AI 中文总结
本文引入道格拉斯加权狄利克雷空间，证明其再生核是Szego核在1预解式下的像，发展相关位势理论，得到关联对称纯跳Hunt过程，经典情形下为缠绕柯西过程。
AI 中文摘要
我们在单位圆盘上引入一类加权狄利克雷空间，称为道格拉斯加权狄利克雷空间，其特征是存在加权狄利克雷积分的道格拉斯型边界表示。该表示在Beurling-Deny和Fukushima的意义下自然诱导出单位圆上的非局部狄利克雷型。我们的主要结果表明，与正则道格拉斯加权狄利克雷型相关联的加权狄利克雷空间的再生核是Szego核在相应1预解式下的像，该预解式表示对超调和权是新的。我们还发展了与道格拉斯加权狄利克雷空间相关的位势理论，用再生核刻画了容量。最后，作为每个道格拉斯权诱导正则非局部狄利克雷型这一事实的应用，结合狄利克雷型的一般理论，我们得到了一个相关的对称纯跳Hunt过程，在经典狄利克雷情形下，该过程是缠绕柯西过程。
英文摘要
We introduce a class of weighted Dirichlet spaces on the unit disk, called Douglas weighted Dirichlet spaces, characterized by the existence of a Douglas type boundary representation of the weighted Dirichlet integral. This representation naturally induces a nonlocal Dirichlet form on the unit circle in the sense of Beurling Deny and Fukushima. Our main result shows that the reproducing kernel of the weighted Dirichlet space associated with a regular Douglas weighted Dirichlet form is the image of the Szego kernel under the corresponding 1 resolvent. This resolvent representation is new for superharmonic weights. We also develop the potential theory associated with Douglas weighted Dirichlet spaces. We characterize the capacity in terms of reproducing kernels. Finally, as an application of the fact that every Douglas weight induces a regular nonlocal Dirichlet form, together with the general theory of Dirichlet forms, we obtain an associated symmetric pure jump Hunt process. In the classical Dirichlet case, this process is the wrapped Cauchy process.
