English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For $n > k \geq 0$, $\lambda >0$, and $p, r>1$, we establish the followingoptimal 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 necessaryconditions \[ \beta < \left\{ \begin{aligned} & k - k/r & & \text{if } \; 0 <\lambda \leq n-k,\\ & n - \lambda - k/r & & \text{if } \; n-k < \lambda,\end{aligned} \right. \] and \[ \frac{n-k}n \frac 1p + \frac 1r + \frac { \beta+ \lambda} n = 2 -\frac kn. \] We call this the optimalHardy-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. Themotivation of working on the above inequality is to provide a unification ofmany known Hardy-Littewood-Sobolev inequalities including the classicalHardy-Littewood-Sobolev inequality when $k=\beta=0$, theHardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1}\times \mathbf R_+^n$ when $k=1$ and $\beta = 0$, and theHardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1}\times \mathbf R_+^n$ with extended kernel when $k=1$ and $\beta \ne 0$. Weshow that the above condition for $\beta$ is sharp. In the unweighted case,namely $\beta=0$, our finding immediately leads to the sharpHardy-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 ofthe Stein-Weiss inequality in the context of $\mathbf R^{n-k} \times \mathbfR^n$.

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