Carleson measures, trees, extrapolation, and T(b) theorems

Abstract

Abstract. The theory of Carleson measures, stopping time arguments, and atomic decompositions has been well-established in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The purpose of this paper is to demonstrate that the two theories are in fact closely related, by taking existing results and reproving them in a unified setting. In particular we give a dyadic version of extrapolation for Carleson measures, as well as a two-sided local dyadic T(b) theorem which generalizes earlier T(b) theorems of David, Journe, Semmes, and Christ.

On general local $Tb$ theorems

“…For doubling measures, one can also consider more general L p type testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele [AHM02], and further studied by Hofmann [Hof07], Auscher and Yang [AY09] and Tan and Yan [TY09]. The most general assumption used in these papers is of the form that…”

“…In [AHM02] a theorem of this type is proved only for very special operators, the so-called perfect dyadic singular integral operators. This was expected to easily generalize for all Calderón-Zygmund operators -but this turned out not to be the case (it being easy, at least).…”

b-weighted dyadic BMO from dyadic BMO and associatedT(b) theorems

“…In effect, this changes the theorem from a b-input case (where T is evaluated at a function b) to a b-output case, where T * (1) is evaluated by a norm which depends on b. This is a natural extension of the previous work, in particular in [2] where the dyadic T (b) theorems are proven and in [13], where sharp growth bounds for b-input theorems are proven. Our approach utilizes b-adapted Haar wavelets as in [5].…”

“…As in [2,13], we restrict ourselves to a finite set of dyadic intervals on the half-line. We fix a large M , and let…”

“…In 1985, David, Journé, and Semmes [7] further develop the theory by by replacing the constant function 1 on which the operator T is evaluated by a function b whose mean is bounded away from zero. These T (b) theorems have been studied in several contexts (see [3,4,12]) and in their 2002 paper [2], Auscher, Hofmann, Muscalu, Tao, and Thiele prove several T (b) theorems in a dyadic setting in the context of Carleson measures and trees. In 2003, Tolsa [15] used the non-doubling T (b) theorem in [12] in his answer to the Painlevé problem and his proof of the semiadditivity of analytic capacity of a compact set in C.…”