论文标题
通过分数差异商的本地规律性$ s $ harmonic功能非常弱
Local Regularity of very weak $s$-harmonic functions via fractional difference quotients
论文作者
论文摘要
本文的目的是给出一个新的证明，表明任何非常弱的$ s $ harmonic函数$ u $ u $ u $ u $ b $ b $都很顺利。作为第一步，我们改善了$ U $的本地总和属性。然后，我们利用量身定制的差异商方法的合适版本，以摆脱整体内核的奇异性，并获得$ h^{s} _ {\ rm loc} $ norm of $ u $的$ h^{s} _ {s} _ {s}的局部线性估计。最后，通过应用更多的标准方法，例如椭圆规律性和Schauder估计，我们达到了$ U $的实际分析性。根据作者的知识，差异商技术是新的。
The aim of this paper is to give a new proof that any very weak $s$-harmonic function $u$ in the unit ball $B$ is smooth. As a first step, we improve the local summability properties of $u$. Then, we exploit a suitable version of the difference quotient method tailored to get rid of the singularity of the integral kernel and gain Sobolev regularity and local linear estimates of the $H^{s}_{\rm loc}$ norm of $u$. Finally, by applying more standard methods, such as elliptic regularity and Schauder estimates, we reach real analyticity of $u$. Up to the authors' knowledge, the difference quotient techniques are new.
