Harmonic analysis in a domain with fractal boundary

考虑具有分形侧边界s的域Ω= dx [O，T]上的初边值问题。在侧边界上的Besov空间B上的算子K往往相对于s上的Besov范数是有界的。我们可以用下面的方法证明从B到B的算子的有界性。(1)利用扩展算子e将定义在S上的函数f扩展到rx [O，T]。(2)用|△f(x)|×δ(x)^n / Ω的积分估计f的Besov范数，其中δ(x)是x到S的距离，n是一个合适的数。(3)代替K的有界性，利用Ω与Ω.We之间的极大函数证明了从Ω上的函数空间到Ω外的函数空间的算子的有界性，证明了算子K的有界性，这对利用双层热势求解Dirichlet问题具有重要意义。

We consider the initial-boundary-value problems in a domain Ω=D×[O,T] with fractal lateral boundary S. It often occurs that an operator K on the Besov space B on the lateral boundary is bounded with respect to the Besov norms on S. We can prove the boundedness of an operator from B to B in the following method.(1)We extend a function f defined on S to R×[O,T] by using an extension operator E.(2)The Besov norm of f is estimated by the integral of |▽f(x)|×δ(x)^n over Ω, where δ(x) is the distance from x to S and n is a suitable number.(3)Instead of the boundedness of K we prove the boundedness of an operator from a function space on Ω to a function space on the outside of Ω by using the maximal functions between Ω and the outside of Ω.We proved the boundedness of an operator K, which is important to solve the Dirichlet problem by using double layer heat potentials.