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 formF(D;α;x)ϕ=F−1(|F(ξ,x)|pαF(ϕ)), where α>0α>0, ϕ   is a function of Lizorkin type and FF denotes the Fourier transform, and F(ξ,x)=f(ξ)+∑kck(x)ξk∈Qp[ξ1,…,ξn]F(ξ,x)=f(ξ)+∑kck(x)ξk∈Qp[ξ1,…,ξn], where f(ξ)f(ξ) is a quasielliptic polynomial of degree d  , and each ck(x)ck(x) is a function from Qpn into QpQp satisfying ‖ck(x)‖L∞<∞‖ck(x)‖L∞<∞. We determine the function spaces where the equations F(D;α;x)u=vF(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.