L∞-BMO boundedness of some pseudo-differential operators.

In this note we study the pseudo-differential operator T a f (x) = ∫ R n e i x · ξ a (x , ξ) f ^ (ξ) d ξ. For 0 ≤ δ < 1 , it is well-known that T a is bounded on L 2 when the symbol a is in the Hörmander class S ρ , δ n (ρ - 1) 2 . But it is not bounded on L 2 or L ∞ in general when a ∈ S ρ , 1 n (ρ - 1) 2 . Here we give a special critical Hörmander class S ρ m (ln 6 L) such that S ρ , δ m ⫋ S ρ m (ln 6 L) ⫋ S ρ , 1 m for any 0 ≤ δ < 1 . We show that the pseudo-differential operator T a is bounded from L ∞ to BMO if 0 ≤ ρ < 1 and a ∈ S ρ n (ρ - 1) 2 (ln 6 L) . This result is an improvement of all known related theorems. [ABSTRACT FROM AUTHOR]