L8-BMO boundedness of some pseudo-differential operators

In this note we study the pseudo-differential operator Ta f (x) = integral (n)(R) e(i x center dot.xi) a(x,xi) (f) over cap (xi)d xi. For 0 <= delta < 1, it is well-known that Ta is bounded on L-2 when the symbol a is in the Hormander class S (n(rho-1) /2)(rho,delta). But it is not bounded on L-2 or L-infinity in general when a is an element of S (n(rho-1) / 2)(rho,1). Here we give a special critical Hormander class S-rho(m) (ln(6) L) such that S-rho,delta(m) (sic) S-rho(m) (ln(6) L) (sic) S-rho,1(m) for any 0 <= delta < 1. We show that the pseudo-differential operator T-a is bounded from L-infinity to BMO if 0 <= rho < 1 and a. S (n(rho-1) /2)(rho) (ln(6) L). This result is an improvement of all known related theorems.