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 $$ P(x,\xi,\gamma)=\frac{x_n}{(|x'-\xi|^2+x_n^2)^{(n+2-\gamma)/2}}, $$ $2\le \gamma < n$, $f\in L^{p}(\partial\mathbb{R}^n_{+})$, $g\in L^{q'}(\mathbb{R}^n_{+})$, and $p,\ q'\in (1,\infty)$ and satisfy $(n-1)/(np)+1/q'+(\alpha+\beta+2-\gamma)/{n}=1$. 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 $\gamma=2$ and $\alpha=\beta=0$. 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.

中文翻译：

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分数泊松核的Stein-Weiss不等式

在本文中，我们用分数泊松核建立了以下Stein-Weiss不等式：\ 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 *}其中$ $ P（x，\ xi，\ gamma）= \ frac {x_n} {（| x'-\ xi | ^ 2 + x_n ^ 2）^ {（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）$，并满足$（n-1）/（np）+ 1 / q'+（\ alpha + \ beta + 2- \ gamma）/ {n} = 1 $。然后我们证明存在Stein-Weiss不等式$（\ star）$的极值，并且极值必须在原点附近呈放射状减小。我们还提供了积分系统正解的正则性和渐近估计，这些积分系统是带有分数Poisson核的Stein-Weiss不等式$（\ star）$的极值的Euler-Lagrange方程。我们的结果是由Hang，Wang和Yan的工作启发的，其中当$ \ gamma = 2 $和$ \ alpha = \ beta = 0 $时，首先建立了Hardy–Littlewood–Sobolev型不等式。用分数泊松核证明Stein-Weiss不等式$（\ star）$的方法是使用Chen，Lu和Tao对分数分数泊松核的Hardy-Littlewood-Sobolev不等式的最新研究，本文是在这个方向上的进一步研究。