Multiplier extension and sampling theorem on Hardy spaces (Q1896479)

The paper studies the transference and extension of Fourier multipliers of \(H_p (T^d)\). Consider \(d = 1\) only. Let \(\varphi (x)\) be a continuous function with compact support, and \(S (\varphi) = \overline{\text{span}}(\{\varphi (x - k)\}_{k \in \mathbb{Z}})\). Define the semi-convolution operator \(\varphi^{*'}: \{c(n)\} \to \sum_n c(n) \varphi (x - n)\). Denote the Fourier multiplier of \(H_p (T)\) and \(H_p (R)\) by \(\widetilde M (p)\), \(M(p)\), respectively, and the restriction of \(\varphi^{*'}\) on \(\widetilde M (p)\) by \(I\). The paper gives two theorems devoted to the transference and extension of \(\widetilde M (p)\). Theorem 1. Let \(0 < p < \infty\), \(\varphi (x)\) be continuous and compactly supported. Assume \[ \int^\infty_{-\infty} |\widehat \varphi (\xi) |^{\min (1,p)} d \xi < \infty. \] Then \(I\) is bounded from \(\widetilde M (p)\) to \(M(p)\). The theorem generalizes the \(L^p\)- result of \textit{E. Berkson} and \textit{T. A. Gillespie} [J. Lond. Math. Soc., II. Ser. 41, No. 3, 472-488 (1990; Zbl 0707.42015)]. Theorem 2. Let \(\varphi (x)\) be continuous and compactly supported. Assume that \(\varphi^{*'}\) is one-to-one from the space of all sequences to \(S (\varphi)\). Then \(I\) has a bounded inverse \(I^{-1}\) from \(M(p) \cap S (\varphi)\) to \(\widetilde M (p)\). Another result is devoted to the sampling theorem on \(H_p\). It says (Theorem 3): Let \(0 < p < \infty\), \(f\) such that \(\text{supp} \widehat f \subset [- {1 \over 2} + \varepsilon, {1 \over 2} - \varepsilon]\). Then \[ C^{-1} \biggl |\bigl \{f(n) \bigr\} \biggr |_{H_p (\mathbb{Z})} \leq |f |_{H_p (\mathbb{R})} \leq C \biggl |\bigl \{f(n) \bigr\} \biggr |_{H_p (\mathbb{Z})}. \]