Stein–Weiss inequalities with the fractional Poisson kernel
Lu Chen
Beijing Institute of Technology, ChinaZhao Liu
Jiangxi Science Technology Normal University, Nanchang, ChinaGuozhen Lu
University of Connecticut, Storrs, USAChunxia Tao
Beijing Normal University, China
Abstract
In this paper, we establish the following Stein–Weiss inequality with the fractional Poisson kernel:
where
, , , and and satisfy . Then we prove that there exist extremals for the Stein–Weiss inequality (), and that the extremals must be radially decreasing about the origin. We also provide the regularity and asymptotic estimates of positive solutions to the integral systems which are the Euler–Lagrange equations of the extremals to the Stein–Weiss inequality () with the fractional Poisson kernel. Our result is inspired by the work of Hang, Wang and Yan, where the Hardy–Littlewood–Sobolev type inequality was first established when and . The proof of the Stein–Weiss inequality () with the fractional Poisson kernel in this paper uses recent work on the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel by Chen, Lu and Tao, and the present paper is a further study in this direction.
Cite this article
Lu Chen, Zhao Liu, Guozhen Lu, Chunxia Tao, Stein–Weiss inequalities with the fractional Poisson kernel. Rev. Mat. Iberoam. 36 (2020), no. 5, pp. 1289–1308
DOI 10.4171/RMI/1167