Bernstein inequality for multivariate functions with smooth Fourier images
UDC 517.5 Let $K$ be a compact set in ${\Bbb R}^n$ with $(O)$-property and let $1\leq p\leq \infty$. Then there exists a constant $C_K< \infty $ independent of $f$ and $\alpha$ such that $$\|D^{\alpha } f \|_p \leq C_K \sup\limits_{\xi \in K } |\xi ^{\alpha} |\, \|f\|_{\mathcal{H}_{p,K,3}...
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| Datum: | 2026 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6386 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1865793407525322752 |
|---|---|
| author | Bang, Ha Huy Huy, Vu Nhat Bang, Ha Huy Huy, Vu Nhat |
| author_facet | Bang, Ha Huy Huy, Vu Nhat Bang, Ha Huy Huy, Vu Nhat |
| author_institution_txt_mv | [
{
"author": "Ha Huy Bang",
"institution": "Inst. Math., Vietnam Acad. Sci. and Technology, Hanoi, Vietnam"
},
{
"author": "Vu Nhat Huy",
"institution": "Hanoi Univ. Sci., Vietnam Nat. Univ., and TIMAS, Thang Long Univ., Hanoi, Vietnam"
}
] |
| author_sort | Bang, Ha Huy |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-02-23T13:26:31Z |
| description | UDC 517.5
Let $K$ be a compact set in ${\Bbb R}^n$ with $(O)$-property and let $1\leq p\leq \infty$. Then there exists a constant $C_K< \infty $ independent of $f$ and $\alpha$ such that $$\|D^{\alpha } f \|_p \leq C_K \sup\limits_{\xi \in K } |\xi ^{\alpha} |\, \|f\|_{\mathcal{H}_{p,K,3}}$$ for all $\alpha \in \mathbb{Z}_+^n$ and $f\in \mathcal{H}_{p,K,3},$ where $\mathcal{H}_{p,K,3}=\big\{f \in L^p({\Bbb R}^n)\colon {\rm supp\,} \widehat f \subset K ,D^{(3,3,\ldots,3)} \widehat{f} \in C({\Bbb R}^n) \big\},$ $\|f\|_{\mathcal{H}_{p,K,3}} = \big\|D^{(3,3,\ldots,3)} \widehat{f}\,\big\|_\infty$, and $\widehat{f}$ is the Fourier transform of $f$. Note that $K$ is said to have the $(O)$-property if there exists a constant $C>0$ such that $$\sup\limits_{{\bf x} \in K} |{\bf x}^{\alpha + e_j} | \geq C\sup\limits_{{\bf x} \in K } |{\bf x}^{\alpha} |$$ for all $\alpha \in \mathbb{Z}_+^n$ and $j=1,2, \ldots ,n$. |
| doi_str_mv | 10.37863/umzh.v74i11.6386 |
| first_indexed | 2026-03-24T03:27:16Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-6386 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:27:16Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-63862026-02-23T13:26:31Z Bernstein inequality for multivariate functions with smooth Fourier images Bernstein inequality for multivariate functions with smooth Fourier images Bang, Ha Huy Huy, Vu Nhat Bang, Ha Huy Huy, Vu Nhat $L^p$- spaces, Bernstein inequality, generalized functions 2010 AMS Subject Classification: 26D10, 46E30 UDC 517.5 Let $K$ be a compact set in ${\Bbb R}^n$ with $(O)$-property and let $1\leq p\leq \infty$. Then there exists a constant $C_K< \infty $ independent of $f$ and $\alpha$ such that $$\|D^{\alpha } f \|_p \leq C_K \sup\limits_{\xi \in K } |\xi ^{\alpha} |\, \|f\|_{\mathcal{H}_{p,K,3}}$$ for all $\alpha \in \mathbb{Z}_+^n$ and $f\in \mathcal{H}_{p,K,3},$ where $\mathcal{H}_{p,K,3}=\big\{f \in L^p({\Bbb R}^n)\colon {\rm supp\,} \widehat f \subset K ,D^{(3,3,\ldots,3)} \widehat{f} \in C({\Bbb R}^n) \big\},$ $\|f\|_{\mathcal{H}_{p,K,3}} = \big\|D^{(3,3,\ldots,3)} \widehat{f}\,\big\|_\infty$, and $\widehat{f}$ is the Fourier transform of $f$. Note that $K$ is said to have the $(O)$-property if there exists a constant $C>0$ such that $$\sup\limits_{{\bf x} \in K} |{\bf x}^{\alpha + e_j} | \geq C\sup\limits_{{\bf x} \in K } |{\bf x}^{\alpha} |$$ for all $\alpha \in \mathbb{Z}_+^n$ and $j=1,2, \ldots ,n$. УДК 517.5 Нерівність Бернштейна для функцій багатьох змінних з гладкими зображеннями Фур’є  Нехай $K$ – компактна множина в ${\Bbb R}^n$, що має $(O)$-властивість і $1\leq p\leq \infty$.  Тоді існує стала $C_K< \infty $, незалежна від $f$ та  $\alpha$, така, що $$ \|D^{\alpha } f \|_p \leq C_K \sup\limits_{\xi \in K } |\xi ^{\alpha} |\, \|f\|_{\mathcal{H}_{p,K,3}}$$ для всіх $\alpha \in \mathbb{Z}_+^n$ і $f\in \mathcal{H}_{p,K,3},$ де $\mathcal{H}_{p,K,3}=\big\{f \in L^p({\Bbb R}^n)\colon {\rm supp\,} \widehat f \subset K ,D^{(3,3,\ldots ,3)} \widehat{f} \in C({\Bbb R}^n) \big\},$ $\|f\|_{\mathcal{H}_{p,K,3}}  = \big\| D^{(3,3,\ldots,3)} \widehat{f}\,\big\|_\infty$ і $\widehat{f}$ є перетворенням Фур'є $f$.  Зауважимо, що $K$ має $(O)$-властивість, якщо існує стала $C>0$ така, що $$\sup\limits_{{\bf x} \in K} |{\bf x}^{\alpha + e_j} | \geq C\sup\limits_{{\bf x} \in  K } |{\bf x}^{\alpha} |$$ для всіх $\alpha \in \mathbb{Z}_+^n$ і $j=1,2, \ldots ,n$.  Institute of Mathematics, NAS of Ukraine 2026-02-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6386 10.37863/umzh.v74i11.6386 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 11 (2022); 1558 - 1570 Український математичний журнал; Том 74 № 11 (2022); 1558 - 1570 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6386/9334 Copyright (c) 2022 Ha Huy Bang, Vu Nhat Huy |
| spellingShingle | Bang, Ha Huy Huy, Vu Nhat Bang, Ha Huy Huy, Vu Nhat Bernstein inequality for multivariate functions with smooth Fourier images |
| title | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_alt | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_full | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_fullStr | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_full_unstemmed | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_short | Bernstein inequality for multivariate functions with smooth Fourier images |
| title_sort | bernstein inequality for multivariate functions with smooth fourier images |
| topic_facet | $L^p$- spaces Bernstein inequality generalized functions 2010 AMS Subject Classification: 26D10 46E30 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6386 |
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