Potential theory in a domain with fractal boundary

我们考虑具有分形边界的区域D中的边值问题。边界上Besov空间上的算子K关于Besov范数有界是经常发生的。我们可以用下列方法证明算子从δD到δD的有界性：(1)利用扩张算子E将定义在δD上的函数推广到R^n；(2)f的Besov范数由(∫_D|▽f(X)|^&lt；Pλ&gt；dx)^&lt；1/P&gt；，其中δ(X)是从x到Dirichlet D的距离。(3)代替K的有界性，我们利用D与D外部之间的极大函数证明了算子F关于适当范数的有界性。我们证明了算子K的有界性，这对于利用双层势解决δ问题是重要的。

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.