Potential-kernels of logarithmic type and their applications

利用对数型势核的一些性质，我们给出了下列著名问题的一个确定解。“满足支配原理的卷积核的全体与亨特卷积核集合的闭包一致吗？在这方面，我们证明了一个势核是谱合成的，如果它满足支配原理。应用势核理论，得到了当给定的势核满足控制原理时，由好的势核构成的预解式是相关联的。根据扫出法的思想，我们提出了一种弧变分法来考察分析能力。在经典调和函数论的研究中，确定非零次调和函数不可积的区域是一个值得注意的问题，对数型势核具有递归半群，这与概率论密切相关。在概率论的研究中，我们得到了Ornstein-Uhlenbeck型过程瞬时性的一个判别准则和关于最优扩散过程的一些结果。

By using some properties of potential-kernels of logarithmic type, we gave a definitive solution of the following well-known problem. "Does the totality of convolution kernels satisfying the domination principle coincide with the closure of the set of Hunt convolution kernels?" In this connection, we proved that a potential-kernel is spectral synthetic if it satisfies the domination principle. Applying to the theory of potential-kernels, we obtain that with a given potential-kernel satisfying the domination principle, its resolvent formed by nice potential-kernels is associated. Suggested by the sweeping-out process, we worked out an arc-variation to investigate the analytic capacity. In the study of the classical harmonic function theory, it is remarkable to determine domains on which non-zero subharmonic functions are not integrable.Potential-kernels of logarithmic type possess recurrent semi-group,s which is closely related with the probability theory. In the study of the probability theory, we obtain a criterion of the transiency of Ornstein-Uhlenbeck type processes and results concerning optimal diffusion processes.