Abstract
For a function ϕ satisfying some suitable growth conditions, consider the following general dispersive equation defined by
where \(\phi (\sqrt { - \Delta } )\) is a pseudo-differential operator with symbol ϕ(∣ξ∣). In the present paper, when the initial data f belongs to Sobolev space, we give the local and global weighted Lq estimate for the global maximal operator \(S_\phi ^{ \ast \ast }\) defined by \(S_\phi ^{ \ast \ast }f(x) = \mathop {\sup }\limits_{t \in \mathbb{R}} \left| {{S_{t,\phi }}f(x)} \right|\), where
is a formal solution of the equation (*).
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Acknowledgments
The authors would like to express their deep gratitude to the referees for their very careful reading, important comments and valuable suggestions.
Funding
This paper is supported by the National Natural Science Foundation of China (Nos. 11871096, 12071473, 11661061, 11761054) and by the Natural Science Foundation of Inner Mongolia (Nos. 2019MS01003, 2021MS01001), and Inner Mongolia University scientific research projects (Nos. NJZY19186, NJZZ21050).
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Ding, Y., Niu, Ym. Weighted Estimates for a Class of Global Maximal Operators Associated with Dispersive Equation. Acta Math. Appl. Sin. Engl. Ser. 38, 187–208 (2022). https://doi.org/10.1007/s10255-022-1071-y
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DOI: https://doi.org/10.1007/s10255-022-1071-y