


Vol 11, No 2 (2019)
- Year: 2019
- Articles: 6
- URL: https://journals.rcsi.science/2070-0466/issue/view/12536
Research Articles
Pseudodifferential Operators and Markov Processes on Adèles
Abstract
In this article a class of Markov processes on the ring of finite adèles of the rational numbers are introduced. A class of non-Archimedean metrics on \(\mathbb{A}_{f}\) are chosen in order to describe this ring as a general polyadic ring and to introduce a family of pseudodifferential operators and parabolic-type equations on \({L^2}(\mathbb{A}_{f})\). The fundamental solutions of these parabolic equations determine transition functions of time and space homogeneous Markov processes on \(\mathbb{A}_{f}\) which are invariant under multiplication by units. Considering the infinite place ℝ, we extend these results to the complete ring of adèles.



Hausdorff Operator on Weighted Lebesgue and Grand Lebesgue p-Adic Spaces
Abstract
For Hausdorff operator of general type defined on p-adic linear space \(\mathbb{Q}_p^n\), we give sufficient conditions of its boundedness in weighted Lebesgue and grand Lebesgue spaces. Sharpness of some these conditions is also established.



Weighted Estimates for Maximal Operators, Riesz Potential Operators and Commutators on p-Adic Lebesgue and Morrey Spaces
Abstract
In this paper, we establish the boundedness of p-adic Hardy-Littlewood maximal operators and p-adic Riesz potential operators on the weighted Lebsegue and Morrey spaces. Moreover, the boundedness for the commutators of p-adic Riesz potential operators on the weighted Morrey spaces with symbols in central BMO space is also given.



Generalized Open Mapping Theorem for X-Normed Spaces
Abstract
The theory of X-normed spaces over non-Archimedean valued fields with valuations of higher rank was introduced by H. Ochsenius and W. H. Schikhof in [9] and further developed in [10–12, 16, 17] and [13]. In order to obtain results like the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem, H. Ochsenius and W. H. Schikhof used 1st countability conditions in the value group of the based field. In this article the author develops a new tool to work with transfinite induction simplifying the techniques employed in X-normed spaces, thus accomplishing a Generalized Baire Category Theorem that allows the proof of an Open Mapping Theorem for X-normed spaces without restrictions on the value group of the based field. Additionally, some contributions to the theory of X-normed spaces are presented regarding quotient spaces.



The Hausdorff Operator on Weighted p-Adic Morrey and Herz Type Spaces
Abstract
In the present article, we come up with the boundedness of Hausdorff operator, defined by means of linear transformation A, on the weighted p-adic Morrey and weighted p-adic Herz type spaces. Also, by imposing some special conditions on A, we discuss the sharpness of the results presented in this article.



Some Problems in the Theory of Approximation of Functions on Locally Compact Vilenkin Groups
Abstract
Some problems in the theory of approximation of complex-valued functions on locally compact Vilenkin groups in the metric of Lp, 1 ≤ p ≤ ∞, by functions with bounded spectrum, are investigated. A description of certain function spaces in terms of the best approximations are obtained and some imbedding theorems are proved.


