Urban Cegrell

Convergence in capacity

In this note we study the convergence of sequences of Monge-Ampere measures {(dd(c)u(s))(n)}, where {u(s)} is a given sequence of plurisubharmonic functions, converging in capacity.

A generalization of the corona theorem in the unit disc

The purpose of this note is to prove a variant of the corona theorem in the unit disc without using the notation of Carleson measure or any particular duality or factorization theorem for analytic functions. The corona theorem was first proved by Carleson [1]. For more background and further references, see Gamelin 1-2] and Garnett I-3, 4]. Notations. D is the unit disc in ~ ; T its boundary; H(D) the analytic functions on D; He(D)={ f sH(D) , sup ~ l f f d a < + ~ } where dcr is the normalized O < r < l Lebesgue measure on T. H ~ (D) = H(O) c~ L ~ (O).

Subextension of plurisubharmonic functions with bounded Monge-Ampere mass.

Abstract Let Ω⋐ C n be a hyperconvex domain. Denote by E 0 (Ω) the class of negative plurisubharmonic functions ϕ on Ω with boundary values 0 and finite Monge–Ampere mass on Ω. Then denote by F (Ω) the class of negative plurisubharmonic functions ϕ on Ω for which there exists a decreasing sequence (ϕ)j of plurisubharmonic functions in E 0 (Ω) converging to ϕ such that sup j ∫ Ω (dd c ϕ j ) n +∞. It is known that the complex Monge–Ampere operator is well defined on the class F (Ω) and that for a function ϕ∈ F (Ω) the associated positive Borel measure is of bounded mass on Ω. A function from the class F (Ω) is called a plurisubharmonic function with bounded Monge–Ampere mass on Ω. We prove that if Ω and Ω are hyperconvex domains with Ω⋐ Ω ⋐ C n and ϕ∈ F (Ω), there exists a plurisubharmonic function ϕ ∈ F ( Ω ) such that ϕ ⩽ϕ on Ω and ∫ Ω (dd c ϕ ) n ⩽∫ Ω (dd c ϕ) n . Such a function is called a subextension of ϕ to Ω . From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampere masses on Ω. To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

The Mathematical Theory of Ito Diffusions on Hypersurfaces, with Applications to NMR Relaxation Problems

A mathematical framework for translational Brownian motion on hypersurfaces is presented, using an imbedding of the surface and Ito diffusions in the ambient space. This includes a survey of Ito calculus and differential geometry. Computational methods for time correlation functions relevant to spin relaxation studies on curved interfaces are given, and explicit calculations of time correlation functions and order parameters for a “Rippled” surface are presented.

A counterexample to the “fine” problem in pluripotential theory

In this paper we prove that pointwise values of the non-regularized pluriharmonic measure are not capacities. This answers the question raised by E. Bedford and U. Cegrell.

Construction of Equilibrium Magnetic Vector Potentials

In [9] we introduced the notion of equilibrium surface current JdSx on the closed Cω smooth surfaces in R3 as a generalization of electric solenoid, and proved their existence. We here show an algorithm for JdSx starting from a given surface current J dSx on Σ.

Representing Measures in the Spectrum of H∞(Ω)

Let Ω be a domain in ℂ n , 0 ∈ Ω and denote by H(Ω) the analytic functions on ΩH∞ (Ω) = H(Ω) ⋂ L∞(Ω) and A(Ω) = H(Ω) ⋂ C(Ω) We denote by M the spectrum (= the multiplicative linear functionals) of H∞(Ω).

Capacities on the Boundary

Let U be an open, bounded and connected subset of ₵n containing zero. Then H(U) (H∞(U)) is the class of (bounded) analytic functions on U and A(U) consists of the functions in H(U) that extends continuously to Ū. If μ is a positive measure on ∂U we define Hp(μ,∂U) (1≤p<+∞) to be the closure of A(U) in Lp(μ,∂U).

Subharmonic Functions in IRn

Let B be the unit ball in ℝn, let F be the restriction to B of all positive superharmonic functions on RB, where R is a fixed number >1. If we take δ to be the Lebesgue measure on B, it is well known that U=B and δ satisfies all the assumptions made in Section III; the “coarse” problem has a positive solution. But much more can be said: the fine problem has a positive solution.

Plurisubharmonic Functions in ℂn — The Monge-Ampère Capacity

Let B be the unit ball in ₵n and let F be the restriction to B of all positive plurisuperharmonic functions on RB, where R is a fixed number >1. We take δ to be the Lebesgue measure on B and form c as in Section III:b. It is then true that F and δ meet all the requirements in Section III:b and we are going to see that F has property 1) and c has property 5); thus 2) and 3) hold true by Propositions III:3 and III:1.