An optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ and its consequences
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arXiv
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| Natura: | Preprint |
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2020
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| _version_ | 1866912965172658176 |
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| author | Ngô, Quôc Anh Nguyen, Quoc-Hung Nguyen, Van Hoang |
| author_facet | Ngô, Quôc Anh Nguyen, Quoc-Hung Nguyen, Van Hoang |
| contents | 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 $\mathbf R^{n-k} \times \mathbf R^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_09868 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | An optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ and its consequences Ngô, Quôc Anh Nguyen, Quoc-Hung Nguyen, Van Hoang Functional Analysis Analysis of PDEs Classical Analysis and ODEs 26D15, 35A23, 42B25 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 $\mathbf R^{n-k} \times \mathbf R^n$. |
| title | An optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ and its consequences |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs 26D15, 35A23, 42B25 |
| url | https://arxiv.org/abs/2009.09868 |