# Hölder a priori estimates for second order tangential operators on CR manifolds

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Hölder a Priori Estimates for Second Order Tangential Operators on CR Manifolds Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. II (2003), pp. 345-378 Ho¨lder a Priori Estimates for Second Order Tangential Operators on CR Manifolds ANNAMARIA MONTANARI Abstract. On a real hypersurface M in Cn+1 of class C2,α we consider a local CR structure by choosing n complex vector fields Wj in the complex tangent space. Their real and imaginary parts span a 2n-dimensional subspace of the real tangent space, which has dimension 2n + 1. If the Levi matrix of M is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations with Cα coefficients, which are not elliptic because they involve second-order differentiation only in the directions of the real and imaginary part of the tangential operators Wj . In particular, our result applies to a class of fully nonlinear PDE’s naturally arising in the study of domains of holomorphy in the theory of holomorphic functions of several complex variables. Mathematics Subject Classification (2000): 35J70, 35H20 (primary), 32W50, 22E30 (secondary). 1. – Introduction In this paper we prove a priori estimates for solutions of the linear subel- liptic equation Hv = f in R2n+1, where (1) H = 2n∑ m, j=1 hmj Zm Zj − λ∂t , the coefficients λ, hmj are α-Ho¨lder continuous and such that hmj = hjm, m, j = 1, . . . , 2n, and (2) 2n∑ m, j=1 hmjηmηj ≥ M 2n∑ j=1 η2j , ∀η = (η1, . . . , η2n) ∈ R2n Investigation supported by University of Bologna. Funds for selected research topics. Pervenuto alla Redazione il 18 marzo 2002 ed in forma definitiva il 7 febbraio 2003. 346 ANNAMARIA MONTANARI for a suitable positive constant M . Here the first order differential operators Zj are (3) Z2l = ∂ ∂yl + ω2l ∂ ∂t , Z2l−1 = ∂ ∂xl + ω2l−1 ∂ ∂t , Z = (Z1, Z2, . . . , Z2n) , where (x1, y1,

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