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.

Full document as PDF file

doi:10.5186/aasfm.2013.3831

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