ON SHARPNESS, APPLICATIONS AND GENERALIZATIONS OF SOME CARLEMAN TYPE INEQUALITIES

Carleman's inequality for Hilbert-Schmidt operators and its generaliza tions for Schatten-von Neumann operator ideals (see [7]) are shown to be sharp in a certain sense. Explicit classes of extremizing operators are found on which the generalized Carleman inequalities turn to asym ptotic equalities. Applications are made to a priori estimation of the solutions of Fredholm and Volterra first- and second kind integral eq uations and to perturbation and error analysis. Some further generaliz ations are considered which extend the applications to singular integr al equations, pseudodifferential equations and analytic functions of o perator argument.