论文标题
通过广义高斯估算
Weak type $(p,p)$ bounds for Schrödinger groups via generalized Gaussian estimates
论文作者
论文摘要
令$ l $为非负性自动伴侣操作员，在$ l^2（x）$上发挥作用，其中$ x $是具有尺寸$ n $的同质类型的空间。假设热运算符$ e^{ - tl} $满足一般的高斯$（p_0，p'_0）$ - 订单$ m $的估计，以$ 1 \ leq p_0 <2 $。众所周知，操作员$（i+l）^{ - s} e^{itl} $在$ l^p（x）$上限制为$ s \ geq n | {1/2} - {1/ p} | $和$ p \ in（p_0，p_0'）$（例如，请参见\ cite {blunck2，bdn，cco，cco，cdly，dn，mi1}）。在本文中，我们研究终点案例$ p = p_0 $，并证明$ s_0 = n \ big | {1 \ fover 2} - {1 \ fos p_0} \ big | $，操作员$（i+l）^{ - {s_0}}}常数$ c> 0 $，独立于$ t $和$ f $，因此开始\ begin {eqnarray*}μ\ left（\ left \ {x：\ big |（i+l）^{ - s_0} e^{ - s_0} e^{itl} f（itl} f（x）f（x） {p_0 \ over 2}）}} \ left（{{\ | f \ | _ {p_0} \averα} \ right）^{p_0}，\ \ \ \ \ t \ in {\ mathbb r} $α> \ big（\ | f \ | _ {p_ {0}}}/μ（x）\ big）^{p_ {0}} $当$μ（x）<\ infty $。我们的结果可以应用于具有粗糙的低阶项的粗糙电势和二阶椭圆算子的Schrödinger运算符，或具有可测量系数的高阶椭圆运算符，尽管通常，他们的半群落无法满足高斯上限。
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 $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order $m$ for some $1\leq p_0 < 2$. It is known that the operator $(I+L)^{-s } e^{itL}$ is bounded on $L^p(X)$ for $s\geq n|{1/ 2}-{1/p}| $ and $ p\in (p_0, p_0')$ (see for example, \cite{Blunck2, BDN, CCO, CDLY, DN, Mi1}). In this paper we study the endpoint case $p=p_0$ and show that for $s_0= n\big|{1\over 2}-{1\over p_0}\big|$, the operator $(I+L)^{-{s_0}}e^{itL} $ is of weak type $(p_{0},p_{0})$, that is, there is a constant $C>0$, independent of $t$ and $f$ so that \begin{eqnarray*} μ\left(\left\{x: \big|(I+L)^{-s_0}e^{itL} f(x)\big|>α\right\} \right)\leq C (1+|t|)^{n(1 - {p_0\over 2}) } \left( {\|f\|_{p_0} \over α} \right)^{p_0} , \ \ \ t\in{\mathbb R} \end{eqnarray*} for $α>0$ when $μ(X)=\infty$, and $α>\big(\|f\|_{p_{0}}/μ(X) \big)^{p_{0}}$ when $μ(X)<\infty$. Our results can be applied to Schrödinger operators with rough potentials and %second order elliptic operators with rough lower order terms, or higher order elliptic operators with bounded measurable coefficients although in general, their semigroups fail to satisfy Gaussian upper bounds.
