Transferring Fourier multipliers from \(S^{2p-1}\) to \(H^{p-1}\)
Given a function \(\varphi\) on \(\widehat{\mathbb{R}}=\mathbb{R}\), consider its restriction \(\varphi_1\) to \(\widehat{\mathbb{T}}=\mathbb{Z}\), or more generally its restriction \(\varphi_\varepsilon\) to \(\varepsilon\mathbb{Z}\subset\mathbb{R}\). A theorem of de Leeuw from 1965 asserts that (1) if \(\varphi\) is a bounded Fourier multiplier on \(L^q(\mathbb{R})\), then \(\varphi_\varepsilon\) is a bounded Fourier multiplier on \(L^q(\mathbb{T})\) and \(\|T_{\varphi_\varepsilon}\|_q\leq\|T_\varphi\|_q\); and (2) if \(\forall\varepsilon>0\), \(T_{\varphi_\varepsilon}\) is bounded on \(L^q(\mathbb{T})\) with \(\limsup_{\varepsilon\to 0}\|T_{\varphi_\varepsilon}\|_q=K<\infty\), then \(T_\varphi\) is bounded on \(L^q(\mathbb{R})\) with \(\|T_\varphi\|_q\leq K\). Here \(T_\varphi\) stands for the Fourier multiplier \(\widehat{T_\varphi f}=\varphi\widehat f\) on \(\mathbb{R}\) or \(\mathbb{T}\), and \(\|T_\varphi\|_q\) for its norm on the corresponding \(L^q\). In the present paper, continuing the results of the first author and \textit{G. I. Gaudry} [J. Funct. Anal. 67, 1-24 (1986; Zbl 0597.43005), Ann. Inst. Fourier 36, 107-136 (1986; Zbl 0589.43003)] and of \textit{F. Ricci} and \textit{R. C. Rubin} [Am. J. Math. 108, 571-588 (1986; Zbl 0613.43005)], the authors prove an analogue (too technical to state here in detail) of de Leeuw's theorem for \(\mathbb{T} \) and \(\mathbb{R}\) replaced by \(S^{2n-1}\) and the Heisenberg group \(H^{n-1}\), respectively.
scientific article; zbMATH DE number 1865524