Downloaded from https://www.cambridge.org/core. 10 Dec 2020 at 18:44:05, subject to the Cambridge Core terms of use. 514 BARNET M. WEINSTOCK For continuous functions on B we introduce the following terminology: Definition 1.1. A continuous function / on B is a weak solution of the tangential Cauchy-Riemann equations if (1.5) f /3" = 0 for all C°° forms 77 of bidegree (n, n -2) in a neighborhood of £. Every smooth solution of the tangential Cauchy-Riemann equations on B satisfies (1.5), and

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... smooth function on B which satisfies (1.5) is a solution of the tangential Cauchy-Riemann equations. These are simple consequences of Stokes' theorem. The second result of this paper (Theorem 4.2 below) is that if M admits a Kâhler metric and (1.3) and (1.4) are satisfied then every weak solution of the tangential Cauchy-Riemann equations on B has a holomorphic extension to D. We recall that (1.4) is always satisfied if M is a Stein manifold by the Serre duality theorem [20], and that every Stein manifold admits a Kâhler metric [10]. Thus, if M is a Stein manifold, every weak solution of the tangential Cauchy-Riemann equations on B has a holomorphic extension to D provided only that D has no compact complementary components. This was proved for the case M = C n in [23]. The techniques used in this paper are potential-theoretic. The theory of the Laplace operator on Riemannian and Kâhler manifolds and the associated boundary-value problems, as developed by Bidal, de Rham, Kodaira, Spencer, Duff, Garabedian and others [3; 6; 7; 9; 18; 21] is used to extend to domains on Kâhler manifolds the arguments used in [22] to study holomorphic extension from the boundary for domains in Euclidean space. The specific results which are required are collected in § 2 and § 3 below. The extendability of smooth (C l ) solutions of the tangential Cauchy-Riemann equations for domains in C n was first shown by Bochner [4] . (See also Martinelli [15].) The extendability of functions satisfying (1.1) was proved by Fichera [8] for domains with connected boundary in C n under the additional hypothesis that / be the boundary value of a function with finite Dirichlet integral. Royden [19] studied the extendability of integrable functions satisfying (1.1) for the case n = 1, i.e., for finite Riemann surfaces. Kohn and Rossi [13] gave conditions for the extendability of smooth functions for more general complex manifolds. The extendability of weak solutions of the tangential Cauchy-Riemann equations has also been treated by Harvey and Lawson [11] by other methods, as part of their general study of boundaries of complex manifolds.