Explicit variational forms for the inverses of integral logarithmic operators over an interval

We introduce explicit and exact variational formulations for the weakly singular and hypersingular operators over an open interval as well as for their corresponding inverses. Contrary to the case of a closed curve, these operators no longer map fractional Sobolev spaces in a dual fashion but degenerate into different subspaces depending on their extensibility by zero. We show that an average and jump decomposition leads to precise coercivity results and characterize the mismatch occurring between associated functional spaces. Through this setting, we naturally define Calderóntype identities with their potential use as preconditioners. Moreover, we provide an interesting relation between the logarithmic operators and one-dimensional Laplace Dirichlet and Neumann problems. This work is a detailed and extended version of the article "Variational Forms for the Inverses of Integral Logarithmic Operators over an Interval" by Jerez-Hanckes and Nédélec [C.R. Acad. Sci. Paris Ser. I, 349 (2011), pp. 547-552].