Regularization of Newtonian functions via weak boundedness of maximal operators

TL;DR

This paper investigates the density of Lipschitz functions in Newtonian spaces on metric measure spaces, emphasizing the role of maximal operators' weak boundedness and absolute continuity of the quasi-norm.

Contribution

It establishes new conditions for the density of Lipschitz functions in Newtonian spaces based on the weak boundedness of maximal operators and the absolute continuity of the quasi-norm.

Findings

Density of Lipschitz functions is achieved under weak boundedness conditions.

Absolute continuity of the quasi-norm is crucial for approximation.

Various sufficient conditions for maximal operator boundedness are provided.

Abstract

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 $p$-Poincar\'e 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 others, that a suitable maximal operator is locally weakly bounded. In particular, various sufficient conditions for such boundedness on rearrangement-invariant spaces are established and applied.