论文标题
$ l^p $空间的schrödinger方程，适用于热核令人满意的泊松类型边界
The Schrödinger equation in $L^p$ spaces for operators with heat kernel satisfying Poisson type bounds
论文作者
论文摘要
令$ l $为非负自动偶像运算符，在$ l^2（x）$上作用，其中$ x $是具有尺寸$ n $的同质类型的空间。在本文中，我们研究了Sharpolev方程的初始值问题的尖锐端点$ l^p $ -sobolev估计，$ i \ partial_t u + l u = 0 $，并向所有$ f \ in l^p（x），1 <p <p <p <p <p <\ infty，$ \ begin {eqnarray*} \ left \ | e^{itl}（i+l）^{ - {σn}} f \ right \ | _ {p} \ leq c（1+ | t |） \ \ \ \ t \ in {\ mathbb r}，\ \ \ \σ\ geq \ geq \ big | {1 \ over 2} - {1 \ over p} \ big |，\ big |，\ end end {eqnarray*} semogeup $ e^{ - tl $ l $ l $ coles os a poiss a poiss type a poiss emperson type a poiss oss oss oss oss oss oss osson poisson poisson poisson。这将先前的结果扩展为\ cite {cdly1}，其中$ l $生成的semigroup $ e^{ - tl} $满足指数衰减。
Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. In this paper, we study sharp endpoint $L^p$-Sobolev estimates for the solution of the initial value problem for the Schrödinger equation, $i \partial_t u + L u=0 $ and show that for all $f\in L^p(X), 1<p<\infty,$ \begin{eqnarray*} \left\| e^{itL} (I+L)^{-{σn}} f\right\|_{p} \leq C(1+|t|)^{σn} \|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ σ\geq \big|{1\over 2}-{1\over p}\big|, \end{eqnarray*} where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound. This extends the previous result in \cite{CDLY1} in which the semigroup $e^{-tL}$ generated by $L$ satisfies the exponential decay.
