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$L^{p} (ℝ ⁿ)$ bounds for commutators of convolution operators

Guoen Hu; Qiyu Sun; Xin Wang — 2002

Colloquium Mathematicae

The $L^{p} (ℝ ⁿ)$ boundedness is established for commutators generated by BMO(ℝⁿ) functions and convolution operators whose kernels satisfy certain Fourier transform estimates. As an application, a new result about the $L^{p} (ℝ ⁿ)$ boundedness is obtained for commutators of homogeneous singular integral operators whose kernels satisfy the Grafakos-Stefanov condition.

The boundedness of two classes of integral operators

Xin Wang; Ming-Sheng Liu — 2021

Czechoslovak Mathematical Journal

The aim of this paper is to characterize the $L^{p} - L^{q}$ boundedness of two classes of integral operators from $L^{p} (𝒰, d V_{α})$ to $L^{q} (𝒰, d V_{β})$ in terms of the parameters $a$ , $b$ , $c$ , $p$ , $q$ and $α$ , $β$ , where $𝒰$ is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

Analytic solutions of a second-order functional differential equation with a state derivative dependent delay

Jian-Guo Si; Xin-Ping Wang — 1999

Colloquium Mathematicae

This paper is concerned with a second-order functional differential equation of the form $x^{''} (z) = x (a z + b x^{'} (z))$ with the distinctive feature that the argument of the unknown function depends on the state derivative. An existence theorem is established for analytic solutions and systematic methods for deriving explicit solutions are also given.