非局部Feller生成元的泊松方程的Schauder估计
Schauder estimates for Poisson equations associated with non-local Feller generators
摘要：
我们展示了如何Hölder估计Feller半群可以用来获得正则性结果的解决方案的泊松方程$Af=g$与（扩展）无穷小生成元$A$的Feller过程。利用变阶Hölder-Zygmund空间描述了$f$的正则性，并建立了Schauder估计.由于Feller半群的Hölder估计在过去几年中得到了深入的研究，我们的结果适用于广泛的Feller过程，例如Lévy过程的随机时变和Lévy驱动的随机微分方程的解。最突出的是，我们建立Schauder估计的泊松方程与分数拉普拉斯变量的顺序。作为一个副产品，我们得到了新的正则性估计半群与稳定的过程。
英文摘要：
We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $Af=g$ associated with the (extended) infinitesimal generator $A$ of a Feller process. The regularity of $f$ is described in terms of Hölder-Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of Lévy processes and solutions to Lévy-driven stochastic differential equations. Most prominently, we establish Schauder estimates for the Poisson equation associated with the fractional Laplacian of variable order. As a by-product, we obtain new regularity estimates for semigroups associated with stable-like processes.
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