Local grand Lebesgue spaces on quasi-metric measure spaces and some applications

We introduce local grand Lebesgue spaces, over a quasi-metric measure space (X,d,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ( X,d, \mu ) $$\end{document}, where the Lebesgue space is “aggrandized” not everywhere but only at a given closed set F of measure zero. We show that such spaces coincide for different choices of aggrandizers if their Matuszewska–Orlicz indices are positive. Within the framework of such local grand Lebesgue spaces, we study the maximal operator, singular operators with standard kernel, and potential type operators. Finally, we give an application to Dirichlet problem for the Poisson equation, taking F as the boundary of the domain.

We introduce a version of grand Lebesgue spaces which we call local grand Lebesgue spaces. The word "local" means that the Lebesgue space is "aggrandized" not everywhere but only at a given set of measure zero. We consider such spaces in the general setting of quasi-metric measure spaces (X , d, μ). To be more precise, let F ⊂ X be a closed set of measure zero, in particular in the Euclidean case we can choose the boundary of a domain as the set F. We introduce the space L p),θ F,a (X , μ) by the norm where the "aggrandizer" a : [0, D] → R + , D = diam(X ), is a suitable function with a(0) = 0. Such local grand Lebesgue spaces in the Euclidean case and F consisting of a unique point where studied in [15]. We also introduce the corresponding vanishing local grand Lebesgue space VL p),θ F,a (X , μ). We prove some properties of such spaces. In particular, we consider relations between such spaces under different choice of the function a and show that two local grand Lebesgue spaces with aggrandizers a and b coincide if the Matuszewska-Orlicz indices of a and b are positive. We also demonstrate that the space L p (X ) is strictly embedded into such local grand Lebesgue space when (X , d, μ) has some regularity properties. If (X , d, μ) is of homogeneous type, within the frameworks of the spaces L p),θ F,a (X ) and VL p),θ F,a (X , μ), we study the maximal operator, singular operators with standard kernel, and potential type operators. Here we used the results of the recent papers [16,17], where there were studied Muckenhoupt weights with singularities at a given set which allowed us to prove the boundedness of these operators via interpolation with respect to weights.
Finally, we give an application to Dirichlet problem for the Poisson equation, taking F as the boundary of the domain, using some pointwise estimates obtained in [18].
The paper is organized as follows. In Sect. 2 we supply necessary preliminaries on quasi-metric measure spaces and the notion of Matuszewska-Orlicz indices. In Sect. 3 we introduce the local grand Lebesgue spaces and show that such spaces coincide for different choices of aggrandizers if their Matuszewska-Orlicz indices are positive. In Sect. 4 we recall the notion of lower Assouad codimension and its connection with power distance functions as Muckenhoupt weights. In Sect. 5 we study the boundedness of the maximal operator, singular operators with standard kernel, and potential operator. In Sect. 6 we give an application to Dirichlet problem for the Laplace equation. Notation:

On quasi-metric measure spaces
By (X , d, μ) we denote a quasi-metric measure space with the quasi-distance d satisfying the standard conditions: and by μ a regular Borel measure. The set X may be bounded or unbounded. We The measure is said to satisfy the growth condition, if there exists N > 0 such that where N is not necessarily an integer. As is well known, extension of various results from the Euclidean case X = ⊆ R n to the general case of (X , d, μ) often depends on the choice of the assumption (2.2) or (2.3) for the measure μ.
We say that (X , d, μ) is N -Ahlfors regular, if there exists N > 0 such that where N is not necessarily an integer. For more details on quasi-metric measure spaces we refer, e.g., to [19,20].

The class G(I) of aggrandizers
Let I = (0, D) with 0 < D ≤ ∞. By L ∞ + (I ) we denote the cone of non-negative functions in L ∞ (I ). We define the class G(I ), following [15], as the set of functions a ∈ L ∞ + (I ) satisfying the conditions: (1) a(t) is continuous in a neighbourhood of the origin and a(0) = 0; and (2) inf t∈(κ,D) a(t) > 0 for every κ ∈ (0, D).
We call functions a ∈ G(I ) aggrandizers.

Matuszewska-Orlicz indices
In Sect. 3 we use the notion of Matuszewska-Orlicz indices m(a) and M(a) of a nonnegative function a introduced in [21], see also [22] where properties of these indices are given in a form convenient for us. Note also that where α ∈ R and β ∈ R + . A non-negative function a on I is called quasi-monotone if there exist α, β ∈ R such that a(t)t −α is almost increasing (a.i.) and a(t)t −β is almost decreasing (a.d.). A quasi-monotone function has finite indices and m(a) = sup{α | a(t)t −α is a.i.} and M(a) = inf{β | a(t)t −β is a.d.}. Everywhere in the sequel, when considering indices of a function, we suppose that it is quasi-monotone near the origin.

