Fourier multipliers for \(L_ p (\mathbb{R}^ n)\) via \(q\)-variation
We give a new sufficient condition for a function \(m\) to be a Fourier multiplier of \(L_p (\mathbb{R}^n)\) via its \(q\)-variation on dyadic rectangles, that is, if \(m\) is of bounded \(q\)-variation uniformly on dyadic rectangles, then \(m\) is a Fourier multiplier for \(L_p (\mathbb{R}^n)\) whenever \(1< p< \infty\) and \(|1/p- 1/2|<1/q\). This solves a problem posed by Coifman, Rubio de Francia and Semmes, who had considered only the one-dimensional case.
scientific article; zbMATH DE number 950237