P-adic Laplacian in local fields. (July 2016) - Li, Yin; Qiu, Hua
Abstract: In this paper a family of multi-dimensional fractional differential operators T α and their corresponding pseudo-differential equations over p -adic fields are investigated. The test function class D ( Q p n ) and distribution class D ′ ( Q p n ) are invariant under the actions of these operators. The p -adic Laplacian Δ p and a fundamental solution of the Laplace equation are constructed. We study the spectral properties of the Laplacian Δ p, and obtain an orthonormal basis of the eigen-functions of this operator in L 2 ( Q p n ) . Furthermore, the Cauchy problems for the wave and heat equations on the p -adic fields related to Δ p are also studied.