L^2-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type
- Series
- Analysis Seminar
- Time
- Wednesday, March 9, 2022 - 14:00 for 1 hour (actually 50 minutes)
- Location
- ONLINE (Zoom link in abstract)
- Speaker
- Carmelo Puliatti – University of the Basque Country, Spain – carmelo.puliatti@ehu.eus
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 H\"older continuity assumption on the coefficients of the matrix $A$.
This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.