Pointwise convergence of the solutions to the initial data for the abstract heat equation

Title: Pointwise convergence of the solutions to the initial data for the abstract heat equation

Abstract: In a recent study, Hartzstein, Torrea, and Viviani characterized all the weights \(v\) for which the solution to the classical heat equation with initial data \(f\), where \(f\in L^p_v(\mathbb{R}^n)\), converges to \(f\) as \(t\to 0\), almost everywhere and for every \(f\in L^p_v(\mathbb{R}^n)\). This work is, of course, in the spirit of Carleson’s program, where similar investigations have been conducted for the Schrödinger operators. In this talk, we will extend the results of Hartzstein et al. to a broader class of operators on metric measure spaces with a volume doubling condition, including \(\phi\)-nonlocal operators, mixed local-nonlocal operators, the Laplacian with a Hardy potential, the Laplace-Beltrami operators, Laplacian on fractals and many others.

This talk, which is based on a recent joint work with Bhimani and Dalai, is going to be a mix of probability, PDE and harmonic analysis.