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In our recent paper (2), the study of the kernel associated with a singular integral led us to another question, relating to the boundary behaviour of the sign of a harmonic function in a half-plane. In this paper, the possible existence of sign oscillations of the Poisson integral P (f) in the half-plane along rays is related to regularity properties of the boundary function f. This allows us to obtain a result of Fatou type for the sign of P (f), under a regularity assumption that we prove to be optimal.

Sur les changements de signe d'une fonction harmonique dans le demi-plan