Marcinkiewicz estimates for solution to fractional elliptic Laplacian equation

Abstract

In this paper, we consider the Marcinkiewicz summability of solutions to the following fractional elliptic problem (−Δ)su=f(x),∈Ω,u>0,∈Ω,u=0,x∈RN∖Ω,where (−Δ)s denotes the fractional Laplacian operator, s∈(0,1), Ω⊂RN is a bounded domain with Lipschitz boundary, f belongs to some Marcinkiewicz space Mm(Ω) with m>1. The main novelty of this paper is actually the fact that the solutions to the above equation are bounded if m>2NN+2s2, instead of m>N2s. The results of this paper are new even for s=1.