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

Nicolas Th. Varopoulos

Displaying similar documents to “A theorem on weak type estimates for Riesz transforms and martingale transforms”

A note on martingale transforms and $A_{p}$ -weights

Rodrigo Bañuelos (1987)

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Conjugate martingale transforms

Ferenc Weisz (1992)

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Characterizations of H₁, BMO and VMO martingale spaces generated by bounded Vilenkin systems via conjugate martingale transforms are studied.

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Richard Gundy (1969)

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BMO and commutators of martingale transforms

Svante Janson (1981)

Annales de l'institut Fourier

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The commutator of multiplication by a function and a martingale transform of a certain type is a bounded operator on $L^{p}$ , $1 < p < \infty$ , if and only if the function belongs to BMO. This is a martingale version of a result by Coifman, Rochberg and Weiss.

Martingale operators and Hardy spaces generated by them

Ferenc Weisz (1995)

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Martingale Hardy spaces and BMO spaces generated by an operator T are investigated. An atomic decomposition of the space $H_{p}^{T}$ is given if the operator T is predictable. We generalize the John-Nirenberg theorem, namely, we prove that the $B M O_{q}$ spaces generated by an operator T are all equivalent. The sharp operator is also considered and it is verified that the $L_{p}$ norm of the sharp operator is equivalent to the $H_{p}^{T}$ norm. The interpolation spaces between the Hardy and BMO spaces are identified by...

Logarithmic inequalities for second-order Riesz transforms and related Fourier multipliers

Adam Osękowski (2013)

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We study logarithmic estimates for a class of Fourier multipliers which arise from a nonsymmetric modulation of jumps of Lévy processes. In particular, this leads to corresponding tight bounds for second-order Riesz transforms on $ℝ^{d}$ .

Carthaginian enlargement of filtrations

Giorgia Callegaro, Monique Jeanblanc, Behnaz Zargari (2013)

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This work is concerned with the theory of initial and progressive enlargements of a reference filtration $𝔽$ F with a random time. We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of Jacod, alternative proofs to results concerning canonical decomposition of an $𝔽$ F -martingale in the enlarged filtrations. Also, we address martingales’ characterization in the enlarged filtrations in terms of martingales in the reference filtration, as...

Harmonic interpolating sequences, $L^{p}$ and BMO

John B. Garnett (1978)

Annales de l'institut Fourier

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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.

An almost-sure estimate for the mean of generalized $Q$ -multiplicative functions of modulus $1$

Jean-Loup Mauclaire (2000)

Journal de théorie des nombres de Bordeaux

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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 $lim_{x_{k} \to x} (\frac{1}{x_{k}} \sum_{n \leq x_{k} - 1} g (n) - \prod_{0 \leq j \leq k} \frac{1}{q_{j}} \sum_{0 \leq a \leq q_{j}} g (a Q_{j})) = 0 μ -a.e .$

On a weak type (1,1) inequality for a maximal conjugate function

Nakhlé Asmar, Stephen Montgomery-Smith (1997)

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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.

A martingale approach to general Franklin systems

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Studia Mathematica

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We prove unconditionality of general Franklin systems in $L^{p} (X)$ , where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

Riesz transforms for the Dunkl Ornstein-Uhlenbeck operator

Adam Nowak, Luz Roncal, Krzysztof Stempak (2010)

Colloquium Mathematicae

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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 < ∞.