\(W^ p\)-spaces and Fourier transform. (English) Zbl 0816.46033

Summary: The spaces \(W^ p_ M\), \(W^ p_{M,a}\), \(W^{\Omega, p}\), \(W^{\Omega, b,p}\), \(W^{\Omega, p}_ M\), \(W^{\Omega, b,p}_{M,a}\) 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: \[ F: W^ p_{M, a}\to W^{\Omega, 1/a,r},\;F: W^{\Omega, b, p}\to W^ r_{M, 1/b},\;F: W^{\Omega, b, p}_{M, a}\to W^{\Omega, 1/a, r}_{M, 1/b}. \] These results will be used in investigating uniqueness classes of certain Cauchy problems in future work.


46F12 Integral transforms in distribution spaces
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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