On weighted mixed-norm Sobolev estimates for some basic parabolic 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 $${\partial }_{t}u=\Delta u-|x{|}^{2}u+f,$$ 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 $({\partial }_t-\Delta {)}^{s}u=f$ and $({\partial }_t-\Delta +|x{|}^2{)}^{s}u=f$, for $0<s<1$.

23 pages. To appear in Communications on Pure and Applied Analysis

References in corpus (1)

• Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation

Cited by in corpus (1)

• Parabolic equations involving Bessel operators and singular integrals