scholarly journals Transference on certain multilinear multiplier operators

2001 ◽  
Vol 70 (1) ◽  
pp. 37-55 ◽  
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.

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

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.

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.

scholarly journals The bilinear Hilbert transform in UMD spaces

2020 ◽  
Vol 378 (3-4) ◽  
pp. 1129-1221 ◽  
Author(s):  
Alex Amenta ◽  
Gennady Uraltsev
Keyword(s):  
Hilbert Transform ◽  
Scale Space ◽  
Banach Lattices ◽  
Lebesgue Spaces ◽  
Time Frequency ◽  
Bilinear Hilbert Transform ◽  
Umd Spaces ◽  
Frequency Scale

Abstract We prove $$L^p$$ L p -bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces $$L^p(\mathbb {R};X)$$ L p ( R ; X ) into outer Lebesgue spaces on the time-frequency-scale space $$\mathbb {R}^3_+$$ R + 3 .

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 Banach-Valued Multilinear Singular Integrals with Modulation Invariance

2020 ◽  
Author(s):  
Francesco Di Plinio ◽  
Kangwei Li ◽  
Henri Martikainen ◽  
Emil Vuorinen
Keyword(s):  
Singular Integrals ◽  
Dimensional Subspace ◽  
Projection Technique ◽  
Function Type ◽  
One Dimensional ◽  
Bilinear Hilbert Transform ◽  
Single Tree ◽  
Umd Spaces ◽  
Space Projection ◽  
Multilinear Singular Integrals

Abstract We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded $L^p$-extension to triples of intermediate $\operatorname{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\operatorname{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\textrm{UMD}$-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the $\textrm{UMD}$-valued bound for bilinear Calderón–Zygmund operators recently obtained by the same authors.

scholarly journals Pointwise convergence of certain continuous-time double ergodic averages

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.

scholarly journals A note on maximal singular integrals with rough kernels

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiao Zhang ◽  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Maximal Operators ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Homogeneous Mapping ◽  
Recent Developments ◽  
Maximal Singular Integrals ◽  
Maximal Singular Integral

Abstract In this note we study the maximal singular integral operators associated with a homogeneous mapping with rough kernels as well as the corresponding maximal operators. The boundedness and continuity on the Lebesgue spaces, Triebel–Lizorkin spaces, and Besov spaces are established for the above operators with rough kernels in $H^{1}({\mathrm{S}}^{n-1})$ H 1 ( S n − 1 ) , which complement some recent developments related to rough maximal singular integrals.

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