ON THE MINIMAL WEAK UPPER GRADIENT OF A BANACH-SOBOLEV FUNCTION ON A METRIC SPACE

Abstract

We prove that every function belonging to a Sobolev-type space N^1,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 = L^p(X), p > 1 [16] or, more general, Orlicz spaces B = L^Ψ (X) [17] with a strictly convex Young function Ψ satisfying a Δ₂ - condition.

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