Global maximal inequality to a class of oscillatory integrals

摘要

In the present paper, we give the global Lq estimates for maximal operators generated by multiparameter oscillatory integral St,Φ, which is defined by St,Φf(x)=(2π)−n∫Rneix⋅ξei(t1ϕ1(|ξ1|)+t2ϕ2(|ξ2|)+⋯+tnϕn(|ξn|))fˆ(ξ)dξ,x∈Rn, where n≥2 and f is a Schwartz function in S(Rn), t=(t1,t2,…,tn), Φ=(ϕ1,ϕ2,…,ϕn), ϕi (i=1,2,3,…,n) is a function on R+→R, which has a suitable growth condition. These estimates are apparently good extensions to the results of Sjölin and Soria (J. Math. Anal. Appl 411:129–143, ) for the multiparameter fractional Schrödinger equation.