Morera theorems via microlocal analysis (Q1923006)

More general Morera theorems state that, if \(y(c)= \int_c fdz=0\) for certain subclasses of closed curves in a region, then \(f\) is holomorphic in that region. The present paper shows Morera theorems for circles passing through the origin, for circles of arbitrary radius and arbitrary center, and for translates of a fixed closed convex curve. The theorems have somewhat different character from known Morera theorems: Here \(y(c)\) is constant for all relevant curves \(c\), not necessarily vanishing. Further, \(f\) is assumed to be homomorphic on a small set, and this yields that \(f\) is holomorphic on a much larger set. (And a strong holomorphy assumption is necessary, as counterexamples (with a spendour as in the classical theory) show.) Finally, the theorems are valid for distributions. The proofs use the microlocal analysis of associated Radon transforms, they are very close to microlocal proofs of support theorems for Radon transform, see e.g. papers due to the first author [J. Math. Anal. Appl. 165, 284-287 (1992; Zbl 0755.44003)] and to I. Boman and the second author [Duke Math. J. 55, 943-948 (1987; Zbl 0645.44001)].