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

Montanari, Annamaria

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 2, pp. 345-378.

On a real hypersurface $M$ in ${ℂ}^{n+1}$ of class $C^{2,\alpha }$ we consider a local CR structure by choosing $n$ complex vector fields $W_j$ 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 $C^{\alpha }$ 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 $W_j.$ 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.

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