Endpoint estimates for multiparameter multipliers of Marcinkiewicz type
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 $O((p')^{{3n}/{2}})$ 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 $L\log^{σ/2}L$, 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.