**GIUSEPPE TOMASSINI**

*
Levi type extremal operators
*

**Pages** 101-116

**Received:** 9 September 1999

**Mathematics Subject Classification (2000):** 32F25

Supported by the Project M.U.R.S.T. "Geometric Properties of Real and Complex Manifolds" and by G.N.S.G.A. of C.N.R.

**Abstract**
We introduce maximal and minimal operators L_{1}, L_{2} (in
the sense of Pucci) for the family F_{W} of the "Levi type
operators"
L_{ a, b} on a domain W of C´R. We
state
some
simple property
for (weak, viscosity) solutions of L_{1}, L_{2}. In
particular we prove a
special form of the maximum principle. As a consequence we obtain that
solutions
of L_{1}, L_{2} satisfy a "weak Hartogs property". We are
also dealing with
the Dirichlet problem for L_{1}, L_{2}. After shown that
solutions of that
provide barriers for the Levi operator we prove that this problem
translates into a
Dirichlet problem for a Bellman equation.

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