Capacity associated to a positive closed current. (Capacité associée à un courant positif fermé.)
Cauchy-Poisson transform and polynomial inequalities
We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in is Hölder continuous then E admits a Szegö type inequality with weight function with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.
Characterization of global Phragmén-Lindelöf conditions for algebraic varieties by limit varieties only
For algebraic surfaces, several global Phragmén-Lindelöf conditions are characterized in terms of conditions on their limit varieties. This shows that the hyperbolicity conditions that appeared in earlier geometric characterizations are redundant. The result is applied to the problem of existence of a continuous linear right inverse for constant coefficient partial differential operators in three variables in Beurling classes of ultradifferentiable functions.
Characterization of nuclear Fréchet spaces in which every bounded set is polar
Commuting Toeplitz operators on the pluriharmonic Bergman space
We prove that two Toeplitz operators acting on the pluriharmonic Bergman space with radial symbol and pluriharmonic symbol respectively commute only in an obvious case.
Comparison of Two Capacities in Cn.
Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions [Book]
Continuité et uniforme continuité du spectre dans les algèbras de Banach
Continuous pluriharmonic boundary values
Let be a bounded hyperconvex domain in and set , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.
Convexity of sublevel sets of plurisubharmonic extremal functions
Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.
Counterexample to Regularity for the Complex Monge-Ampère Equation.