Maz'ya's $Φ$-inequalities on domains
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929427345047552 |
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| author | Stolyarov, Dmitriy |
| author_facet | Stolyarov, Dmitriy |
| contents | We find necessary and sufficient conditions on the function $Φ$ for the inequality $$\Big|\int_ΩΦ(K*f)\Big|\lesssim \|f\|_{L_1(\mathbb{R}^d)}^p$$ to be true. Here $K$ is a positively homogeneous of order $α- d$, possibly vector valued, kernel, $Φ$ is a $p$-homogeneous function, and $p=d/(d-α)$. The domain $Ω\subset \mathbb{R}^d$ is either bounded with $C^{1,β}$ smooth boundary for some $β> 0$ or a halfspace in $\mathbb{R}^d$. As a corollary, we describe the positively homogeneous of order $d/(d-1)$ functions $Φ\colon \mathbb{R}^d \to \mathbb{R}$ that are suitable for the bound $$\Big|\int_ΩΦ(\nabla u)\Big|\lesssim \int_Ω|Δu|.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14052 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maz'ya's $Φ$-inequalities on domains Stolyarov, Dmitriy Classical Analysis and ODEs We find necessary and sufficient conditions on the function $Φ$ for the inequality $$\Big|\int_ΩΦ(K*f)\Big|\lesssim \|f\|_{L_1(\mathbb{R}^d)}^p$$ to be true. Here $K$ is a positively homogeneous of order $α- d$, possibly vector valued, kernel, $Φ$ is a $p$-homogeneous function, and $p=d/(d-α)$. The domain $Ω\subset \mathbb{R}^d$ is either bounded with $C^{1,β}$ smooth boundary for some $β> 0$ or a halfspace in $\mathbb{R}^d$. As a corollary, we describe the positively homogeneous of order $d/(d-1)$ functions $Φ\colon \mathbb{R}^d \to \mathbb{R}$ that are suitable for the bound $$\Big|\int_ΩΦ(\nabla u)\Big|\lesssim \int_Ω|Δu|.$$ |
| title | Maz'ya's $Φ$-inequalities on domains |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2407.14052 |