H1-L1Boundedness of Pseudo-differential Operators with Forbidden Amplitudes

In this note, we consider a pseudo-differential operator T-a defined as T(a)f(x) = integral(Rn) integral(Rn) e(a)(2 pi i(x - y)center dot xi)(x, y, xi)f(y)d xi dy. If 0 <= rho, delta < 1 and a(x, y, xi) satisfies that sup(y,xi epsilon Rn) (1 + vertical bar xi vertical bar)(n(1-rho)+rho N - delta M) vertical bar vertical bar del(N)(xi)del(M)(y) a(center dot, y, xi) vertical bar vertical bar L-infinity(R-n) < +infinity 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 rho = 1 or delta = 1.