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This article aims to formulate the main technical input for the constructionof a solution functor in a still hypothetical $p$-adic analytic Riemann-Hilbertcorrespondence. Our approach relies on a novel period sheaf$\mathbb{B}_{\text{la}}^{+}$, which is a certain ind-Banach completion of$\mathbb{B}_{\text{inf}}$ along the kernel of Fontaine's map $\theta$. Werelate the ind-continuous $\mathbb{B}_{\text{la}}^{+}$-linear endomorphisms ofa corresponding period structure sheaf to the sheaf of infinite orderdifferential operators D-cap introduced by Ardakov-Wadsley. Locally on thecotangent bundle, this yields a definition of a solution functor. The mainresult computes that significant information about a perfect complex can bereconstructed out of its solutions, which hints strongly towards areconstruction theorem for a large category of complexes of D-cap-modules.

On a solution functor for D-cap-modules via $p$-adic Hodge theory