No 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: \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\in L^{q^{\prime}}(\mathbb{R}^{n}_{+})} , {g\in L^{p}(\partial\mathbb{R}^{n}_{+})} , and {p,q^{\prime}\in(0,1)} , {\beta<\frac{1-n}{p^{\prime}}} or {\alpha<-\frac{n}{q}} , {\lambda>0} satisfying \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 {\mathbb{R}^{n}_{+}} .

In this paper, we establish the existence of extremals for two kinds of Stein-Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein-Weiss inequalities with full weights in Theorem 1.1 and the Stein-Weiss inequalities with horizontal weights in Theorem 1.4. Different from the proof of the analogous inequality in Euclidean spaces given by Lieb [26] using Riesz rearrangement inequality which is not available on the Heisenberg group, we employ the concentration compactness principle to obtain the existence of the maximizers on the Heisenberg group. Our result is also new even in the Euclidean case because we don't assume that the exponents of the double weights in the Stein-Weiss inequality (1.1) are both nonnegative (see Theorem 1.3 and more generally Theorem 1.5). Therefore, we extend Lieb's celebrated result of the existence of extremal functions of the Stein-Weiss inequality in the Euclidean space to the case where the exponents are not necessarily both nonnegative (see Theorem 1.3). Furthermore, since the absence of translation invariance of the Stein-Weiss inequalities, additional difficulty presents and one cannot simply follow the same line of Lions' idea to obtain our desired result. Our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities (Theorem 1.5).

In this paper, we establish the following reverse Stein-Weiss inequality, namely the reversed weighted Hardy-Littlewood-Sobolev inequality, in R n :We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler-Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein-Weiss and reverse Stein-Weiss inequalities on the ndimensional sphere S n by using the stereographic projections. Our proof of the reverse Stein-Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy-Littlewood-Sobolev inequalities.

In this paper, we establish the following Stein-Weiss inequality with the fractional Poisson kernel (see Theorem 1.1):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 established when γ = 2 and α = β = 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.Keywords: Existence of extremal functions; Stein-Weiss inequality; Poisson kernel; Hardy inequality in high dimensions.2010 MSC. 35B40, 45G15.