On the connection between weighted norm inequalities, commutators and real interpolation

Abstract

We show that the class of weights w for which the Calderón operator is bounded on L p (w) can be used to develop a theory of real interpolation which is more general and exhibits new features when compared to the usual variants of the Lions-Peetre methods. In particular we obtain extrapolation theorems (in the sense of Rubio de Francia's theory) and reiteration theorems for these methods. We also consider interpolation methods associated with the classes of weights for which the Calderón operator is bounded on weighted Lorentz spaces and obtain similar results. We extend the commutator theorems associated with the real method of interpolation in several directions. We obtain weighted norm inequalities for higher order commutators as well as commutators of fractional order. One application of our results gives new weighted norm inequalities for higher order commutators of singular integrals with multiplications by BMO functions. We also introduce analogs of the space BMO in order to consider the relationship between commutators for Calderón type operators and their corresponding classes of weights.

Calderon weights as Muckenhoupt weights

“…Results about extrapolation theorems for Calderón weights appear in [1]. Note that, unlike for the usual Hardy-Littlewood maximal operator, we obtain nontrivial L 1 -weighted inequalities starting from L p -weighted inequalities.…”

“…Given 1 ≤ p < ∞, it is said that a nonnegative measurable function w defined in (0, +∞) is a Calderón weight of the class C p (see [1]), and we write w ∈ C p , if S is bounded on L p (w), or, equivalently, if P and Q are both bounded on L p (w). For p > 1 it is known that w ∈ C p if and only if there exists C > 0 such that for all t > 0 it holds that The case p = 1 is easier to describe: w ∈ C 1 if and only if (1.1) Sw(x) ≤ Cw(x) a.e.…”

“…Similarly, [w] C 1 will denote the best constant in (1.1). It is known that [w] Cp is essentially equal to the norm of the Calderón operator on L p (w), in the sense that the quotient of both quantities is bounded above and below by constants depending only on p. For all these definitions and results, see [1] and [13]. The Calderón operator plays a significant role in the theory of real interpolation.…”

“…The Calderón operator plays a significant role in the theory of real interpolation. Such theory associated with Calderón weights is developed in [1].…”

Commutators in real interpolation with quasi‐power parameters

“…The higher order commutator theorem was proved by Rochberg for the complex interpolation [16], and by Milman for the real interpolation [14]. Recently, these results were extended to the real interpolation with the Calderón weights [1]. In addition, Milman and Rochberg presented a comparison between the commutator results of the real and complex interpolation methods, and emphasized the role of internal cancellation in these results [15].…”