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As an application of the residue functions corresponding to the lc-measuresdeveloped by the authors, the proof of the injectivity theorem on compactK\"ahler manifolds for plt pairs by Matsumura is improved in this article toallow multiplier ideal sheaves of plurisubharmonic functions with neat analyticsingularities in the coefficients of the relevant cohomology groups. With theuse of a refined version of analytic adjoint ideal sheaves, a plan towards asolution to the generalised version of Fujino's conjecture (i.e. an injectivitytheorem on compact K\"ahler manifolds for lc pairs with multiplier idealsheaves) is laid down and, in addition to the result for plt pairs, a proof forlc pairs in dimension 2, which is also an improvement of Matsumura's result, isgiven.

On an injectivity theorem for log-canonical pairs with analytic adjoint ideal sheaves