An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators

In this article we prove the existence and uniqueness of a (weak) solution $u$ in $L_p((0,T),{\mathrm{\Lambda }}_{\gamma +m})$ to the Cauchy problem \begin{align} \notag &\frac{\partial u}{\partial t}(t,x)=\psi(t,i\nabla)u(t,x)+f(t,x),\quad (t,x) \in (0,T) \times $\mathbf{R}$^d \label{main eqn} & u(0,x)=0, \end{align} where $d\in \mathbb{N}$, $p \in (1,\infty]$,$\gamma,m \in (0,\infty)$,$\Lambda_{\gamma+m}$ is the Lipschitz space on ${\mathbf{R}}^d$ whose order is $\gamma +m$, $f \in L_p\left((0,T) , \Lambda_{\gamma} \right)$,and$\psi(t,i\nabla)$ is a time measurable pseudo-differential operator whose symbol is $\psi (t,\xi )$, i.e. $$\psi (t,i\mathrm{\nabla })u(t,x)={\text{cF}}^{-1}[\psi (t,\xi )\text{cF}[u(t,\cdot )](\xi )](x),$$ with the assumptions \begin{align*} \Re[\psi(t,\xi)] \leq -\nu|\xi|^{\gamma}, \end{align*} and \begin{align*} |D_{\xi}^{\alpha}\psi(t,\xi)|\leq\nu^{-1}|\xi|^{\gamma-|\alpha|}. \end{align*} Furthermore, we show \begin{align} \label{e 1028 1} \int_0^T |u(t,\cdot)|^p_{\Lambda_{\gamma+m}} dt \leq N \int_0^T |f(t,\cdot)|^p_{\Lambda_{m}} dt, \end{align} where $N$ is a positive constant depending only on $d$, $p$, $\gamma$, $\nu$, $m$, and $T$, The unique solvability of equation (\ref{main eqn}) in $L_p$-H"older space is also considered. More precisely, for any $f\in L_p((0,T);C^{n+\alpha })$, there exists a unique solution $u \in L_p((0,T);C^{\gamma+n+\alpha}($\mathbf{R}$^d))$ to equation (\ref{main eqn}) and for this solution $u$, \begin{align} \label{e 1029 1} \int_0^T |u(t,\cdot)|^p_{C^{\gamma+n+\alpha}}dt \leq N \int_0^T |f(t,\cdot)|^p_{C^{n+\alpha}}dt, \end{align} where $n \in $\mathbb{Z}$+$,$\alpha \in (0,1)$,and$\gamma+\alpha \notin $\mathbb{Z}$+$.