On Jodeit's multiplier extension theorems (Q1343319)

Let \(\Lambda\) be a Fourier multiplier for \(L^p(\mathbb{R})\), i.e., a bounded function such that \({\mathcal F}^{- 1}\Lambda({\mathcal F} f)\in L^p(\mathbb{R})\) whenever \(f\in L^p(\mathbb{R})\) (\(\mathcal F\) denoting the Fourier transformation), and \(m_n\) be a sequence of Fourier multipliers of \(L^p(\mathbb{T})\). From \(\Lambda\) and \(m_n\) one constructs the sequence \(w_n\) of functions on \(\mathbb{R}\) given by \(w_n(\xi)= \sum_{k\in \mathbb{Z}} \Lambda(\xi- k) m_n(k)\). This paper is concerned with the properties of \(w_n\). In particular, if \(\Lambda\) has compact support, and the maximal operator \(f\mapsto \sup_n|{\mathcal F}^{- 1} m_n{\mathcal F} f|\) is bounded on \(L^p(\mathbb{T})\), then the maximal operator \(f\mapsto \sup_n |{\mathcal F}^{- 1} w_n {\mathcal F} f|\) is bounded on \(L^p(\mathbb{R})\), and interesting results are obtained when \(\Lambda\) does not have compact support. Many of the results of this paper can also be proved by ``transference'' methods [see the second author, \textit{M. Paluszynski} and \textit{G. Weiss}, ``Transference couples and their applications to convolution and maximal operators'', in Interaction Between Functional Analysis, Harmonic Analysis, and Probability, edited by Nigel Kalton, Elias Saab, and Stephen Montgomery-Smith; Marcel Dekker (1995)].