Rubio de Francia's Extrapolation Theorem and Sparse Bounds

Abstract

Sparse bounds have proved to be a powerful tool which simplify the understanding of many important operators in harmonic analysis. The existence of a sparse bound for an operator implies the weighted boundedness of the operator for weights within the Muckenhoupt classes. As in the example of Calderón-Zygmund operators, this implication can prove weighted boundedness with the best possible dependency on the characteristic constant of the Muckenhoupt weight. In this paper, we prove a partial converse for sub-additive operators by showing that weighted boundedness, with an arbitrary a priori control of the operator norm in terms of the characteristic constants of the weight, implies the existence of sparse form bounds. This result implies that a near-optimal dependency of the operator norm on characteristic constants of the weights follows from the weighted boundedness itself, rather than relying on the particular structure of the sub-additive operator. Moreover, it allows Rubio de Francia's extrapolation theorem to be extended to the weak-type end-point for sub-linear operators and leads to an interpolation result for sparse form bounds for analytic families of operators. As an application, we obtain new sparse form bounds for rough singular integral operators.

Theorem 1.6 is not true, as the example of the iterated maximal function, which is not weak-type $(1,1)$, shows. The error occurs in (26) and (31), as we cannot bound the $A_s$ characteristic constant of the weight by $\Omega$, as is required by (9). The author thanks Emiel Lorist for kindly pointing this out and for sharing a weaker but correct version of Theorem 1.6