Equivalence of function-based and distribution-based analyticity in mixed characteristic

Establish that for any mixed-characteristic Banach pair (B,B^+) satisfying the paper’s hypotheses and any solid complex C in the derived category D(B[G]′), the two definitions of h-analytic vectors—one constructed via analytic functions C^{h-an}(G,B) and the other via analytic distributions D^{h-an}(G,B)′—are canonically isomorphic for all h without imposing nuclearity assumptions on C.

Background

The paper introduces analytic vectors for solid B[G]′-modules and complexes using distribution algebras D^{h-an}(G,B)′ and D_{h-an}(G,B)′. For nuclear objects, the authors identify these with a construction using rings of analytic functions C^{h-an}(G,B) (via binomial expansions), thereby matching the classical Q_p theory. In the characteristic 0 case (B,B^+)=(Q_p,Z_p), Rodrigues Jacinto–Rodríguez Camargo proved that the two definitions (functions vs. distributions) agree for general complexes without nuclearity hypotheses. The authors extend the framework to mixed characteristic but note that beyond the nuclear case, it is unknown whether the two approaches coincide in this broader setting.

Clarifying this equivalence would unify the analytic vector theory across function- and distribution-based constructions in mixed characteristic and remove additional assumptions currently needed for the comparison.