The aim of this paper is to study the \(L^p\) boundedness of certain maximal and singular integral operators that act by integration along variable homogeneous curves in the plane, of the form \(\Gamma^{\alpha}_u(t):=(t,u\cdot[t]^{\alpha})\), where \(\alpha>0\) and the notation \([t]^{\alpha}\) stands for either \(|t|^{\alpha}\) or \({\mathrm{sgn}}(t)|t|^{\alpha}\).
More precisely, let \(u: {\mathbb{R}}^2 \to {\mathbb{R}}\) be a measurable function and \(0<\varepsilon_0 \leq\infty\). The following objects are considered:
(1) the (\(\varepsilon_0\)-truncated) maximal operator along \(\Gamma^{\alpha}_u\), defined by \[ \mathcal{M}_{u,\varepsilon_0}^{(\alpha)}f(x,y) =\sup_{0<\varepsilon<\varepsilon_0}\frac1{2\varepsilon} \int^{\varepsilon}_{-\varepsilon} |f(x-t,y-u(x,y)[t]^{\alpha})|dt, \]
(2) the (\(\varepsilon_0\)-truncated) Hilbert transform along \(\Gamma^{\alpha}_u\), given by \[ \mathcal{H}_{u,\varepsilon_0}^{(\alpha)}f(x,y) ={\mathrm{p.v.}}\int^{\varepsilon_0}_{-\varepsilon_0} f(x-t,y-u(x,y)[t]^{\alpha})\frac{dt}{t}. \]
This research was motivated by the so-called Zygmund conjecture, which asks whether the Lipschitz regularity of \(u\) suffices to guarantee any non-trivial \(L^p\) bounds for the maximal operator \(\mathcal{M}_{u,\varepsilon_0}:=\mathcal{M}_{u,\varepsilon_0}^{(1)}\), provided \(\varepsilon_0\) is small enough depending on \(\|u\|_{\mathrm{Lip}}\).
The authors obtain two main results when \(\alpha \neq 1\). The first one shows \(\mathcal{M}_{u,\varepsilon_0}^{(\alpha)}\) is bounded on \(L^p\) for some \(\varepsilon_0=\varepsilon_0(\|u\|_{\mathrm{Lip}})>0\) and every \(1 < p \leq 2\) provided \(u\) is Lipschitz. Moreover, an alternative proof of a result of G. Marletta and F. Ricci [Studia Math. 130, No. 1, 53–65 (1998; Zbl 0921.42014)] is also given. The second one says \(\mathcal{H}_{u}^{(\alpha)}:=\mathcal{H}_{u,\infty}^{(\alpha)}\) is bounded on \(L^p\) for all \(1< p <\infty\), under the assumption that \(u\) is a measurable function and is constant in the second variable. The proofs of the results rely on stationary phase methods, \(TT^{\ast}\) arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.