Hardy spaces and theTb theorem

Abstract

It is well-known that Calderon-Zygmund operators T are bounded on Hp for $$\frac{n}{{n + 1}}< p \leqslant 1$$ 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 Hp 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 LP, 1 < p < ∞, boundedness of the Littlewood-Paley g function associated to a para-accretive function is generalized to the case of p ≤ 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.