2026/07/23 by Odysseas Bakas, Ioannis Parissis
Mathematics · #math.CA #math.FA
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 R2,n-multipliers, a class that contains all multipliers of bounded Vq(ℝ⊗ n)-variation for 1≤ q<2. The class \mathcal R2,n is a multiparameter generalization, introduced in this paper, of the \mathcal R2-multipliers of Coifman, Rubio de Francia, and Semmes. We show that \mathcal R2,n-multiplier operators locally map Llog^3(n-1)/2+1/2L into L1,∞, 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 Lp(\mathbb Rn)→ Lp(\mathbb Rn) operator norms of such multiplier operators as p → 1+. The proof of our Llog^3(n-1)/2+1/2L-to-L1,∞ result combines a vector-valued endpoint estimate for the multipliers with an implicit square function characterization of Llogσ/2L, 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.