A transference theorem and its application (Q6566721)

Given \(m\in L^{\infty}(\mathbb{R}^{n})\) continuous, consider maximal operators of the form\N\begin{align*}\NA_{m}^\ast : f\mapsto \underset{0<t<\infty}{\sup}\left|\int _{\mathbb{R}^{n}} \widehat{f}(\xi)m(t\xi)\exp(i\langle\cdot,\xi\rangle)d\xi\right|, \\\N\widetilde{A}_{m}^\ast: g\mapsto \underset{0<t<\infty}{\sup}\left|\sum _{k\in\mathbb{Z}^{n}} \widehat{g}(k)m(tk)\exp(i\langle\cdot,k\rangle)\right|,\N\end{align*}\Nthat are bounded, respectively, on \(L^{2}(\mathbb{R}^{n})\) and \(L^{2}(\mathbb{T}^{n})\). For \(p \in (0,1)\), the authors prove the following transference theorem: \(A_{m}^\ast\) is bounded from \(H^{p}(\mathbb{R}^{n})\) to \(L^{p,\infty}(\mathbb{R}^{n})\) if and only if \(\widetilde{A}_{m}^\ast\) is bounded from \(H^{p}(\mathbb{T}^{n})\) to \(L^{p,\infty}(\mathbb{T}^{n})\).\N\NThis allows them, in particular, to transfer their previous result on a maximal operator associated with fractional wave equations (namely \(m(\xi):= \Phi(|\xi|) \exp(i|\xi|^{\alpha})|\xi|^{-\beta}\) where \(\Phi\) is a cutoff equal to \(0\) near \(0\); see [\textit{Z. Liu} et al., J. Geom. Anal. 34, No. 6, Paper No. 184, 35 p. (2024; Zbl 1541.41017)] from the context of compact manifolds to the Euclidean context.\N\NBesides usual techniques for \(H^{p}\) and \(L^{p,\infty}\) analysis (atomic decompositions and level sets decompositions), the key ingredient in the proof is the pointwise control of (truncated versions of) \(\widetilde{A}_{m}^\ast\) by \(A_{m}^\ast\), up to a friendly error term. This result from [\textit{D. Chen} and \textit{D. Fan}, Stud. Math. 131, No. 2, 189--204 (1998; Zbl 1050.42010)] is included in the present paper as Lemma 2.4 for completeness.