Maximal functions along surfaces on product domains

In this paper, we study the L ^p boundedness for a class of maximal functions along surfaces in ℝⁿ × ℝ^m of the form {(φ₁(|u|)u′, φ₂(|v|)v′) : (u,v) ∈ ℝ_n × ℝ_m}.We prove that such maximal functions are bounded on L ^p for all 2 ≤ p < ∞ provided that the functions φ₁ and φ₂ satisfy certain oscillatory estimates of van der Corput type.