Maximal directional operators along algebraic varieties
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2018
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| _version_ | 1866917779945291776 |
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| author | Di Plinio, Francesco Parissis, Ioannis |
| author_facet | Di Plinio, Francesco Parissis, Ioannis |
| contents | We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is an extension of the works by Córdoba, for one-dimensional manifolds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of $\sqrt{\log N}$ on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new $L^2$-estimates for Kakeya-type maximal functions with tubes pointing along polynomial directions. Our proof technique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1807_08255 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Maximal directional operators along algebraic varieties Di Plinio, Francesco Parissis, Ioannis Classical Analysis and ODEs 42B25 (Primary) 42B20 (Secondary) We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is an extension of the works by Córdoba, for one-dimensional manifolds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of $\sqrt{\log N}$ on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new $L^2$-estimates for Kakeya-type maximal functions with tubes pointing along polynomial directions. Our proof technique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl. |
| title | Maximal directional operators along algebraic varieties |
| topic | Classical Analysis and ODEs 42B25 (Primary) 42B20 (Secondary) |
| url | https://arxiv.org/abs/1807.08255 |