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

TL;DR

This paper establishes a new optimal Hardy-Littlewood-Sobolev inequality on product spaces, unifying and extending classical results, with sharp conditions and implications for related inequalities.

Contribution

The paper introduces an optimal Hardy-Littlewood-Sobolev inequality on 0 and its sharp conditions, unifying several known inequalities.

Findings

Established the optimal inequality with sharp conditions.

Unified multiple known Hardy-Littlewood-Sobolev inequalities.

Derived the sharp range for the unweighted case 0

Abstract

For $n > k \geq 0$, $\lambda >0$, and $p, r>1$, we establish the following optimal Hardy-Littlewood-Sobolev inequality \[ \Big| \iint_{\mathbf R^n \times \mathbf R^{n-k}} \frac{f(x) g(y)}{ |x-y|^\lambda |y"|^\beta} dx dy \Big| \lesssim \| f \| _{L^p(\mathbf R^{n-k})} \| g\| _{L^r(\mathbf R^n)} \] with $y = (y', y") \in \mathbf R^{n-k} \times \mathbf R^k$ under the two necessary conditions \[ \beta < \left\{ \begin{aligned} & k - k/r & & \text{if } \; 0 < \lambda \leq n-k,\\ & n - \lambda - k/r & & \text{if } \; n-k < \lambda, \end{aligned} \right. \] and \[ \frac{n-k}n \frac 1p + \frac 1r + \frac { \beta + \lambda} n = 2 -\frac kn. \] We call this the optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$. The existence of an optimal pair for this new inequality is also studied. The motivation of working on the above inequality is to provide a unification of many known Hardy-Littewood-Sobolev inequalities including the classical Hardy-Littewood-Sobolev inequality when $k=\beta=0$, the Hardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1} \times \mathbf R_+^n$ when $k=1$ and $\beta = 0$, and the Hardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1} \times \mathbf R_+^n$ with extended kernel when $k=1$ and $\beta \ne 0$. We show that the above condition for $\beta$ is sharp. In the unweighted case, namely $\beta=0$, our finding immediately leads to the sharp Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ with the optimal range $$0<\lambda<n-k/r,$$ which has not been observed before, even in the case $k=1$. As one of many consequences, we give a short proof of the Stein-Weiss inequality in the context of $\mathbf R^{n-k} \times \mathbf R^n$.