$W^ p$-spaces and Fourier transform

Abstract:

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