A Transference Theorem and Its Application

Abstract

In this article, we establish a transference between the n-dimensional Euclidean space \( \mathbb {R} ^{n}\) and the n-torus \(\mathbb {T}^{n}\) about the \(H^{p}-L^{p,\infty }\) boundedness of maximal multipliers. As an application, we obtain that the maximal oscillatory integral \(S_{\alpha ,\beta }^{*}\) is bounded from \( H^{p}\left( \mathbb {R} ^{n}\right) \) to \(L^{p,\infty }\left( \mathbb {R} ^{n}\right) \) under the sharp relation among \(\alpha ,\beta \) and p.