A sharp estimate for the Hilbert transform along finite order lacunary sets of directions
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| Format: | Preprint |
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2017
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| _version_ | 1866908056386797568 |
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| author | Di Plinio, Francesco Parissis, Ioannis |
| author_facet | Di Plinio, Francesco Parissis, Ioannis |
| contents | Let $D$ be a nonnegative integer and ${\mathbfΘ}\subset S^1$ be a lacunary set of directions of order $D$. We show that the $L^p$ norms, $1<p<\infty$, of the maximal directional Hilbert transform in the plane $$ H_{\mathbfΘ} f(x):= \sup_{v\in {\mathbfΘ}} \Big|\mathrm{p.v.}\int_{\mathbb R }f(x+tv)\frac{\mathrm{d} t}{t}\Big|, \qquad x \in {\mathbb R}^2, $$ are comparable to $(\log\#{\mathbfΘ})^\frac{1}{2}$. For vector fields $\mathsf{v}_D$ with range in a lacunary set of of order $D$ and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field $\mathsf{v}_D$, $$ H_{\mathsf{v}_D,1} f(x):= \mathrm{p.v.} \int_{ |t| \leq 1 } f(x+t\mathsf{v}_D(x)) \,\frac{\mathrm{d} t}{t}, $$ is $L^p$-bounded for all $1<p<\infty$. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1704_02918 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | A sharp estimate for the Hilbert transform along finite order lacunary sets of directions Di Plinio, Francesco Parissis, Ioannis Classical Analysis and ODEs 42B20 Let $D$ be a nonnegative integer and ${\mathbfΘ}\subset S^1$ be a lacunary set of directions of order $D$. We show that the $L^p$ norms, $1<p<\infty$, of the maximal directional Hilbert transform in the plane $$ H_{\mathbfΘ} f(x):= \sup_{v\in {\mathbfΘ}} \Big|\mathrm{p.v.}\int_{\mathbb R }f(x+tv)\frac{\mathrm{d} t}{t}\Big|, \qquad x \in {\mathbb R}^2, $$ are comparable to $(\log\#{\mathbfΘ})^\frac{1}{2}$. For vector fields $\mathsf{v}_D$ with range in a lacunary set of of order $D$ and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field $\mathsf{v}_D$, $$ H_{\mathsf{v}_D,1} f(x):= \mathrm{p.v.} \int_{ |t| \leq 1 } f(x+t\mathsf{v}_D(x)) \,\frac{\mathrm{d} t}{t}, $$ is $L^p$-bounded for all $1<p<\infty$. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele. |
| title | A sharp estimate for the Hilbert transform along finite order lacunary sets of directions |
| topic | Classical Analysis and ODEs 42B20 |
| url | https://arxiv.org/abs/1704.02918 |