Sharp estimates for\bar \partial on convex domains of finite typeon convex domains of finite type

Let Ω be a bounded convex domain in C _n , with smooth boundary of finite typem. The equation is solved in Ω with sharp estimates: iff has bounded coefficients, the coefficients of our solutionu are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging toL ^p(Ω),p≥1. We solve the -equation by means of integral operators whose kernels are not based on the choice of a “good” support function. Weighted kernels are used; in order to reflect the geometry ofbΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω.     

Boundary behavior of<Emphasis Type="Italic">μ</Emphasis>-homeomorphisms

We provide conditions on the complex dilatation of a homeomorphismf of the upper half plane ℍ into ℂ, which guarantee thatf(ℍ) is a proper subset of ℂ and, in case wheref(ℍ) is a Jordan domain, thatf has a homeomorphic extension onto .     

Asymptotic values of strongly normal functions

Letf be meromorphic in the open unit discD and strongly normal; that is,

Compact Hankel operators with conjugate analytic symbols

In 1986 S. Axler [3] proved that forf∈L _a ² the Hankel operator is compact if and only iff is in the little Bloch space {itB}{in0}. In this note we show that the same is true for , 1<p<∞. Moreover we prove that is ⋆-compact if and only if as |z|→1⁻.     

Quasiconformal extension of biholomorphic mappings in several complex variables

Letf(z, t) be a subordination chain fort ∈ [0, α], α>0, on the Euclidean unit ballB inC ⁿ. Assume thatf(z) =f(z, 0) is quasiconformal. In this paper, we give a sufficient condition forf to be extendible to a quasiconformal homeomorphism on a neighbourhood of . We also show that, under this condition,f can be extended to a quasiconformal homeomorphism of onto itself and give some applications. Partially supported by Grant-in-Aid for Scientific Research (C) no. 14540195 from Japan Society for the Promotion of Science, 2004.     

Closures of finitely generated ideals in Hardy spaces

LetH ^∞ be the algebra of bounded analytic functions in the unit diskD. LetI=I(f ₁,...,f _N) be the ideal generated byf ₁,...,f _N∈H ^∞ andJ=J(f ₁,...,f _N) the ideal of the functionsf∈H ^∞ for which there exists a constantC=C(f) such that |f(z)|≤C(|f ₁ (z)|+...;+|f _N (z)|),z∈D. It is clear that , but an example due to J. Bourgain shows thatJ is not, in general, in the norm closure ofI. Our first result asserts thatJ is included in the norm closure ofI ifI contains a Carleson-Newman Blaschke product, or equivalently, if there existss>0 such that

Some remarks on the identity between a variational integral and its relaxed functional

Summary Letf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] be measurable inx and convex inz. It is proved, by an example, that even iff verifies a condition as|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) with 1<p<q,a∈L _loc ^s (R ⁿ),s>1, the functional that isL ¹(Ω)-lower semicontinuous onW ^1,1(Ω), does not agree onW ^1,1(Ω) with its relaxed functional in the topologyL ¹(Ω) given by inf

---

Riassunto Siaf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) con 1<p<q,a∈L _loc ^s (R ⁿ),s>1, il funzionale , che èL ¹(Ω)-semicontinuo inferiormente suW ^1,1(Ω), non coincide suW ^1,1(Ω) con il suo funzionale rilassato nella topologiaL ¹(Ω) definito da inf

     

The algebraic closure of a function

The algebraic operations on a set A which admit (as a homomorphism) a particular mapf∈A ^A, admit all powers off and certain other maps as well. The algebraic closure off, , is the totality of all maps which admit the algebraic operations thatf does. The purpose of this paper is to characterize in terms off, both forA finite and forA infinite. This research was supported in part by N.R.C. Operating Grants A7213 and A8094 from the National Research Council of Canada. Presented by B. Jónsson     

Natural extensions of holomorphic motions

We consider an arbitrary real analytic family X_z, , over the closed unit disc , of real analytic plane Jordan curves X_z. Ifj _e ^iθ ,e ^iθ ∋ ∂D, is an arbitrary real-analytic family of orientation-reversing homeomorphisms of fixingX _e ^iθ pointwise, we show that there is a unique holomorphic motion of extending the given motion of Jordan curves and consistent with the given family of involutions. If these generalized reflections are defined using the barycentric extension construction of Douady-Earle-Nag, then the resulting extension method for holomorphic motions of X is natural, that is Moebius-invariant and continuous with respect to variation of the given motion of X₀.     

Quasiconformal and harmonic mappings between Jordan domains

Let Ω and Ω₁ be Jordan domains, let μ ∈ (0, 1], and let be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂Ω, ∂Ω₁ ∈ C ^1,μ , then f is Lipschitz; (b) if f is q.c., ∂Ω, ∂Ω₁ ∈ C ^1,μ and Ω₁ is convex, then f is bi-Lipschitz; and (c) if Ω is the unit disk, Ω₁ is convex, and ∂Ω₁ ∈ C ^1,μ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ^∞. These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365–372, 2002).      

