论文标题
加权$ L_P $ - 抛物线偏微分方程与时间可测量的伪差异操作员
A weighted $L_p$-regularity theory for parabolic partial differential equations with time measurable pseudo-differential operators
论文作者
论文摘要
我们获得以下cauchy问题的存在，独特性和规律性估计\ begin {equation} \ label {ab eqn} \ begin {case} \ partial_t u（t，x）=ψ（t，-i \ nabla）u（t，x）+f（t，x，x），\ quad＆（t，x）\ in（0，t）\ times \ times \ times \ times \ mathbb {r}^d，\ \ \ \ \ \ \ \ \ \ \ \ \ \ \（0，x）= 0，x） \ end {equation} in（muckenhoupt）加权$ l_p $ - 带有时间测量的pseudo-differential operators \ begin \ begin {equation} \ label {ab op}ψ（t，-i \ nabla）u（t，x）：= \ Mathcal {f}^{ - 1} \ left [ψ（t，\ cdot）\ Mathcal {f} [f} [f} [u]（t，\ cdot）\ right]（x）（x）。 \ end {equation}更确切地说，我们发现符号$ψ（t，ξ）$的足够条件（尤其取决于符号相对于$ξ$的平滑度），以确保方程在（muckenhoupt）中得到充分含量（Muckenhoupt）加权$ L_P $ -Spaces。在这里，符号$ψ（t，ξ）$仅与$ t $相对于$ t $来测量，而相对于$ξ$的$ψ（t，ξ）$的足够平滑度的特征是每个重量的属性。特别是，我们证明存在正常数$ n $，以便对于任何解决方案$ u $ to equation，\ begin {equation} \ label {ab est} \ int_0^t \ int _ {\ mathbb {r}^d} |（-Δ） \ int _ {\ mathbb {r}^d} | f（t，x）|^p（t^2 + | x |^2）^{α/2} \ Mathrm {d} x \ Mathrm {d} \ label {ab est 2} \ int_0^t \ left（\ int _ {\ Mathbb {r}^d} |（-ux） t^{α_1} \ mathrm {d} t \ leq n \ int_0^t \ left（\ int _ {\ mathbb {\ mathbb {r}^d} | f（t，x，x）|^p | x | x | x |^|^|^|^{α_2} {α_2}} \ MATHRM {D} t^{α_1} \ mathrm {d} t，\ end {qore}其中$ p，q \ in（1，\ infty）$，$ -d-1 <α<（d+1）（p-1）（p-1）$，$ -1 <α_1<α_1<q-1 <q-1 $，$ -dd <q-d <α_2<α_2<d（p-1） $ψ（t，-i \ nabla）$。
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)=ψ(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} ψ(t,-i\nabla)u(t,x):=\mathcal{F}^{-1}\left[ψ(t,\cdot)\mathcal{F}[u](t,\cdot)\right](x). \end{equation} More precisely, we find sufficient conditions of the symbol $ψ(t,ξ)$ (especially depending on the smoothness of the symbol with respect to $ξ$) to guarantee that equation is well-posed in (Muckenhoupt) weighted $L_p$-spaces. Here the symbol $ψ(t,ξ)$ is merely measurable with respect to $t$, and the sufficient smoothness of $ψ(t,ξ)$ with respect to $ξ$ 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} |(-Δ)^{γ/2} u(t,x) |^p (t^2 + |x|^2)^{α/2} \mathrm{d}x\mathrm{d}t \leq N\int_0^T \int_{\mathbb{R}^d} |f(t,x)|^p (t^2 + |x|^2)^{α/2} \mathrm{d}x\mathrm{d}t \end{equation} and \begin{equation} \label{ab est 2} \int_0^T \left(\int_{\mathbb{R}^d} |(-Δ)^{γ/2} u(t,x) |^p |x|^{α_2} \mathrm{d}x \right)^{q/p} t^{α_1}\mathrm{d}t \leq N\int_0^T \left(\int_{\mathbb{R}^d} |f(t,x) |^p |x|^{α_2} \mathrm{d}x \right)^{q/p} t^{α_1}\mathrm{d}t, \end{equation} where $p,q\in(1,\infty)$, $-d-1<α< (d+1)(p-1)$, $-1 < α_1 < q-1$, $-d <α_2< d(p-1)$, and $γ$ is the order of the operator $ψ(t,-i\nabla)$.
