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Global Calderón-Zygmund theory for fractional Laplacian type equations

Sun-Sig Byun, Kyeong Bae Kim, Deepak Kumar

Abstract

We establish several fine boundary regularity results of weak solutions to non-homogeneous $s$-fractional Laplacian type equations. In particular, we prove sharp Calderón-Zygmund type estimates of $u/d^s$ depending on the regularity assumptions on the associated kernel coefficient including VMO, Dini continuity or the Hölder continuity, where $u$ is a weak solution to such a nonlocal problem and $d$ is the distance to the boundary function of a given domain. Our analysis is based on point-wise behaviors of maximal functions of $u/d^s$.

Global Calderón-Zygmund theory for fractional Laplacian type equations

Abstract

We establish several fine boundary regularity results of weak solutions to non-homogeneous -fractional Laplacian type equations. In particular, we prove sharp Calderón-Zygmund type estimates of depending on the regularity assumptions on the associated kernel coefficient including VMO, Dini continuity or the Hölder continuity, where is a weak solution to such a nonlocal problem and is the distance to the boundary function of a given domain. Our analysis is based on point-wise behaviors of maximal functions of .

Paper Structure

This paper contains 11 sections, 25 theorems, 359 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Comparison estimates
  4. The case of discontinuous coefficients
  5. Localized problems.
  6. Proof of Theorem \ref{['thm.vmo']}
  7. The case of continuous coefficients
  8. Localized problems
  9. Proof of Main Theorems
  10. General kernel coefficients
  11. Some sharpness results

Key Result

theorem 1

Let $u\in W^{s,2}( B_{1})\cap L^1_{2s}(\bbR^n)$ be a weak solution to eq: thm. For any $q\in[2_*,n/s)$ and $\rho_0>0$, there is a constant $\delta=\delta(n,s,\Lambda,q)$ such that if $A$ is $(\delta,\rho_0)$-vanishing in $B_1\times B_1$, then we have for some constant $c=c(n,s,\Lambda,\alpha,q,\rho_0)$.

Theorems & Definitions (64)

  • definition 1
  • remark 1
  • definition 2
  • theorem 1
  • remark 2
  • remark 3
  • definition 3
  • theorem 2
  • corollary 1
  • remark 4
  • ...and 54 more