψ and such a space with the weight ψ, perturbed by a logarithmic factor. Both the embeddings are sharp.


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 papers 5, 6 .Results on weighted estimations of Hardy operators in Lebesgue spaces may be found in the abovementioned books.In the papers 7-9 the weighted boundedness of the Hardy type operators was studied in Morrey spaces.
In this paper we study multi-dimensional weighted Hardy operators where α ≥ 0, in the so called complementary Morrey spaces.The one-dimensional case will include the versions adjusted for the half-axis R 1 , so that in the sequel R n with n 1 may be read either as R 1 or R 1 .
The classical Morrey spaces L p,λ Ω , Ω ⊆ R n , defined by the norm where B x, r Ω ∩ B x, r , are well known, in particular, because of their usage in the study of regularity properties of solutions to PDE; see for instance the books 10-12 and references therein.There are also known various generalizations of the classical Morrey spaces L p,λ , and we refer for instance to the surveying paper 13 .One of the direct generalizations is obtained by replacing r λ in 1.3 by a function ϕ r , usually satisfying some monotonicity type conditions.We also denote it as L p,ϕ Ω without danger of confusion.Such spaces appeared in 14, 15 and were widely studied in 16,17 where admission of the multiplier ψ r vanishing at r 0 controls the growth of the norm f L p Ω\B x 0 , r .Such spaces were also studied in some later papers, and we refer for instance to 18 .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 books 2-4 and references given there.
We also prove a new property for the generalized complementary Morrey spaces; see Theorem 3.1, by showing that the spaces L p,ψ {0} Ω over bounded domains Ω are embedded between the weighted Lebesgue space L p 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 in 19 .
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 Hardytype 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.

Definitions
Let Ω be an open set in R n , Ω ⊆ R n and diam Ω, 0 < ≤ ∞, B x, r {y ∈ R n : |x − y| < r} and B x, r B x, r ∩ Ω.Let also ψ r be a continuous function nonnegative on 0, with ψ 0 : lim r → 0 ψ r 0 which may arbitrarily grow as r → ∞.Let also 1 ≤ p < ∞.
In the case of power factor ψ r ≡ r λ , λ > 0, we also denote without danger of confusion.
The space L p, ψ {x 0 } Ω is nontrivial in the case of any locally bounded function ψ r 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: M p, ψ f; x 0 , r : ψ r Ω\ B x 0 , r f y p dy.

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.

On Belongness of Radial Functions to the Space
The proof of Lemma 2.3 is obvious.By means of this lemma we easily obtain the following corollary.

2.5
In the case n γp 0, the same holds with r min{n γp,0} replaced by | ln r|.When Ω R n , the necessary and sufficient conditions are n γp < 0 and sup r>0 r n γp ψ r < ∞.
As is known, there may be given sufficient numerical inequalities for the validity of the integral condition in 2.4 in terms of Matuszewska-Orlicz indices m u and M u 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 classes Z α,β , we put these notions in the appendix.
Let now u be a nonnegative function in Bary-Stechkin class, namely, Then r u p t t n−1 dt ∼ r n u p r , 0 < r < , so that the condition sup 0<r< r n ψ r u p r < ∞ is sufficient for u u |x − x 0 | to be in L p, ψ {x 0 } Ω .Note also that 2.6 is equivalent to the inequalities pM u 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 .

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 L p Ω, : The class W 0 0, , used in this theorem, is defined in the appendix; see Definitions A.1-A.2 therein.In the case where ψ r is a power function, Theorem 3.1 was proved in 19 .
, diam Ω, and let ψ be absolutely continuous and where ψ ε t ψ t / ln A/t 1 ε , A > .The left-hand side embedding in 3.2 is strict, when ψ satisfies the condition The right-hand side embedding in 3.2 is strict, when ψ satisfies the doubling condition ψ 2r ≤ Cψ r , (in particular, if ψ r r λ , λ ≥ 0), there exists a function g 0 g 0 x such that Proof.Without loss of generality, we may take x 0 0 for simplicity, supposing that 0 ∈ Ω.where 3.9 so that

3.11
where the last integral converges when ε > 0 since The strictness of the embeddings: the corresponding counterexamples are

3.12
Calculations for the function f 0 , which is obviously not in L p Ω, ψ |y| , are easy.Indeed, where the right-hand side is bounded by 3.3 .
In the case of the function g 0 we have for every ε > 0. However, for small r ∈ 0, δ/2 , where δ dist 0, ∂Ω , we obtain

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 modular M p,ψ f; x 0 , r with x 0 0. These estimates are of independent interest and also crucial for our study of the Hardy operators in the complementary Morrey space.

Journal of Function Spaces and Applications
, where C > 0 does not depend on f and r ∈ 0, ∞ and Proof.We use the dyadic decomposition as follows: where Since there exists a β such that t β v t is almost increasing, we observe that on B k r .Applying this in 4.3 and making use of the H ölder inequality with the exponent p/s ≥ 1, we obtain

4.5
Hence On the other hand we have s/p p,ψ f; 0, t v t ψ S/p t dt.

4.7
The function 1/ ψ t M p,ψ f; 0, t is decreasing, and the function 1/ v t is almost decreasing after multiplication by some power functions.Therefore,
where C > 0 does not depend on r > 0 and f.
Proof.We use the corresponding dyadic decomposition: where B k 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 where C may depend on β but does not depend on r and f.Applying the H ölder inequality with the exponent p/s, we get

4.13
On the other hand, the integral on the right-hand side of 4.10 can be estimated as follows: which completes the proof.

