Membership of the invariant density in W^{1,2}

Determine whether the invariant density h_\varepsilon constructed for the self-consistent transfer operator belongs to W^{1,2}([0,1]).

Background

The main theorem constructs, for sufficiently small coupling strength, a unique fixed-point density h_\varepsilon in a bounded subset \mathcal D of W^{1,1}([0,1]) and proves exponential convergence to this fixed point. The proof uses a smaller invariant set \bar B contained in a space with a stronger W^{1,2}-type regularity bound for the iterates used in the contraction argument.

The authors explicitly note that the fixed point is known to lie in \mathcal D and that \bar B\subset\mathcal D, but they do not establish that h_\varepsilon itself belongs to the stronger W^{1,2} space. Thus, determining this regularity is left unresolved by the paper.