Abstract
Consider the general dispersive equation defined by
where ϕ(√−Δ) is a pseudo-differential operator with symbol ϕ(|ξ|). In this paper, for ϕ satisfying suitable growth conditions and the radial initial data f in Sobolev space, we give the local and global L q estimate for the maximal operator S ϕ * defined by S ϕ *f(x) = sup0<t<1 |S t,ϕ f(x)|, where S t,ϕ f is the solution of equation (*). These estimates imply the a.e. convergence of the solution of equation (*).
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Acknowledgements
The authors would like to express their deep gratitude to the referees for their very careful reading. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11371057, 11471033, 11571160, 11661061), the Inner Mongolia University Scientific Research Projects (No. NJZZ16234), and the Natural Science Foundation of Inner Mongolia (No. 2015MS0108).
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Ding, Y., Niu, Y. Maximal estimate for solutions to a class of dispersive equation with radial initial value. Front. Math. China 12, 1057–1084 (2017). https://doi.org/10.1007/s11464-017-0654-z
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DOI: https://doi.org/10.1007/s11464-017-0654-z