## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

### Sobolev type complex-valued functions on $p$-adic fields and metric spaces

Abstract

In the talk we shall discuss different approaches to define Sobolev spaces $W_q^k$. Some of the approaches have advantage of possible generalization to domains different from standard $\mathbb{R}^n$ or its subsets, e.g. fractal sets or metric spaces, or to fractional $k$.
We shall present a definition of Sobolev type spaces on the $p$-adic field. This definition utilizes Steklov mean operator. Connection with Vladimirov pseudo-differential operator will be described, and comparison with the Hai{\l}asz definition of Sobolev type space will be given. Sobolev type embedding theorems will be presented.
Possible generalization to the case of metric measure space with doubling condition will be discussed.