Smooth approximations preserving asymptotic Lipschitz bounds. arXiv:2409.01772
Preprint, arXiv:2409.01772 [math.FA] (2024).
Summary: 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.
MSC:
| 46B28 | Spaces of operators; tensor products; approximation properties |
| 46G05 | Derivatives of functions in infinite-dimensional spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 46T20 | Continuous and differentiable maps in nonlinear functional analysis |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 49J52 | Nonsmooth analysis |
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