Theory of generalized Bessel potential space and functional completion

Shishkina, Elina; Ekincioglu, Ismail; Keskin, Cansu

doi:10.1007/s40590-023-00514-2

Cited by 1 publication

(1 citation statement)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Bessel potentials were initially studied (in the real case) by Nachman and Smith, see [17,18]. For further details, the author can refer to [4,8,9,[20][21][22], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

p-adic Bessel $$\alpha $$-potentials and some of their applications

Torresblanca-Badillo,

Ospino,

Arias

2024

J. Pseudo-Differ. Oper. Appl.

View full text Add to dashboard Cite

In this article, we will study a class of pseudo-differential operators on p-adic numbers, which we will call p-adic Bessel $$\alpha $$ α -potentials. These operators are denoted and defined in the form $$\begin{aligned} (\mathcal {E}_{\varvec{\phi },\alpha }f)(x)=-\mathcal {F}^{-1}_{\zeta \rightarrow x}\left( \left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\} \right] ^{-\alpha }\widehat{f}(\zeta )\right) , \text { } x\in {\mathbb {Q}}_{p}^{n}, \ \ \alpha \in \mathbb {R}, \end{aligned}$$ ( E ϕ , α f ) ( x ) = - F ζ → x - 1 max { 1 , | ϕ ( | | ζ | | p ) | } - α f ^ ( ζ ) , x ∈ Q p n , α ∈ R , where f is a p-adic distribution and $$\left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\}\right] ^{-\alpha }$$ max { 1 , | ϕ ( | | ζ | | p ) | } - α is the symbol of the operator. We will study some properties of the convolution kernel (denoted as $$K_{\alpha }$$ K α ) of the pseudo-differential operator $$\mathcal {E}_{\varvec{\phi },\alpha }$$ E ϕ , α , $$\alpha \in \mathbb {R}$$ α ∈ R ; and demonstrate that the family $$(K_{\alpha })_{\alpha >0}$$ ( K α ) α > 0 determines a convolution semigroup on $$\mathbb {Q}_{p}^{n}$$ Q p n . Furthermore, we will introduce new types of Feller semigroups, and explore new Markov processes and non-homogeneous initial value problems on p-adic numbers.

show abstract