Off-Diagonal Two-Weight Boundedness and Compactness of Commutators of Singular Integrals on Lebesgue Spaces

The main objects of this work are certain commutators that are applied in harmonic analysis. Such a commutator is formed by commuting a singular integral operator and a pointwise multiplication operator. Commutators and their boundedness or compactness have been applied for example in factorizations for Hardy spaces and in the Jacobian problem.

The focus of this work is to provide characterizations of boundedness and compactness of commutators that map between two distinct weighted Lebesgue spaces. By a characterization we mean that we aim to give both sufficient and necessary conditions for the mentioned two properties. Characterizations are given in terms of the pointwise multiplier. Weighted Lebesgue spaces consist of functions that are p-integrable with respect to a weight function, where p is greater than 1. The weight functions in question are Muckenhoupt weights tied to the exponent p. Beyond the results of this work, these weights are also tied to many important weighted estimates in harmonic analysis.

The domain and codomain of the commutator may be distinct for two non-exclusive reasons: the weights might be different or the exponents might be different. In the Euclidean setting, the characterization of boundedness of commutators in the context where just the weights may be different was pioneered by Bloom in 1985 for the Hilbert transform. A distinctive feature is that the characterisation is given in terms of a single third weight, constructed from the given two weights. This feature is present all over this work as well. The result of Bloom was later revived and generalised by Holmes, Lacey and Wick in 2016. The compactness in the same context was characterized by Lacey and Li a couple of years later. The boundedness and compactness of commutators are characterised by functions of bounded and vanishing weighted mean oscillations, respectively. In contrast to these results, the present work focuses on the context where also the exponents are different. This is what we call the off-diagonal setting. This setting is further divided to the sub-diagonal setting, where the exponent of the domain is smaller than that of the codomain, and the super-diagonal setting, where the mutual order of the exponents is the opposite.

In the first article, we provide the characterizations of both boundedness and compactness of commutators in the sub-diagonal setting. With a resemblance to the on-diagonal results, the sub-diagonal boundedness and compactness of commutators may be characterised by functions of bounded and vanishing fractional weighted mean oscillations, respectively. We also define the new Bloom weight in the off-diagonal setting. This weight is also used in the articles of this thesis. To provide the necessary conditions, we use the approximate weak factorisation recently developed by Hytönen. One of the strengths of this technique is that it applies to all singular integral operators that satisfy a quite general non-degeneracy condition. For the sufficiency parts, we apply the method of bounding commutators by applicable sparse operators. The origins of sparse domination are attributed to Lerner's work in the 2010's. These main techniques persist through all articles in this work.

In the second article, we provide the characterization of boundedness of commutators in the super-diagonal setting. The correct necessary condition for super-diagonal boundedness, when both weights are equally one, was only recently given by Hytönen. When two possibly different weights are involved, we demonstrate that a new condition based on the sharp weighted maximal function is needed. In contrast to the case studied by Hytönen, the sufficiency of the new condition is not classical and thus also needs demonstrating.

Throughout history, these characterization results have been generalised by allowing more general kernels and more general underlying spaces. In the third article, we engage in the latter by stepping away from the Euclidean setting of the first and second articles. Namely, we extend the boundedness characterization of the first article to spaces of homogeneous type. These spaces support a large part of the harmonic analysis that is possible in the Euclidean setting. An important difference is that the underlying measure is merely doubling and not necessarily translation-invariant.