Morera theorem for holomorphic Hp spaces in the Heisenberg group.

The classical theorem of Morera characterizes holomorphic functions of one complex variable in terms of integral conditions, e.g., a continuous function / in the unit disk D is holomorphic if and only if \f(z)dz = 0 for all Lipschitz Jordan curves γ <= D (or all y squares with sides parallel to axes in D or allcircles in D). In 1972 Zalcman [ZI] (see also [Z2]) showed that to characterize entire functions among /e C(i?) it is enough to integrate over all circles of two fixed well chosen radii, or over squares of a fixed size [ZI] (but rotated in every possible direction). These results were put in perspective in [BZ 1], [BZ 2] by showing that they were really questions in Symmetrie spaces, so that, for instance, if / is a continuous function in the unit ball B of <C, then one can see whether / is holomorphic by verifying that J /ω = 0 for all (n, n — 1) differential forms ω with constant coefficients, <r( ) all Moebius transformations σ, and a single relatively compact domain Ω, provided that B \ is connected, dQ is at least Lipschitz but not everywhere real analytic [B]. For n — l, no special condition on dQ is required if we assume /e L(B) [AI], [A2]. A surprising recent result [BG1], [BG2] indicates that one may also consider Euclidean motions,