Doubling property and vanishing order of Steklov eigenfunctions

The paper is concerned with the doubling estimates and vanishing order of the Steklov eigenfunction on the boundary of a smooth boundary domain $\mathbb R^n$. The eigenfunction is given by a Dirichlet-to-Neumann map. We improve the doubling property shown by Lin and Bellova \cite{BL}. Furthermore, we show that the vanishing order of Steklov eigenfunction is everywhere less than $C\lambda$ where $\lambda$ is the Steklov eigenvalue and $C$ depends only on $\Omega$.