\(p\)-adic Laplacian in local fields (Q289571)

Let \(\mathbb{Q}_p\) be the non-Archimedean valued field of \(p\)-adic numbers (\(p\) a prime number). The main purpose of this paper is the construction and study of the \(p\)-adic Laplacian on \(\mathbb{Q}_{p}^{n}\). In order to get it, a key point is to define derivative operators on \(\mathbb{Q}_{p}^{n}\) such that the test function class \({\mathcal D} (\mathbb{Q}_{p}^{n})\) and its distribution class \({\mathcal D} '(\mathbb{Q}_{p}^{n})\) are invariant under the action of these operators. By \({\mathcal D} (\mathbb{Q}_{p}^{n})\), we mean the linear space of all complex-valued locally constant functions with compact supports defined on \(\mathbb{Q}_{p}^{n}\) and equipped with a certain uniform convergence topology.NEWLINENEWLINEFor \(n =1\), \textit{W.-Y. Su} gave in [Approximation Theory Appl. 4, No. 2, 119--129 (1988; Zbl 0665.35077); Sci. China, Ser. A 35, No. 7, 826--836 (1992; Zbl 0774.43006)] a definition of derivatives and integrals of fractional orders, denoted by \(T^{s}\) \((s \in \mathbb{R}\)), for general locally compact Vilenkin groups, by using pseudodifferential operators that satisfy the required invariance property. \(T^{s}\) is a pseudodifferential operator with the symbol \(\big<\xi\big>^{s}\), defined by the formula NEWLINE\[NEWLINE T^{s} (\varphi) = \left(\big<\xi\big>^{s} \varphi^{\wedge} \right)^{\vee} \quad (\varphi \in \mathcal D(\mathbb{Q})),NEWLINE\]NEWLINE where \(^{\wedge}\) and \(^{\vee}\) are the Fourier transform and the inverse Fourier transform, respectively.NEWLINENEWLINEIn this paper, the authors extend the definition of these fractional differential operators to the multidimensional space \(\mathbb{Q}_{p}^{n}\). Then they have the machinery to construct the Laplacian \(\Delta_p\) on \(\mathbb{Q}_{p}^{n}\) and a fundamental solution of the associated Laplace equation. The Laplacian \(\Delta_p\) is the operator defined by NEWLINE\[NEWLINE \Delta_p f(x) = \sum_{j=1}^{n} T_{x_j}^{2} f(x) \quad (f \in {\mathcal D}' (\mathbb{Q}_{p}^{n})), NEWLINE\]NEWLINE where \(T_{x_j}^{2}\) is the \(j\)-th partial differential operator for \(s =2\).NEWLINENEWLINESince the domain of the Laplacian \(\Delta_p\) is dense in the complex Hilbert space \(L^2(\mathbb{Q}_{p}^{n})\), \(\Delta_p\) admits a continuous extension to \(L^2(\mathbb{Q}_{p}^{n})\). It is proved in the paper hat \(\Delta_p\) is a non-negative self-adjoint operator on \(L^2(\mathbb{Q}_{p}^{n})\) having an orthonormal basis of eigenfunctions. Also, the solutions of the associated \(p\)-adic Cauchy problems for the wave and heat equations are investigated.