\(L^p\) bounds of maximal operators along variable planar curves in the Lipschitz regularity (Q826486)

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 additionally that \(U\) is Lipschitz, for \(\varepsilon_0>0\) satisfying \(\gamma(2\varepsilon_0)\le 1/4\|U\|_{\mathrm{Lip}}\,\), the two above estimates with \(\infty\) now replaced by \(\varepsilon_0\), hold for \(1<p\le2\). The assumptions imposed on \(\gamma\), in addition to continuity and \(\gamma(0)=0\), are twofold: (1) qualitative -- \(\gamma\) is either odd or even, sufficiently smooth and increasing on \((0,\infty)\); (2) quantitative -- (i) \(\inf_{t>0}|(\gamma^\prime/\gamma^{\prime\prime})^\prime(t)|>0\), (ii) \(\inf_{t>0}|t^j\gamma^{(j)}(t)/\gamma(t)|>0\) for \(j=1,2\), (iii) \(\sup_{t>0}|t^j\gamma^{(j)}(t)/\gamma(t)|<\infty\) for \(j=1,\ldots,N\), and \(N\) sufficiently large. Theorems A and B generalize an earlier result of \textit{S. Guo} et al. [Proc. Lond. Math. Soc. 115, 177--219 (2017; Zbl 1388.42039)], known in the particular case of \(\gamma(t)\) being either \(|t|^\alpha\) or \(\mathrm{sgn}(t)|t|^\alpha\), where \(\alpha>0\) and \(\alpha\neq1\) (and with the same assumptions on \(U\)). The technique used by the authors in the proofs of Theorems A and B is quite different from that in the paper of Guo et al., and relies, roughly, on a variable coefficient local smoothing estimate, the Littlewood-Paley theory, and a bootstrapping argument. Beforementioned Zygmund's and Stein's conjectures are related to \(\gamma(t)=t\). The introductory section of the paper contains a discussion of partial progress toward understanding these two conjectures.