Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We consider an abstract CR manifold equipped with a strictly positive definite Levi form, which defines a pseudo-Hermitian metric on the manifold. On such a manifold it is possible to define a natural sums of squares sub-Laplacian operator. We use Bochner identities to obtain Cordes–Friedrichs type inequalities on such manifolds where the L 2 norm of the Hessian tensor of a function is controlled by the L 2 norm of the sub-Laplacian of the function with a sharp constant for the inequality. By perturbation we proceed to develop a Cordes–Nirenberg type theory for non-divergence form equations on CR manifolds. Some applications are given to the regularity of p-Laplacians on CR manifolds.


Heisenberg Group Levi Form Paneitz Operator Cordes Condition Hessian Tensor 
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© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.Rutgers UniversityPiscatawayUSA
  2. 2.Department of MathematicsUniversity of PittsburghPittsburghUSA

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