Reverse Stein–Weiss Inequalities on the Upper Half Space and the Existence of Their Extremals

Abstract

Abstract The purpose of this paper is four-fold. First, we employ the reverse weighted Hardy inequality in the form of high dimensions to establish the following reverse Stein–Weiss inequality on the upper half space: ∫ ℝ + n ∫ ∂ ⁡ ℝ + n | x | α | x - y | λ f ( x ) g ( y ) | y | β d y d x ≥ C n , α , β , p , q ′ ∥ f ∥ L q ′ ⁢ ( ℝ + n ) ∥ g ∥ L p ⁢ ( ∂ ⁡ ℝ + n ) \int_{\mathbb{R}^{n}_{+}}\int_{\partial\mathbb{R}^{n}_{+}}\lvert x|^{\alpha}|x% -y|^{\lambda}f(x)g(y)|y|^{\beta}\,dy\,dx\geq C_{n,\alpha,\beta,p,q^{\prime}}\|% f\/\|_{L^{q^{\prime}}(\mathbb{R}^{n}_{+})}\|g\|_{L^{p}(\partial\mathbb{R}^{n}_% {+})} for any nonnegative functions f ∈ L q ′ ⁢ ( ℝ + n ) {f\in L^{q^{\prime}}(\mathbb{R}^{n}_{+})} , g ∈ L p ⁢ ( ∂ ⁡ ℝ + n ) {g\in L^{p}(\partial\mathbb{R}^{n}_{+})} , and p , q ′ ∈ ( 0 , 1 ) {p,q^{\prime}\in(0,1)} , β < 1 - n p ′ {\beta<\frac{1-n}{p^{\prime}}} or α < - n q {\alpha<-\frac{n}{q}} , λ > 0 {\lambda>0} satisfying n - 1 n ⁢ 1 p + 1 q ′ - α + β + λ - 1 n = 2 . \frac{n-1}{n}\frac{1}{p}+\frac{1}{q^{\prime}}-\frac{\alpha+\beta+\lambda-1}{n}% =2. Second, we show that the best constant of the above inequality can be attained. Third, for a weighted system analogous to the Euler–Lagrange equations of the reverse Stein–Weiss inequality, we obtain the necessary conditions of existence for any positive solutions using the Pohozaev identity. Finally, in view of the stereographic projection, we give a spherical form of the Stein–Weiss inequality and reverse Stein–Weiss inequality on the upper half space ℝ + n {\mathbb{R}^{n}_{+}} .