On the one-sided ergodic Hilbert transform

2007 ◽  
pp. 21-39 ◽  
Author(s):  
Idris Assani ◽  
Michael Lin
Keyword(s):  
Hilbert Transform ◽  
The One

Uniform estimates for families of singular integrals and double Fourier series

1986 ◽  
Vol 41 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Elena Prestini
Keyword(s):  
Fourier Series ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Double Fourier Series ◽  
Variable Coefficients ◽  
Partial Sums ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Almost Everywhere ◽  
The One

AbstractIt is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

MAXIMAL SEMIGROUP SYMMETRY AND DISCRETE RIESZ TRANSFORMS

2015 ◽  
Vol 100 (2) ◽  
pp. 216-240
Author(s):  
TOSHIYUKI KOBAYASHI ◽  
ANDREAS NILSSON ◽  
FUMIHIRO SATO
Keyword(s):  
Hilbert Transform ◽  
Riesz Transform ◽  
Riesz Transforms ◽  
Dimensional Case ◽  
One Dimensional ◽  
Linearly Independent ◽  
The One ◽  
The Hilbert Transform ◽  
Multiplier Operators

We raise a question of whether the Riesz transform on $\mathbb{T}^{n}$ or $\mathbb{Z}^{n}$ is characterized by the ‘maximal semigroup symmetry’ that the transform satisfies. We prove that this is the case if and only if the dimension is one, two or a multiple of four. This generalizes a theorem of Edwards and Gaudry for the Hilbert transform on $\mathbb{T}$ and $\mathbb{Z}$ in the one-dimensional case, and extends a theorem of Stein for the Riesz transform on $\mathbb{R}^{n}$. Unlike the $\mathbb{R}^{n}$ case, we show that there exist infinitely many linearly independent multiplier operators that enjoy the same maximal semigroup symmetry as the Riesz transforms on $\mathbb{T}^{n}$ and $\mathbb{Z}^{n}$ if the dimension $n$ is greater than or equal to three and is not a multiple of four.

scholarly journals On the p-norm of the truncated Hilbert transform

1988 ◽  
Vol 38 (3) ◽  
pp. 413-420 ◽  
Author(s):  
W. McLean ◽  
D. Elliott
Keyword(s):  
Hilbert Transform ◽  
Positive Measure ◽  
Integral Operators ◽  
Finite Hilbert Transform ◽  
Measurable Set ◽  
Symmetry Properties ◽  
Lebesgue Measurable Set ◽  
The One ◽  
The Hilbert Transform

The p-norm of the Hilbert transform is the same as the p-norm of its truncation to any Lebesgue measurable set with strictly positive measure. This fact follows from two symmetry properties, the joint presence of which is essentially unique to the Hilbert transform. Our result applies, in particular, to the finite Hilbert transform taken over (−1, 1), and to the one-sided Hilbert transform taken over (0, ∞). A related weaker property holds for integral operators with Hardy kernels.

scholarly journals DYADIC TRIANGULAR HILBERT TRANSFORM OF TWO GENERAL FUNCTIONS AND ONE NOT TOO GENERAL FUNCTION

2015 ◽  
Vol 3 ◽  
Author(s):  
VJEKOSLAV KOVAČ ◽  
CHRISTOPH THIELE ◽  
PAVEL ZORIN-KRANICH
Keyword(s):  
Singular Integral ◽  
Hilbert Transform ◽  
Integral Form ◽  
General Function ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Bilinear Hilbert Transform ◽  
The One ◽  
Special Case ◽  
Dyadic Model

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate $L^{p}$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.

scholarly journals A special form of solution to half-wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hyungjin Huh
Keyword(s):  
Transport Equation ◽  
Hilbert Transform ◽  
Special Form ◽  
Wave Equations ◽  
Nonlinear Transport ◽  
One Dimensional ◽  
Nonlinear Transport Equation ◽  
Half Wave ◽  
The One ◽  
Nonlocal Nonlinear

<p style='text-indent:20px;'>We investigate a special form of solution to the one-dimensional half-wave equations with particular forms of nonlinearities. Using the special form of solution involving Hilbert transform, the half-wave equations reduce to nonlocal nonlinear transport equation which can be solved explicitly.</p>

scholarly journals The one-sided ergodic Hilbert transform in Banach spaces

2010 ◽  
Vol 196 (3) ◽  
pp. 251-263 ◽  
Author(s):  
Guy Cohen ◽  
Christophe Cuny ◽  
Michael Lin
Keyword(s):  
Banach Spaces ◽  
Hilbert Transform ◽  
The One

On the a.s. convergence of the one-sided ergodic Hilbert transform

2009 ◽  
Vol 29 (6) ◽  
pp. 1781-1788 ◽  
Author(s):  
CHRISTOPHE CUNY
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Unitary Operator ◽  
Finite Measure ◽  
Positive Answer ◽  
Positive Contraction ◽  
Finite Measure Space ◽  
The One

AbstractWe show that for T a Dunford–Schwartz operator on a σ-finite measure space (X,Σ,μ) and f∈L1(X,μ), whenever the one-sided ergodic Hilbert transform ∑ n≥1(Tnf/n) converges in norm, it converges μ-a.s. A similar result is obtained for any positive contraction of some fixed Lp(X,Σ,μ), p>1. Applying our result to the case where T is the (unitary) operator induced by a measure-preserving (invertible) transformation, we obtain a positive answer to a question of Gaposhkin.

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