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We show that the boundary values of holomorphic functions in the unit ball of Cn (n > 1) are characterized by the vanishing of their contour integrals over certain closed curves. The classical Morera theorem states that if a continuous function f in some region Q in the complex plane C1 has zero contour integral around any triangle in Q then f is holomorphic in Q. Recently there have appeared some interesting variations of this theorem. The theme of these variations involves changing the class of curves used to test for holomorphy. M. L. Agranovskii and R. E. Val'skii [2] have shown that if Q is the unit ball then it is enough for f to have vanishing nonnegative moments with respect to G-translates of a fixed smooth curve -Y, where G is the Moebius group of biholomorphisms of Q. In detail, the required conditions are Zj f(z) dz = o (m = O,1, 2,1...). Here g(-y) denotes the image of the curve -y by the biholomorphism g. In [1] Agranovskii shows that if f satisfies certain growth conditions, then it is enough to test the zero-moments alone, i.e. it is enough for f to have zero contour integrals over G-translates of -y. Agranovskii also gives a similar result for the case where Q is a bounded symmetric domain in Cn. In C. Berenstein's paper [3] we find, in the case where Q is the unit ball in Cn and -y is not real analytic, that the same result holds for arbitrary f (i.e. without assuming growth conditions). There are simple examples which show that it is not enough to test for holomorphy using only the family of circles of a fixed radius without assuming a priori growth conditions. On the other hand, L. Zalcman [7] has shown that a 'two-radius' family of circles detects holomorphy in certain regions. J. Globevnik [4] has formulated conditions for holomorphy using rotation invariant families of curves. In this note we will examine boundary values of holomorphic functions in the unit ball in Cn (n > 1) and show that they too can be characterized by the vanishing of their contour integrals along curves. We will use two preliminary results. Received by the editors March 7, 1986 and, in revised form, September 29, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 43A85, 32A10, 30E99.

A boundary analogue of Morera’s theorem in the unit ball of 𝐶ⁿ