A fractional Gehring lemma, with applications to nonlocal equations
A fractional Gehring lemma, with applications to nonlocal equations
复制标题
分数格林引理及其在非局部方程中的应用
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Y. Sire
中科院分区：
文献类型：
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作者：
Tuomo Kuusi;G. Mingione;Y. Sire
We describe a fractional version of the classical Gehring lemma. As a consequence, new self-improving regularity properties of solutions to integrodifferential equations emerge. 1. The classical Gehring lemma The Gehring lemma [7, 9] is a fundamental tool in modern nonlinear analysis, with crucial implications in several different fields, ranging from nonlinear elliptic and parabolic equations to the calculus of variations, from quasiconformal geometry to stability issues [2, 6, 11]. Its ultimate essence relies on a basic, self-improving property of certain kind of inequalities, called reverse Holder type inequalities. This can be described as follows: if one can control the L-means of a given function f ∈ L, at all scales, with similar L-means, and p > q, then the function f is necessarily better than just being in L. Starting from the original work of Gehring, there have been several different versions of this result; see [9] for a panorama. The following one, involving reverse inequalities with increasing support, can be for instance found in [8]. Theorem 1.1. Let f ∈ Lploc(Ω), p > 1 be a non-negative function such that the following reverse Holder type inequality holds whenever B is a ball in the open subset Ω ⊂ R: (∫
