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

Abstract

In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators $\mathcal{A}(t)$ of arbitrary order ${\hat{I}}^3\in (0,\infty )$. It is assumed that $\cA(t)$ is merely measurable with respect to the time variable. The unique solvability of the equation $$ \frac{\partial u}{\partial t}=\cA u-λu+f, \quad (t,x)\in \fR^{d+1} $$ and the $L_{q}(\fR,L_{p})$-estimate $$ \|u_{t}\|_{L_{q}(\fR,L_{p})}+\|(-Δ)^{γ/2}u\|_{L_{q}(\fR,L_{p})} +λ\|u\|_{L_{q}(\fR,L_{p})}\leq N\|f\|_{L_{q}(\fR,L_{p})} $$ are obtained for any $\hat{I}»>0$ and $p,q\in (1,\infty )$.