Asymptotical behavior of one class of p-adic singular Fourier integrals

We study the asymptotical behavior of the p-adic singular Fourier integralsJπα,m;φ(t)=〈fπα;m(x)χp(xt),φ(x)〉=F[fπα;mφ](t),|t|p→∞,t∈Qp, where fπα;m∈D′(Qp)fπα;m∈D′(Qp) is a quasi associated homogeneous   distribution (generalized function) of degree πα(x)=|x|pα−1π1(x) and order m  , πα(x)πα(x), π1(x)π1(x), and χp(x)χp(x) are a multiplicative, a normed multiplicative, and an additive characters of the field QpQp of p  -adic numbers, respectively, φ∈D(Qp)φ∈D(Qp) is a test function, m=0,1,2,…m=0,1,2,…, α∈Cα∈C. If Reα>0Reα>0 the constructed asymptotics constitute a p-adic version of the well-known Erdélyi lemma. Theorems which give asymptotic expansions of singular Fourier integrals are the Abelian type theorems. In contrast to the real case, all constructed asymptotics have the stabilization property.