Lipschitz Invariance of Critical Exponents on Besov Spaces

In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms.Under mild conditions,the critical exponent of Besov spaces of certain selfsimilar sets coincides with the walk dimension,which plays an important role in the analysis on fractals.As an application,we show examples having different critical exponents are not Lipschitz equivalent.