Smooth approximations preserving asymptotic Lipschitz bounds

TL;DR

This paper proves that Lipschitz functions on Banach spaces can be approximated by smooth cylindrical functions with controlled asymptotic Lipschitz constants, enabling density results in metric Sobolev and BV spaces.

Contribution

It establishes a method to approximate Lipschitz functions with smooth cylindrical functions while preserving asymptotic Lipschitz bounds, with applications in functional analysis.

Findings

Smooth cylindrical functions are dense in energy in metric Sobolev spaces.

Approximation preserves asymptotic Lipschitz constants.

Results apply to weighted Banach spaces.

Abstract

The goal of this note is to prove that every real-valued Lipschitz function on a Banach space can be pointwise approximated on a given $\sigma$-compact set by smooth cylindrical functions whose asymptotic Lipschitz constants are controlled. This result has applications in the study of metric Sobolev and BV spaces: it implies that smooth cylindrical functions are dense in energy in these kinds of functional spaces defined over any weighted Banach space.