Marcinkiewicz estimates for solution to fractional elliptic Laplacian equation

In this paper, we consider the Marcinkiewicz summability of solutions to the following fractional elliptic problem {(-Delta)(s)u = f(x), is an element of Omega, u > 0, is an element of Omega, u = 0, x is an element of R-N \ Omega, where (-Delta)(s) denotes the fractional Laplacian operator, s is an element of (0, 1),Omega subset of R-N is a bounded domain with Lipschitz boundary, f belongs to some Marcinkiewicz space M-m(Omega) with m > 1. The main novelty of this paper is actually the fact that the solutions to the above equation are bounded if m > (2N/N+2s)(2), instead of m > N/2S. The results of this paper are new even for s = 1. (C) 2019 Elsevier Ltd. All rights reserved.