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

Open Access article
 Printed: Sep 2000
  • Brian P. Kelly
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Let $\T$ denote the unit circle in the complex plane, and let $X$ be a Banach space that satisfies\break Burkholder's UMD condition. Fix a natural number, $N \in \N$. Let $\od$ denote the reverse lexicographical order on $\Z^N$. For each $f \in L^1 (\T^N,X)$, there exists a strongly measurable function $\wt{f}$ such that formally, for all $\bfn \in \Z^N$, $\Dual{{\wt{f}}} (\bfn) = -i \sgn_\od (\bfn) \Dual{f} (\bfn)$. 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 UMD spaces, we prove that the associated maximal operator satisfies a weak-type $(1,1)$ inequality with a constant independent of the dimension~$N$.
MSC Classifications: 42A61 show english descriptions Probabilistic methods 42A61 - Probabilistic methods

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