Weighted Boundedness of Certain Sublinear Operators in Generalized Morrey Spaces on Quasi-Metric Measure Spaces Under the Growth Condition

We prove weighted boundedness of Calderón–Zygmund and maximal singular operators in generalized Morrey spaces on quasi-metric measure spaces, in general non-homogeneous, only under the growth condition on the measure, for a certain class of weights. Weights and characteristic of the spaces are independent of each other. Weighted boundedness of the maximal operator is also proved in the case when lower and upper Ahlfors exponents coincide with each other. Our approach is based on two important steps. The first is a certain transference theorem, where without use homogeneity of the space, we provide a condition which insures that every sublinear operator with the size condition, bounded in Lebesgue space, is also bounded in generalized Morrey space. The second is a reduction theorem which reduces weighted boundedness of the considered sublinear operators to that of weighted Hardy operators and non-weighted boundedness of some special operators.


Introduction
We study weighted boundedness of certain sublinear operators in generalized Morrey spaces L p,ϕ (X ) defined on quasi-metric measure spaces (X , d, μ). We do not suppose that (X , d, μ) is homogeneous, i.e. we assume that it satisfies the growth condition μB(x, r ) ≤ cr ν , 0 < r < diam X ≤ ∞, ν > 0. (1.1) The study includes Calderón-Zygmund singular operators with standard kernel, the corresponding maximal singular operator and the standard maximal operator. In the case of the singular and maximal singular operators we obtain results on weighted boundedness in the generalized Morrey spaces, under the only assumption that the measure satisfies the growth condition (1.1).
Sublinear operators under consideration are supposed to satisfy the following two conditions: (1) they are bounded in L p (X ), (2) they satisfy a certain size condition, related to the exponent of the growth condition.
For the study of sublinear operators of singular type in the space L p (X ) under the growth condition (2.2) we refer to [25]. There are known results on the boundedness of such operators in L p (X ) under the growth condition more general than (2.2), where r ν is replaced by a given dominant λ(x, r ), see [14] and [15].
We consider the generalized Morrey spaces defined by the norm (1.2) where is any subset in X . Introduction of helps to unite local and global Morrey spaces. For a sublinear operator T satisfying the conditions (1) and (2), we study the boundedness of the weighted operators For classical and generalized Morrey spaces and their applications we refer, for instance, to the books [8,19,27,36,38,39] and the overview paper [28].
Singular-type operators under more general growth condition (in the sense of [14] and [15]) were studied in [21] and [40]. In [21] the operators T were studied in the nonweighted case, while in [40] they were considered in the weighted space L p,k (X , w) of specific form, which goes back to [20], namely in the case Note that in this case, the Morrey space L p,k (X , w) is in fact the non-weighted classical In this paper we study sublinear operators satisfying the properties (1) and (2) in weighted generalized Morrey spaces on quasi-metric measure space (X , d, μ), under the "classical" growth condition (2.2). We consider "radial" weights w(x) = v[d(x, x 0 )], x 0 ∈ X and the function v belongs to some class V + V − , see its definition in Sect. 2.3.
Our main results are as follows. First we show that the known way of transference of L p -boundedness to Morreyboundedness under the size condition, may be proved without using homogeneity of the space, see Transference Theorem in Sect. 3.1. More precisely, we show that the condition with ν from (1.1), imposed on the function ϕ(x, r ) defining the Morrey space, guarantees that any sublinear operator with the size condition, bounded in L p (X ), is also bounded in the Morrey space L p,ϕ (X ).
Moreover, under the only growth condition, we are able to efficiently estimate the Morrey modular of T f via that of f , see Theorem 3.8, which leads to the boundedness result in Morrey spaces in Theorem 3.9.
Further, we provide a certain pointwise estimate for weighted singular, maximal singular and maximal operators, with above mentioned radial weights, via non-weighted such operators plus the following operators: weighted Hardy operators, certain nonweighted operators which may be considered as a kind of hybrids of Hardy operators and potential operators, see Reduction Theorem 3.11.
Since the estimate in this theorem is pointwise, it reduces the weighted boundedness of the weighted singular, maximal singular and maximal operators in any Banach functions spaces with lattice properties to the boundedness of non-weighted operators, weighted Hardy operators with the same weight and some specific "hybrids". In this paper we use this estimate for the case of the generalized Morrey space L p,ϕ (X ).
As a separate result of interest we show that some of those hybrids are dominated by the modified maximal operator (modification concerns the use of the growth condition), see Theorem 3.5.
This reduction and the above mentioned Transference Theorem together with the L p results [25], allow us to obtain a result on the weighted boundedness of the weighted singular, maximal singular and maximal operators in the spaces L p,ϕ (X ) as given in Theorem 3.20. To this end, we obtain conditions for the weighted boundedness of Hardy operators in the spaces L p,ϕ (X ) under the only growth condition for (X , d, μ).
The paper is organized as follows. In Sect. 2 we provide necessary information on quasi-metric measure spaces (X , d, μ) together with definition of the space L p,ϕ (X ) and define the class of weights. Sect. 3 contains our main results. In Sect. 3.1 we prove the above mentioned Transference Theorem for an arbitrary sublinear operator with the size condition. In Sect. 3.2 we pass to weighted operators and prove the above mentioned Reduction Theorem containing the pointwise estimate of weighted operators. Section 3.3 starts with a result of weighted boundedness of Hardy operators in generalized Morrey spaces L p,ϕ (X ). This allows us to apply Transference and Reduction Theorems to obtain conditions on the weight and the function ϕ(x, r ), insuring the weighted boundedness of singular, maximal singular and maximal operators in the spaces L p,ϕ (X ). In Corollary 3.21, where we take ϕ(x, r ) = r λ for simplicity, we give sufficient conditions for the validity of those conditions in terms of Matuszewska-Orlicz indices of the weight. Finally, in Sect. 1 (Appendix), for reader's convenience, we provide necessary information for Matuszewska-Orlicz indices. The author thanks the anonymous referees for their careful reading of the paper, and useful comments.

