Parabolic-type pseudodifferential equations with elliptic symbols in dimension 3 over p-adics

Abstract

In this paper we deal with the operator defined as $$f(\partial ,\alpha )\phi : = \mathcal{F}_{\xi \to x}^{ - 1} \left( {\left| {f(\xi )} \right|_p^\alpha \mathcal{F}_{x \to \xi } \phi } \right)$$ , where f(ξ) is an elliptic quadratic form of dimension 3 over ℚ p . We study the Cauchy problem associated that operator, and find the fundamental solution and some properties of it, using the techniques given by Kochubei.