Maximal estimate for solutions to a class of dispersive equation with radial initial value

doi:10.1007/s11464-017-0654-z

Front. Math. China

2017, Vol. 12

Issue (5) : 1057-1084 https://doi.org/10.1007/s11464-017-0654-z

RESEARCH ARTICLE

Maximal estimate for solutions to a class of dispersive equation with radial initial value

Yong DING¹, Yaoming NIU^2,¹(

)

¹. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing 100875, China
². Faculty of Mathematics, Baotou Teachers’ College, Baotou 014030, China

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Abstract

Consider the general dispersive equation defined by (∗) ${i ∂ t u + ϕ (− Δ) u = 0, (x, t) ∈ ? n × ?, u (x, 0) = f (x), f ∈ ℓ (? n),$ 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) = sup_0<t<1|S_t,φf(x)|, where S_t,φfis the solution of equation (∗). These estimates imply the a.e. convergence of the solution of equation (∗).

Keywords Dispersive equation maximal operator local estimate global estimate

Corresponding Author(s): Yaoming NIU

Issue Date: 30 September 2017

Cite this article:

Yong DING,Yaoming NIU. Maximal estimate for solutions to a class of dispersive equation with radial initial value[J]. Front. Math. China, 2017, 12(5): 1057-1084.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0654-z
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I5/1057

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