A Transference Theorem and Its Application

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