论文标题
$ \ mathbf r^{n-k} \ times \ mathbf r^n $及其后果上的最佳耐磨木材 - 索伯夫不等式
An optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ and its consequences
论文作者
论文摘要
对于$ n> k \ geq 0 $，$λ> 0 $，$ p，r> 1 $，我们建立以下最佳hardy-littlewood-sobolev不平等\ [\ big | \ iint _ {\ Mathbf r^n \ times \ Mathbf r^{n-k}}} \ frac {f（x）g（y）} {| x-y |^λ| y |^β} |^β} dx dx dx dx dx d d x dy \ big | g \ |。 0<λ\ leq n-k，\\＆n-λ-k/r＆＆\ text {if} \; n-k <λ，\ end {对齐} \ right。 \]和\ [\ frac {n -k} n \ frac 1p + \ frac 1r + \ frac {β +λ} n = 2 - \ frac kn。 \]我们将其称为$ \ mathbf r^{n-k} \ times \ mathbf r^n $的最佳Hardy-Littlewood-Sobolev不等式。还研究了这种新不平等的最佳对的存在。处理上述不平等的动机是提供许多已知的Hardy-littewood-Sobolev不平等现象，包括经典的Hardy-Littewood-Soboleve不平等现象时，当$ k =β= 0 $时$β= 0 $，以及上半部空间上的hardy-littewood-sobolev $ \ mathbf r^{n-1} \ times \ times \ mathbf r _+^n $，当$ k = 1 $和$β\ ne 0 $时，带有扩展的内核。我们表明，上述$β$的条件很清晰。在未加权的情况下，即$β= 0 $，我们的发现立即导致$ \ Mathbf r^{n-k} \ times \ times \ Mathbf r^n $的尖锐强硬的木材 - 索布尔，并带有最佳范围$ 0 <λ<λ<n-k/r，$ $，$ k = 1 $ k = 1 $ k = 1 $ k = 1 $ k =作为许多后果之一，我们在$ \ mathbf r^{n-k} \ times \ mathbf r^n $的上下文中简短证明了Stein-Weiss不等式。
For $n > k \geq 0$, $λ>0$, 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 $y = (y', y") \in \mathbf R^{n-k} \times \mathbf R^k$ 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 $k=β=0$, the Hardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1} \times \mathbf R_+^n$ when $k=1$ and $β= 0$, 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 $β\ne 0$. We show that the above condition for $β$ is sharp. In the unweighted case, namely $β=0$, our finding immediately leads to the sharp Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ with the optimal range $$0<λ<n-k/r,$$ 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 $\mathbf R^{n-k} \times \mathbf R^n$.
