6 JESUS BASTERO, MARIO MILMAN AND FRANCISCO J. RUIZ
give analogous estimates for operators acting on spaces of homogeneous type. Our
results also give weighted norm inequalities for commutators of other classical oper-
ators of Harmonic Analysis. For background on these developments in the setting of
Lorentz spaces we refer to [Sw]. It is relevant to mention here the recent work [Pe]
on weighted norm inequalities for commutators which complements our work here.
The direct connection between the two subjects treated in this paper is developed
in the last part of the paper. Recall that weighted norm inequalities for singular
integral operators, commutator estimates for singular integrals with multiplication
by BMO functions are closely related to the connection between Ap weights and
BMO. In our context these considerations also lead us to consider variants of the
space BMO in the context of the weights Cp. Thus, in analogy with the classical
theory of Ap weights, the set of logarithms of Bp weights gives BMO type spaces.
We also construct its predual, via a suitable atomic theory.
Applications play an important role in our development. Throughout the paper
we consider several applications and examples relating our results to singular in-
spaces, Tent spaces, Hardy-Sobolev spaces, approximation
theory, Schatten ideals, Dirichlet spaces, etc. Concrete new estimates (weighted and
unweighted) for commutators are given in different contexts including estimates for
Jacobians of maps and other operations with sufficient cancellations.
We shall now review the organization of the paper. In §2 we study the Calderon
weights Cp and compare them with Kalugina weights and quasipower weights. In
§3 we study the real interpolation spaces associated with Calderon weights, paying
special attention to reiteration and extrapolation results. In §4 we briefly consider
interpolation methods associated with Bp weights, in §5 we consider a connection
between Rubio de Francia's theory and the theory of extrapolation of Jawerth-
Milman, in §6 we discuss some applications to the study of other scales of function
spaces including Lorentz spaces, weighted Tent spaces, ideals of operators, Hardy
spaces, Hardy-Sobolev spaces, Dirichlet spaces, in §7 we develop our theory of
commutators, in §8 we discuss generalized commutators including those of fractional
order, in §9 we extend our results to the setting of Quasi-Banach spaces. In §10
we discuss some applications of our results to singular integrals and to results
related with compensated compactness, in §11 we introduce the space BMO and
H1 type of spaces associated to the weights under study and show in §12 duality
and interpolation results for these spaces as well as new applications illustrating
the connection between the two parts of the paper.
In conclusion we should mention that there also several commutator theorems
associated with the complex method of interpolation and the interested reader
should consult [RW], [CCMS], [R], as well as the unified theories developed in
[CCS] and [CKMR] and their references.
Throughout the paper we shall follow the notation and terminology of [BL].
We would like to thank the referee for her/his comments
and for helpful suggestions to improve the presentation of the paper.