We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact Kähler manifolds. We characterize toric qpsh functions and give necessary and sufficient conditions for them to have finite (weighted) energy, both in terms of the associated convex function in R n , and through the integrability properties of its Legendre transform. We characterize Log-Lipschitz convex functions on the Delzant polytope, showing that they correspond to toric qpsh functions which satisfy a certain exponential integrability condition. In the particular case of dimension one, those Log-Lipschitz convex functions of the polytope correspond to Hölder continuous toric quasisubharmonic functions.
In this talk we will consider Poletsky-Stessin Hardy spaces of holomorphic functions, denoted by H p u (Ω) (Poletsky, Stessin; 2008) on hyperconvex domains Ω. These classes are associated with a continuous, negative, plurisubharmonic exhaustion function u and the Monge-Ampére measure (Demailly, 1985) generated from this exhaustion function. In the first part of this talk we will consider the Poletsky-Stessin Hardy spaces on domains in the complex plane bounded by an analytic Jordan curve. We will give the boundary value characterization, factorization and approximation results that are analogous to classical Hardy space case. Moreover we will show the differences between the Poletsky-Stessin Hardy spaces and the classical Hardy spaces as far as the composition operators on these spaces are concerned. In the second part we will consider the Poletsky-Stessin Hardy spaces on domains in C n for n > 1. We will give some results on boundary behavior of Poletsky-Stessin Hardy spaces on polydisc, complex ellipsoids and strongly convex domains in C n , n > 1.
We study Poletsky-Stessin Hardy spaces that are generated by continuous, subharmonic exhaustion functions on a domain ⊂ C, that is bounded by an analytic Jordan curve. Different from Poletsky and Stessin's work these exhaustion functions are not necessarily harmonic outside of a compact set but have finite Monge-Ampère mass. We have showed that functions belonging to Poletsky-Stessin Hardy spaces have a factorization analogous to classical Hardy spaces and the algebra A( ) is dense in these spaces as in the classical case; however, contrary to the classical Hardy spaces, composition operators with analytic symbols on these Poletsky-Stessin Hardy spaces need not always be bounded
We study Poletsky-Stessin Hardy spaces on complex ellipsoids in C n . Different from one variable case, classical Hardy spaces are strictly contained in Poletsky-Stessin Hardy spaces on complex ellipsoids so boundary values are not automatically obtained in this case. We have showed that functions belonging to Poletsky-Stessin Hardy spaces have boundary values and they can be approached through admissible approach regions in the complex ellipsoid case. Moreover, we have obtained that polynomials are dense in these spaces. We also considered the composition operators acting on Poletsky-Stessin Hardy spaces on complex ellipsoids and gave conditions for their boundedness and compactness.
In this study we extend the concepts of m-pluripotential theory to the Riemannian superspace formalism. Since in this setting positive supercurrents and tropical varieties are closely related, we try to understand the relative capacity notion with respect to the intersection of tropical hypersurfaces. Moreover, we generalize the classical quasicontinuity result of Cartan to m-subharmonic functions of Riemannian spaces and lastly we introduce the indicators of m-subharmonic functions and give a geometric characterization of their Newton numbers.
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