ON THE MINIMAL WEAK UPPER GRADIENT OF A BANACH-SOBOLEV FUNCTION ON A METRIC SPACE
MARCELINA MOCANU Department of Mathematics and Informatics, Faculty of Sciences, "Vasile Alecsandri" University of Bacău, Spiru Haret 8, Bacău 600114, Romania, e-mail: mmocanu@ub.ro
We prove that every function belonging to a Sobolev-type space N1,B(X) on a metric measure space X has a B-weak upper gradient in B that is pointwise minimal μ - almost everywhere, provided that the Banach function space B has a strictly convex and strictly monotone norm. This result generalizes corresponding known results involving Lebesgue spaces B = Lp(X), p > 1 [16] or, more general, Orlicz spaces B = LΨ (X) [17] with a strictly convex Young function Ψ satisfying a Δ2 - condition.
