Endpoint estimates for multiparameter multipliers of Marcinkiewicz type

Abstract

In this paper we prove sharp endpoint estimates for multiparameter Marcinkiewicz multiplier operators. More precisely, this result is a consequence of a more general theorem for multiparameter ${\mathcal{R}}_{2,n}$-multipliers, a class that contains all multipliers of bounded ${\mathcal{V}}_q({\mathbb{R}}^{\otimes n})$-variation for $1\le q<2$. The class ${\mathcal{R}}_{2,n}$ is a multiparameter generalization, introduced in this paper, of the ${\mathcal{R}}_2$-multipliers of Coifman, Rubio de Francia, and Semmes. We show that ${\mathcal{R}}_{2,n}$-multiplier operators locally map $L{\log }^{3(n-1)/2+1/2}L$ into $L^{1,\infty }$, and that this estimate is best possible, extending the corresponding one-parameter result of Tao and Wright to arbitrarily many parameters. We also establish the sharp bound $O((p^{\prime }{)}^{3n/2})$ for the $L^p({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)$ operator norms of such multiplier operators as $p\to 1^{+}$. The proof of our $L{\log }^{3(n-1)/2+1/2}L$-to-$L^{1,\infty }$ result combines a vector-valued endpoint estimate for the multipliers with an implicit square function characterization of $L{\log }^{\ddot{I}ƒ/2}L$, obtained via duality from the Chang-Wilson-Wolff inequality. The latter produces, at each iterative step, auxiliary proxy functions that are fed into an intermediate one-parameter vector-valued weak-$(1,1)$ estimate.

32 pages. Submitted for publication