Definition 3.1 Let
be an open set in X and F ⊂ X be a closed non-empty set with μF = 0. For a function a ∈ L ∞ + (I ), we define the local grand Lebesgue space L In particular, we may choose the boundary of as the set F (assuming that μ(∂ ) = 0). In the case the set F consists of a single point x 0 ∈ , we use the notation Note that the space L ∞),θ F,a ( ) contains, in general, unbounded functions. Let, for simplicity, X = R n and be a bounded domain in R n . Then It is worth pointing out that a p = ∞-version of the usual well-known grand spaces was studied in [23]. The embedding holds, whenever a ∈ L ∞ + (I ). To ensure that the local grand space L p),θ F,a ( , μ) is larger than L p ( , μ), we should restrict the choice of functions a(t), mainly by the assumption that a(0) = 0. In most of the statements, we assume that a ∈ G(I ).
The norm (3.1) is equivalent to for every s ∈ (0, D). The norm in the Lemma 3.2 is written as f L p),θ F,a; ( ,μ) to underline dependence on the range for ε.
Proof The proof is straightforward.
The corresponding vanishing space VL

Basic properties
The spaces L p),θ F,a ( , μ) do not depend much on the "properly chosen" aggrandizer a, as shown in Theorem 3.4. First, we prove the following lemma on embedding between such spaces.

Lemma 3.3
The following is valid: (2) If the function a is almost increasing near the origin, and F 1 and F 2 are closed non-empty sets such that Proof To show (1) it suffices to use Lemma 3.2. The proof of (2) follows from noticing that δ F 2 (x) ≤ δ F 1 (x) and then it remains to use the fact that a is almost increasing.

Theorem 3.4 Let a and b be quasi-monotone functions on
up to equivalence of norms.
Proof It suffices to refer to (3.2) and the fact that, for an arbitrarily small η > 0, there are constants c(η) and C(η) such that One can be interested in the strict embedding By Lemma 3.3 (2) it suffices to consider the case F = {x 0 }. We start with the following lemma, where neither growth nor doubling condition are assumed.
Proof Suppose that u ∈ L p ( , μ). Then We can use the norm in equivalence form (3.2) so we can replace by B(x 0 , r ). Using dyadic decomposition, we obtain which completes the proof.

Power of distances as Muckenhoupt weights
In the study of operators of grand local Lebesgue spaces we will use the important Proposition 4.3. We now recall some definitions and introduce the corresponding notations.

Definition 4.1 (Lower Assouad codimension) Let F ⊂ X and
The lower Assouad codimension, denoted by N F , is the supremum of all ν ≥ 0 for which there exists a constant C ≥ 1 such that for every x ∈ F and all 0 < r < R < 2 diam(X ).

Definition 4.2 (Muckenhoupt weights)
For 1 < p < ∞, the class A p (X ) is defined as the set of all weights w on X such that where the supremum is taken with respect to all balls B in X . For p = 1, the class  (X , d, μ) be homogeneous, F be a closed non-empty set, and 1 < p < ∞. Then

Proposition 4.4 Let (X , d, μ) be homogeneous and F be a closed non-empty set. Then
where δ F (x) = d(x, F) and N F is the lower Assouad codimension of the set F.
In the study of operators in Sect. 5 we will base ourselves on Propositions 4.3 and 4.4. Note that, in applications, when the set F is supplied with a positive measure ν, the Proposition 4.5 is more practical. To this end, we need the following definition.
By F s = {F} we denote the family of sets F ⊂ X satisfying the conditions: (1) F is closed; (2) μF = 0; and (3) the set F ⊂ X is s-Ahlfors regular with respect to the s-dimensional Hausdorff measure.

Proposition 4.5 ([16, Theorem 7]) Let (X , d, μ) be a homogeneous quasi-metric measure space N -Ahlfors regular and F ∈ F s , with s < N . Then
In the case where X is N -Ahlfors regular and F ∈ F s , then Proposition 4.3 turns into Proposition 4.5. Non-trivial examples of fractal spaces X and sets F may be found in [16].
Statements of the type of lemma below are known, see for instance [27, Lemma 3.5].

