Sharp reversed Hardy–Littlewood–Sobolev inequality with extension kernel
Volume 271 / 2023
Abstract
In this paper, we prove the following reversed Hardy–Littlewood–Sobolev inequality with extension kernel: $$\int _{\mathbb R_+^n}\int _{\partial\mathbb R^n_+}\frac{x_n^\beta}{|x-y|^{n-\alpha}}f(y)g(x)\,dy\,dx\geq C_{n,\alpha ,\beta ,p}\|f\|_{L^{p}(\partial\mathbb R_+^n)}\|g\|_{L^{q’}(\mathbb R_+^n)} $$ for any nonnegative functions $f\in L^{p}(\partial\mathbb R_+^n)$ and $g\in L^{q’}(\mathbb R_+^n)$, where $n\geq 2$, $p, q’\in (0,1)$, $\alpha \gt n$, $0\leq \beta \lt \frac{\alpha-n}{n-1}$, $p \gt \frac{n-1}{\alpha -1-(n-1)\beta}$ are such that $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q’}-\frac{\alpha +\beta -1}{n}=1$. We prove the existence of extremal functions for the above inequality. Moreover, in the conformal invariant case, we classify all the extremal functions and hence derive the best constant via the method of moving spheres. It is quite surprising that the extremal functions do not depend on $\beta $. Finally, we derive the sufficient and necessary conditions for existence of positive solutions to the Euler–Lagrange equations by using Pohozaev identities.
