Upper bounds for the exponent of Hölder mappings in problems of approximation and extension

Abstract.

Under some suitable assumptions on Banach spaces X and Y, we give upper bounds for the exponent α(B _X , Y) so that uniformly continuous functions from the closed unit ball B _X of X to Y can be uniformly approximated by β-Hölder functions for β <α (B _X , Y). For Banach lattices X with some lower $p < \infty$ estimate, our bound gives the exact value of the critical exponent α(B _X , l _q ), which was known for X = l _p . We notice that such estimates give upper bounds in the setting of isomorphic extension of Hölder maps.