Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators
Journal article, 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
regularization
Lipschitz function
maximal operator
Banach function lattice
density of Lipschitz functions
Author
Lukas Maly
Chalmers, Mathematical Sciences, Analysis and Probability Theory
University of Jyväskylä
Journal dAnalyse Mathematique
0021-7670 (ISSN) 15658538 (eISSN)
Vol. 134 1 1-54Subject Categories
Mathematical Analysis
DOI
10.1007/s11854-018-0001-7