Sobolev Embeddings for Fractional Hajłasz-Sobolev Spaces in the Setting of Rearrangement Invariant Spaces

We obtain symmetrization inequalities in the context of Fractional Hajłasz-Sobolev spaces in the setting of rearrangement invariant spaces and prove that for a large class of measures our symmetrization inequalities are equivalent to the lower bound of the measure.


Introduction
Let us consider a metric measure space (Ω, d, μ) where μ is a Borel measure on (Ω, d) such 0 < μ(B) < ∞, for every ball B in Ω.We will always assume μ(Ω) = ∞ and μ({x}) = 0 for all x ∈ Ω.Let X be a rearrangement invariant (r.i.) space on Ω (see Section 2.2.1 below).In this paper, we introduce the fractional Hajłasz-Sobolev spaces M s,X (Ω) for s > 0, and we will focus on understanding the relation between Sobolev embeddings theorems for spaces M s,X (Ω) and the growth properties of the measure μ.
Let s > 0 and let X be a r.i.space on Ω.We say that f ∈ M s,X (Ω), if f ∈ X, and there exits a non-negative measurable function g ∈ X such that |f (x) − f (y)| ≤ d(x, y) s (g(x) + g(y)) μ − a.e.x, y ∈ Ω. ( A function g satisfying (1) will be called a s−gradient of f .We denote by D s (f ) the collection of all s−gradients of f .The homogeneous Hajłasz-Sobolev space Ṁs,X (Ω) consists of all functions f ∈ X for which f Ṁs,X (Ω) = inf g∈D s (u) g X is finite.The Hajłasz-Sobolev space M s,X (Ω) is Ṁs,X (S) ∩ X equipped with the norm f M s,X (Ω) = f X + f Ṁs,X (Ω) .
It is well known that the lower bound for the growth of the measure implies Sobolev embedding theorems for Hajłasz-Sobolev spaces M 1,p (see [11] and [12]).
The converse problem, i.e. when the embedding implies a lower bound for the growth of the measure, has been considered by several authors (see [10,18,[20][21][22][23] and the references quoted therein).In the recent paper [2], R. Alvarado, P. Górka and P. Hajłasz show that in fact if (3) holds with q = αp/(α − p), then lower bound for the growth (2) holds.The purpose of this paper is to obtain an analogous result for M s,X spaces.This will be done by obtaining pointwise estimates between the special difference f * * (t)−f * (t) (called the oscillation 2 of f ) and the function g (see Theorem 1 below), i.e. we will see that for a wide range of measures, condition (2) implies for every f ∈ M s,L 1 +L ∞ and g ∈ D s (f ).Moreover, if 0 < s ≤ 1, then (4) implies (2).
1 μ is said to be doubling provided there exists a constant C > 0 such that μ(2B) ≤ Cμ(B) for all balls B ⊂ Ω. 2 Here f * is the decreasing rearrangement of f , f * * (t) = 1 t t 0 f * (s)ds, for all t > 0, (see Section 2.2).
Symmetrization inequalities imply Sobolev inequalities in the setting of rearrangement invariant spaces.Indeed, from (4) we obtain: for any r.i.space X with upper Boyd3 index ᾱX < 1, we have where c = c(s, α, X).
Notice that we avoid one common drawback of the usual approaches to Sobolev inequalities which require the choice of specific norms before one starts the analysis.Instead, we work with pointwise symmetrization inequalities which are *universal* and it is the inequalities themselves that select the *correct* spaces.
For example, in the particular case of X = L p (see Corollary 1 below) we obtain that if 1 > s/α > 1 p , then4 On the other hand, since p < p * s we have that .

Remark 2
The technique to obtain Sobolev oscillation type inequalities has been developed by M. Milman and J. Martín (see [27,28] and [29]) and provide a considerable simplification in the theory of embeddings of Sobolev spaces based on rearrangement invariant spaces.
The paper is organized as follows.In Section 2, we introduce the notation and the standard assumptions used in the paper, in Section 3, we will obtain oscillation type inequalities for spaces M s,X , we will see that they are equivalent to the lower bound for the growth of the measure and will obtain Sobolev type embedding of M s,X into a rearrangement invariant spaces.Finally, in the Appendix A we will give some properties of the measures we will be working with.

Preliminaries
In this section we establish some further notation and background information and we provide more details about metrics spaces and r.i.spaces that we will working with.

