Fourier multipliers for Besicovitch spaces
The notion of Fourier multiplier \(M\) for the space \(L^q(\mathbb{R}^s)\), where \(1<q<\infty\), is extended to the case of a Fourier multiplier \(\widetilde M\) for the space \(B^q_{\text{ap}}(\mathbb{R}^s,\Lambda)\) of Besicovitch almost periodic functions with spectrum \(\Lambda\), where \(\widetilde M= (M(\lambda))_{\lambda\in \Lambda}\), where the functions from \(B^q_{\text{ap}}(\mathbb{R}^s,\Lambda)\) are treated as tempered distributions. There are also defined Sobolev-Besicovitch spaces \(H^{m,q}_{\text{ap}}(\mathbb{R}^s,\Lambda)\) of order \(m\) and type \(H\) and Sobolev-Besicovitch spaces \(W^{m,q}_{\text{ap}}(\mathbb{R}^s,\Lambda)\) of order \(m\) and type \(W\), where \(1<q<\infty\) and \(m\in\mathbb{R}\) in the first case, \(m\in\mathbb{N}\) in the second one. If \(1<q<\infty\) and \(N\in\mathbb{N}\), \(N\geq 1\), there is proved the identity \(H^{N,q}_{\text{ap}}(\mathbb{R}^s, \Lambda)= W^{N,q}_{\text{ap}}(\mathbb{R}^s, \Lambda)\).
scientific article; zbMATH DE number 1270170
scientific article