The Jensen envelope is plurisubharmonic on all manifolds

Abstract

The Jensen envelope $J\phi$ of an upper semicontinuous function $\phi$ on a complex manifold X is defined at $x\in X$ as the infimum of $\mu(\phi)$ over all Jensen measures $\mu$ centred at x. The Poisson envelope $P\phi$ is defined by using only the boundary measures of analytic discs centred at x. One of the main open problems in the theory of disc functionals is whether the Poisson envelope is plurisubharmonic on an arbitrary manifold. This is equivalent to the two envelopes being equal, so plurisubharmonicity of $J\phi$ is a necessary condition for $P\phi$ to be plurisubharmonic. We prove that the Jensen envelope is plurisubharmonic, with no assumptions on the manifold X. Hence $J\phi$ is the largest plurisubharmonic function smaller than $\phi$. We also show that the Poisson envelope is plurisubharmonic if and only if boundary measures of analytic discs are dense among Jensen measures.