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 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.

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

Distributional estimates for the bilinear Hilbert transform

2006 ◽  
Vol 16 (4) ◽  
pp. 563-584 ◽  
Author(s):  
Dmitriy Bilyk ◽  
Loukas Grafakos
Keyword(s):  
Hilbert Transform ◽  
Bilinear Hilbert Transform

scholarly journals Bilinear Hilbert Transform on Measure Spaces

2005 ◽  
Vol 11 (4) ◽  
pp. 459-470 ◽  
Author(s):  
O. Blasco ◽  
M. Carro ◽  
T. A. Gillespie
Keyword(s):  
Hilbert Transform ◽  
Bilinear Hilbert Transform ◽  
Measure Spaces

scholarly journals Endpoint bounds for the bilinear Hilbert transform

2015 ◽  
Vol 368 (6) ◽  
pp. 3931-3972 ◽  
Author(s):  
Francesco Di Plinio ◽  
Christoph Thiele
Keyword(s):  
Hilbert Transform ◽  
Bilinear Hilbert Transform

scholarly journals Inversion theorem for bilinear Hilbert transform

2008 ◽  
Vol 19 (5) ◽  
pp. 317-325
Author(s):  
Aneta Bučkovska ◽  
Stevan Pilipović ◽  
Mirjana Vuković
Keyword(s):  
Hilbert Transform ◽  
Inversion Theorem ◽  
Bilinear Hilbert Transform

scholarly journals Variational bounds for a dyadic model of the bilinear Hilbert transform

2013 ◽  
Vol 57 (1) ◽  
pp. 105-119 ◽  
Author(s):  
Yen Do ◽  
Richard Oberlin ◽  
Eyvindur Ari Palsson
Keyword(s):  
Hilbert Transform ◽  
Bilinear Hilbert Transform ◽  
Variational Bounds ◽  
Dyadic Model

On the Ergodic Averages and the Ergodic Hilbert Transform

1995 ◽  
Vol 47 (2) ◽  
pp. 330-343
Author(s):  
L. M. Fernández-Cabrera ◽  
F. J. Martín-Reyes ◽  
J. L. Torrea
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Dual Problem ◽  
Finite Measure ◽  
Positive Function ◽  
Ergodic Averages ◽  
Unified Approach ◽  
Measure Preserving Transformation ◽  
Finite Measure Space ◽  
Abstract Method

AbstractLet T be an invertible measure-preserving transformation on a σ-finite measure space (X, μ) and let 1 < p < ∞. This paper uses an abstract method developed by José Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the positive measurable functions v such that the limit of the ergodic averages or the ergodic Hilbert transform exist for all f ∈ Lp(νdμ). As a corollary, we obtain that both problems are equivalent, extending to this setting some results of R. Jajte, I. Berkson, J. Bourgain and A. Gillespie. We do not assume the boundedness of the operator Tf(x) = f(Tx) on Lp(νdμ). However, the method of Rubio de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the ergodic maximal operator and the ergodic Hilbert transform are bounded from LP(νdμ) into LP(udμ). We also study and solve the dual problem.

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