A weighted $L_p$-regularity theory for parabolic partial differential equations with time measurable pseudo-differential operators

Abstract

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem \begin{equation}\label{ab eqn} \begin{cases} \partial_t u(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{cases} \end{equation} in (Muckenhoupt) weighted $L_p$-spaces with time-measurable pseudo-differential operators \begin{equation} \label{ab op} \psi(t,-i\nabla)u(t,x):=\mathcal{F}^{-1}\left[\psi(t,\cdot)\mathcal{F}[u](t,\cdot)\right](x). \end{equation} More precisely, we find sufficient conditions of the symbol $\psi(t,\xi)$ (especially depending on the smoothness of the symbol with respect to $\xi$) to guarantee that equation is well-posed in (Muckenhoupt) weighted $L_p$-spaces. Here the symbol $\psi(t,\xi)$ is merely measurable with respect to $t$, and the sufficient smoothness of $\psi(t,\xi)$ with respect to $\xi$ 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 the equation, \begin{equation} \label{ab est} \int_0^T \int_{\mathbb{R}^d} |(-\Delta)^{\gamma/2} u(t,x) |^p (t^2 + |x|^2)^{\alpha/2} \mathrm{d}x\mathrm{d}t \leq N\int_0^T \int_{\mathbb{R}^d} |f(t,x)|^p (t^2 + |x|^2)^{\alpha/2} \mathrm{d}x\mathrm{d}t \end{equation} and \begin{equation} \label{ab est 2} \int_0^T \left(\int_{\mathbb{R}^d} |(-\Delta)^{\gamma/2} u(t,x) |^p |x|^{\alpha_2} \mathrm{d}x \right)^{q/p} t^{\alpha_1}\mathrm{d}t \leq N\int_0^T \left(\int_{\mathbb{R}^d} |f(t,x) |^p |x|^{\alpha_2} \mathrm{d}x \right)^{q/p} t^{\alpha_1}\mathrm{d}t, \end{equation} where $p,q\in(1,\infty)$, $-d-1<\alpha < (d+1)(p-1)$, $-1 < \alpha_1 < q-1$, $-d <\alpha_2< d(p-1)$, and $\gamma$ is the order of the operator $\psi(t,-i\nabla)$.