Metric Spaces
Let (Ω, d) be a metric space.As usual a ball B in Ω with a center x and radius r > 0 is a set B = B(x, r) := {y ∈ Ω; d(x, y) < r}.Throughout the paper by a metric measure space we mean a triple (Ω, d, μ), where μ is a Borel measure on (Ω, d) such 0 < μ(B) < ∞, for every ball B in Ω, we also assume that μ(Ω) = ∞ and μ({x}) = 0 for all x ∈ Ω.
We will say that a measure μ is α−lower bounded if there are b, α > 0 such that for all x ∈ Ω and r > 0.
For simplicity we assume in what follows that μ(B(x, r)) ≥ r α .
In what follows we will, furthermore, assume that the measure μ is continuous, i.e. μ satisfies that the map r → μ(B(x, r)) is continuous 5 or that μ is doubling, i.e. there exists a constant C D such that, for all x ∈ Ω and for all r > 0, we have that Notice that in both cases there is a constant c = c μ ≥ 1 such that given t > 0, for all x ∈ Ω, there is a positive number r(x) such that In the doubling case, given x ∈ Ω, consider r 0 (x) = sup {r : μ(B(x, r)) < t} and take r such that r < r 0 (x) < 2r, then In what follows we call these measures c−almost continuous 6 .

Background on Rearrangement Invariant Spaces
For measurable functions f : Ω → R, the distribution function of f is given by It is easy to see that for any measurable set Since f * is decreasing, the function f * * , defined by is also decreasing and, moreover, Remark 3 An elementary computation shows that t and that the function t → t (f * * (t) − f * (t)) is increasing.Moreover, it is well known and easy to see

Rearrangement Invariant Spaces
We recall briefly the basic definitions and conventions we use from the theory of rearrangement-invariant (r.i.) spaces and refer the reader to [6,25], for a complete treatment.We say that a Banach function space X = X(Ω) on (Ω, d, μ) is rearrangement-invariant (r.i.) space, if g ∈ X implies that all μ− measurable functions f with the same decreasing rearrangement function with respect to the measure μ, i.e. such that f * = g * , also belong to X, and, moreover, f X = g X .For any r.i.space X(Ω) we have with continuous embedding.A r.i.space X(Ω) can be represented by an r.i.space on the interval (0, μ(Ω)), with Lebesgue measure, X = X(0, μ(Ω)), such that for every f ∈ X.A characterization of the norm • X is available (see [6, Theorem 4.10 and subsequent remarks]).Typical examples of r.i.spaces are the L p -spaces, Lorentz spaces and Orlicz spaces.
The associated space X (Ω) of X(Ω) is the r.i.space of all measurable functions h for which the r.i.norm given by is finite.Note that by the definition (10), the generalized Hölder inequality holds.
Classically conditions on r.i.spaces are given in terms of the Hardy defined by The boundedness of these operators on r.i.spaces can be described in terms of the so called Boyd indices 7 defined by where h X (s) denotes the norm of the compression/dilation operator E s on X, defined for s > 0, by E s f (t) = f * ( t s ).For example if X = L p with p > 1, then ᾱX = α X = 1 p .It is well known that (see [26], and [33])

Symmetrization Inequalities and Embeddings for Fractional Hajłasz-Sobolev Spaces
The method of proof of the following theorem follows the ideas of [30, Theorem 2] (see also [31]).
Theorem 1 Let (Ω, d, μ) be a metric measure space such that μ is c−almost continuous and α−lower bounded.Let s > 0, f ∈ M s,L 1 +L ∞ and g ∈ D s (f ).Let 0 < p ≤ 1.Then, for all t > 0, we have where C = C(c, p) is a constant that just depends on c and p. 15) is obvious, otherwise let notice that by ( 9) the set A is not empty.Given x ∈ A, since μ is c−almost continuous, there is a radius r(x) such that 2t ≤ μ(B(x, r(x))) ≤ 2ct.
Theorem 2 Let (Ω, d, μ) be a metric measure space such that μ is c−almost continuous.
Remark 4 In Kalis' 2007 PhD thesis at FAU (see also [20]) was proved that Sobolev embeddings estimates for Hömander vector fields imply a lower bound for the growth of the measure.
Theorem 3 provides us the following Sobolev embedding result for Fractional Hajłasz-Sobolev spaces.
Theorem 3 Let (Ω, d, μ) be a metric measure space such that μ is c−almost continuous and α−lower bounded.Let X be a r.i.space.Let f ∈ M s,X and g ∈ D s (f ).
Remark 5 In the classical setting, embedding theorems for W 1,p (R n ) have different behavior when < n, p = n,or p > n.The point is that In the metric-measure context the counterpart condition (25) is provided by the lower bound for the growth of the measure (5).Notice also the different character on the embeddings for p < n, p = n, or p > n in the r.i.context is done by the role of the Boyd index.
Corollary 1 Under conditions of Theorem 3, in the particular case of X = L p , (0 < p < ∞), we obtain