APPROXIMATE DIFFERENTIABILITY IN NEWTONIAN SPACES BASED ON BANACH FUNCTION SPACES
MARCELINA MOCANU Department of Mathematics and Informatics,
Faculty of Sciences, "Vasile Alecsandri" University of Bacău,
157 Calea Mărășești, 600115 Bacău, ROMANIA,
e-mail: mmocanu@ub.ro
In this note we investigate the approximate differentiability of Newtonian functions on a doubling metric measure space. The Newtonian space under consideration consists of functions belonging to a rearrangement invariant Banach function space E and possesing an upper gradient which also belongs to E. Our main tools are a Poincaré inequality and a noncentered maximal operator, both defined via the Banach function space E. Under our assumptions, considering for a Newtonian function u an upper gradient g belonging to the given Banach function space E, it turns out that a Hajłasz gradient of u is a constant mul-tiple of the maximal function M_{E}g, which is Borel measurable and finite almost eve-rywhere.
