Derivative estimates for eigenfunctions with accessible boundaries and application to diffustions

Abstract

We consider an eigenvalue problem on a bounded interval and establish some uniform bounds on the first derivatives of the L2-normalized eigenfunctions in terms of the eigenvalues. Such bounds are established for singular boundaries of regular type and most cases of exit type, according to Feller's boundary classification. Implications of such bounds for diffusion semigroups are discussed and an appli¬cation is made to a nronertv of conditional diffusions.