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where the integrals are taken in the sense of the extended principal value of Definition 1.1. This problem, suggested by W. Feller, arises in the theory of stochastic processes. However, it is interesting from a purely analytic point of view, and this paper will be concerned only with that aspect. A probabilistic interpretation of our result will be found in a forthcoming paper by Feller. For a precise definition of the operators ß and ß* see §1. The equation

The boundary value problems and semigroups associated with certain integro-differential operators