On weighted mixed-norm Sobolev estimates for some basic parabolic equations

TL;DR

This paper establishes new global weighted Sobolev and mixed-norm estimates for various parabolic equations, including classical, degenerate, and fractional nonlocal types, with convergence results for singular integrals.

Contribution

It introduces novel weighted Sobolev and mixed-norm estimates for a broad class of parabolic equations, including fractional and degenerate cases, extending classical results.

Findings

Weighted estimates for classical heat and harmonic oscillator equations

Weighted mixed-norm estimates for fractional nonlocal equations

Almost everywhere convergence of singular integrals in evolution equations

Abstract

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=\Delta u-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-\Delta)^su=f$ and $(\partial_t-\Delta+|x|^2)^su=f$, for $0<s<1$.