论文标题
由对称的非本地迪里奇形式产生的半群的紧凑性，具有无限系数
Compactness of semigroups generated by symmetric non-local Dirichlet forms with unbounded coefficients
论文作者
论文摘要
令$（\ e，\ f）$为$ l^2（\ r^d; \ d x）$的对称的非本地dirichlet，由$ \ e（f，g）= \ iint _ {\ iint _ {\ r^d \ r^d \ r^d \ r^r^d}定义 (f(y)-f(x))(g(x)-g(y)){W(x,y)}\, J(x,\d y)\,\d x, \quad f,g\in \F,$$ where $J(x,\d y)$ is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and $ W（x，y）$被视为加权（无限）功能。我们在$ l^2（\ r^d; \ d x）$上建立了相关的马尔可夫半群$（p_t）_ {t \ ge0} $的紧凑性和非紧凑性的尖锐标准。特别是，我们证明，如果$ j（x，\ d y）= | x-y |^{ - d-d-α} \，\ d y $，\ d y $，则$α\ in（0,2）$，$$ w（x，x，y）= \ begin {case {case}（case}（1+| x | x |） （1+ | x |）^q+（1+ | y |）^q，\＆| x-y | \ geq 1 \ end {cases} $ 4，$ p \ in [0，\ infty）$和$ q \ in [0，α）$，然后$（p_t）_ {t \ ge0} $ compact us compact us compact of $ p_ {p_t）这表明$（\ e，\ f）$的紧凑性很大程度上取决于加权功能的增长$ W（x，y）$仅以$ | x-y | <1 $。我们的方法基于建立$（\ e，\ f）$的基本超级庞加莱不平等。我们的一般结果工作 即使跳跃内核$ j（x，\ d y）$是堕落的或是单数 关于Lebesgue度量。
Let $(\E,\F)$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^2(\R^d;\d x)$ defined by $$\E(f,g)=\iint_{\R^d\times \R^d} (f(y)-f(x))(g(x)-g(y)){W(x,y)}\, J(x,\d y)\,\d x, \quad f,g\in \F,$$ where $J(x,\d y)$ is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and $W(x,y)$ is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup $(P_t)_{t\ge0}$ on $L^2(\R^d;\d x)$. In particular, we prove that if $J(x,\d y)=|x-y|^{-d-α}\,\d y$ with $α\in (0,2)$, and $$W(x,y)= \begin{cases} (1+|x|)^p+(1+|y|)^p, \ & |x-y|< 1 \\ (1+|x|)^q+(1+|y|)^q, \ & |x-y|\geq 1 \end{cases}$$ with $p\in [0,\infty)$ and $q\in [0,α)$, then $(P_t)_{t\ge0}$ is compact, if and only if $p>2$. This indicates that the compactness of $(\E,\F)$ heavily depends on the growth of the weighted function $W(x,y)$ only for $|x-y|<1$. Our approach is based on establishing the essential super Poincaré inequality for $(\E,\F)$. Our general results work even if the jumping kernel $J(x,\d y)$ is degenerate or is singular with respect to the Lebesgue measure.
