# 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