Global $L^{2}$-boundedness of a New Class of Rough Fourier Integral Operators

Abstract

. In this paper, we investigate the L 2 boundedness of Fourier integral operator T φ,a with rough symbol a ∈ L ∞ S mρ and rough phase φ ∈ L ∞ Φ 2 which satisfies (cid:12)(cid:12) { x : |∇ ξ φ ( x, ξ ) − y | ≤ r } (cid:12)(cid:12) ≤ C ( r n − 1 + r n ) for any ξ, y ∈ R n and r > 0. We obtain that T φ,a is bounded on L 2 if m < ρ ( n − 1) / 2 − n/ 2 when 0 ≤ ρ ≤ 1 / 2 or m < − ( n + 1) / 4 when 1 / 2 ≤ ρ ≤ 1. When ρ = 0 or n = 1, the condition of m is sharp. Moreover, the maximal wave operator is a special class of T φ,a which is studied in this paper. Thus, our main theorem substantially extends and improves some known results about the maximal wave operator.