A note on the almost everywhere convergence to initial data for some evolution equations

TL;DR

This paper extends recent characterizations of initial data spaces ensuring almost everywhere convergence of the heat equation to more complex evolution equations involving the harmonic oscillator and Ornstein-Uhlenbeck operators.

Contribution

It demonstrates that the weighted Lebesgue spaces guaranteeing convergence for the heat equation are also optimal for related parabolic equations with harmonic oscillator and Ornstein-Uhlenbeck operators.

Findings

Weighted Lebesgue spaces are optimal for convergence in these equations.

The same initial data spaces apply to heat-diffusion equations with harmonic oscillator.

Convergence properties are established for these more general evolution equations.

Abstract

The weighted Lebesgue spaces of initial data for which almost everywhere convergence of the heat equation holds was only very recently characterized. In this note we show that the same weighted space of initial data is optimal for the heat--diffusion parabolic equations involving the harmonic oscillator and the Ornstein--Uhlenbeck operator.