Local smoothing for operators failing the cinematic curvature condition

[摘要] In this paper, we examine a class of averaging operators which exhibit local smoothing. That is, viewed as a function of space and time variables, the operators yield more smoothing than the fixed-time estimates. Sogge showed in a more general setting that if these operators satisfy a cinematic curvature condition, they will exhibit some local smoothing [C.D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991) 231-251]. Here we translate this condition into the setting of averaging operators in the plane. We prove that cinematic curvature is not necessary for local smoothing to occur, exhibiting a class of operators which fail the cinematic curvature condition but still satisfy a local smoothing estimate. Furthermore, the amount of local smoothing exhibited by these operators is strictly less than that conjectured for operators satisfying the cinematic curvature condition. (c) 2005 Elsevier Inc. All rights reserved.

[发布日期] 2006-12-01

[关键词] harmonic analysis;Fourier integral operators;averaging operators