Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains
Plurisubharmonic extension in non-degenerate analytic polyhedra.
Plurisubharmonic functions on compact sets
Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in ℂⁿ. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.
Plurisubharmonic functions with logarithmic singularities
To a plurisubharmonic function on with logarithmic growth at infinity, we may associate the Robin functiondefined on , the hyperplane at infinity. We study the classes , and (respectively) of plurisubharmonic functions which have the form and (respectively) for which the function is not identically . We obtain an integral formula which connects the Monge-Ampère measure on the space with the Robin function on . As an application we obtain a criterion on the convergence of the Monge-Ampère...
Plurisubharmonic saddles
A certain linear growth of the pluricomplex Green function of a bounded convex domain of at a given boundary point is related to the existence of a certain plurisubharmonic function called a “plurisubharmonic saddle”. In view of classical results on the existence of angular derivatives of conformal mappings, for the case of a single complex variable, this allows us to deduce a criterion for the existence of subharmonic saddles.
Poisson kernel and Green function of balls for complex hyperbolic Brownian motion
The aim of this paper is to give a description of the Poisson kernel and the Green function of balls in the complex hyperbolic space. The description is in terms of the hypergeometric function and unitary spherical harmonics in ℂⁿ.
Poisson kernel characterization of Reifenberg flat chord arc domains
Poisson kernels and pluriharmonic -functions on homogeneous Siegel domains.
Polar sets for supersolutions of degenerate elliptic equations.
Polynomial diffeomorphisms of C2: currents, equilibrium measure and hyperbolicity.
Polynomial growth harmonic functions on complete Riemannian manifolds.
In this paper, we give a sharp estimate on the dimension of the space of polynomial growth harmonic functions with fixed degree on a complete Riemannian manifold, under various assumptions.
Positive Harmonic Functions on Nontangentially Accessible Domains.
Positive Toeplitz operators between the pluriharmonic Bergman spaces
We study Toeplitz operators between the pluriharmonic Bergman spaces for positive symbols on the ball. We give characterizations of bounded and compact Toeplitz operators taking a pluriharmonic Bergman space into another for in terms of certain Carleson and vanishing Carleson measures.
Potential spaces on fractals
We introduce potential spaces on fractal metric spaces, investigate their embedding theorems, and derive various Besov spaces. Our starting point is that there exists a local, stochastically complete heat kernel satisfying a two-sided estimate on the fractal considered.
Potential theory and Lefschetz theorems for arithmetic surfaces
Potential theory for quasilinear elliptic equations.
Potential Theory for Schrödinger operators on finite networks.
We aim here at analyzing the fundamental properties of positive semidefinite Schrödinger operators on networks. We show that such operators correspond to perturbations of the combinatorial Laplacian through 0-order terms that can be totally negative on a proper subset of the network. In addition, we prove that these discrete operators have analogous properties to the ones of elliptic second order operators on Riemannian manifolds, namely the monotonicity, the minimum principle, the variational treatment...
Potential theory of quasiminimizers.
Potentials in pluripotential theory
Potentials of a Markov process are expected suprema
Expected suprema of a function f observed along the paths of a nice Markov process define an excessive function, and in fact a potential if f vanishes at the boundary. Conversely, we show under mild regularity conditions that any potential admits a representation in terms of expected suprema. Moreover, we identify the maximal and the minimal representing function in terms of probabilistic potential theory. Our results are motivated by the work of El Karoui and Meziou (2006) on the max-plus decomposition...