Some Notes on Endpoint Estimates for Pseudo-differential Operators

We study the pseudo-differential operator T-a f (x) = integral(n)(R) e(ix.xi) a(x, xi) (f) over cap (xi) d xi, where the symbol a is in the Hormander class S-rho,1(m) or more generally in the rough Hormander class L-infinity S-rho(m) with m is an element of R and rho is an element of [0,1]. It is known that T-a is bounded on L-1 (R-n) for m < n (rho - 1). In this paper, we mainly investigate its boundedness properties when m is equal to the critical index n(p - 1). For any 0 <= p <= 1, we construct a symbol a is an element of S-rho,1(n(rho-1)), such that T-a is unbounded on L-1, and furthermore, it is not of weak type (1,1) if rho = 0. On the other hand, we prove that T-a is bounded from H-1 to L-1 if 0 <= rho < 1 and construct a symbol a is an element of S-1,1(0), such that T-a is unbounded from H-1 to L-1. Finally, as a complement, for any 1 < p < infinity, we give an example a is an element of S-0,1(-1/p), such that T-a is unbounded on L-p(R).