Kodai Mathematical Journal

$L^2$ continuity of the Calderón type commutator for the Littlewood-Paley operator with rough variable kernel

Yanping Chen, Zhendong Niu, and Liwei Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For $b \in Lip(\mathbf{R}^{n})$, the Calderón type commutator for the Littlewood-Paley operator with variable kernel is defined by $$\mu_{\Omega,1;b}(f)(x) = \left( \int_{0}^{\infty}\left| \frac{1}{t^2}\int_{\vert x-y\vert \leq t}\frac{\Omega(x, x-y)}{\vert x - y \vert ^{n-1}} (b(x) - b(y))f(y) dy\right|^{2} \frac{dt}{t}\right)^{1/2}.$$ By giving a method based on Littlewood-Paley theory, Fourier transform and the spherical harmonic development, we prove the $L^{2}$ norm inequalities for the rough operators $\mu_{\Omega,1;b}$ with $\Omega(x,z^{\prime}) \in L^{\infty}(\mathbf{R}^{n} \times L^{q}(S^{n-1})\left(q \gt \frac{2(n-1)}{n} \right)$ satisfying certain cancellation conditions.

Note

The research is supported by NSF of China (Grant: 11471033), NCET of China (Grant: NCET- 11-0574), the Fundamental Research Funds for the Central Universities (FRF-BR-16-011A).

Article information

Source
Kodai Math. J., Volume 40, Number 3 (2017), 405-420.

Dates
Received: 30 May 2016
Revised: 24 November 2016
First available in Project Euclid: 31 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1509415223

Digital Object Identifier
doi:10.2996/kmj/1509415223

Mathematical Reviews number (MathSciNet)
MR3718490

Zentralblatt MATH identifier
06827096

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory

Keywords
commutator Littlewood-Paley operator Fourier transform variable kernel

Citation

Chen, Yanping; Niu, Zhendong; Wang, Liwei. $L^2$ continuity of the Calderón type commutator for the Littlewood-Paley operator with rough variable kernel. Kodai Math. J. 40 (2017), no. 3, 405--420. doi:10.2996/kmj/1509415223. https://projecteuclid.org/euclid.kmj/1509415223


Export citation