The Endpoint Estimate for Fourier Integral Operators

Let T a,ϕ be a Fourier integral operator defined by the oscillatory integral T a φ u ( x ) = 1 ( 2 π ) n ∫ ℝ n e i φ ( x ξ ) a ( x ξ ) u ^ ( ξ ) d ξ where a ∈ S ϱ δ m and ϕ ∈ Φ 2 , satisfying the strong non-degenerate condition. It is shown that if 0 < ϱ ≤ 1, 0 ≤ δ < 1 and ≤ ϱ 2 − n 2 , then T a,ϕ is a bounded operator from L ∞ (ℝ n ) to BMO(ℝ n ).