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 \begin{document} $ \partial_tu=\Delta u+f, $ \end{document} the harmonic oscillator evolution equation \begin{document}$\partial_tu=\Delta u-|x|^2u+f, $ \end{document} 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