A necessary condition for multipliers of weak type (1,1) (Q2739259)

The author provides several conditions on a Fourier multiplier \(\varphi \) under which the Fourier multiplier operator \(f\mapsto T_{\varphi }(f)=(\varphi \widehat{f})^{\vee }\) fails to be bounded from \(L^{1}\) to weak-\(L^{1}\) on Euclidean space. The main theorem says that if \(\varphi \) is a bounded continuous function such that there is a sequence \(a_{j}\) and \(C>0\) such that (i) \(\lim_{j\to \infty }|\langle a,a_{j}\rangle |=\infty \), (ii) \(|\varphi (a_{j})|\geq C\) and (iii) \(\lim_{j\to \infty }\varphi (x\pm a_{j})=0\) whenever \(x\neq \lambda a_{j}\) then \(T_{\varphi }\) fails to be of weak-type \((1,1)\). The proof is based first on a transference theorem that allows one to reduce to a question of boundedness for an associated multiplier defined on the integers. If such a multiplier is of weak type \((1,1)\) then it is automatically continuous from \(L^{1}\) to \(L^{2}\) on the torus. The arithmetic properties of the \(a_{j}\) enable one to construct a sequence of \(\Lambda (2)\) sets, that is, sets upon which \(\|f\|_{2}<K\|f\|_{1}\) when \(\widehat{f}\) is supported on such a set, along which values \(\varphi (b_{j})\) associated to compositions of \(\varphi \) with suitable bounded transformations are too big. One concludes from this that the transferred multiplier fails to be bounded from \(L^{1}\) to \(L^{2}\). As applications it is shown that certain rational function multipliers are not of weak-type \((1,1)\).