Stein–Weiss inequalities with the fractional Poisson kernel
Stein–Weiss inequalities with the fractional Poisson kernel
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DOI：
10.4171/rmi/1167
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发表时间：
2018-07
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通讯作者：
Lu Chen;Zhaoli Liu;Guozhen Lu;Chunxia Tao
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作者：
Lu Chen;Zhaoli Liu;Guozhen Lu;Chunxia Tao
本文用分数阶泊松核（见定理1.1）建立了如下的Stein-Weiss不等式：\begin{equation}\label{int1} \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{equation}其中$P(x,\xi,\gamma)=\frac{x_n}{(|x'-\xi|^2+x_n^2)^{\frac{n+2-\gamma}{2}}}$, $2\le \gamma<n$, $f\in L^{p}(\partial\mathbb{R}^n_{+})$, $g\in L^{q'}(\mathbb{R}^n_{+})$和$p,\ q'\in (1,\infty)$满足$\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}+\frac{\alpha+\beta+2-\gamma}{n}=1$。然后我们证明了Stein-Weiss不等式（0.1）存在极值，并且极值必须在原点附近呈径向递减（见定理1.5）。我们还提供了积分系统正解的正则性和渐近估计，这些积分系统是具有分数泊松核的Stein-Weiss不等式（0.1）的极值的Euler-Lagrange方程（见定理1.7和1.8）。我们的结果受到了Hang， Wang和Yan的工作的启发[0]，他们首先建立了Hardy-Littlewood-Sobolev型不等式（$\gamma=2$和$\alpha=\beta=0$）（见（1.5））。本文用分数阶泊松核证明了Stein-Weiss不等式（0.1），利用了我们最近关于分数阶泊松核[18]的Hardy-Littlewood-Sobolev不等式的证明，是在这个方向上的进一步研究。
In this paper, we establish the following Stein-Weiss inequality with the fractional Poisson kernel (see Theorem 1.1): \begin{equation}\label{int1} \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{equation} where $P(x,\xi,\gamma)=\frac{x_n}{(|x'-\xi|^2+x_n^2)^{\frac{n+2-\gamma}{2}}}$, $2\le \gamma<n$, $f\in L^{p}(\partial\mathbb{R}^n_{+})$, $g\in L^{q'}(\mathbb{R}^n_{+})$ and $p,\ q'\in (1,\infty)$ and satisfy $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}+\frac{\alpha+\beta+2-\gamma}{n}=1$. Then we prove that there exist extremals for the Stein-Weiss inequality (0.1) and the extremals must be radially decreasing about the origin (see Theorem 1.5). 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 (0.1) with the fractional Poisson kernel (see Theorems 1.7 and 1.8). Our result is inspired by the work of Hang, Wang and Yan [29] where the Hardy-Littlewood-Sobolev type inequality was first establishedmwhen $\gamma=2$ and $\alpha=\beta=0$ (see (1.5)). The proof of the Stein-Weiss inequality (0.1) with the fractional Poisson kernel in this paper uses our recent work on the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel [18] and the present paper is a further study in this direction.