Reduced inequalities for vector-valued functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914681821593600 |
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| author | Hytönen, Tuomas P. |
| author_facet | Hytönen, Tuomas P. |
| contents | Building on the notion of convex body domination introduced by Nazarov, Petermichl, Treil, and Volberg, we provide a general principle of bootstrapping bilinear estimates for scalar-valued functions into vector-valued versions with a reduced right-hand side involving iterated norms of a pointwise dot product $\vec f(x)\cdot\vec g(y)$ instead of the product of lengths $|\vec f(x)| |\vec g(y)|$ that would result from a naïve extension of the scalar inequality. On the way, we study connections between convex body domination and tensor norms. In order to cover the full regime of $L^p$ norms, also with $p<1$, that naturally arise in bilinear harmonic analysis, we develop a framework in general quasi-normed spaces. A key application is a vector-valued Kato-Ponce inequality (or fractional Leibnitz rule) with a reduced right-hand side, which we obtain as a soft corollary of the known scalar-valued version and our general bootstrapping method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10630 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reduced inequalities for vector-valued functions Hytönen, Tuomas P. Functional Analysis 15A69, 26D15, 47G10 Building on the notion of convex body domination introduced by Nazarov, Petermichl, Treil, and Volberg, we provide a general principle of bootstrapping bilinear estimates for scalar-valued functions into vector-valued versions with a reduced right-hand side involving iterated norms of a pointwise dot product $\vec f(x)\cdot\vec g(y)$ instead of the product of lengths $|\vec f(x)| |\vec g(y)|$ that would result from a naïve extension of the scalar inequality. On the way, we study connections between convex body domination and tensor norms. In order to cover the full regime of $L^p$ norms, also with $p<1$, that naturally arise in bilinear harmonic analysis, we develop a framework in general quasi-normed spaces. A key application is a vector-valued Kato-Ponce inequality (or fractional Leibnitz rule) with a reduced right-hand side, which we obtain as a soft corollary of the known scalar-valued version and our general bootstrapping method. |
| title | Reduced inequalities for vector-valued functions |
| topic | Functional Analysis 15A69, 26D15, 47G10 |
| url | https://arxiv.org/abs/2402.10630 |