具有格林函数的Hunt过程的调和函数的Hölder连续性
Hölder continuity of harmonic functions for Hunt processes with Green function
摘要：
设$(X,\mathcal W)$是平衡空间$1\in \mathcal W$，或等价地，设$\mathcal W$是局部紧空间$X$上Hunt过程的过剩函数的集合，$\mathcal W$具有可数基，使得$\mathcal W$分离点，$\mathcal W$中的每个函数是其连续子变量的上确界，并且存在严格正连续的$u,v\in \mathcal W$，使得$u/v\to 0$在无穷远处。我们假设存在一个$X$的格林函数$G>0$，一个$X$的度量$ρ$和一个具有加倍性质和温和上衰减的递减函数$g\colon[0,\infty)\to (0,\infty]$，使得$G\approx g\circρ$和半径为$r$的球的容量近似为$1/g(r)$ .证明了有界调和函数是Hölder连续的，如果常数函数$1$是调和的且跳出球允许多项式估计.后者被证明，如果缩放不变Harnack不等式成立。
英文摘要：
Let $(X,\mathcal W)$ be a balayage space, $1\in \mathcal W$, or - equivalently - let $\mathcal W$ be the set of excessive functions of a Hunt process on a locally compact space $X$ with countable base such that $\mathcal W$ separates points, every function in $\mathcal W$ is the supremum of its continuous minorants and there exist strictly positive continuous $u,v\in \mathcal W$ such that $u/v\to 0$ at infinity. We suppose that there is a Green function $G>0$ for $X$, a metric $ρ$ on $X$ and a decreasing function $g\colon[0,\infty)\to (0,\infty]$ having the doubling property and a mild upper decay such that $G\approx g\circρ$ and the capacity of balls of radius $r$ is approximately $1/g(r)$. It is shown that bounded harmonic functions are Hölder continuous, if the constant function $1$ is harmonic and jumps out of balls admit a polynomial estimate. The latter is proven if scaling invariant Harnack inequalities hold.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
