Characterization of Hölder-on-bounded-sets continuity for C_f: l^p(E)→bv_q(E) when α≤1<p/q

Develop necessary and sufficient conditions on the generator f: E→E under which the composition operator C_f: l^p(E)→bv_q(E) is Hölder continuous on bounded sets with exponent α in the regime α≤1<p/q, providing an analogue of the results established for the cases α≥p/q and α≤p/q≤1.

Background

For mappings C_f: l^p(E)→bv_q(E), the paper establishes that when 1≤p≤q and α≥p/q, C_f is Hölder on bounded sets with exponent α if and only if f is Hölder with the same exponent on bounded sets. When 0<α≤p/q≤1, the authors give sufficient (but not necessary) conditions ensuring Hölder continuity on bounded sets.

The unresolved regime is α≤1<p/q (i.e., p>q). While isolated examples show nontrivial behavior can occur (e.g., with quadratic maps in commutative normed algebras), a general analogue of the characterization theorems is not known.