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

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    * Corresponding author 
The first author was supported by grants 11471251 and 11271293 from National Natural Science Foundation of China. The second and third authors were supported by grant MTM2015-66157-C2-1-P from Government of Spain.
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  • 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

    $ \partial_tu=\Delta u+f, $

    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 $0 < s < 1$.

    Mathematics Subject Classification: Primary: 35K10, 35B45, 42B37; Secondary: 58J35, 42B20.


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