Henri Martikainen

On general local Tb theorems

In this paper, local Tb theorems are studied both in the doubling and non-doubling situation. We prove a local Tb theorem for the class of upper doubling measures. With such general measures, scale invariant testing conditions are required (L^{\infty} or BMO). In the case of doubling measures, we also modify the general non-homogeneous method of proof to yield a new proof of the local Tb theorem with L^2 type testing conditions.

Square functions with general measures

where st is a kernel satisfying for some α > 0 that |st(x, y)| t tαλ(x, t) + |x− y|αλ(x, |x− y|) and |st(x, y)− st(x, z)| |y − z| tαλ(x, t) + |x− y|αλ(x, |x− y|) whenever |y − z| < t/2. We use the ∞ metric on R. If Q ⊂ R is a cube with sidelength (Q), we define the associated Carleson box Q̂ = Q× (0, (Q)). In this note we will prove the following theorem: 1.1. Theorem. Assume that there exists a function b ∈ L∞(μ) such that ∣∣∣ ˆ Q b(x) dμ(x) ∣∣∣ μ(Q)

Local Tb theorem with L 2 testing conditions and general measures: square functions

Local Tb theorems with Lp type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. In the non-homogeneous world local Tb theorems have only been proved assuming scale invariant (L∞ or BMO) testing conditions. In this paper, for the first time, we overcome these obstacles in the non-homogeneous world, and prove a nonhomogeneous local Tb theorem with L2 type testing conditions. This paper is in the setting of the vertical and conical square functions defined using general measures and kernels. On the technique side, we demonstrate a trick of inserting Calderón–Zygmund stopping data of a fixed function into the construction of the twisted martingale difference operators. This built-in control of averages is an alternative to Carleson embedding.

Some obstacles in characterising the boundedness of bi-parameter singular integrals

The famous T1 theorem for classical Calderón–Zygmund operators is a characterisation for their boundedness in

Note about square function estimates and uniformly rectifiable measures

L^{2}

Representation of bi-parameter singular integrals by dyadic operators

L2. In the bi-parameter case, on the other hand, the current T1 theorem is merely a collection of sufficient conditions. This difference in mind, we study a particular dyadic bi-parameter singular integral operator, namely the full mixed bi-parameter paraproductP, which is precisely the operator responsible for the outstanding problems in the bi-parameter theory. We make several remarks about P, the common theme of which is to demonstrate the delicacy of the problem of finding a completely satisfactory product T1 theorem. For example, P need not be unconditionally bounded if it is conditionally bounded—a major difference compared to the corresponding one-parameter model operators. Moreover, currently the theory even lacks a characterisation for the potentially easier unconditional boundedness. The product BMO condition is sufficient, but far from necessary: we show by example that unconditional boundedness does not even imply the weaker rectangular BMO condition.

A new approach to non-homogeneous local

We aim to showcase the wide applicability and power of the big pieces and suppression methods in the theory of local Tb theorems. The setting is new: we consider conical square functions with cones