Preliminaries on Quasi-Metric Measure Spaces
Basics on quasi-metric measure spaces may be found e.g. in [7] and [13]. Below we provide necessary definitions which we use in the paper.
Let (X , d, μ) be a quasi-metric measure space with measure μ and quasi-distance d: Everywhere in the sequel we suppose that the following properties of (X , d, μ) hold: (1) all balls are open sets; (2) the spheres S(x, r ) := {y ∈ X : d(y, x) = r } have zero measure for all x and r ; ( The set (X , d, μ) is said to satisfy the growth condition if there exist a constant A > 0 and exponent ν > 0, which is fractional in general, such that where x ∈ X and r ∈ (0, ). For more general notion of the growth condition, i.e. with a given dominant of measure of balls we refer to [14] and [15]. In this paper we use the growth condition of the form (2.2). We say that (X , d, μ) is regular, if the measure satisfies the lower and upper Ahlfors conditions with coinciding exponents, i.e.
Estimates of the type provided by the lemma below are known but we give its short proof for completeness of presentation.
Lemma 2.2 below provides a certain replacement of the formula of passage to polar coordinates used in the case X = R n . This lemma is a simplified version of more general estimates proved in [32].
Let be an arbitrary set of points in X . We use the uniform doubling condition In the sequel we use the abbreviations: a.i. = almost increasing and a.d. = almost decreasing Lemma 2.2 [32,Lemmas 2.5 and 2.8] Let (X , d, μ) satisfy the growth condition (2.2), L(ξ, t) be a non-negative function on × (0, ), 0 < ≤ ∞, a.i. in t uniformly in x ∈ and the doubling condition (2.5) be satisfied. Then where ξ ∈ , x ∈ (0, ), a ∈ R and 0 < r < ≤ ∞, whenever the right hand side of these estimates exists or not.

Generalized Morrey Spaces L p,' (X)
The generalized Morrey spaces are defined by the norm: The spaces defined by the norm are often called generalized local Morrey spaces. The spaces defined by the norm (2.8) are correspondingly called generalized global Morrey spaces. Both may be united in a single approach by the localization applied with respect to an arbitrary set ⊆ X , not just with respect to the case = {x 0 } of an isolated point. That is, one can estimate the Morrey-regularity of functions f at an arbitrary given subset of X , with admission of the extremal cases = X and = {x 0 }, x 0 ∈ X . The corresponding space defined by the norm will be denoted by L p,ϕ (X ). The principal estimates on which the proofs in this paper are based, are pointwise, see Sect. 3.2.
Everywhere in the sequel we suppose that ϕ(x, r ) is a positive measurable function on × (0, ), = diam X , 0 < ≤ ∞, and the following à priori assumptions hold: ( In the sequel we use the notation (2.13) For classical Morrey spaces L p,λ (R n ), as is known, |x| is global or local centered at the origin, respectively. We shall deal with the corresponding "model" function in the general setting of quasi-metric measure spaces with growth condition.
To this end we introduce the assumption that: Uniform Zygmund conditions hold: where 0 < r < , x ∈ and c does not depend on x and r .

