  The purpose of this paper is to establish a Lagrangian potential theory,
analogous to the classical pluripotential theory, and to define and study a
Lagrangian differential operator of Monge-Ampere type. This development is new
even in ${\bf C}^n$. However, it applies quite generally -- perhaps most
importantly to symplectic manifolds equipped with a Gromov metric.
  The Lagrange Monge-Ampere operator is an explicit polynomial on ${\rm
Sym}^2(TX)$ whose principle branch defines the space of Lag-harmonics.
Interestingly the operator depends only on the Laplacian and the SKEW-Hermitian
part of the Hessian. The Dirichlet problem for this operator is solved in both
the homogeneous and inhomogeneous cases. The homogeneous case is also solved
for each of the other branches.
  This paper also introduces and systematically studies the notions of
Lagrangian plurisubharmonic and harmonic functions, and Lagrangian convexity.
An analogue of the Levi Problem is proved. In ${\bf C}^n$ there is another
concept, Lag-plurihamonics, which relate in several ways to the harmonics on
any domain. Parallels of this Lagrangian potential theory with standard
(complex) pluripotential theory are constantly emphasized.
