pp-adic Laplacian in local fields

In this paper a family of multi-dimensional fractional differential operators TαTα and their corresponding pseudo-differential equations over pp-adic fields are investigated. The test function class D(Qpn) and distribution class D′(Qpn) are invariant under the actions of these operators. The pp-adic Laplacian ΔpΔp and a fundamental solution of the Laplace equation are constructed. We study the spectral properties of the Laplacian ΔpΔp, and obtain an orthonormal basis of the eigen-functions of this operator in L2(Qpn). Furthermore, the Cauchy problems for the wave and heat equations on the pp-adic fields related to ΔpΔp are also studied.