Theorem 2.3
Let (X , d, μ) satisfy the growth condition and ϕ(x, r ) satisfy the Zygmund condition (2.14). Then On the right hand side we can apply the inequality (2.6) with L(x, r ) = ϕ(x, r ). Note that the condition (2.5) of Lemma 2.2 is satisfied, being easily derived from (2.12). By (2.6) we obtain due to (2.14). Let d(x, x 0 ) > 2kr. By the triangle inequality we have d(y, where the inequality (2.6) is applicable and we can proceed as in the previous case.
Let w(y) be an arbitrary weight on (X , d, μ), i.e. μ-a.e. positive function in L 1 loc (X ). We define the weighted generalized Morrey space L p,ϕ ,w (X ) as the space of functions with the finite norm (2.15)

Classes V + and V − of Radial Weights
The following classes of weight functions were introduced in [30], see also [26].
Note that for power weights we have The following lemma provides sufficient conditions for functions to belong to the classes V + and V − .
for some c > 0, then v ∈ V − . In particular, where it is assumed that < ∞. In the case = ∞, the statement holds with log A t replaced by log e max{t, 1 t }.
Definition 3.1 Let 1 < p < ∞. A sublinear operator T will be called p-admissible singular-type operator, if: (1) T satisfies the size condition of the form where ν comes from the growth condition (2.2) ; (2) T is bounded in L p (X , d, μ).

Remark 3.2 Usually the size condition is defined in the form
which insures (3.1). In the main theorem of this section, i.e. in the transference of L pboundedness to Morrey-boundedness, the form (3.1) of the size condition is sufficient for our goals.
First of all we keep in mind singular-type operators as p-admissible operators in view of Theorem 3.3. To be precise we define the singular operator T as where the kernel K (x, y) satisfies the conditions for some σ > 0, and assuming, as usual (see, for instance, [25]) that the operator T is bounded in L 2 (X ).
The following is known.  [14] and [15] in a more general setting) Let (X , d, μ) satisfy the growth condition (2.2), 1 < p < ∞. The singular operator T and the maximal singular operator T with a standard kernel, if bounded in L 2 (X ), are bounded in L p (X ). (3.8) and the following "hybrids"

As other examples we mention the Hardy-type operators
of Hardy and potential operators, where 0 < γ ≤ ν. Note that K γ,ν γ =ν = H and K γ,ν γ =ν = H. Operators (3.9) arise in the sequel in the reduction of weighted boundedness of weighted singular operators in Morrey spaces to the boundedness of non-weighted singular operators, see Sect. 3.2. The operators K γ,ν and K γ,ν are p-admissible operators as follows from Lemmas 3.4 and Theorem 3.5, taking Remark 3.6 into account.

Lemma 3.4 Let
x ∈ X \supp f and 0 < γ ≤ ν. Then the operators K γ,ν and K γ,ν satisfy the size condition: In the theorem below we use the modified maximal operator The operator K γ,ν is dominated by the operator M N as shown in the next theorem. (X , d, μ) satisfy the growth condition (2.2) and 0 < γ ≤ ν. Then

Theorem 3.5 Let
where A is the constant from the growth condition (2.2).
Proof of Theorem 3.8 is based on the following crucial lemma. (X , d, μ) satisfy the growth condition (2.2), 1 ≤ p ≤ ∞, and ∈ R. Then

Lemma 3.7 Let
where C does not depend on f , x ∈ X and r ∈ 0, 2 .
Proof The inequality (3.13), is proved by the known trick. We have where we choose β > max 0, ν p . It is easy to check that 1 r β ≤ cβ r dt t 1+β , with c = 2 β 2 β − 1 when 0 < r < 2 and < ∞; in the case = ∞ this holds with c = 1 and ≤ replaced by = . Then (B(x,s)) Since (ν − β) p < ν, by Lemma 2.1 we then obtain (X , d, μ) satisfy the growth condition (2.2) and = diam X ≤ ∞, let 1 < p < ∞ and T be a p-admissible sublinear operator of singular-type. Then (3.14) for every f ∈ L p loc (X ), where C does not depend on x ∈ X , r ∈ (0, ) and f .

