$L^p$-theory for fractional gradient PDE with VMO coefficients

TL;DR

This paper establishes $L^p$ estimates for fractional derivatives in elliptic PDEs with VMO coefficients, extending classical regularity results to fractional orders.

Contribution

It introduces $L^p$ regularity results for fractional elliptic PDEs with VMO coefficients, expanding the scope of known elliptic regularity theory.

Findings

Proved $L^p$ estimates for fractional derivatives of solutions.

Extended classical regularity results to fractional elliptic equations.

Demonstrated the applicability of VMO coefficients in fractional PDE analysis.

Abstract

In this paper, we prove $L^p$ estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are $VMO$. In particular, our work extends the optimal regularity known in the second order elliptic setting to a spectrum of fractional order elliptic equations.