Median-type John-Nirenberg space in metric measure spaces
Kim Myyryl\"ainen

TL;DR
This paper extends the John-Nirenberg space concept to metric measure spaces using medians instead of averages, establishing inequalities and equivalences under certain conditions.
Contribution
It introduces a median-based John-Nirenberg space in metric measure spaces and proves key inequalities and boundary behavior results.
Findings
Established local and global John-Nirenberg inequalities.
Proved the equivalence of integral and median-type spaces under chaining assumptions.
Developed a Calderón-Zygmund decomposition and good-$\lambda$ inequality for medians.
Abstract
We study the so-called John-Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John-Nirenberg inequalities, which give weak type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calder\'{o}n-Zygmund decomposition and a good- inequality for medians. A John-Nirenberg inequality up to the boundary is proven by using chaining arguments. As a consequence, the integral-type and the median-type John-Nirenberg spaces coincide under a Boman-type chaining assumption.
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Nonlinear Partial Differential Equations · Advanced Mathematical Physics Problems
