scholarly journals On spline collocation and the Hilbert transform

2015 ◽  
Vol 31 (1) ◽  
pp. 89-95
Author(s):  
SANDA MICULA ◽  
Keyword(s):  
Integral Operator ◽  
Fourier Analysis ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Hilbert Transform ◽  
Projection Operator ◽  
Spline Collocation ◽  
The Hilbert Transform

In this paper we examine a relationship between the spline collocation projection operator πn and the Hilbert singular integral operator H0. We use Fourier analysis to prove that under certain conditions, a commutator property holds between the two operators. More specifically, we show that for u ∈ Ht, ||(πnH0 − H0πn)u||t ≤ Chλ||u||s (where h = 1/n), for some t, s and λ ∈ R.

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 $\mathbb{R}$.

On a singular integral operator

1988 ◽  
Vol 43 (3) ◽  
pp. 199-200
Author(s):  
K Kh Boimatov ◽  
G Dzhangibekov
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral

scholarly journals Precompact Sets, Boundedness, and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Wei Wang ◽  
Jingshi Xu
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Singular Integrals ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Bmo Function

We give sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Then we obtain the boundedness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces. Finally, we discuss the compactness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces.

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