Abstract
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.
Cuvinte cheie
metric measure space
approximate differentiability
Banach function space
upper gradi-ent
Newtonian space
Poincaré inequality
maximal operator