2019/04/01 by Ismael Gutiérrez García, García, Ismael Gutiérrez, Anselmo Torresblanca-Badillo +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Number Theory (math.NT) #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1904.01968
openalex publication_date 2019/04/01 · openalex created_date 2019/04/11 · openalex updated_date 2026/07/28
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 Jα, α>n) are of the form (Jα)(x)=Fξ→ x-1[ (max\1,||ξ||p\)-α\widehatφ(ξ)] , φ∈ Dℚpn), x∈ℚpn. 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∈ℚ pn,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 -Jα is not a function negative definite, we show that the operator -Jα satisfies the positive maximum principle on C0(ℚpn). Moreover, we will show that the closure -Jα of the operator -Jα is single-valued and generates a strongly continuous, positive, contraction semigroup T(t) on C0(ℚpn). On the other hand, we will show that the operator -Jα is m-dissipative and is the infinitesimal generator of a C0-semigroup of contractions T(t), t>= 0, on L2(ℚpn). The latter will allow us to show that for f∈ L1([0,T):L2(ℚpn)), the function u(t)=T(t)u0+∫\nolimits0tT(t-s)f(s)ds, 0<=t <=T, is the mild solution of the initial value problem (∂ u)/(∂ t)(x,t)=-Jαu(x,t)+f(t) & t>0, x∈ ℚpn u(x,0)=u0∈ L2(ℚpn).