Regularization of Newtonian functions on metric spaces via weak boundedness of maximal operators
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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-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 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.
Newtonian space, Sobolev-type space, metric measure space, upper gradient, Banach function lattice, rearrangement-invariant space, maximal operator, Lipschitz function, regularization, weak boundedness, density of Lipschitz functions
IdentifiersURN: urn:nbn:se:liu:diva-105614OAI: oai:DiVA.org:liu-105614DiVA: diva2:708829