Sharpening the gap between $L^{1}$ and $L^{2}$ norms
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916314019266560 |
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| author | Ivanisvili, Paata Stone, Yonathan |
| author_facet | Ivanisvili, Paata Stone, Yonathan |
| contents | We refine the classical Cauchy--Schwartz inequality $\|X\|_{1} \leq \|X\|_{2}$ by demonstrating that for any $p$ and $q$ with $q>p>2$, there exists a constant $C=C(p,q)$ such that $\|X\|_1 \leq 1 - C \Big{(}\|X\|_p^p - 1\Big{)}^{\frac{q-2}{q-p}}\Big{(}\|X\|_q^q - 1\Big{)}^{\frac{2-p}{q-p}}$ holds true for all Borel measurable random variables $X$ with $\|X\|_{2}=1$ and $\|X\|_{p}<\infty$. We illustrate two applications of this result: one for biased Rademacher sums and another for exponential sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04835 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharpening the gap between $L^{1}$ and $L^{2}$ norms Ivanisvili, Paata Stone, Yonathan Probability Classical Analysis and ODEs Differential Geometry Functional Analysis 39B62 We refine the classical Cauchy--Schwartz inequality $\|X\|_{1} \leq \|X\|_{2}$ by demonstrating that for any $p$ and $q$ with $q>p>2$, there exists a constant $C=C(p,q)$ such that $\|X\|_1 \leq 1 - C \Big{(}\|X\|_p^p - 1\Big{)}^{\frac{q-2}{q-p}}\Big{(}\|X\|_q^q - 1\Big{)}^{\frac{2-p}{q-p}}$ holds true for all Borel measurable random variables $X$ with $\|X\|_{2}=1$ and $\|X\|_{p}<\infty$. We illustrate two applications of this result: one for biased Rademacher sums and another for exponential sums. |
| title | Sharpening the gap between $L^{1}$ and $L^{2}$ norms |
| topic | Probability Classical Analysis and ODEs Differential Geometry Functional Analysis 39B62 |
| url | https://arxiv.org/abs/2407.04835 |