The Hardy–Lorentz spaces Hp,q(Rn)

Abstract

We deal with the Hardy-Lorentz spaces H p,q (R n ) where 0 < p � 1, 0 < q � 1. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them. The real variable theory of the Hardy spaces represents a fruitful set- ting for the study of maximal functions and singular integral operators. In fact, it is because of the failure of these operators to preserve L 1 that the Hardy space H 1 assumes its prominent place in harmonic analysis. Now, for many of these operators, the role of L 1 can just as well be played by H 1,∞ , or Weak H 1 . However, although these operators are amenable to H 1 -L 1 and H 1,∞ -L 1,∞ estimates, interpolation between H 1 and H 1,∞ has not been available. Similar considerations apply to H p and Weak H p for 0 < p < 1. The purpose of this paper is to provide an interpolation result for the Hardy-Lorentz spaces H p,q , 0 < p ≤ 1, 0 < q ≤ ∞, including the case of Weak H p as an end point for real interpolation. Since in this context neither truncations are available nor reiteration applies, the atomic decomposition will be the key ingredient in dealing with interpolation. The paper is organized as follows. The Lorentz spaces, including criteria that ensure membership in L p,q , 0 < p < ∞, 0 < q ≤ ∞, are discussed in Section 1. In Section 2 we show that distributions in H p,q have an atomic de- composition in terms of H p atoms with coefficients in an appropriate mixed norm space. An interesting application of this decomposition is to H p,q -L p,∞ estimates for Calderon-Zygmund singular integral operators, p < q ≤ ∞. Also, by manipulating the different levels of the atomic decomposition, we show that, for 0 < q1 < q < q2 ≤ ∞, H p,q is an intermediate space between H p,q1 and H p,q2 . This result applies to Calderon-Zygmund singular integral