A failing in the Calderon-Zygmund theory of Dirichlet problems for linear equations with discontinuous coefficients

Abstract

This note is devoted to the Calderon-Zygmund theory for linear differential operators with discontinuous coefficients. It is known that the theory holds if the datum $f(x)$, in (1.1) belongs to the Lebesgue space $L^m\Omega$, with $1<m<\frac{2N}{N+2}$ (see [6]). In this paper we prove that the theory fails if $m>\frac{N}{2}$.