Potential analysis

在容量密度条件下，我们将字符化统一域、内部统一域和约翰域作为边界锤原则和谐波度量估计的条款。我们研究了非集成内核的绑定行为，并产生了Fatou型理论和Littlewood型理论。我们展示了内部统一域的3G特性。我们构建了马丁边界和拓扑边界硬币的领域，但仍然是克兰斯顿-麦康奈尔不平等性，而且作为一个结果，3G-不平等性不能保持在尺寸上大于或等于3。“我们展示了边界哈克原则和卡尔森估计之间的平等。”在欧几里德空间的一个光滑的地方。我们展示了P-谐波函数的边界哈纳克原则。我们给了Carleson estomate for p-harmonic function。我们给出了p-Dirichlet解决方案的条件，以便成为持有者连续性的边界函数。我们研究了约翰领域的马丁边界。根据约翰·康斯坦特，我们估计了一个极简的马丁边界点超过了拓扑边界点的数目。

・Under the capacity density condition, we characterize uniform domains, inner uniform domains and John domains in terms of the boundary Hamack principle and estimates of harmonic measure.・We study the bounday behavior of non-integrable kernel and derive the Fatou type theorem and Littlewood type theorem.・We show the 3G-inequality for inner uniform domains. We construct a domain whose Martin boundary and topological boundary coincide, and yet the Cranston-McConnell inequality, and as a result, the 3G- inequality fail to hold in case the dimension is greater than or equal 3.・We show the equivalence between the boundary Hamack principle and the Carleson estimate.・For a smooth doimain in the Euclid space. we show the boundary Harnack principle for p-harmonic func- tion.・We give the Carleson estomates for p-harmonic functions.・We give conditions for the p-Dirichlet solution of a Holder continuos boundary function to be Hoder continuos up tp the boundary.・We study the Martin boundary of a John domain. By the John constant, we estimate the number of minimal Martin boundary points over a topological boundary.