On some sharp Landau--Kolmogorov--Nagy type inequalities in Sobolev spaces of multivariate functions
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| Format: | Preprint |
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2023
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| _version_ | 1866909537887322112 |
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| author | Babenko, V. F. Babenko, V. V. Kovalenko, O. V. Parfinovych, N. V. |
| author_facet | Babenko, V. F. Babenko, V. V. Kovalenko, O. V. Parfinovych, N. V. |
| contents | For a function $f$ from the Sobolev space $W^{1,p}(C)$ ($C\subset\mathbb{R}^d$ is an open convex cone), a sharp inequality that estimates $\| f\|_{L_{\infty}}$ via the $L_{p}$-norm of its gradient and a seminorm of the function is obtained. With the help of this inequality, a sharp inequality is proved, which estimates the ${L_{\infty}}$-norm of the Radon--Nikodym derivative of a charge defined on Lebesgue measurable subsets of $C$ via the $L_p$-norm of the gradient of this derivative and a seminorm of the charge. In the case, when $C=\mathbb{R}_+^m\times \mathbb{R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the ${L_{\infty}}$-norm of a mixed derivative of a function $f\colon C\to \mathbb{R}$ using its ${L_{\infty}}$-norm and the $L_p$-norm of the gradient of the function's mixed derivative. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_06188 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On some sharp Landau--Kolmogorov--Nagy type inequalities in Sobolev spaces of multivariate functions Babenko, V. F. Babenko, V. V. Kovalenko, O. V. Parfinovych, N. V. Functional Analysis 26D10, 41A17, 41A44 For a function $f$ from the Sobolev space $W^{1,p}(C)$ ($C\subset\mathbb{R}^d$ is an open convex cone), a sharp inequality that estimates $\| f\|_{L_{\infty}}$ via the $L_{p}$-norm of its gradient and a seminorm of the function is obtained. With the help of this inequality, a sharp inequality is proved, which estimates the ${L_{\infty}}$-norm of the Radon--Nikodym derivative of a charge defined on Lebesgue measurable subsets of $C$ via the $L_p$-norm of the gradient of this derivative and a seminorm of the charge. In the case, when $C=\mathbb{R}_+^m\times \mathbb{R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the ${L_{\infty}}$-norm of a mixed derivative of a function $f\colon C\to \mathbb{R}$ using its ${L_{\infty}}$-norm and the $L_p$-norm of the gradient of the function's mixed derivative. |
| title | On some sharp Landau--Kolmogorov--Nagy type inequalities in Sobolev spaces of multivariate functions |
| topic | Functional Analysis 26D10, 41A17, 41A44 |
| url | https://arxiv.org/abs/2307.06188 |