Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators
Artikel i vetenskaplig tidskrift, 2018

Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among other facts, the fact that a suitable maximal operator is locally weakly bounded. In particular, various sufficient conditions for such boundedness on quasi-Banach function lattices (and rearrangement-invariant spaces, in particular) are established and applied.

metric measure space

Newtonian space

Sobolev-type space

weak boundedness

rearrangement-invariant space

upper gradient


Lipschitz function

maximal operator

Banach function lattice

density of Lipschitz functions


Lukas Maly

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Jyväskylän Yliopisto

Journal dAnalyse Mathematique

0021-7670 (ISSN)

Vol. 134


Matematisk analys