Multipliers for weighted \(L^p\)-spaces, transference, and the \(q\)-variation of functions
Let \(m\) be a Fourier multiplier on \(L^p\) which is of bounded \(q\)-variation on each dyadic interval \(\pm [2^j, 2^{j+1}]\). It is a theorem of \textit{R. Coifman, J. L. Rubio de Francia} and \textit{S. Semmes} [C. R. Acad. Sci., Paris, Sér. I 306, No. 8, 351-354 (1988; Zbl 0643.42010)] that such multipliers are bounded on \(L^p(R)\) whenever \(1 < p < \infty\) and \(|1/p - 1/2|< 1/q\). The authors extend this for \(p \geq 2\) to weighted \(L^p\) spaces; in fact under these conditions these multipliers are bounded on \(L^p_w(R)\) whenever \(w\) is an \(A_{p/2}\) weight. The \(q=1\) case was previously established by \textit{D. S. Kurtz} [Trans. Am. Math. Soc. 259, 235-254 (1980; Zbl 0436.42012)]. This result is then transferred to the integers. The methods are modeled on those in \textit{R. Coifman}, \textit{J. L. Rubio de Francia}, and \textit{S. Semmes} (loc. cit.) and involve an interpolation with the \(p=2\) endpoint which is implicitly in that paper.
scientific article; zbMATH DE number 1236340
scientific article