Partial Differential Equations seminar
Speaker: Anup Biswas (IISER Pune)
Host: Mayukh Mukherjee
Title: Pointwise convergence of the solutions to the initial data for the abstract heat
equation
Time, day and date: 2:30:00 PM, Tuesday, March 11
Venue: Ramanujan Hall
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 Laplacia-Beltrami operators, Laplacian on fractals and many others.
This talk is based on a recent joint work with Bhimani and Dalai.