For a function Ф satisfying some suitable growth conditions,consider the following general dispersive equation defined by{i■tu+Ф(√-△)u=0,u(x,0)=f(x),(x,t)∈R^(n)× R,f∈S(R^(n) where Ф(√-△)is a pseudo-diff...For a function Ф satisfying some suitable growth conditions,consider the following general dispersive equation defined by{i■tu+Ф(√-△)u=0,u(x,0)=f(x),(x,t)∈R^(n)× R,f∈S(R^(n) where Ф(√-△)is a pseudo-differential operator with symbol Ф(|ξ|).In the present paper,when the initial data f belongs to Sobolev space,we give the local and global weighted L^(q) estimate for the global maximal operator S^(**)Ф defined by S^(**)Фf(x)=sup_(t∈R)|S_(t,Ф)f(x)|,where S_(t,Ф)f(x)=(2π)^(-n)∫_(R^(n)e^(ix·ζ+itФ(|ζ+|)f(ζ)dζ is a formal solution of the equation(*).展开更多
基金supported by the National Natural Science Foundation of China(Nos.11871096,12071473,11661061,11761054)by the Natural Science Foundation of Inner Mongolia(Nos.2019MS01003,2021MS01001)Inner Mongolia University scientific research projects(Nos.NJZY19186,NJZZ21050)。
文摘For a function Ф satisfying some suitable growth conditions,consider the following general dispersive equation defined by{i■tu+Ф(√-△)u=0,u(x,0)=f(x),(x,t)∈R^(n)× R,f∈S(R^(n) where Ф(√-△)is a pseudo-differential operator with symbol Ф(|ξ|).In the present paper,when the initial data f belongs to Sobolev space,we give the local and global weighted L^(q) estimate for the global maximal operator S^(**)Ф defined by S^(**)Фf(x)=sup_(t∈R)|S_(t,Ф)f(x)|,where S_(t,Ф)f(x)=(2π)^(-n)∫_(R^(n)e^(ix·ζ+itФ(|ζ+|)f(ζ)dζ is a formal solution of the equation(*).
