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

Abstract

For $n>k\ge 0$, $\lambda >0$, and $p,r>1$, we establish the following optimal Hardy-Littlewood-Sobolev inequality \[ \Big| \iint_{\mathbf R^n \times \mathbf R^{n-k}} \frac{f(x) g(y)}{ |x-y|^λ|y"|^β} dx dy \Big| \lesssim \| f \| _{L^p(\mathbf R^{n-k})} \| g\| _{L^r(\mathbf R^n)} \] with $y=(y^{\prime },y")\in {\mathbf{R}}^{n-k}\times {\mathbf{R}}^k$ under the two necessary conditions \[ β< \left\{ \begin{aligned} & k - k/r & & \text{if } \; 0 < λ\leq n-k,\\ & n - λ- k/r & & \text{if } \; n-k < λ, \end{aligned} \right. \] and \[ \frac{n-k}n \frac 1p + \frac 1r + \frac { β+ λ} n = 2 -\frac kn. \] 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 ${\mathbf{R}}^{n-k}\times {\mathbf{R}}^n$.

29 pages, 2 figures, accepted by Journal d'Analyse Mathématique

References in corpus (1)

• Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications