Accretive system Tb-theorems on nonhomogeneous spaces

Abstract

We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calderón-Zygmund operator on L 2 (µ). We do not assume any kind of doubling condition on the measure µ, so we are in the nonhomogeneous situation. Another interesting difference from the theorem of Christ is that we allow the operator to send the functions of our accretive system into the space bounded mean oscillation (BMO) rather than L ∞ . Thus we answer positively a question of Christ as to whether the L ∞ -assumption can be replaced by a BMO assumption.We believe that nonhomogeneous analysis is useful in many questions at the junction of analysis and geometry. In fact, it allows one to get rid of all superfluous regularity conditions for rectifiable sets. The nonhomogeneous accretive system theorem represents a flexible tool for dealing with Calderón-Zygmund operators with respect to very bad measures.

Boundedness and applications of singular integrals and square functions: a survey

“…In practice, this program was quite difficult to carry out, especially in the general situation in which the underlying measure may be non-doubling. Some extensions of either local or global Tb theorems to the non-doubling setting have been obtained by David [41] and by Nazarov, Treil and Volberg [79,80]. The latter, especially, played a useful role in Tolsa's solution of the Painlevé problem and the related Vitushkin conjecture concerning the semi-additivity of analytic capacity.…”

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

“…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.…”

BCR algorithm and the T(b) Theorem

“…It lead to the solution of the Vitushkin's conjecture by G. David [D] or to a proof of the semiadditivity of analytic capacity (Painlevé problem) by X. Tolsa [T]. Those solutions required similar T (b) theorems but in non-homogeneous spaces as developed by G. David [D], and S. Nazarov, S. Treil and A. Volberg [NTV1,NTV2,V].…”