A new proof of the restriction theorem for weak type \((1,1)\) multipliers on \(\mathbb{R}^ n\) (Q1815238)

Suppose that \(m\in L^\infty ({\mathbf R}^n),\) and that \(m_\varepsilon (\xi) = m (\varepsilon \xi)\) for all \(\varepsilon\) in \({\mathbf R}^+\) and \(\xi \) in \({\mathbf R}^n\). Denote by \(T_m^{\mathbf R}\) the Fourier multiplier operator on \(L^2 ({\mathbf R}^n)\) such that \((T_m^{\mathbf R} f)\widehat{\phantom{w}} =m\widehat{f}\), and by \(T_m^{\mathbf T}\) the corresponding operator on \(L^2 ({\mathbf T}^n)\), viz, \((T_m^{\mathbf T} f) \widehat{\phantom{w}} = m|_{{\mathbf Z}^n} \widehat{f}\), provided this makes sense. Denote by \(|||T|||\) the possibly infinite weak type \((1,1)\) ``norm'' of an operator \(T\), i.e., the smallest constant \(C\) such that \(|\{ x: \;|Tf(x)|> \lambda\}|\leq C |f|_1 / \lambda,\) for all \(\lambda\) in \({\mathbf R}^+\). In the paper under review, it is shown that, if \(m\) is continuous at the points of \({\mathbf Z}^n\), so that \(T_m^{\mathbf T}\) is well-defined, then \(|||T_m^{\mathbf T}|||\leq |||T_m^{\mathbf R}|||\) (this was known for some \(m\) by results of \textit{A. P. Calderón} [Proc. Natl. Acad. Sci. USA 59, 349-353 (1968; Zbl 0185.21806)] and for general \(m\), but with a constant possibly greater than 1, by results of \textit{N. Asmar, E. Berkson} and \textit{J. Bourgain} [Stud. Math. 108, No. 3, 291-299 (1994; Zbl 0827.42008)]. It is also shown that, if \(m \in C({\mathbf R}^n), \) then \(|||T_m^{\mathbf R}|||=\sup \{ |||T_{m_\varepsilon}^{\mathbf T}|||\colon \varepsilon \in {\mathbf R}^+\}\). It should be remarked that similar results were proved by \textit{C. E. Kenig} and \textit{P. A. Tomas} [Stud. Math. 68, 79-83 (1980; Zbl 0442.42013)] for weak type \((p,p)\) operators. It should also be noted that, for Fourier multiplier operators, the weak type \((1,1)\) norm is the limit, as \(p\) approaches \(1+\), of the weak type \((p,p)\) norms. These theorems may also be proved using this fact.