Proof
We split the function f into the parts supported in a neighbourhood of the point x and outside it, in the usual way: where r > 0, and by the sublinearity of the operator T we have ,r )) .
By the assumption 2) in Definition 3.1, we obtain (B(x,2kr)) . (3.16) To estimate T f 2 , we make use of the assumption 1) from Definition 3.1: By the triangle inequality (2.1) it is easy to check that the conditions z ∈ B(x, r ) and y ∈ X \ B(x, 2kr) imply that Therefore, where the right-hand side does not depend on z, so that d(x, y) ν and then applying Lemma 3.7, we get (3.17) The simpler direct estimate (3.16) for T f 1 L p (B(x,r )) , as can be easily seen, is dominated by the estimate of similar form: which yields (3.14).
Theorem 3.9 (Transference Theorem). Let (X , d, μ) satisfy the growth condition (2.2) and 1 < p < ∞. Let also Then any sublinear operator T satisfying the size condition Proof The proof of this theorem is prepared by the pointwise estimate of Theorem 3.8: since it remains to pass to supremum in (3.14).
Note that in [22] there was studied the boundedness of the Hardy operator H in local Morrey spaces L p,ϕ x 0 (X ) and local vanishing Morrey spaces under other assumptions on the function ϕ and the triplet (X , d, μ).

Remark 3.10
The condition (3.19) is not needed in the case where ϕ(x, r ) does not depend on x or if contains a finite number of points.

Reduction of Boundedness of Weighted Singular Integral Operators with Size Condition and the Weighted Maximal Operator to the Weighted Boundeness of Hardy Operators
In this section we consider integral operators, in general of singular type: K (x, y) f (y)dμ(y) (3.20) under the only assumption that its kernel K (x, y) satisfies the size condition |K (x, y)| ≤ cd(x, y) −ν , (3.21) where ν comes from the growth condition. Note that in this section in fact we even do not need to know that ν comes from the growth condition, since in the proof of the pointwise estimate in the theorem below we use only properties of weights of the classes V ± , the fact that the operator T has the size condition with some ν > 0 and do not use at all any information about (X , d, μ).
Our goal is to study the boundedness of such operators in weighted Morrey spaces in the non-weighted space L p,ϕ (X ). We shall study the operator wT 1 w with "radial" weights The pointwise estimate of Theorem 3.11 shows that for any Banach function space with lattice property over an arbitrary quasi-metric measure space (X , d, μ), the boundedness of the weighted operator wT 1 w with w(y) = v[d(y, x 0 )], v ∈ V + V − , x 0 ∈ X , is reduced to the non-weighted boundedness of the operator T , boundedness of the weighted Hardy operators w H 1 w , wH 1 w and non-weighted boundedness of simple operators K γ,ν and K γ,ν .
We provide also a similar reduction for the weighted maximal operator. Then and and for some C > 0 and α > 0, then

28)
when v ∈ V + and

29)
when v ∈ V − , whereᾱ is the least integer greater or equal to α and the sumᾱ −1 m=1 should be omitted in the caseᾱ = 1.

Proof
We assume that f (x) > 0, x ∈ X , without loss of generality. By the size condition we have (3.30) Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
A function v(t) positive on (0, ) is called quasi-monotone near the origin if there exist numbers α, β ∈ R such that v(t) t α is a.i. and v(t) t β is a.d. in a neighborhood of the origin. In the case = ∞ it is called quasi-monotone at infinity if there exist a, b ∈ R such that v(t) t a is a.i. and v(t) t b is a.d. in a neighborhood of infinity. Functions quasi-monotone at the origin and infinity have finite Matuszewska-Orlicz indices at the origin and infinity, respectively. These indices are defined as follows:  x α is a.i. and M 0 (v) = inf β > 0 : v(x) x β is a.d. .
If v is quasi-monotone at infinity, then x a is a.i. and M ∞ (v) = inf b > 0 : v(x) x b is a.d. .