An $L_q(L_p)$-theory for parabolic pseudo-differential equations: Calderón-Zygmund approach

Abstract: In this paper we present a Calder\'{o}n-Zygmund approach for a large class of parabolic equations with pseudo-differential operators $\mathcal{A}(t)$ of arbitrary order $\gamma\in(0,\infty)$. It is assumed that $\cA(t)$ is merely measurable with respect to the time variable. The unique solvability of the equation $$ \frac{\partial u}{\partial t}=\cA u-\lambda u+f, \quad (t,x)\in \fR^{d+1} $$ and the $L_{q}(\fR,L_{p})$-estimate $$ |u_{t}|{L{q}(\fR,L_{p})}+|(-\Delta)^{\gamma/2}u|{L{q}(\fR,L_{p})} +\lambda|u|{L{q}(\fR,L_{p})}\leq N|f|{L{q}(\fR,L_{p})} $$ are obtained for any $\lambda > 0$ and $p,q\in (1,\infty)$.