Strong Hypercontractivity and Relative Subharmonicity

A Hermitian metric, g, on a complex manifold, M, together with a smooth probability measure, μ, on M determine minimal and maximal Dirichlet forms, QD and Qmax, given by Q(f)=∫Mg(gradf(z), gradf(z))dμ(z). QD is the form closure of Q on C∞c(M) and Qmax is the form closure of Q on C1b(M). The corresponding operators, AD and Amax, generate semigroups having standard hypercontractivity properties in the scale of Lp spaces, p>1, when the corresponding form, Q, satisfies a logarithmic Sobolev inequality. It was shown by the author (1999, Acta Math.182, 159–206) that the semigroup e−tAD has even stronger hypercontractivity properties when restricted to certain holomorphic subspaces of Lp. These results are extended here to Amax. When (M, g) is not complete it is necessary that the elliptic differential operator Amax degenerate on the boundary of M. A second proof of these strong hypercontractive inequalities for both AD and Amax is given, which depends on an extension of the submean value property of subharmonic functions. The Riemann surface for z1/n and the weighted Bergman spaces in the unit disc are given as examples.