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

Abstract

We study the asymptotical behavior of the p-adic singular Fourier integrals J π α , m ; φ ( t ) = 〈 f π α ; m ( x ) χ p ( x t ) , φ ( x ) 〉 = F [ f π α ; m φ ] ( t ) , | t | p → ∞ , t ∈ Q p , where f π α ; m ∈ D ′ ( Q p ) is a quasi associated homogeneous distribution (generalized function) of degree π α ( x ) = | x | p α − 1 π 1 ( x ) and order m, π α ( x ) , π 1 ( x ) , and χ p ( x ) are a multiplicative, a normed multiplicative, and an additive characters of the field Q p of p-adic numbers, respectively, φ ∈ D ( Q p ) is a test function, m = 0 , 1 , 2 , … , α ∈ C . If Re α > 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.