Mathematics – Classical Analysis and ODEs
Scientific paper
1999-11-19
Math. Z. 236 (2001), no. 3, 461-489
Mathematics
Classical Analysis and ODEs
23 pages, 5 figures
Scientific paper
Let T be an oscillatory integral operator on L^2(R) with a smooth real phase function S(x,y). We prove that, in all cases but the one described below, after localization to a small neighborhood of the origin the norm of T decays like N^{-d/2} as the frequency N->infty, where d is the Newton decay rate introduced by Phong and Stein, which is determined by the Newton diagram of S(x,y). For real analytic phase functions this result was proved by Phong and Stein. For smooth phase functions the best known results so far contained a loss of epsilon in the exponent (implicit in the work of Seeger). The exceptional case mentioned above happens when the Taylor series of the mixed partial derivative S''(x,y) factorizes in R[[x,y]] as U(x,y)(y-f(x))^k, where U(x,y) is a unit, k\ge 2, and f(x) is a series in R[[x]] of the form Cx+(higher order), C\ne 0. In this case we prove an estimate with a loss of of a power of log N.
No associations
LandOfFree
Sharp L^2 bounds for oscillatory integral operators with C^\infty phases does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Sharp L^2 bounds for oscillatory integral operators with C^\infty phases, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Sharp L^2 bounds for oscillatory integral operators with C^\infty phases will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-445701