Potential theory in a domain with fractal boundary

研究具有分形边界的D域上的边值问题。边界上的贝索夫空间上的算子K是相对于贝索夫范数有界的。我们可以用下面的方法证明从δD到δD的算子的有界性。(1)利用扩展算子e将δ d上定义的函数扩展到R^n (2) f的Besov范数估计为（∫_D |△f(x)|^<Pλ>dx）^<1/P>，其中δ(x)是x到δ d的距离。(3)代替K的有界性，利用D与D外的极大函数证明了算子F在合适的范数下从D到D外的有界性，证明了算子K的有界性，这对利用双层势解决Dirichlet问题具有重要意义。

We consider the boundary-value problems in a domain D with fractal boundary. It often occurs that an operator K on the Besov space on the boundary is bounded with respect to the Besov norms. We can prove the boundedness of an operator from δD to δD in the following method.(1) We extend a function defined on δD to R^n by using an extension operator E.(2) The Besov norm of f is estimated by (∫_D |▽f(x)|^<Pλ>dx)^<1/P>, where δ(x) is the distance from x to δD.(3) Instead of the boundedness of K we prove the boundedness of an operator F from D to the outside of D with respect to suitable norms by using the maximal functions between D and the outside of D.We proved the boundedness of an operator K, which is important to solve the Dirichlet problem by using double layer potentials.