Quantitative versions of unique continuation are proved for finite sums of eigenfunctions of the Laplacian on compact Riemannian manifolds. The results include a lower bound for the order of vanishing, a growth estimate for the supremum on compact balls, and a gradient bound. For real analytic metrics, an upper bound for the Hausdorff measure of the zero set is derived.

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a 2 A'/2 at isolated singular points. We do not know if the radius a 3 A-1/4 of Theorem 1.2 can be replaced by any a 3 A-e , where e < !, to give an upper bound 7!"(NnB(p, a 3 A-e)) < a 4 A'/2-e. However, both of our theorems follow from certain basic Carleman inequalities. Since the first theorem is sharp, there appears to be a serious obstruction to improving the second theorem via the current techniques. Covering M by order A'/2 balls of radius a 3 r'/4, we immediately deduce Corollary 1.3. 7!" N < asA3/4. Yau has conjectured that 7!" N < a 6 A'/2. We proved this conjecture [5] when M is a real analytic manifold with real analytic metric. For differentiable manifolds, Hardt and Simon [8] established the estimate 7!" N < a 7 exp(agJX log A). In particular, Corollary 1.3 gives an upper bound of polynomial growth. Some time ago [2], Bruning derived the complementary lower bound 7!" N > a 9 A'/2 in Yau's conjecture. Recently, Nadirashvili [10] announced the upper bound , 7!' N < alOAlog(1 + A). A brief sketch of the proof was provided. In the case when M is only differentiable, his method seems substantially different from ours. 2. CARLEMAN INEQUALITIES In his seminal work [3] on unique continuation for first-order differential equations in two independent variables, Carleman derived certain weighted integral inequalities. His method of proof relied upon integral representations via fundamental solutions. We develop related inequalities using a rather different approach, which involves repeated partial integration and estimates of commutators. These commutator functions may be interpreted as the Gaussian curvatures of conformally flat metrics. Consequently, our estimates are reminiscent of the standard Bochner formulas [7]. However, we eschew this differential geometry viewpoint in favor of a more elementary outlook. Consider the weighted Hilbert space L' (P~ , e- dx dy) of complex-valued functions. Here g is a bounded open subset of the complex plane, and ¢ is a smooth real-valued function. Our primary concern involves smooth compactly supported functions u E c;:(g) , a subset of L 2 (g, e- dxdy). One has the basic first-order differential operators au = au/a z = t.(a/ax-iO/ay)u and its companion 8u = au/az = t.(a/ax + ia/ay)u. In our weighted Hilbert space, the adjoint of 8 is 8*. A calculation verifies the formula 8* v =-e a (e- v) .

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