Pointwise convergence for the heat equation on tori $\mathbb T^n$ and waveguide manifold $\mathbb T^n \times \mathbb R^m$

TL;DR

This paper characterizes the weighted Lebesgue spaces on tori and waveguide manifolds where heat equation solutions converge pointwise to initial data, also exploring boundedness of maximal operators.

Contribution

It provides a complete characterization of these spaces for pointwise convergence and maximal operator boundedness on complex manifolds.

Findings

Identifies specific weighted Lebesgue spaces ensuring pointwise convergence.

Characterizes boundedness of maximal operators on these manifolds.

Provides tools for analyzing heat equation behavior on complex geometries.

Abstract

We completely characterize the weighted Lebesgue spaces on the torus $\mathbb T^n$ and waveguide manifold $\mathbb T^n \times \mathbb R^m$ for which the solutions of the heat equation converge pointwise (as time tends to zero) to the initial data. In the process, we also characterize the weighted Lebesgue spaces for the boundedness of maximal operators on the torus and waveguide manifold, which may be of independent interest.