scholarly journals Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform

2008 ◽  
Vol 28 (5) ◽  
pp. 1453-1464 ◽  
Author(s):  
CIPRIAN DEMETER
Keyword(s):  
Harmonic Analysis ◽  
Ergodic Theory ◽  
Hilbert Transform ◽  
Positive Behavior ◽  
Multilinear Operators ◽  
Similar Range

AbstractWe consider multilinear averages in ergodic theory and harmonic analysis and prove their divergence in some range of Lp spaces. This contrasts with the positive behavior exhibited by these averages in a different range, as proved in Demeter et al [Maximal multilinear operators. Trans. Amer. Math. Soc.360(9) (2008), 4989–5042]. We also prove that the trilinear Hilbert transform is unbounded in a similar range of Lp spaces. The principle underlying these constructions is stated, setting the stage for more general results.

scholarly journals Norm variation of ergodic averages with respect to two commuting transformations

2017 ◽  
Vol 39 (3) ◽  
pp. 658-688 ◽  
Author(s):  
POLONA DURCIK ◽  
VJEKOSLAV KOVAČ ◽  
KRISTINA ANA ŠKREB ◽  
CHRISTOPH THIELE
Keyword(s):  
Harmonic Analysis ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Quantitative Result ◽  
Square Function ◽  
Two Dimensional ◽  
Ergodic Averages ◽  
Multilinear Singular Integrals

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.

Odometer actions on G-measures

1991 ◽  
Vol 11 (2) ◽  
pp. 279-307 ◽  
Author(s):  
Gavin Brown ◽  
Anthony H. Dooley
Keyword(s):  
Harmonic Analysis ◽  
Ergodic Theory ◽  
Finite Groups ◽  
Riesz Product ◽  
Direct Sum ◽  
New Methods ◽  
Product Construction ◽  
Ergodic Properties ◽  
Systematic Development

AbstractThe introduction of results from harmonic analysis leads to new methods in the study of the ergodic properties of measures under the action of the direct sum of finite groups. We take the first steps in a systematic development of part of ergodic theory based on the formalism of the Riesz product construction.

Randall Dougherty and Alexander S. Kechris. The complexity of antidifferentiation. Advances in mathematics, vol. 88 (1991), pp. 145–169. - Ferenc Beleznay and Matthew Foreman. The collection of distal flows is not Borel. American journal of mathematics, vol. 117 (1995), pp. 203–239. - Ferenc Beleznay and Matthew Foreman. The complexity of the collection of measure-distal transformations. Ergodic theory and dynamical systems, vol. 16 (1996), pp. 929–962. - Howard Becker. Pointwise limits of subsequences and sets. Fundamenta mathematicae, vol. 128 (1987), pp. 159–170. - Howard Becker, Sylvain Kahane, and Alain Louveau. Some complete sets in harmonic analysis. Transactions of the American Mathematical Society, vol. 339 (1993), pp. 323–336. - Robert Kaufman. PCA sets and convexity Fundamenta mathematicae, vol. 163 (2000), pp. 267–275). - Howard Becker. Descriptive set theoretic phenomena in analysis and topology. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematical Sciences Research Institute publications, vol. 26, Springer-Verlag, New York, Berlin, Heidelberg, etc., 1992, pp. 1–25.

2001 ◽  
Vol 7 (3) ◽  
pp. 385-388
Author(s):  
Gabriel Debs
Keyword(s):  
New York ◽  
Dynamical Systems ◽  
Harmonic Analysis ◽  
Ergodic Theory ◽  
Set Theory ◽  
American Mathematical Society ◽  
And Topology ◽  
Mathematical Sciences ◽  
Descriptive Set ◽  
The Continuum

scholarly journals Distributional estimates for multilinear operators

2005 ◽  
Author(s):  
◽  
Dmytro Bilyk
Keyword(s):  
Distribution Function ◽  
Exponential Decay ◽  
Hilbert Transform ◽  
Finite Measure ◽  
Maximal Operators ◽  
Characteristic Functions ◽  
Multilinear Operators ◽  
Logarithmic Factor ◽  
Bilinear Hilbert Transform ◽  
Invariant Spaces

We prove that if a multilinear operator and all its adjoints map L1 x x L1 to L1/m,oo, then the distribution function of the operator applied to characteristic functions of sets of finite measure has exponential decay at infinity. These estimates are based only on the boundedness properties and not the specific structure of the operator. The result applies to multilinear Calderon-Zygmund operators and several maximal operators. We have also obtained similar distributional estimates for the bilinear Hilbert transform: . . . . . . . .These estimates refect the exponential decay of the distribution function at infinity and also, up to a logarithmic factor, cover the endpoint cases of the region treated by Lacey and Thiele. Distributional estimates of this type also imply the boundedness of the operator on other rearrangement invariant spaces, in particular, the local exponential integrability.

scholarly journals Boundedness of the Hilbert transform on Besov spaces

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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