Integrability of Green potentials in fractal domains

Abstract

We prove L^q-inequalities for the gradient of the Green potential (Gf) in bounded, connected NTA-domains in Rⁿ, n≥2. These domains may have a highly non-rectifiable boundary and in the plane the set of all bounded simply connected NTA-domains coincides with the set of all quasidiscs. We get a restriction on the exponent q for which our inequalities are valid in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.