Abstract
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.
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Acknowledgements
The authors are grateful to professor Xavier Tolsa for providing us the above example and to the referee for his/her useful suggestions to improve the quality of the paper.
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J. Martín is partially supported by Grants PID2020-113048GB-I00 and PID2020-114167GB-I00 both funded by MCIN/AEI/10.13039/501100011033
W.A. Ortiz is partially supported by Grant 2017SGR395 (AGAUR, Generalitat de Catalunya)
Appendix : A
Appendix : A
1.1 A.1 Description of Measures μ Satisfying that the Map \(r\rightarrow \mu (B(x,r))\) is Continuous
Let (Ω,d) be a metric measure space, we say that a measure μ is metrically continuous with respect to metric d if all x ∈Ω and all r > 0 it holds that
where AΔB stands for a symmetric difference of sets A,B ⊂Ω and is defined as follows: AΔB := (A∖B) ∪ (B∖A).
The following lemma collects some basic facts about continuity of a measure with respect to the metric (see [9] and [1] for the proof).
Lemma 1
Let (Ω,d,μ) be a metric space with a Borel regular measure μ. Then the following hold:
-
(i)
If μ is continuous with respect to the metric d, then the map \(x\rightarrow \mu (B(x,r))\) is continuous in d.
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(ii)
If for every x ∈Ω and every r > 0 it holds that μ(∂B(x,r)) = 0, then μ is continuous with respect to the metric d.
-
(iii)
If for every x ∈Ω the function \(r\rightarrow \mu (B(x,r))\) is continuous, then μ is continuous with respect to the metric d.
It is easy to see that if we take \(\mathbb {R}^{n}\) with Lebesgue measure (or with an absolutely continuous measure respect to the Lebesgue measure) with the Euclidean distance, then this measure is metrically continuous. In fact we have more (see [9, Proposition 2.1]) if (Ω,d,μ) and (Ω,d,ν) are metric measure spaces then if μ ≺≺ ν and ν is metrically continuous, then μ is metrically continuous too.
An important example is the following (see [32]).
Lemma 2
Let μ be a nonnegative Radon measure on \(\mathbb {R}^{n}\). Assume that for any point \(p\in \mathbb {R}^{n}\), μ({p}) = 0, then we choose the coordinate axes in such a way that μ(∂Q) = 0 for all cubes Q with sides parallel to the axes. In particular the function \(\ell \rightarrow \mu (Q(x,\ell ))\) where ℓ denotes the length of the edge and x is the center of Q, is continuous.
1.2 A.2 An Example of an α −lower Bounded Measure which is not c −almost Continuous
Consider \(\mathbb {R}^{2}\) with the distance \(d_{\infty }(x,y)=\max \limits \left \{ \left | x\right | ,\left | y\right | \right \} \) and the measure μ = Lebesgue measure in the plane + length measure in vertical axis + length measure in vertical straight line passing through (1,0). It is easy to see that
however, is not c −almost continuous.
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Martín, J., Ortiz, W.A. Sobolev Embeddings for Fractional Hajłasz-Sobolev Spaces in the Setting of Rearrangement Invariant Spaces. Potential Anal 59, 1191–1204 (2023). https://doi.org/10.1007/s11118-022-10006-z
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DOI: https://doi.org/10.1007/s11118-022-10006-z