A theorem on weak type estimates for Riesz transforms and martingale transforms

Nicolas Th. Varopoulos

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Let $(z_{ν})$ be a sequence in the upper half plane. If $1 < p \leq \infty$ and if $y_{ν}^{1 / p} f (z_{ν}) = a_{ν}, ν = 1, 2, ... (*)$ has solution $f (z)$ in the class of Poisson integrals of $L^{p}$ functions for any sequence $(a_{ν}) \in ℓ^{p}$ , then we show that $(z_{ν})$ is an interpolating sequence for $H^{\infty}$ . If $f (z_{ν}) = a_{ν}$ , $ν = 1, 2, ...$ has solution in the class of Poisson integrals of BMO functions whenever $(a_{ν}) \in ℓ^{\infty}$ , then $(z_{ν})$ is again an interpolating sequence for $H^{\infty}$ . A somewhat more general theorem is also proved and a counterexample for the case $p \leq 1$ is described.

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Let $Q = {(Q_{k})}_{k \geq 0}, Q_{0} = 1, Q_{k + 1} = q_{k} Q_{k}, q_{k} \geq 2$ , be a Cantor scale, $𝐙_{Q}$ the compact projective limit group of the groups $𝐙 / Q_{k} 𝐙$ , identified to $\prod_{0 \leq j \leq k - 1} 𝐙 / q_{j} 𝐙$ , and let $μ$ be its normalized Haar measure. To an element $x = {a_{0}, a_{1}, a_{2}, \dots}, 0 \leq a_{k} \leq q_{k + 1} - 1$ , of $𝐙_{Q}$ we associate the sequence of integral valued random variables $x_{k} = \sum_{0 \leq j \leq k} a_{j} Q_{j}$ . The main result of this article is that, given a complex $𝐐$ -multiplicative function $g$ of modulus $1$ , we have $lim_{x_{k} \to x} (\frac{1}{x_{k}} \sum_{n \leq x_{k} - 1} g (n) - \prod_{0 \leq j \leq k} \frac{1}{q_{j}} \sum_{0 \leq a \leq q_{j}} g (a Q_{j})) = 0 μ -a.e .$

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In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of $H^{p}$ spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.

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We propose a definition of Riesz transforms associated to the Ornstein-Uhlenbeck operator based on the Dunkl Laplacian. In the case related to the group ℤ ₂ it is proved that the Riesz transform is bounded on the corresponding $L^{p}$ spaces, 1 < p < ∞.