Norm variation of ergodic averages with respect to two commuting transformations

We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform.

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Pointwise convergence of certain continuous-time double ergodic averages

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.45 ◽

2021 ◽

pp. 1-11

Author(s):

MICHAEL CHRIST ◽

POLONA DURCIK ◽

VJEKOSLAV KOVAČ ◽

JORIS ROOS

Keyword(s):

Recent Work ◽

Hilbert Transform ◽

Continuous Time ◽

Pointwise Convergence ◽

Probability Space ◽

Singular Integrals ◽

Ergodic Averages ◽

Almost Everywhere Convergence ◽

Almost Everywhere ◽

Multilinear Singular Integrals

Abstract We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting $\mathbb {R}$ -actions, coming from a single jointly measurable measure-preserving $\mathbb {R}^2$ -action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.

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Boundedness of the Hilbert transform on Besov spaces

Carpathian Mathematical Publications ◽

10.15330/cmp.12.2.443-450 ◽

2020 ◽

Vol 12 (2) ◽

pp. 443-450

Author(s):

A. Maatoug ◽

S.E. Allaoui

Keyword(s):

Integral Operator ◽

Harmonic Analysis ◽

Singular Integral Operator ◽

Singular Integral ◽

Hilbert Transform ◽

Besov Spaces ◽

Singular Integrals ◽

Modern Theory ◽

The Hilbert Transform ◽

Positive Results

The Hilbert transform along curves is of a great importance in harmonic analysis. It is known that its boundedness on $L^p(\mathbb{R}^n)$ has been extensively studied by various authors in different contexts and the authors gave positive results for some or all $p,1<p<\infty$. Littlewood-Paley theory provides alternate methods for studying singular integrals. The Hilbert transform along curves, the classical example of a singular integral operator, led to the extensive modern theory of Calderón-Zygmund operators, mostly studied on the Lebesgue $L^p$ spaces. In this paper, we will use the Littlewood-Paley theory to prove that the boundedness of the Hilbert transform along curve $\Gamma$ on Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ can be obtained by its $L^p$-boundedness, where $ s\in \mathbb{R}, p,q \in ]1,+\infty[ $, and $\Gamma(t)$ is an appropriate curve in $\mathbb{R}^n$, also, it is known that the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ are embedded into $L^p(\mathbb{R}^n)$ spaces for $s >0$ (i.e. $B^{s}_{p,q}(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n), s>0)$. Thus, our result may be viewed as an extension of known results to the Besov spaces $ B^{s}_{p,q}(\mathbb{R}^n)$ for general values of $s$ in $\mathbb{R}$.

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Transference on certain multilinear multiplier operators

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700002263 ◽

2001 ◽

Vol 70 (1) ◽

pp. 37-55 ◽

Cited By ~ 15

Author(s):

Dashan Fan ◽

Shuichi Sato

Keyword(s):

Hilbert Transform ◽

Hardy Spaces ◽

Singular Integrals ◽

Multilinear Operators ◽

Lebesgue Spaces ◽

Bilinear Hilbert Transform ◽

Multiplier Operators ◽

Multilinear Singular Integrals

AbstractWe study DeLeeuw type theorems for certain multilinear operators on the Lebesgue spaces and on the Hardy spaces. As applications, on the torus we obtain an analog of Lacey—Thiele's theorem on the bilinear Hilbert transform, as well as analogies of some recent theorems on multilinear singular integrals by Kenig—Stein and by Grafakos—Torres.

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Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0187 ◽

2021 ◽

Vol 11 (1) ◽

pp. 72-95

Author(s):

Xiao Zhang ◽

Feng Liu ◽

Huiyun Zhang

Keyword(s):

Hilbert Transform ◽

Besov Spaces ◽

Singular Integrals ◽

Riesz Transform ◽

Morrey Spaces ◽

Riesz Transforms ◽

Rough Kernels ◽

Lebesgue Spaces ◽

Variation Operators

Abstract This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

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