H1–L1 Boundedness of Pseudo-differential Operators with Forbidden Amplitudes.

In this note, we consider a pseudo-differential operator T a defined as T a f (x) = ∫ R n ∫ R n e 2 π i (x - y) · ξ a (x , y , ξ) f (y) d ξ d y. If 0 ≤ ρ , δ < 1 and a (x , y , ξ) satisfies that sup y , ξ ∈ R n (1 + | ξ |) n (1 - ρ) + ρ N - δ M ‖ ∇ ξ N ∇ y M a (· , y , ξ) ‖ L ∞ (R n) < + ∞ for any N ≤ n + 1 and M ≤ 1 , then we show that the pseudo-differential operator T a is bounded from H 1 to L 1 . However, this result is not true if ρ = 1 or δ = 1 . [ABSTRACT FROM AUTHOR]