Advances in Differential Equations

Boundedness of singular integrals and their commutators with BMO functions on Hardy spaces

The Anh Bui and Xuan Thinh Duong

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Abstract

In this paper, we establish sufficient conditions for a singular integral $T$ to be bounded from certain Hardy spaces $H^p_L$ to Lebesgue spaces $L^p$, $0 < p \le 1$, and for the commutator of $T$ and a BMO function to be weak-type bounded on Hardy space $H_L^1$. We then show that our sufficient conditions are applicable to the following cases: (i) $T$ is the Riesz transform or a square function associated with the Laplace--Beltrami operator on a doubling Riemannian manifold, (ii) $T$ is the Riesz transform associated with the magnetic Schr\"odinger operator on a Euclidean space, and (iii) $T = g(L) $ is a singular integral operator defined from the holomorphic functional calculus of an operator $L$ or the spectral multiplier of a non-negative self-adjoint operator $L$.

Article information

Source
Adv. Differential Equations Volume 18, Number 5/6 (2013), 459-494.

Dates
First available in Project Euclid: 14 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1363266254

Mathematical Reviews number (MathSciNet)
MR3086462

Zentralblatt MATH identifier
1275.42019

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 35B65: Smoothness and regularity of solutions 35K05: Heat equation 42B25: Maximal functions, Littlewood-Paley theory 47B38: Operators on function spaces (general) 58J35: Heat and other parabolic equation methods

Citation

Bui, The Anh; Duong, Xuan Thinh. Boundedness of singular integrals and their commutators with BMO functions on Hardy spaces. Adv. Differential Equations 18 (2013), no. 5/6, 459--494. https://projecteuclid.org/euclid.ade/1363266254.


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