The Neumann function and the <i>L</i> ^<i>p</i> Neumann problem in chord-arc domains

Abstract We construct the Neumann function in a 1-sided chord-arc domain (i.e., a uniform domain with an Ahlfors regular boundary), and establish size and Hölder continuity estimates up to the boundary. We then obtain a Kenig-Pipher type theorem, in which L p solvability of the Neumann problem is shown to yield solvability in L q for 1 &lt; q &lt; p , and in the Hardy space H 1 , in 2-sided chord-arc domains, under suitable background hypotheses.