Oscillatory integral operators with homogeneous polynomial phases in several variables

• Published in: Journal of functional analysis Vol. 244; no. 2; pp. 444 - 487
• Main Authors: Greenleaf, Allan, Pramanik, Malabika, Tang, Wan
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.03.2007
• Subjects: Decay estimates; Newton polyhedron; Oscillatory integral operators; Polynomial phase function; Oscillatory integral operators; Decay estimates; Polynomial phase function; Newton polyhedron
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2006.11.005

We obtain L 2 decay estimates in λ for oscillatory integral operators T λ whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions. The decay rates obtained are optimal in the case of ( 2 + 2 ) -dimensions for any m, while in higher dimensions the result is sharp for m sufficiently large. The proof for large m follows from essentially algebraic considerations. For cubics in ( 2 + 2 ) -dimensions, the proof involves decomposing the operator near the conic zero variety of the determinant of the Hessian of the phase function, using an elaboration of the general approach of Phong and Stein [D.H. Phong, E.M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. (2) 140 (1994) 703–722].