An Lq(Lp)-Theory for Parabolic Pseudo-Differential Equations: Calderón-Zygmund Approach

In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators 𝒜(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {A}(t)$\end{document} of arbitrary order γ∈(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma \in (0,\infty )$\end{document}. It is assumed that (t) is merely measurable with respect to the time variable. The unique solvability of the equation ∂u∂t=𝒜u−λu+f,(t,x)∈Rd+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\partial u}{\partial t}=\mathcal{A}u-\lambda u+f, \quad (t,x)\in \mathbf{R}^{d+1} $$\end{document} and the Lq(R,Lp)-estimate ∥ut∥Lq(R,Lp)+∥(−Δ)γ/2u∥Lq(R,Lp)+λ∥u∥Lq(R,Lp)≤N∥f∥Lq(R,Lp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\|u_{t}\|_{L_{q}(\mathbf{R},L_{p})}+\|(-{\Delta})^{\gamma/2}u\|_{L_{q}(\mathbf{R},L_{p})} +\lambda\|u\|_{L_{q}(\mathbf{R},L_{p})}\leq N\|f\|_{L_{q}(\mathbf{R},L_{p})} $$\end{document} are obtained for any λ > 0 and p,q∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p,q\in (1,\infty )$\end{document}.