Maximal functions along surfaces in product spaces (Q819721)

Let \(\Gamma: \mathbb R^n \times \mathbb R^m \mapsto \mathbb R\) be a measurable function which is radial with respect to both variables, that is, \[ \Gamma(y_1, y_2)= \Gamma(|y_1|, |y_2|)\qquad \text{for all}\;y_1\in \mathbb R^n\;\text{and}\;y_2\in \mathbb R^m. \] The author investigates the \(L^p\)-boundedness of the maximal operator \[ \begin{multlined} M_\Gamma f(x_1, x_2, x_3)= \\ \sup_{r_1, r_2 >0}\bigg\{\frac 1{r_1^n r_2^m} \int_{|y_2|\leq r_2} \int_{|y_1|\leq r_1} \big|f\big(x_1-y_1, x_2-y_2, x_3-\Gamma(|y_1|, |y_2|)\big)\big|\, dy_1 dy_2\bigg\},\end{multlined} \] where \(x_1, y_1 \in \mathbb R^n\), \(x_2, y_2 \in \mathbb R^m\), and \(x_3\in \mathbb R\). Applying the result in [\textit{J. Duoandikoetxea}, Ann. Inst. Fourier 36, No. 4, 185--206 (1986; Zbl 0568.42011)], the author proves that, under certain conditions on \(\Gamma\), the operator \(M_\Gamma\) is bounded on \(L^p(\mathbb R^n \times \mathbb R^m \times \mathbb R)\) for \(p>1\) and \(n, m \geq 1\).