Dilatation estimates for quasiconformal extensions

LetD andD′ be ring domains inB ⁿ , withS ⁿ⁻¹ as one boundary component, and let be a homeomorphism which isK-quasiconformal inD and withf(S ⁿ⁻¹)=S ⁿ⁻¹. According to a result of Gehringf÷S ⁿ⁻¹ admits an extension which is quasiconformal inB ⁿ . We find here an upper bound for the dilatation ofg in terms ofn, K, and modD. This work was started during a visit to Université de Paris, financed by a cultural exchange program between France and Finland.     

Second derivatives of circle packings and conformal mappings

William Thurston conjectured that the Riemann mapping functionf from a simply connected region Ω onto the unit disk can be approximated as follows. Almost fill Ω with circles of radius ɛ packed in the regular hexagonal pattern. There is a combinatorially isomorphic packing of circles in . The correspondencef _ɛ of ɛ-circles in Ω with circles of varying radii in should converge tof after suitable normalization. This was proved in [RS], and in [H] an estimate was obtained which led to an approximation of |f′| in terms off _ɛ ; namely, |f′| is the limit of the ratio of the radii of a target circle off _ɛ to its source circle. In the present paper we show how to approximatef′ andf″ in terms off _ɛ . Explicit rates for the convergence tof, f′, andf″ are obtained. In the special case of convergence to |f′|, the estimate in this paper improves the previously known estimate.     

Multigrid methods for the computation of singular solutions and stress intensity factors II: Crack singularities

We consider the Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks. Multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed. The convergence rate for the stress intensity factors is whenfεL ²(Ω) and whenfεH ¹(Ω). The convergence rate in the energy norm is in the first case and in the second case. The costs of these multigrid methods are proportional to the number of elements in the triangulation. The general case wherefεH ^m (Ω) is also discussed. The work of the first author was partially supported by NSF under grant DMS-96-00133     

Central limit theorem for integrated square error of kernel estimators of spherical density

LetX ₁,…,X _n be iid observations of a random variableX with probability density functionf(x) on the q-dimensional unit sphere Ω_q in R^q+1,q ⩾ 1. Let be a kernel estimator off(x). In this paper we establish a central limit theorem for integrated square error off _n under some mild conditions.     

On a conditioned Brownian motion and a maximum principle on the disk

Bounds for the 3G-expression∫ ^Ω G(x,z)G(z,y)d,z/G(x,y) play a fundamental role in potential theory. Here,G(x,y) is the Green function for the Laplace problem with zero dirichlet boundary conditions on Ω. The 3G-formula equals , the expected lifetime for a Brownian motion starting in that is killed on exiting ω and conditioned to converge to and to be stopped at . Although it was shown by probabilistic methods for bounded (simply connected) 2d-domains that ifx ε δΩ, then the supremum ofy \at E _x ^y is assumed for somey at the boundary, the analogous question remained open forx in the interior. Here we are able to give an answer in the case thatB ⊂ ℝ is the unit disk. The dependence of this quantity on the positions ofx andy is investigated, and it is shown that indeed E _x ^y (\gt_\om) is maximized on by opposite boundary points. The result also gives an answer to a number of questions related to the best constant for the positivity-preserving property of some elliptic systems. In particular, it confirms a, relationE _x ^y (\gt_\om) with a ‘sum of inverse eigenvalues’ that was conjectured recently by Kawohl and Sweers.     

Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Let be a separable Hilbert space, an open convex subset, and f: a smooth map. Let Ω be an open convex set in with , where denotes the closure of Ω in . We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ , where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L ²-space under the flow of a class of kinetic transport equations so called BGK model.      

Boundary behavior of the pluricomplex Green function

Let Ω be a bounded domain inC ⁿ . This paper deals with the study of the behavior of the pluricomplex Green functiong _Ω(z, w) when the polew tends to a boundary pointw ₀ of Ω. We find conditions on Ω which ensure that lim _w→wo g _Ω (z, w)=0, uniformly with respect toz on compact subsets of . Our main result is Theorem 5; it gives a sufficient condition for the above property to hold, formulated in terms of the existence of a plurisubharmonic peak function for Ω atw ₀ which satisfies a certain growth condition.     

Boundary functions on a bounded balanced domain

We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.      

Failure of Plais-Smale condition and blow-up analysis for the critical exponent problem inR 2

Let Ω be a bounded smooth domain inR ². Letf:R→R be a smooth non-linearity behaving like exp{s ²} ass→∞. LetF denote the primitive off. Consider the functionalJ:H ₀ ¹ (Ω)→R given by

Problem with Critical Sobolev Exponent and with Weight

The authors consider the problem: -div(p▽u) = uq-1 λu, u > 0 inΩ, u = 0 on (?)Ω, whereΩis a bounded domain in Rn, n≥3, p :Ω→R is a given positive weight such that p∈H1 (Ω)∩C(Ω),λis a real constant and q = 2n/n-2, and study the effect of the behavior of p near its minima and the impact of the geometry of domain on the existence of solutions for the above problem.