An optimal Hardy-Littlewood-Sobolev inequality on ${\mathbf{R}}^{n-k}\times {\mathbf{R}}^n$ and its consequences

Quôc Anh Ngô, Quoc-Hung Nguyen, Van Hoang Nguyen

For $n>k\ge 0$, $\lambda >0$, and $p,r>1$, we establish the following optimal Hardy-Littlewood-Sobolev inequality \[ \Big| \iint_{\mathbf R^n \times \mathbf R^{n-k}} \frac{f(x) g(y)}…