We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem ∂tu(t,x)=ψ(t,-i∇)u(t,x)+f(t,x),(t,x)∈(0,T)×Rd,u(0,x)=0,x∈Rd,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu(t,x)=\psi (t,-i\nabla )u(t,x)+f(t,x), &{} \quad (t,x)\in (0,T)\times {\mathbb {R}}^d,\\ u(0,x)=0, &{} \quad x\in {\mathbb {R}}^d, \end{array}\right. } \end{aligned}$$\end{document}in (Muckenhoupt) weighted Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}-spaces with time-measurable pseudo-differential operators ψ(t,-i∇)u(t,x):=F-1ψ(t,·)F[u](t,·)(x).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \psi (t,-i\nabla )u(t,x):={\mathcal {F}}^{-1}\left[ \psi (t,\cdot ){\mathcal {F}}[u](t,\cdot )\right] (x). \end{aligned}$$\end{document}More precisely, we find sufficient conditions of the symbol ψ(t,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} (especially depending on the smoothness of the symbol with respect to ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}) to guarantee that equation (0.1) is well-posed in (Muckenhoupt) weighted Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}-spaces. Here the symbol ψ(t,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is merely measurable with respect to t, and the sufficient smoothness of ψ(t,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} with respect to ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is characterized by a property of each weight. In particular, we prove the existence of a positive constant N such that for any solution u to equation (0.1), ∫0T∫Rd|(-Δ)γ/2u(t,x …