Abstract:We give a Bellman-function proof of the dimension-free estimate \[ \Big\| \vec{R} f \Big\|_{L^p(\Omega;\,\ell^2)} \lesssim (p-1) \,\|f\|_{L^p(\Omega)}, \qquad 2\le p<\infty, \] for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube $\Omega=\{-1,1\}^n$, as well as for locally compact abelian groups, in particular $\Omega=\mathbb{Z}^n$. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of $\Omega$, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for $1<p<2$ such a dimension-free bound is known to be false.
From: Komla Domelevo [view email]
[v1]
Thu, 18 Jun 2026 14:30:49 UTC (15 KB)
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