ABSTRACT. We prove a T b theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound µ(B(x, r)) ≤ Cr d . Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.

We prove a dyadic representation theorem for bi-parameter singular integrals. That is, we represent certain bi-parameter operators as rapidly decaying averages of what we call bi-parameter shifts. A new version of the product space T 1 theorem is established as a consequence.2010 Mathematics Subject Classification. 42B20. Key words and phrases. Haar shift, bi-parameter singular integral, T 1 theorem, nonhomogeneous analysis.The author is supported by the Academy of Finland through the project "L p methods in harmonic analysis".

Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom type inequality for iterated commutators of linear bi-parameter singular integrals. We prove that if T is a bi-parameter singular integral satisfying the assumptions of the bi-parameter representation theorem, thenHere Ap stands for the bi-parameter weights in R n × R m and bmo(ν) is a suitable weighted little BMO space. We also simplify the proof of the known first order case.2010 Mathematics Subject Classification. 42B20.