Approximation of uniformly continuous mappings in Orlicz spaces.

Sylvain Delpech

We estimate the best exponent α for which every uniformly continuous function, from the unit ball, B_{M}, of a reflexive Orlicz functions space, endowed with the Luxemburg norm ∥ . ∥_{M}, into any normed vector space (Y, ∥ . ∥_{Y}), can be approximated by α-Hölder functions. We also discuss the sharpness of our estimate. Then we prove that this best α, defined by :

α(B_{M}, Y) = sup\{α ∈ [0, 1], \overline{H^{α}(B_{M}, Y)}^{∥ . ∥_{∞}} = UC(B_{M}, Y)\},

where H^{α}(B_{M}, Y) is the set of α-Hölder functions between B_{M} and Y, UC(B_{M}, Y) is the set of uniformly continuous functions between B_{M} and Y, may sometimes be only a supremum and not a maximum.

0Mathematics Subject Classification. 41A30,46E30