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First we establish a weighted Reilly formula for differential forms on asmooth compact oriented Riemannian manifold with boundary. Then we give twoapplications of this formula when the manifold satisfies certain geometricconditions. One is a sharp lower bound for the first positive eigenvalue of theSteklov eigenvalue problem on differential forms investigated by Belishev andSharafutdinov (2008) and Karpukhin (2019). The other one is a comparison resultbetween the spectrum of this Steklov eigenvalue problem and the spectrum of theHodge Laplacian on the boundary of the manifold. Besides, at the end we discussan open problem for differential forms analogous to Escobar's conjecture (1999)for functions.

A weighted Reilly formula for differential forms and sharp Steklov eigenvalue estimates