\(L^p\) bounds of maximal operators along variable planar curves in the Lipschitz regularity
type: Item
instance of: scholarly article
title: \(L^p\) bounds of maximal operators along variable planar curves in the Lipschitz regularity
author: Naijia Liu
author: Liang Song
author: Haixia Yu
published in: Journal of Functional Analysis
publication date: 2021-01-04T00:00:00Z
review text: This research is motivated by Zygmund's and Stein's conjectures [\textit{A. Zygmund}, Colloq. Math. 16, dedie a Franciszek Leja, 199--204 (1967; Zbl 0156.06301); \textit{E. M. Stein}, in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 196--221 (1987; Zbl 0718.42012)] on \(L^p\) regularity of a maximal operator and a singular integral operator along a variable curve on the plane. Given a continuous function \(\gamma\colon\mathbb R\to\mathbb R\) with \(\gamma(0)=0\), a measurable \(U\colon \mathbb R^2\to\mathbb R\), and \(0<\varepsilon_0\le\infty\), let \[ H^{\varepsilon_0}_{U,\gamma}f(x_1,x_2):=\mathrm{p.v.}\,\int^{\varepsilon_0}_{-\varepsilon_0}f\big(x_1-t,x_2-U(x_1,x_2)\gamma(t)\big)\frac{dt}t \] be the Hilbert transform and \[ M^{\varepsilon_0}_{U,\gamma}f(x_1,x_2):=\sup_{0<\varepsilon<\varepsilon_0}\frac1{2\varepsilon}\int^{\varepsilon}_{-\varepsilon} \big|f\big(x_1-t,x_2-U(x_1,x_2)\gamma(t)\big)\big|dt, \] be the corresponding maximal operator along the variable plane curve \(t\to (t,U(x_1,x_2)\gamma(t))\). Given a \(C^\infty\) function \(\psi\colon\mathbb R\to\mathbb R\) supported in \(1/2\le|t|\le2\), such that \(0\le\psi\le1\) and \(\sum_{k\in\mathbb Z}\psi_k(t)=1\) for \(t\neq0\), where \(\psi_k(t)=\psi(2^{-k}t)\), let \(P^{(2)}_k\) denote the Littlewood-Paley projection in the second variable corresponding to \(\psi_k\), \[ P^{(2)}_k f(x_1,x_2)=\int^{\infty}_{-\infty}f(x_1,x_2-z)\check{\psi}_k(z)\,dz; \] here \(\check{\psi}_k\) stands for the inverse Fourier transform of \(\psi_k\). The first of the two main results of the paper, Theorem A, says that under suitable assumptions on \(\gamma\), for any \(p>2\) it holds \[ \|H^{\infty}_{U,\gamma} P^{(2)}_k f\|_{L^p(\mathbb R^2)}\lesssim_{p,\gamma}\|P^{(2)}_k f\|_{L^p(\mathbb R^2)} \] and \[ \|M^{\infty}_{U,\gamma} f\|_{L^p(\mathbb R^2)}\lesssim_{p,\gamma}\|f\|_{L^p(\mathbb R^2)}, \] uniformly on \(f\in L^p(\mathbb R^2)\) and \(k\in\mathbb Z\). The second main result, Theorem B, says that assuming additi…
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 7290157
zbMATH Keywords: Hilbert transform
zbMATH Keywords: maximal function
zbMATH Keywords: local smoothing estimate
zbMATH Keywords: variable curve
reviewed by: Krzysztof Stempak
MaRDI profile type: Publication
full work available at URL: j.jfa.2020.108888
OpenAlex ID: W3111384948
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DOI: 10.1016/J.JFA.2020.108888
description: scientific article; zbMATH DE number 7290157