Sharp reversed Hardy-Littlewood-Sobolev inequality on \(\mathbf{R}^{n}\) (Q2014284)

Let \(p,r\in(0,1)\) and \(\lambda>0\) such that \(1/p+1/r-2/\lambda=2\). Then, there exists a positive constant \(C(n,p,r)\) such that for any non-negative functions \(f\in L^p(\mathbb{R}^n)\) and \(g\in L^r(\mathbb{R}^n)\) the inequality \[ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\,f(x)|x-y|^{\lambda}g(y) dxdy\geq C(n,p,r)\|f\|_{L^p(\mathbb{R}^n})\|g\|_{L^r(\mathbb{R}^n)} \] holds. This is the reversed Hardy-Littlewood-Sobolev (HLS) inequality found by \textit{J. Dou} and \textit{M. Zhu} [Int. Math. Res. Not. 2015, No. 19, 9696--9726 (2015; Zbl 1329.26033)]. Here, the authors give an alternative proof of this inequality, more concise than that of Dou and Zhu [loc. cit.] and their proof also gives an explicit bound from below for the constant \(C(n,p,r)\). They also deal with the existence of optimal functions for the inequality in full generality of the parameters. It is shown that up to a translation all optimal functions are radially symmetric and strictly increasing. In the diagonal case \(p=r=2n/(2n+\lambda)\), the sharp constants are explicitly computed.