A proof of the Fefferman-Stein-Strömberg inequality for the sharp maximal functions

Abstract

Abstract.We give another proof of a theorem of Strömberg for the FeffermanStein sharp maximal functions. Our method is based on a decomposition lemma which is due to the arguments of Carleson, Garnett and Jones for the functions of BMO, and it is valid for a two-weight setting under a condition which is equivalent to the Aoo condition in the case of equal weights.

A pointwise estimate for the local sharp maximal function with applications to singular integrals

“…Then Garnett and Jones [11] obtained a dyadic reformulation of Carleson's result with a different proof. Fujii [9] observed that almost the same proof as in [11] yields actually a decomposition of an arbitrary locally integrable function: given a cube Q, we have…”

“…It was noted by Fujii [9] without the proof that replacing f Q in (1.4) by a median value m f (Q), one can obtain a similar decomposition but with the local sharp maximal function instead of f # Q (x). Indeed, certain modifications of the proof in [9] yield such a variant of (1.4) with a control of g by M # λ f and with (1.5) replaced by…”

On pointwise and weighted estimates for commutators of Calderón–Zygmund operators

“…This lemma is based on several known ideas. The first idea is an estimate by oscillations over a sparse family (see [11,16,22]) and the second idea is an augmentation process (see Section 2.1).…”

Predual Spaces of Morrey Spaces with Non-doubling Measures

“…We remark that the condition like (4) appears in [4]. We are mainly interested in the function f ∈ C p q (µ) such that M(f ) = 0.…”