论文标题
非架构的广义贝塞尔潜力及其应用
Non-archimedean generalized Bessel potentials and their applications
论文作者
论文摘要
本文介绍了一类伪差异操作员\ begin {equation*}（\ mathcal {a}^αφ）（x）= \ mathcal {f}^{ - 1} _ {ξ\ rightarrow x} \ left（\ left [\ max \ {| \ boldsymbolar {1}（||ξ|| _ {p}）|，| \ boldsymbolar 6 {2}（||吗？ \ end {equation*} $φ\ in \ mathcal {d}（\ mathbb {q} _ {p}^{n}）$和$α\ in \ mathbb {c} $;这里$ \ left [\ max \ {| \boldsymbolψ_{1}（||ξ|| _ {p}）|，| \ boldsymbolar 6 {2}（||吗？这些操作员可以看作是$ P $ - 美国环境中贝塞尔电位的概括。我们表明，一家人$ \ left（k_α\右）_ {α> 0} $的卷积内核附加到广义的贝塞尔势$ \ mathcal {a}^α$，$α$，$α> 0 $，确定$ \ mathbb {q} _ _ {q} _ _ {p} $的卷积半径。施加某些条件，我们有$k_α$，$α> 0 $，是$ \ mathbb {q} _ {p}^{n} $的概率度量。此外，我们将研究与操作员$ \ MATHCAL {A}^α$的绿色功能相对应的某些属性，并且我们表明，与这些操作员自然相关的热方程式描述了随着时间的推移在给定区域中的冷却（或热量损失）。
This article describes a class of pseudo-differential operators \begin{equation*} (\mathcal{A}^αφ)(x)=\mathcal{F}^{-1}_{ξ\rightarrow x}\left(\left[\max\{|\boldsymbolψ_{1}(||ξ||_{p})|,|\boldsymbolψ_{2}(||ξ||_{p})|\}\right]^{-α}\widehatφ(ξ)\right), \end{equation*} $φ\in \mathcal{D}(\mathbb{Q}_{p}^{n})$ and $α\in\mathbb{C}$; here $\left[\max\{|\boldsymbolψ_{1}(||ξ||_{p})|,|\boldsymbolψ_{2}(||ξ||_{p})|\}\right]^{-α}$ is the symbol of the operator $\mathcal{A}^α$. These operators can be seen as a generalization of the Bessel potentials in the $p$-adic context. We show that the family $\left(K_α\right)_{α>0}$ of convolution kernels attached to generalized Bessel potentials $\mathcal{A}^α$, $α>0$, determine a convolution semigroup on $\mathbb{Q}_{p}^{n}$. Imposing certain conditions we have that $K_α$, $α>0$, is a probability measure on $\mathbb{Q}_{p}^{n}$. Moreover, we will study certain properties corresponding to the Green function of the operator $\mathcal{A}^α$ and we show that heat equations, naturally associated to these operators, describes the cooling (or loss of heat) in a given region over time.
