A Dimension-Free Weak-Type Estimate for Operators on UMD-Valued Functions

Kelly, Brian P.

Canadian mathematical bulletin, Tome 43 (2000) no. 3, pp. 355-361

Résumé

Let $\mathbb{T}$ denote the unit circle in the complex plane, and let $X$ be a Banach space that satisfies Burkholder’s $\text{UMD}$ condition. Fix a natural number, $N\,\in \,\mathbb{N}$ . Let $\mathcal{P}$ denote the reverse lexicographical order on ${{\mathbb{Z}}^{N}}$ . For each $f\,\in \,{{L}^{1}}({{\mathbb{T}}^{N}},X)$ , there exists a strongly measurable function $\tilde{f}$ such that formally, for all $\mathbf{n}\,\in \,{{\mathbb{Z}}^{N}},\,\hat{\tilde{f}}\,\left( \mathbf{n} \right)\,=\,-i\,{{sgn }_{\mathcal{P}}}\left( \mathbf{n} \right)\hat{f}\left( \mathbf{n} \right)$ . In this paper, we present a summation method for this conjugate function directly analogous to the martingale methods developed by Asmar and Montgomery-Smith for scalar-valued functions. Using a stochastic integral representation and an application of Garling’s characterization of $\text{UMD}$ spaces, we prove that the associated maximal operator satisfies a weak-type (1, 1) inequality with a constant independent of the dimension $N$ .

Détail

DOI : 10.4153/CMB-2000-042-x

Mots-clés : 43A17, 60H30, 46B09