An L q (L p )-Theory for Parabolic Pseudo-Differential Equations: Calderón-Zygmund Approach

In this paper we present a Calderon-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 \(p,q\in (1,\infty )\).