Sharp reversed Hardy-Littlewood-Sobolev inequality on R^n

Tiêu đềSharp reversed Hardy-Littlewood-Sobolev inequality on R^n
Loại công bốJournal Article
Năm xuất bản2017
Tác giảNgô, QAnh, Nguyen, VH
Tạp chíIsrael Journal of Mathematics
Tóm tắt

This is the first in our series of papers that concerns Hardy-Littlewood-Sobolev (HLS) type inequalities. In this paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\mathbf R^n$
\int_{\mathbf R^n} \int_{\mathbf R^n} f(x) |x-y|^\lambda g(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p(\mathbf R^n)}\, \|g\|_{L^r(\mathbf R^n)}
\]for any non-negative functions $f\in L^p(\mathbf R^n)$, $g\in L^r(\mathbf R^n)$, and $p,r\in (0,1)$, $\lambda > 0$ such that $1/p + 1/r -\lambda /n =2$. We will also explore some estimates for $\mathscr C_{n,p,r}$ and the existence of optimal functions for the above inequality, which will shed light on some existing results in literature.