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 $\gamma \in (0,\infty )$. It is assumed that (t) is merely measurable with respect to the time variable. The unique solvability of the equation

$$\frac{\partial u}{\partial t}=\mathcal{A}u-\lambda u+f, \quad (t,x)\in \mathbf{R}^{d+1} $$

and the L _q (R,L _p )-estimate

$$\|u_{t}\|_{L_{q}(\mathbf{R},L_{p})}+\|(-{\Delta})^{\gamma/2}u\|_{L_{q}(\mathbf{R},L_{p})} +\lambda\|u\|_{L_{q}(\mathbf{R},L_{p})}\leq N\|f\|_{L_{q}(\mathbf{R},L_{p})} $$

are obtained for any λ > 0 and $p,q\in (1,\infty )$.