# Research

### Weighted inequalities

We have made several contributions to the currently active area of *sharp* weighted inequalities for singular integrals. Sharpness means that we want to find the optimal quantitative dependence of the norm bounds on the weight in question, elaborating on the classical qualitative theory of Muckenhoupt and others.

A breakthrough in this area was Hytönen's sharp weighted bound for general Calderón-Zygmund operators, based on a new dyadic representation theorem for these operators. Martikainen has extended this representation to bi-parameter operators. Kairema investigates weighted inequalities for positive operators in the context of metric spaces.

*Tb* theorems

*Tb* theorems typically describe the boundedness of an operator *T* on a large class of functions (like *L*^{2}) in terms of its action just on one or a few special functions, often denoted by *b*. We are interested in obtaining very general forms of such theorems, applicable to ever wider classes of problems. Among the recent highlights is the *Tb* theorem for non-doubling measures on metric spaces by Hytönen and Martikainen, and its further vector-valued extension by Martikainen. Also the weighted inequalities often employ a similar proof technique as the *Tb* theorems.

### Harmonic Analysis on metric spaces

One of our research themes is the extension of Euclidean methods using dyadic cubes to abstract metric spaces. Random dyadic cubes in a metric space were built by Hytönen and Martikainen, whereas Hytönen and Kairema constructed a finite collection of adjacent dyadic systems with useful properties.

Another theme is “cleaning up” the theory by eliminating extra assumptions on the space that often appear in the vast literature. We prefer to work in a geometrically doubling (in some sense: finite-dimensional) space with an upper doubling measure. (The latter means that the measure itself may be non-doubling, but it should be dominated by a doubling quantity.) While we sometimes work with a doubling measure, it is our paradigm that all further assumptions (like non-emptyness of annuli, absence of point masses, ...) should be unnecessary for most questions of Harmonic Analysis. This has been confirmed in many instances.

### Vector-valued Harmonic Analysis

Hytönen's earlier research was concentrated in this area, which still offers interesting problems. The standard set-up is the class of Banach space with the UMD (*unconditionality of martingale differences*) property, which is known to be necessary and sufficient for the vector-valued extension of several classical results of Harmonic Analysis involving cancellation. A more recent theme is the understanding of certain *maximal inequalities* for vector-valued functions, which has been systematically studied by Kemppainen.

### Differential operators and functional calculus

This theme has been mainly studied in Hytönen's collaboration with McIntosh and Portal.