On compact embeddings in $\mathbf{L^p}$ and fractional spaces
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911087346057216 |
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| author | Bueno, Hamilton Medeiro, Aldo Miyagaki, Olimpio Pereira, Gilberto |
| author_facet | Bueno, Hamilton Medeiro, Aldo Miyagaki, Olimpio Pereira, Gilberto |
| contents | The study of the fractional Laplacian operator $(-Δ)^s$ in $\mathbb{R}^N$ with Dirichlet boundary conditions gained enormous momentum through its identification with a Neumann operator in $\mathbb{R}^N\times (0, \infty)=\mathbb{R}^{N+1}_+$, a method mainly introduced by Caffarelli and Silvestre. Since then, several other operators have been studied using this method. In general, a crucial question is attached to this method: the embedding (in the trace sense) on the ground space $L^q(\mathbb{R}^{N})$ is compact? This question is very important when dealing with problems of existence of solutions. This paper aims to answer this question for some operators.
Passing to an abstract setting, let $X,Y$ be Hilbert spaces and $\mathcal{A}\colon X\to X'$ a continuous and symmetric elliptic operator. We suppose that $X$ is dense in $Y$ and that the embedding $X\subset Y$ is compact. In this paper we show some consequences of this setting for the study of the fractional operator attached to $\mathcal{A}$ in the extension setting $Ω\times(0,\infty)$ or $\mathbb{R}^{N+1}_+$. Being more specific, we will give some examples where the embedding of the extension domain into $L^2(Ω)$ is compact, even in the case $Ω=\mathbb{R}^N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On compact embeddings in $\mathbf{L^p}$ and fractional spaces Bueno, Hamilton Medeiro, Aldo Miyagaki, Olimpio Pereira, Gilberto Functional Analysis Analysis of PDEs 47B02, 47B07, 35R11 The study of the fractional Laplacian operator $(-Δ)^s$ in $\mathbb{R}^N$ with Dirichlet boundary conditions gained enormous momentum through its identification with a Neumann operator in $\mathbb{R}^N\times (0, \infty)=\mathbb{R}^{N+1}_+$, a method mainly introduced by Caffarelli and Silvestre. Since then, several other operators have been studied using this method. In general, a crucial question is attached to this method: the embedding (in the trace sense) on the ground space $L^q(\mathbb{R}^{N})$ is compact? This question is very important when dealing with problems of existence of solutions. This paper aims to answer this question for some operators. Passing to an abstract setting, let $X,Y$ be Hilbert spaces and $\mathcal{A}\colon X\to X'$ a continuous and symmetric elliptic operator. We suppose that $X$ is dense in $Y$ and that the embedding $X\subset Y$ is compact. In this paper we show some consequences of this setting for the study of the fractional operator attached to $\mathcal{A}$ in the extension setting $Ω\times(0,\infty)$ or $\mathbb{R}^{N+1}_+$. Being more specific, we will give some examples where the embedding of the extension domain into $L^2(Ω)$ is compact, even in the case $Ω=\mathbb{R}^N$. |
| title | On compact embeddings in $\mathbf{L^p}$ and fractional spaces |
| topic | Functional Analysis Analysis of PDEs 47B02, 47B07, 35R11 |
| url | https://arxiv.org/abs/2504.10196 |