Sobolev space and Dirichlet space associated with symmetric Markov process on a local field
The author is focused on a counterpart of the Sobolev space based on Albeverio, Karwowski and Yasuda's additive process on a finite extension \(K\) of \(Q_p\) characterized by a nonnegative nonincreasing sequence \(A=\{a(m)\}\). The counterpart \({\mathcal F}_{r,p}^{(A)}\) of the Sobolev space \(W^{r,p}(\mathbb{R}^n)\) is obtained as the image of the map \(V_p^{(A)}:L^p(K;\mu)\to L^p(K;\mu)\), where \(V_p^{(A)}\) denotes an operator associated with Albeverio, Karwowski and Yasuda's additive process determined by \(A=\{a(m)\}\) and \(\mu\) denotes the Haar measure on \(K\). In Section 2, a sufficient condition is given for the refinement \({\mathcal F}_{r,2}^{(A)}\) of the Dirichlet space to be isometric to the domain \({\mathcal F}^{(B)}\) of the Dirichlet form associated with another sequence \(B=\{b(m)\}\). This isometry enables to find a purely probabilistic characterization of \((r, 2)\)-polar set in the case with non-integer \(r\). In Section 3, a characterization of the kernel \(V_r^{(A)}\) by the method of Fourier transform, is given. As a consequence, there are many cases where the composite of two kernels \(V_r^{(A)}\) and \(V_r^{(B)}\) provides us with another kernel \(V_r^{(C)}\) corresponding to some sequence \(C=\{c(m)\}\).
scientific article; zbMATH DE number 5014108
scientific article