Upper bound of the counting function of Steklov eigenvalues

We study the counting function of Steklov eigenvalues on compact manifolds with boundary and obtain its upper bound involving the leading term of Weyl’s law. Our estimate can be viewed as a weakened version of Pólya’s Conjecture in the Steklov case on general manifolds. As a byproduct, we also obtain a description about the decay behavior of Steklov eigenfunctions near the boundary.