H-log spaces of continuous functions, potentials, and elliptic boundary value problems

In these notes we study a family of Banach spaces, denoted $, D^{0,,\al}(\Ov),,$ $\text{al}\in {\mathbb{R}}^{+},$ and called H-log spaces. For $0<\text{la}\le 1,$ one has $ C^{0,,\la}(\Ov)\subset D^{0,,\al}(\Ov) \subset,C(\Ov),,$ with compact embedding. These spaces present the following “intermediate” regularity behavior. Solutions $u$ of second order linear elliptic boundary value problems, under “external forces” $,f\in, D^{0,,\al}(\Ov),$forsome$,\al>,1,,$satisfy$,\na^2,u\in, D^{0,,\al-,1}(\Ov),$\mathrm{.}Thisresultisoptimal,since$,\na^2,u\in, D^{0,,\beta}(\Ov),,$forsome$,\beta >,\al-1,,$ is false in general. We present a preliminary study on this subject.