Quasi-invariant and pseudo-differentiable measures with values in non-Archimedean fields on a non-Archimedean Banach space
This extensive paper treats integration in non-Archimedean Banach spaces, which is an interesting part of non-Archimedean analysis because of its possible influence in the applications. The work contains the non-Archimedean counterparts of many definitions and results of integration theory in classical Banach spaces over the real or complex field. Frequently, the non-Archimedean formulations and proofs differ from the classical ones. Let \(K\) be a locally compact non-Archimedean and non-trivially valued complete field and let \(K_s\) be a finite algebraic extension of the \(s\)-field \(\mathbb{Q}_s\) of \(s\)-adic numbers for some prime number \(s\). Let \(X\) be a non-Archimedean Banach space over \(K\). All the measures (on \(X\)) considered in the paper have values in the field \(K_s\). Firstly, the author introduces weak distributions and characteristic functionals (called Fourier--Stieltjes transformations) of measures and investigates their most important properties. He gives the non-Archimedean analogues of the Bochner--Kolmogorov and Minlos--Sazonov theorem, which provide descriptions of the characteristic functionals of certain measures. Then the author studies quasi-invariant measures and proves several theorems on the quasi-invariance of a measure relative to a (linear or nonlinear) transformation \(X \rightarrow X\) on the Banach space \(X\). For this end, he gives some results about products of measures, among them the non-Archimedean version of the classical Kakutani theorem. Finally, the author defines pseudo-differentiability of measures. The theory previously developed in the paper is applied now to prove the existence of quasi-invariant pseudo-differentiable measures, to give criteria for pseudo-differentiability of measures, and to show results about convergence of measures with account for their quasi-invariance and pseudo-differentiability. To carry out the above objectives, the author uses basic central tools of functional analysis, such as the notions of convex compact set and compact operator and the Hahn--Banach and the Alaoglu--Bourbaki theorem. The assumption of local compactness on \(K\) gives him the possibility of using these notions and theorems in the non-Archimedean case. For non-locally compact ground fields, however, the situation changes strongly and becomes typically non-Archimedean with respect to compactness and duality theory. I think that it would be interesting for a future work to investigate the validity of the contents of the present paper in the case of non-Archimedean Banach spaces over arbitrary non-Archimedean and non-trivially valued complete fields.
scientific article; zbMATH DE number 2174275