Sharp Hardy's Inequalities in Hilbert Spaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909094084870144 |
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| author | Dimitrov, Dimitar K. Gadjev, Ivan Ismail, Mourad E. H. |
| author_facet | Dimitrov, Dimitar K. Gadjev, Ivan Ismail, Mourad E. H. |
| contents | We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy's inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^2\,dx\leq d(a,b)\,\int_a^b [f(x)]^2 dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2\leq d_n\,\sum_{k=1}^{n}a_k^2. $$ The exact constant $d(a,b)$ and the precise rate of convergence of $d_n$ are established and the extremal function and the ``almost extremal'' sequence are found. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08172 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp Hardy's Inequalities in Hilbert Spaces Dimitrov, Dimitar K. Gadjev, Ivan Ismail, Mourad E. H. Classical Analysis and ODEs Primary 26D10, 26D15, Secondary 33D45 We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy's inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^2\,dx\leq d(a,b)\,\int_a^b [f(x)]^2 dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2\leq d_n\,\sum_{k=1}^{n}a_k^2. $$ The exact constant $d(a,b)$ and the precise rate of convergence of $d_n$ are established and the extremal function and the ``almost extremal'' sequence are found. |
| title | Sharp Hardy's Inequalities in Hilbert Spaces |
| topic | Classical Analysis and ODEs Primary 26D10, 26D15, Secondary 33D45 |
| url | https://arxiv.org/abs/2306.08172 |