𝑊^{𝑝}-spaces and Fourier transform

Abstract

The spaces W M p , W M , a p , W Ω , p , W Ω , b , p , W M Ω , p , W M , a Ω , b , p W_M^p,W_{M,a}^p,{W^{\Omega ,p}},{W^{\Omega ,b,p}},W_M^{\Omega ,p},W_{M,a}^{\Omega ,b,p} generalizing the spaces of type W due to Gurevich (also given by Friedman, and Gelfand and Shilov) are investigated. Here M, Ω \Omega are certain continuous increasing convex functions, a, b are positive constants and 1 ≤ p > ∞ 1 \leq p > \infty . The Fourier transformation F is shown to be a continuous linear mapping as follows: F : W M , a p → W Ω , 1 / a , r , F : W Ω , b , p → W M , 1 / b r , F : W M , a Ω , b , p → W M , 1 / b Ω , 1 / a , r F:W_{M,a}^p \to {W^{\Omega ,1/a,r}},F:{W^{\Omega ,b,p}} \to W_{M,1/b}^r,F:W_{M,a}^{\Omega ,b,p} \to W_{M,1/b}^{\Omega ,1/a,r} . These results will be used in investigating uniqueness classes of certain Cauchy problems in future work.