Singular Integrals on Product Spaces Related to the Carleson Operator

2006 ◽  
Vol 58 (1) ◽  
pp. 154-179 ◽  
Author(s):  
Elena Prestini
Keyword(s):  
Fourier Series ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Double Fourier Series ◽  
Variable Coefficients ◽  
Product Spaces ◽  
One Dimensional ◽  
Carleson Operator ◽  
Two Phases ◽  
The Hilbert Transform

AbstractWe prove Lp(T2) boundedness, 1 < p ≤ 2, of variable coefficients singular integrals that generalize the double Hilbert transform and present two phases that may be of very rough nature. These operators are involved in problems of a.e. convergence of double Fourier series, likely in the role played by the Hilbert transform in the proofs of a.e. convergence of one dimensional Fourier series. The proof due to C.Fefferman provides a basis for our method.

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 Generalization of the Hardy-Littlewood theorem on Fourier series

2021 ◽  
Vol 104 (4) ◽  
pp. 49-55
Author(s):  
S. Bitimkhan ◽  
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Lebesgue Space ◽  
Fourier Coefficients ◽  
Double Fourier Series ◽  
Function Spaces ◽  
Dimensional Case ◽  
One Dimensional ◽  
Littlewood Theorem ◽  
The One

In the theory of one-dimensional trigonometric series, the Hardy-Littlewood theorem on Fourier series with monotone Fourier coefficients is of great importance. Multidimensional versions of this theorem have been extensively studied for the Lebesgue space. Significant differences of the multidimensional variants in comparison with the one-dimensional case are revealed and the strengthening of this theorem is obtained. The Hardy-Littlewood theorem is also generalized for various function spaces and various types of monotonicity of the series coefficients. Some of these generalizations can be seen in works of M.F. Timan, M.I. Dyachenko, E.D. Nursultanov, S. Tikhonov. In this paper, a generalization of the Hardy-Littlewood theorem for double Fourier series of a function in the space L_qϕ(L_q)(0,2π]^2 is obtained.

Energy Counterexamples in Two Weight Calderón–Zygmund Theory

2019 ◽  
Author(s):  
Eric T Sawyer ◽  
Chun-Yen Shen ◽  
Ignacio Uriarte-Tuero
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Hilbert Transform ◽  
The Other ◽  
Energy Conditions ◽  
Convolution Kernel ◽  
One Dimensional ◽  
T1 Theorem ◽  
The Hilbert Transform

Abstract We show that the energy conditions are not necessary for boundedness of Riesz transforms in dimension $n\geq 2$. In dimension $n=1$, we construct an elliptic singular integral operator $H_{\flat } $ for which the energy conditions are not necessary for boundedness of $H_{\flat }$. The convolution kernel $K_{\flat }\left ( x\right ) $ of the operator $H_{\flat }$ is a smooth flattened version of the Hilbert transform kernel $K\left ( x\right ) =\frac{1}{x}$ that satisfies ellipticity $ \vert K_{\flat }\left ( x\right ) \vert \gtrsim \frac{1}{\left \vert x\right \vert }$, but not gradient ellipticity $ \vert K_{\flat }^{\prime }\left ( x\right ) \vert \gtrsim \frac{1}{ \vert x \vert ^{2}}$. Indeed the kernel has flat spots where $K_{\flat }^{\prime }\left ( x\right ) =0$ on a family of intervals, but $K_{\flat }^{\prime }\left ( x\right ) $ is otherwise negative on $\mathbb{R}\setminus \left \{ 0\right \} $. On the other hand, if a one-dimensional kernel $K\left ( x,y\right ) $ is both elliptic and gradient elliptic, then the energy conditions are necessary, and so by our theorem in [30], the $T1$ theorem holds for such kernels on the line. This paper includes results from arXiv:16079.06071v3 and arXiv:1801.03706v2.

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 $\mathbb{R}$.

The n-Dimensional Hilbert Transform of Distributions, Its Inversion and Applications

1990 ◽  
Vol 42 (2) ◽  
pp. 239-258 ◽  
Author(s):  
O. P. Singh ◽  
J. N. Pandey
Keyword(s):  
Hilbert Transform ◽  
One Dimension ◽  
Distribution Space ◽  
One Dimensional ◽  
Schwartz Distribution ◽  
The Hilbert Transform

Pandey and Chaudhary [13] recently developed the theory of Hilbert transform of Schwartz distribution space (DLp)',p > 1 in one dimension using Parseval's types of relations for one dimensional Hilbert transform [17] and noted that their theory coincides with the corresponding theory for the Hilbert transform developed by Schwartz [16] by using the technique of convolution in one dimension.The corresponding theory for the Hilbert transform in n-dimension is considerably harder and will be successfully accomplished in this paper.

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