  We introduce new boundary integral operators which are the exact inverses of
the weakly singular and hypersingular operators for the Laplacian on flat
disks. Moreover, we provide explicit closed forms for them and prove the
continuity and ellipticity of their corresponding bilinear forms in the natural
Sobolev trace spaces. This permit us to derive new Calder\'on-type identities
that can provide the foundation for optimal operator preconditioning in
Galerkin boundary element methods.
