  We prove sharp $L^p$ estimates for the Steklov eigenfunctions on compact
manifolds with boundary in terms of their $L^2$ norms on the boundary. We prove
it by establishing $L^p$ bounds for the harmonic extension operators as well as
the spectral projection operators on the boundary. Moreover, we derive lower
bounds on the size of nodal sets for a variation of the Steklov spectral
problem. We consider a generalized version of the Steklov problem by adding a
non-smooth potential on the boundary but some of our results are new even
without potential.
