In the context of the study of the Ohsawa–Takegoshi L^2 extension theorem, a "qualitative" extension theorem in terms of adjoint ideal sheaves (i.e. extension result without L^2 estimates but with some form of local L^2 condition ensured) is proved on compact Kähler manifolds via harmonic theory and residue computations. The arguments are adapted from those in the study of the injectivity theorem by Chan–Choi–Matsumura. The result guarantees, in particular, the existence of extensions of line-bundle-valued holomorphic top forms over the union of any log-canonical centres of the same codimension to top forms over the ambient space under suitable positivity assumptions.
On two subsurfaces of a Riemann surface divided by a p-Weil-Petersson curve γ, we consider the spaces of harmonic functions whose p-Dirichlet integrals are finite in the complementary domains of γ. By requiring the coincidence of boundary values on γ, we establish a correspondence between the harmonic functions in these Banach spaces. We analyze the operator arising from this correspondence via the composition operator acting on the Banach space of p-Besov functions on the unit circle.
