escobar proved a sharp sobolev inequality for the embedding of $w^{1,2}(x^{n+1})$ into $l^{2n/(n-1)}(\partial x)$ by exploiting the conformal properties of the laplacian in x and the normal derivative along the boundary. more recently, an alternative proof was given by using a dirichlet-to-neumann operator along the boundary and its close relationship to the 1/2-power of the laplacian. in this talk, i describe a new relationship between the conformally covariant fractional powers of the laplacian due to graham--zworski and higher-order dirichlet-to-neumann operators in the interior, and use it to prove sharp sobolev inequalities for embeddings of $w^{k,2}$. other consequences of this relationship, such as a surprising maximum principle for the conformal 3/2-power of the laplacian, will also be discussed.