On Polynomial Carleson operators along quadratic hypersurfaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910565553668096 |
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| author | Anderson, Theresa C. Maldague, Dominique Pierce, Lillian B. Yung, Po-Lam |
| author_facet | Anderson, Theresa C. Maldague, Dominique Pierce, Lillian B. Yung, Po-Lam |
| contents | We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by $(y,Q(y))\subseteq \mathbb{R}^{n+1}$, for an arbitrary non-degenerate quadratic form $Q$, admits an a priori bound on $L^p$ for all $1<p<\infty$, for each $n \geq 2$. This operator takes the form of a polynomial Carleson operator of Radon-type, in which the maximally modulated phases lie in the real span of $\{p_2,\ldots,p_d\}$ for any set of fixed real-valued polynomials $p_j$ such that $p_j$ is homogeneous of degree $j$, and $p_2$ is not a multiple of $Q(y)$. The general method developed in this work applies to quadratic forms of arbitrary signature, while previous work considered only the special positive definite case $Q(y)=|y|^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_15865 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Polynomial Carleson operators along quadratic hypersurfaces Anderson, Theresa C. Maldague, Dominique Pierce, Lillian B. Yung, Po-Lam Classical Analysis and ODEs 42B20, 43A50, 42B25, 44A12 We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by $(y,Q(y))\subseteq \mathbb{R}^{n+1}$, for an arbitrary non-degenerate quadratic form $Q$, admits an a priori bound on $L^p$ for all $1<p<\infty$, for each $n \geq 2$. This operator takes the form of a polynomial Carleson operator of Radon-type, in which the maximally modulated phases lie in the real span of $\{p_2,\ldots,p_d\}$ for any set of fixed real-valued polynomials $p_j$ such that $p_j$ is homogeneous of degree $j$, and $p_2$ is not a multiple of $Q(y)$. The general method developed in this work applies to quadratic forms of arbitrary signature, while previous work considered only the special positive definite case $Q(y)=|y|^2$. |
| title | On Polynomial Carleson operators along quadratic hypersurfaces |
| topic | Classical Analysis and ODEs 42B20, 43A50, 42B25, 44A12 |
| url | https://arxiv.org/abs/2211.15865 |