Estimates of the Besov norms on the fractal boundary and applications

Abstract

Let $u$ be a $\lambda$-Hölder continuous function on the closure of a bounded domain $D$ with fractal boundary $\partial D$. We estimate the Besov norm of the restriction of $u$ to $\partial D$ by the $L^p\left(D\right)$-norm of $\left|\nabla u\right(y\left)\right|$ dist $(y,\partial D{)}^{\lambda }$ for an adequate $\lambda >0$. We apply it to the boundedness of operators related to the double layer potentials on the Besov spaces on $\partial D$.