The paper under review deals with the question of the \(L^p\) boundedness of a class of Carleson type maximal operators with rough kernels, i.e., “standard” kernels with no a priori smoothness assumptions. Namely, let \(\Omega\) be a measurable function on \(\mathbb R^n\backslash \{0\}\), homogeneous of degree 0 with zero average on \(S^{n-1}\), viz.
\[ \Omega(tx)=\Omega(x) \quad \text{for any} \quad x \in \mathbb R^n\backslash \{0\} \quad \text{and} \;t>0, \]
\(\Omega \in L^1(S^{n-1})\) and
\[ \int_{S^{n-1}} \Omega(x') \,d\sigma (x')=0. \]
Let \(Q_\lambda(r)=\sum_{2\leq k\leq d} \lambda_k r^k\), \(\lambda=(\lambda_2,\dots, \lambda_d)\in \mathbb R^{d-1}\). Then the Carleson type maximal operator \({\mathcal T}^*\) associated to the real-valued polynomial \(Q_\lambda(r)\) is defined by
\[ {\mathcal T}^*(f)(x)=\sup_{\lambda} |T_\lambda (f)(x)|, \]
where \(T_\lambda\) is an oscillatory integral given by
\[ T_\lambda(f)(x)=\int_{\mathbb R^n} e^{iQ_\lambda(|y|)}K(y)f(x-y)\,dy, \]
with \(K(y)=\Omega(y) / |y|^n\).
The authors show that the mapping \(f\mapsto{\mathcal T}^*(f)\) is bounded in \(L^p(\mathbb R^n)\), \(1<p<\infty\), if \(\Omega \in H^1(S^{n-1})\), where \(H^1(S^{n-1})\) is the Hardy space on \(S^{n-1}\). The lengthy proof is based on an idea of linearizing maximal operators, Stein-Wainger’s \(TT^*\) method and the Calderón-Zygmund rotation method (where the rotation method was used to overcome the lack of smoothness of the kernel).
This result is, in the sense of removing the smoothness assumption on the kernel, an improvement on a result from E. M. Stein and S. Wainger [Math. Res. Lett. 8, No. 5–6, 789–800 (2001; Zbl 0998.42007)].