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We show that a CR function of class C, 0≤k<∞, on a tube submanifold of C holomorphically extends to the convex hull of the submanifold. The extension and all its derivatives through order k are shown to have nontangential pointwise boundary values on the original tube submanifold. The C-norm of the extension is shown to be no bigger than the C-norm of the original CR function. 1. Definitions and main results. Recently, Boivin and Dwilewicz [BD] have generalized Bochner’s Tube Theorem by showing that continuous CR functions on a tube submanifold of C holomorphically extend to its convex hull. In this paper, we generalize this result to CR functions of class C, for k a nonnegative integer. The technique we use (analytic discs) is different than the one used in [BD], and in the case of bounded CR functions provides an estimate on the norm of the extension. For any submanifold S in C = R, let C(S) be the space of complexvalued functions on S with continuous derivatives through order k. We will identify an element in C(S) with the restriction of an element of C(R) to S. For f ∈ C(S), let ‖f‖k(S) = sup t∈S ∑

The holomorphic extension of $C^k$ CR functions on tube submanifolds