Analysis Seminar

\(L^2\)-Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type

Carmelo Puliatti, University of the Basque Country

Friday, February 18th, 2022
1:00 PM - 2:00 PM
Zoom ID 573 239 4086

We consider a uniformly elliptic operator \(L_A\) in divergence form associated with an \((n+1)\times(n+1)\)-matrix \(A\) with real, bounded, and possibly non-symmetric coefficients. If a proper \(L^1\)-mean oscillation of the coefficients of \(A\) satisfies suitable Dini-type assumptions, we prove the following: if \(\mu\) is a compactly supported Radon measure in \(\mathbb{R}^{n+1}\), \(n \geq 2\), and \(T_\mu f(x)=\int \nabla_x\Gamma_A (x,y)f(y)\, d\mu(y)\) denotes the gradient of the single layer potential associated with \(L_A\), then \(1+ \|T_\mu\|_{L^2(\mu)\to L^2(\mu)}\approx 1+ \|\mathcal R_\mu\|_{L^2(\mu)\to L^2(\mu)},\) where \(\mathcal R_\mu\) indicates the \(n\)-dimensional Riesz transform. This makes possible to obtain direct generalization of some deep geometric results, initially obtained for \(\mathcal R_\mu\), which were recently extended to \(T_\mu\) under a Holder continuity assumption on the coefficients of the matrix \(A\).

This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.

Event contact: dan dot geba at rochester dot edu