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

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 $${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u-\mathrm{\mid }x{\mathrm{\mid }}^{2}u+f,$$ 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 $({\mathrm{\partial }}_t-\mathrm{\Delta }{)}^{s}u=f$ and $({\mathrm{\partial }}_t-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{s}u=f$, for $0<s<1$.