  First we establish a weighted Reilly formula for differential forms on a
smooth compact oriented Riemannian manifold with boundary. Then we give two
applications of this formula when the manifold satisfies certain geometric
conditions. One is a sharp lower bound for the first positive eigenvalue of the
Steklov eigenvalue problem on differential forms investigated by Belishev and
Sharafutdinov (2008) and Karpukhin (2019). The other one is a comparison result
between the spectrum of this Steklov eigenvalue problem and the spectrum of the
Hodge Laplacian on the boundary of the manifold. Besides, at the end we discuss
an open problem for differential forms analogous to Escobar's conjecture (1999)
for functions.
