Maximal functions along surfaces on product domains

Abstract

In this paper, we study the L ^p boundedness for a class of maximal functions along surfaces in ℝⁿ × ℝ^m of the form

$$ \{ (\varphi _1 (|u|)u',\varphi _2 (|v|)v'):(u,v) \in \mathbb{R}^n \times \mathbb{R}^m \} . $$

We prove that such maximal functions are bounded on L ^p for all 2 ≤ p < ∞ provided that the functions ϕ ₁ and ϕ ₂ satisfy certain oscillatory estimates of van der Corput type.

Резуме

В работе установлена L ^p-ограниченностя класса максималяных функций вдоля поверхностей в ℝⁿ × ℝ^m вида

$$ \{ \theta _1 (|u|)u',\theta _2 (|v|)v'):(u,v) \in \mathbb{R}^n \times \mathbb{R}^m \} . $$

.