On operators in local grand Lebesgue spaces
We start with the simple statement. Concerning the boundedness of operators in grand local Lebesgue spaces, we note that one of the principal technical tools, i.e., the Stein-Weiss interpolation theorem, remains valid in the general setting of arbitrary quasi-metric measure spaces, see [28,Cor. 2.2] or [29, 17 and 120 pp.].
is a continuous linear operator. Then

Maximal operator
Recall that the maximal operator is defined as (5.1) where the constant C does not depend on the weight w or the function f ; see, e.g. [30].
In the case p = ∞, for the space L ∞ (X , w) we have the following lemma, where

Lemma 5.3 The maximal operator M is bounded in L ∞ (X , w) if and only if
Proof Sufficiency: Necessity: (X , d, μ) be homogeneous, F be a closed non-empty set,

Theorem 5.4 Let
where the constants C do not depend on ε. The result now follows from Theorem 5.1. Arguments for p = ∞ are similar, being based on Lemma 5.3 and Proposition 4.4.

Singular operators
We consider the Calderón-Zygmund singular operator K with the standard kernel, as defined in [19] and [20, p. 502]. Let k : X × X \ {(x, x) | x ∈ X } → R be a measurable function satisfying the conditions: d(x, y)) , x, y ∈ X , x = y; and |k(x 1 , y) − k(x 2 , y)| + |k(y, d(x 2 , y)) , Here ω is a positive, nondecreasing function on (0, ∞) satisfying the 2 -condition (i.e., ω(2t) ≤ cω(t) for all t > 0 and some c > 0 independent of t) and the Dini condition 1 0 We also assume as well that for some p 0 , 1 < p 0 < ∞, and all f ∈ L p 0 (X ) the limit exists a.e. and that the operator K is bounded in L p 0 (X ).
The following proposition was proved in [31], see also [20,Theorem A,p. 503]. (X , d, μ) be a space of homogeneous type and 1 < p < ∞. If w ∈ A p (X ), then the operator K is bounded in L p (X , w 1/ p ).

Proposition 5.5 Let
Note that the boundedness of the Calderón-Zygmund operators in L p (X , w) follows from the boundedness of the maximal operator M in L p (X , w) and [L p (X , w)] by the known arguments, see e.g. [32,Lemma 3.2]. Proof We apply the interpolation Proposition 5.2 with p 0 = p 1 = p, w 0 (x) ≡ 1, and w p 1 (x) = δ F (x) pε 0 , corresponding to the cases ε = 0 and ε = ε 0 in the definition of the norm of the space. We also keep in mind that one can replace by ε 0 , according to Lemma 3.2. The boundedness of the Calderón-Zygmund operator, at the endpoints, follows from Proposition 4.3 (taking ε 0 < N F/p ) and the boundedness of the operator K in the space L p (X , w 1/ p ), w ∈ A p (X ), see Proposition 5.5. Note that, in the case of "nice" boundaries, the codimension is always positive and equals to n − 1 in case of Lipschitz boundary.
For a given η > 0, we say that μ satisfies a reverse doubling condition if The following proposition was proved in [17, Theorem 4.1], see also [33] for the optimal exponent for the A p constant in the Euclidean case.

Proposition 5.8
Let α > 0. Assume that the reverse doubling condition (5.4) holds with the exponent η = α and that there exists Q > α such that μB ≥ crad(B) Q for all balls B ⊂ X . Let ∅ = F ⊂ X be a closed set, Proof The proof follows closely the one given in Theorem 5.6. We apply the interpolation Proposition 5.2 with p 0 = p, The boundedness of the Riesz potential operator at the endpoints is given in Proposition 5.8, under the choice ε 0 < N F/p for the L p 1 (X , w 1/ p 1 1 ) → L q 1 (X , σ 1/q 1 1 )-boundedness. This completes the proof.

An application to Dirichlet problem
In this section we give an application to Dirichlet problem basing ourselves on some results from [18] for classical weighted Lebesgue spaces. Recall that the Dirichlet problem is given by − u = f , , The operator K ji is defined as where M is the maximal operator and K * ji is defined in (6.3).
We now introduce the Sobolev grand local space W In the following theorem, the set F may be any closed non-empty set with N F > 0 in , although the most interesting case seems to be when F = ∂ . Theorem 6.2 Let ⊂ R n be a bounded C 2 domain, 1 < p < ∞, F ⊂ be a closed non-empty set, N F > 0, θ > 0, a ∈ G(I ), and m(a) > 0. If u is the solution of problem (6.1), then there exists a constant C depending only on n and such that u W 2, p),θ F,a ( ) ≤ C f L p),θ F,a ( ) .

(6.4)
Proof The result follows immediately from the pointwise inequalities from Proposition 6.1 and the boundedness of the maximal and the maximal singular operators. The boundedness of the maximal operator is given in Theorem 5.4 and the boundedness of the maximal singular operator follows from the known estimate where K * is the maximal singular operator related with the singular operator K , see [34,Theorem 2,p. 205].