Pointwise Estimations
The proof of our main result of this Section given in Theorems 5.3 and 5.6 is prepared by the following Theorems 5.1 and 5.2 on the pointwise estimates of the Hardy-type operators.

The Case of the Operator H α w
The L p → L q -boundedness of the multidimensional Hardy operators within the frameworks of Lebesgue spaces the 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 in 7, 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 ϕ 0 0 or ψ 0 0 admit the value p 1.

5.5
The operator and V t is the same as in 5.1 .Under this condition,

5.7
Proof.From the estimate 5.2 of Theorem 5.1 we have

5.8
Hence, calculating H α w f L p,ψ
Remark 5.4.Note that The following corollary gives sufficient conditions for the boundedness of the operator H α w in terms of the Matuszewska-Orlicz indices of the function ψ and the weight w we 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 ψ r ≥ Cr αpq/q−p −n 5.10 in the case α / 0. The operator

5.11
The condition min{m V , m ∞ V } > 0, is guaranteed by the inequalities

5.12
In the power case ψ r r λ , λ > 0, and w r r μ , the conditions 5.10 -5.11 reduce to 13 conditions 5.13 are also necessary for the operator H α w to be bounded from Proof.We have to check that the condition sup r>0 W r < ∞ of Theorem 5.3 holds under the assumptions 5.10 -5.11 .From the inequality min{m V , m ∞ V } > 0 it follows that

5.14
We represent ψ q/p as ψ q/p ψ q/p −1 ψ and by 5.10 obtain which is bounded in view of the second assumption in 5.11 , by the property A.4 .The sufficiency of the conditions 5.13 in the case of power functions is then obvious since m ψ m ∞ ψ λ and m w m ∞ w μ in this case.Let us show the necessity of these conditions.From the boundedness {0} R n with ψ r r λ , λ > 0, and w r r μ , by standard homogeneity arguments with the use of the dilation operator Π δ f x : f δx it is easily derived that the conditions 1 ≤ p < n λ /α, 1/q 1/p − α/ n λ necessarily hold, via the relation 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 ∈ L p,ψ {0} R n by Corollary 2.4, so that this condition is necessary for the operator H α w with w r μ to be defined on the space L p,ψ {0} R n .

The Case of the Operator
where V is the same as in 5.3 .

5.18
The operator and then

5.20
Proof.From the estimate 5.4 we obtain , 5.21 from which 5.20 follows.
As above, we provide also sufficient conditions for the boundedness of the operator H β w in 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 conditions 5.18 .The operator and the condition 5.10 holds; the assumption max{M V , M ∞ V } < 0 is guaranteed by the conditions

5.23
In the case ψ r r λ , λ > 0, and w r r μ , the conditions 5.22 and 5.10 reduce to Proof.We have to find, in terms of the Matuszewska-Orlicz indices, conditions sufficient for the validity of 5.19 .For the latter we have by the assumption max{M V , M ∞ V } < 0. With ψ q/p ρ ψ q/p −1 ρ ψ ρ and the condition 5.10 we arrive at W r ≤ Cψ r ∞ r d / ψ , where the boundedness of the righthand side is guaranteed by the condition min{m ψ , m ∞ ψ } > 0.
The proof for the case of power functions is similar to that in Corollary 5.5.

Journal of Function Spaces and Applications
For a function ϕ ∈ W, the numbers are known as the Matuszewska-Orlicz type lower and upper indices of the function ϕ r .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 ϕ x need 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, and the following formulas are valid: The proof of the following statement may be found in 21 , 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 ∈ W 0 .It is evidently true also for ϕ ∈ W and all β, γ ∈ R 1 , in view of formulas A.3 .
We define the following subclass in W 0 : A.8

A.2. ZBS Classes and MO Indices of Weights at Infinity
Following Section 2.2 in 29 , we use the following notation.Let −∞ < α < β < ∞.We put Ψ We say that a continuous function ϕ in 0, ∞ is in the class W 0,∞ R 1 if its restriction to 0, 1 belongs to W 0, 1 and its restriction to 1, ∞ belongs to W ∞ 1, ∞ .For functions in W 0,∞ R 1 the notation H α w f x : |x| α−n w |x| |y|<|x| x 0 ∈ Ω, are known as generalized local Morrey spaces Hardy type operators 1.1 in the spaces L p,ϕ Ω , L p,ϕ loc Ω have been studied in 7-9 .The norm in Morrey spaces controls the smallness of the integral B x,r |f y | p dy over small balls B x, r and also a possible growth of this integral for r → ∞ in the case Ω is unbounded .There are also known spaces L p,ψ {x 0 } Ω , called complementary Morrey spaces, with the norm controlling possible growth, as r → 0, of the integral Ω\B x balls.Such spaces have sense in the local setting only.It is introduced in 16, 17 that the space L p,ψ {x 0 } Ω is defined as the space of all functions f ∈ L p loc Ω \ {x 0 } with the finite norm