Estimates for the (Formula Presented) operator are used to derive estimates for the Neumann operator in weighted Hilbert spaces. The technique is similar to that used to prove regularity of solutions of elliptic partial differential equations. A priori estimates are first obtained for smooth compactly supported forms and these estimates are then extended by suitable approximation results. These estimates are applied to give new bounds for the reproducing kernels in the subspaces of entire functions. © 1976 American Mathematical Society.