  In study of eigenvalue problems, a classical problem is the Stekloff
eigenvalue problem. There are many estimates of the first non- zero Stekloff
eigenvalue, including a sharp estimate on surfaces, obtained by Escobar in "The
geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal. 150
(1997)". In this paper we are interested to study this problem in weighted
context. Various estimates are obtained, including a sharp estimate on
surfaces, similar to that Escobar obtained.
