Boundedness properties of modified averaging operators and geometrically doubling metric spaces

Abstract We characterize the geometrically doubling condition of a metric space in terms of the uniform $$L^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -boundedness of superaveraging operators, where uniform refers to the existence of bounds independent of the measure being considered.