Boundedness of averaging operators on geometrically doubling metric spaces

Abstract

We prove that averaging operators are uniformly bounded on $L^1$ for all geometrically doubling metric measure spaces, with bounds independent of the measure. From this result, the $L^1$ convergence of averages as $r\to 0$ immediately follows.

arXiv comment7 pages, to appear in Ann. Acad. Sci. Fenn. Math