Analysis Seminar: "L2-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type"

Abstract: We consider a uniformly elliptic operator LA in divergence form associated with a matrix A with real, bounded, and possibly non-symmetric coefficients. If a proper L1-mean oscillation of the coefficients of A satisfies suitable Dini-type assumptions, we prove the following: if μ is a compactly supported Radon measure in Rn+1, n≥2, the L2(μ)-operator norm of the gradient of the single layer potential Tμ associated with LA is comparable to the L2-norm of the n-dimensional Riesz transform Rμ, modulo an additive constant. This makes possible to obtain direct generalization of some deep geometric results, initially obtained for the Riesz transform, which were recently extended to Tμ under a Hölder continuity assumption on the coefficients of the matrix A. This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.

Host: Walton Green