Increase in Sobolev norm induced by composite with normal contraction on a ultrametric space

It is well-known in the theory of Dirichlet form theory that every Dirichlet form and its domain provide us with a counterpart of $H^1$-space and the space possesses non-increasing property of $H^1$-norm under normal contraction, i.e., the $H^1$-norm of composite of normal contractions with function in the domain does not exceed the $H^1$-norm of the function without composite of normal contraction. Accordingly, it might be expected that so does such counterpart of the Sobolev space initiated by M. Fukushima and the speaker on the basis of transition semigroup kernels. In this talk, we find a counter-example to such a non-increasing property of the probabilistic Sobolev norm, i.e., the composite of normal contraction with some function in such Sobolev space induces increase in the norm.