Journal d'Analyse Mathématique

, Volume 134, Issue 1, pp 1–54 | Cite as

Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators

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Abstract

Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolevtype 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.

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Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland

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