JournalsrmiVol. 36, No. 5pp. 1289–1308

Stein–Weiss inequalities with the fractional Poisson kernel

  • Lu Chen

    Beijing Institute of Technology, China
  • Zhao Liu

    Jiangxi Science Technology Normal University, Nanchang, China
  • Guozhen Lu

    University of Connecticut, Storrs, USA
  • Chunxia Tao

    Beijing Normal University, China
Stein–Weiss inequalities with the fractional Poisson kernel cover
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Abstract

In this paper, we establish the following Stein–Weiss inequality with the fractional Poisson kernel: \begin{align*} (\star)\qquad \int_{\mathbb{R}^n_{+}} \int_{\partial\mathbb{R}^n_{+}} |\xi|^{-\alpha} f(\xi) &\,P(x,\xi,\gamma)\, g(x)\, |x|^{-\beta}\, d\xi\, dx \\ &\leq C_{n,\alpha,\beta,p,q'}\, \|g\|_{L^{q'}(\mathbb{R}^n_{+})}\, \|f\|_{L^p(\partial \mathbb{R}^{n}_{+})}, \end{align*} where

P(x,ξ,γ)=xn(xξ2+xn2)(n+2γ)/2,P(x,\xi,\gamma)=\frac{x_n}{(|x'-\xi|^2+x_n^2)^{(n+2-\gamma)/2}},

2γ<n2\le \gamma < n, fLp(R+n)f\in L^{p}(\partial\mathbb{R}^n_{+}), gLq(R+n)g\in L^{q'}(\mathbb{R}^n_{+}), and p, q(1,)p,\ q'\in (1,\infty) and satisfy (n1)/(np)+1/q+(α+β+2γ)/n=1(n-1)/(np)+1/q'+(\alpha+\beta+2-\gamma)/{n}=1. Then we prove that there exist extremals for the Stein–Weiss inequality ()(\star), 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 ()(\star) 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 γ=2\gamma=2 and α=β=0\alpha=\beta=0. The proof of the Stein–Weiss inequality ()(\star) 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