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The Besov capacity in metric spaces

Juho Nuutinen — 2016

Annales Polonici Mathematici

We study a capacity theory based on a definition of Hajłasz-Besov functions. We prove several properties of this capacity in the general setting of a metric space equipped with a doubling measure. The main results of the paper are lower bound and upper bound estimates for the capacity in terms of a modified Netrusov-Hausdorff content. Important tools are γ-medians, for which we also prove a new version of a Poincaré type inequality.

Fractional Maximal Functions in Metric Measure Spaces

Toni Heikkinen;  Juha Lehrbäck;  Juho Nuutinen;  Heli Tuominen — 2013

Analysis and Geometry in Metric Spaces

We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

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