Maximal Theorems for the Directional Hilbert Transform on the Plane
| dc.creator | Lacey, Michael T | |
| dc.creator | Li, Xiaochun | |
| dc.date | 2003-10-21 | |
| dc.date | 2004-08-15 | |
| dc.date.accessioned | 2026-07-07T05:02:10Z | |
| dc.date.available | 2026-07-07T05:02:10Z | |
| dc.description | For a Schwartz function $f$ on the plane and a non-zero $v\in\ZR^2$ define the Hilbert transform of $f$ in the direction $v$ to be $$ H_vf(x)=\text{p.v.}\int_\ZR f(x-vy) \frac{dy}y $$ Let $ζ$ be a Schwartz function with frequency support in the annulus $1\le| ξ|\le2$. We prove that the maximal operator $$ \sup_{\abs v=1}\abs{H_vζ* f} $$ maps $L^2$ into weak $L^2$, and $L^p$ into $L^p$ for $p>2$. The $L^2$ estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series, especially the recent proof given by Lacey and Thiele. | |
| dc.description | Substantially revised with 23 pages, 8 figures, and 14 references | |
| dc.identifier | https://arxiv.org/abs/math/0310346 | |
| dc.identifier | http://arxiv.org/abs/math/0310346 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68950 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Maximal Theorems for the Directional Hilbert Transform on the Plane | |
| dc.type | text |