Maximal operators in nondoubling metric measure spaces

TL;DR

This thesis investigates the properties and boundedness of maximal operators in nondoubling metric measure spaces, classifying their inequalities, boundedness conditions, and interrelations with function spaces like BMO.

Contribution

It provides a comprehensive classification of maximal operator inequalities, boundedness from Lorentz spaces, and interrelations with BMO spaces in nondoubling metric measure spaces.

Findings

Classified interrelations of strong, weak, and restricted weak type inequalities.

Characterized boundedness of maximal operators between Lorentz spaces.

Illustrated configurations of maximal function finiteness in nondoubling spaces.

Abstract

This is a revised version of the doctoral dissertation of the same title, written under the supervision of Professor Krzysztof Stempak in 2019. For general (possibly nondoubling) metric measure spaces various properties of the associated maximal operators, centered $\mathcal{M}^{\rm c}$ and noncentered $\mathcal{M}$, are investigated. Chapter 1 is the introduction to the topic. In Chapter 2 the classification of possible interrelations between the occurrences of strong, weak, and restricted weak type inequalities for both $\mathcal{M}^{\rm c}$ and $\mathcal{M}$ simultaneously is given. In Chapter 3 a similar analysis for the so-called modified maximal operators is performed. Chapter 4 is devoted to studying the boundedness of $\mathcal{M}^{\rm c}$ from $L^{p,q}$ to $L^{p,r}$. In particular, for each fixed $p \in (1, \infty)$ the classification of possible shapes of the sets \[ \Big\{ \Big( \frac{1}{q},\frac{1}{r} \Big) \in [0,1] \times [0,1] : \mathcal{M}^{\rm c} \text{ is bounded from } L^{p,q} \text{ to } L^{p,r} \Big\} \] is given for the class of spaces satisfying a mild support assumption $\mu(X \setminus {\rm supp}(\mu)) = 0$. The main result of Chapter 5 is the classification of possible interrelations between the spaces ${\rm BMO}^p$, $p \in [1,\infty)$. In Chapter 6 a dichotomy regarding the finiteness of maximal functions associated with doubling spaces is tested in general setting. As a result, each of the four configurations regarding its occurrence or not for $\mathcal{M}^{\rm c}$ and $\mathcal{M}$ is illustrated with a suitably chosen nondoubling space. Finally, Appendix contains a new elementary proof of the interpolation theorem for Lorentz spaces with the first parameter fixed and the second parameter varying among its natural range of admissibility.