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

Bibliographie

[1] [1] Asmar, N., Berkson, E. and Gillespie, T. A., Distributional control and generalized analyticity. Integral Equations Operator Theory 14 (1991), 311–341. Google Scholar

[2] [2] Asmar, N., Kelly, B. P., and Montgomery-Smith, S., A note on UMD spaces and transference in vector-valued function spaces. Proc. Edinburgh Math. Soc. 39 (1996), 485–490. Google Scholar

[3] [3] Asmar, N. and Montgomery-Smith, S., Dimension-free estimates for conjugate maximal functions and pointwise convergence. Studia Math., to appear. Google Scholar

[4] [4] Berkson, E., Gillespie, T. A. and Muhly, P. S., Generalized analyticity in UMD spaces. Ark. Mat. 27 (1989), 1–14. Google Scholar

[5] [5] Bourgain, J., Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21 (1983), 163–168. Google Scholar

[6] [6] Burkholder, D., A geometric condition that implies the existence of certain singular integrals of Banach-spacevalued functions. In: Proceedings, Conference on Harmonic Analysis in Honor of A. Zygmund, Chicago, 1981 (eds.W. Becker et al.),Wadsworth, Belmont, CA, 1983, 270–286. Google Scholar

[7] [7] Burkholder, D., Gundy, R. F. and Silverstein, M. L., A maximal characterization of the class Hp. Trans. Amer. Math. Soc. 157 (1971), 137–153. Google Scholar

[8] [8] Doob, J. L., Stochastic Processes. Wiley Publications in Statistics, New York, 1990. Google Scholar

[9] [9] Garling, D. J. H., Brownian motion and UMD spaces. In: Probability and Banach spaces, Springer Lecture Notes in Math. 1221 (1986), 36–49. Google Scholar

[10] [10] Helson, H., Conjugate series in several variables. Pacific J. Math. 9 (1959), 513–523. Google Scholar

[11] [11] Kelly, B. P., Distributional controlled representations acting on vector-valued functions spaces. Doctoral Dissertation, University of Missouri, 1994. Google Scholar

[12] [12] Zygmund, A., Trigonometric Series. 2nd edition (2 vols.), Cambridge University Press, 1959 Google Scholar