Hardy spaces and the Tb theorom

It is well-known that Calderon-Zygmund operators T are bounded on H-p for (n)/(n+1) < p less than or equal to 1 provided T* (1) = 0. In this article, it is shown that if T* (b) = 0, where b is a para-accretive function, T is bounded from the classical Hardy space H-p to a new Hardy space H-b(p). To develop an H-b(p) theory, a discrete Calderon-type reproducing formula and Plancherel-Polya-type inequalities associated to a para-accretive function are established. Moreover, David, Journe, and Semmes' result [9] about the L-p, 1 < p < infinity, boundedness of the Littlewood-Paley g function associated to a para-accretive function is generalized to the case of p less than or equal to 1. A new characterization of the classical Hardy spaces by using more general cancellation adapted to para-accretive functions is also given. These results complement the celebrated Calderon-Zygmund operator theory.