  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 $0<s<1$.
