Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k}
  \times \mathbf R^n$ and its consequences   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 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 \[ \frac{n-k}n \frac 1p + \frac 1r + \frac { \beta + \lambda} n
= 2 -\frac kn. \] Remarkably, there is no upper bound for $\lambda$, which is
quite different from the case with the weight $|y|^{-\beta}$, commonly known as
Stein-Weiss inequalities. We also show that the above condition for $\beta$ is
sharp. Apparently, the above inequality includes the classical
Hardy-Littlewood-Sobolev inequality when $k=0$ and the HLS inequality on the
upper half space $\mathbf R_+^n$ when $k=1$. In the unweighted case, namely
$\beta=0$, our finding immediately leads to the sharp HLS inequality on
$\mathbf R^{n-k} \times \mathbf R^n$ with the \textit{optimal} range
$$0<\lambda<n-k/r,$$ which has not been observed before, even for the case
$k=1$. Improvement to the Stein-Weiss inequality in the context of $\mathbf
R^{n-k} \times \mathbf R^n$ is also considered. The existence of an optimal
pair for this new inequality is also studied.

22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: funct-an