Annales Academić Scientiarum Fennicć
Mathematica
Volumen 38, 2013, 727-745

MINIMAL WEAK UPPER GRADIENTS IN NEWTONIAN SPACES BASED ON QUASI-BANACH FUNCTION LATTICES

Lukás Malý

Linköping University, Department of Mathematics
SE-581 83 Linköping, Sweden; lukas.maly 'at' liu.se

Abstract. Properties of first-order Sobolev-type spaces on abstract metric measure spaces, so-called Newtonian spaces, based on quasi-Banach function lattices are investigated. The set of all weak upper gradients of a Newtonian function is of particular interest. Existence of minimal weak upper gradients in this general setting is proven and corresponding representation formulae are given. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.

2010 Mathematics Subject Classification: Primary 46E35; Secondary 30L99, 46E30.

Key words: Newtonian space, upper gradient, weak upper gradient, Banach function lattice, quasi-normed space, metric measure space.

Reference to this article: L. Malý: Minimal weak upper gradients in Newtonian spaces based on quasi-Banach function lattices. Ann. Acad. Sci. Fenn. Math. 38 (2013), 727-745.

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doi:10.5186/aasfm.2013.3831