Pseudo-differential operators with semi-quasielliptic symbols over p-adic fields

In this article, we study pseudo-differential operators with a semi-quasielliptic symbols, which are extensions of operators of the form F ( D ; α ; x ) ϕ = F − 1 ( | F ( ξ , x ) | p α F ( ϕ ) ) , where α > 0 , ϕ is a function of Lizorkin type and F denotes the Fourier transform, and F ( ξ , x ) = f ( ξ ) + ∑ k c k ( x ) ξ k ∈ Q p [ ξ 1 , … , ξ n ] , where f ( ξ ) is a quasielliptic polynomial of degree d, and each c k ( x ) is a function from Q p n into Q p satisfying ‖ c k ( x ) ‖ L ∞ < ∞ . We determine the function spaces where the equations F ( D ; α ; x ) u = v have solutions. We introduce the space of infinitely pseudo-differentiable functions with respect to a semi-quasielliptic operator. By using these spaces we show the existence of a regularization effect for certain parabolic equations over p-adics.