Variational forms for the inverses of integral logarithmic operators over an interval. http://dx.doi.org/10.1016/j.crma.2011.01.016

Abstract

We present explicit and exact variational formulations for the weakly singular and hypersingular operators over an interval as well as for their corresponding inverses. By decomposing the solutions in symmetric and antisymmetric parts, we precisely characterize the associated Sobolev spaces. Moreover, we are able to define novel Calderón-type identities in each case.