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Non-Archimedean Pseudo-Differential Operators With Bessel Potentials (1904.01968v1)

Published 1 Apr 2019 in math.NT and math.FA

Abstract: In this article, we study a class of non-archimedean pseudo-differential operators associated via Fourier transform to the Bessel potentials. These operators (which we will denote as $J{\alpha },$ $\alpha >n$) are of the form (J{\alpha })(x)=\mathcal{F}{\xi \rightarrow x}{-1}\left[ (\max{1,||\xi ||{p}}){-\alpha }\widehat{\varphi }(\xi )\right] ,\text{ } \varphi \in \mathcal{D}\mathbb{Q}{p}{n}),\text{ } x\in\mathbb{Q}{p}{n}. We show that the fundamental solution $Z(x,t)$ of the $p-$adic heat equation naturally associated to these operators satisfies $Z(x,t)<= 0,x\in\mathbb{Q} {p}{n},t>0. So this equation describes the cooling (or loss of heat) in a given region over time. Unlike the archimedean classical theory, although the operator symbol -J{\alpha } is not a function negative definite, we show that the operator -J{\alpha } satisfies the positive maximum principle on C{0}(\mathbb{Q}{p}{n}). Moreover, we will show that the closure \overline{-J{\alpha }} of the operator -J{\alpha } is single-valued and generates a strongly continuous, positive, contraction semigroup {T(t)} on C{0}(\mathbb{Q}{p}{n}). On the other hand, we will show that the operator -J{\alpha } is m-dissipative and is the infinitesimal generator of a C{0}-semigroup of contractions T(t), t>= 0, on L{2}(\mathbb{Q}_{p}{n}). The latter will allow us to show that for f\in L{1}([0,T):L{2}(\mathbb{Q}_{p}{n})), the function u(t)=T(t)u_{0}+\int\nolimits_{0}{t}T(t-s)f(s)ds,\text{ \ \ }0<=t <=T, is the mild solution of the initial value problem \frac{\partial u}{\partial t}(x,t)=-J{\alpha }u(x,t)+f(t) & t>0\text{,\ } x\in \mathbb{Q}{p}{n} \ u(x,0)=u{0}\in L{2}(\mathbb{Q}_{p}{n})\text{.}

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