Euclidean Unit Ball Research Articles

On some classes of bounded univalent mappings in several complex variables

Let Bn be the Euclidean unit ball in Cn. In this paper, we study several properties of strongly starlike mappings of order α (0 < α < 1) and bounded convex mappings on Bn. We prove that K-quasiregular strongly starlike mappings of order α on Bn have continuous and univalent extensions to \({\overline{B}^n}\). We show that bounded convex mappings on Bn are strongly starlike of some order α. We give a coefficient estimate for K-quasiregular strongly starlike mappings of order α on Bn. Finally, we give examples of strongly starlike mappings of order α and bounded convex mappings on Bn.

Homeomorphic extension of strongly spirallike mappings in ℂ n

In this paper we consider a proper subclass ŜAn of the full class of spirallike mappings on the Euclidean unit ball Bn in ℂn with respect to a given linear operator A. We use the method of subordination chains to obtain an upper growth result for ŜAn, and we obtain various examples of mappings in the same class of normalized biholomorphic mappings on the unit ball Bn in ℂn. We also prove that the class ŜAn is compact, while the full class of spirallike mappings with respect to a linear operator need not be compact in dimension n ⩾ 2, even when the operator is diagonal. This is one of the motivations for considering the class ŜAn. Finally we prove that if f is a quasiregular strongly spirallike mapping on Bn such that |[Df(z)]−1Af(z)| is uniformly bounded on Bn, then f extends to a homeomorphism of ℝ2n onto itself. In addition, if A+A* = 2aIn for some a > 0, this extension is also quasiconformal on ℝ2n.

Convex Subordination Chains in Several Complex Variables

Abstract.In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in ℂn. We also obtain a sufficient condition for injectivity off(z/‖z‖, ‖z‖) onBn\ ﹛0﹜, wheref(z,t) is a convex subordination chain over (0, 1).

Extension of convex mappings of order α of the unit disk in C to convex mappings of the unit ball in Cn

For a convex mapping f of order α ∈ [ 0 , 1 ) of the unit disk in the complex plane C , we consider conditions on the parameter β ∈ [ 0 , 1 / 2 ] so that the mapping F ( z ) = ( f ( z 1 ) , [ f ′ ( z 1 ) ] β z ˆ ) , z ˆ = ( z 2 , … , z n ) , is a convex mapping of the Euclidean unit ball in C n .

The Geometry of L0

AbstractSuppose that we have the unit Euclidean ball in ℝn and construct new bodies using three operations — linear transformations, closure in the radial metric, and multiplicative summation defined by We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in L0 that naturally extends the corresponding properties of Lp-spaces with p ≠ 0, and show that the procedure described above gives exactly the unit balls of subspaces of L0 in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in L0, and prove several facts confirming the place of L0 in the scale of Lp-spaces.

A class of Loewner chain preserving extension operators

We consider operators that extend locally univalent mappings of the unit disk Δ in C to locally biholomorphic mappings of the Euclidean unit ball B of C n . For such an operator Φ, we seek conditions under which e t Φ ( e − t f ( ⋅ , t ) ) , t ⩾ 0 , is a Loewner chain on B whenever f ( ⋅ , t ) , t ⩾ 0 , is a Loewner chain on Δ. We primarily study operators of the form [ Φ G , β ( f ) ] ( z ) = ( f ( z 1 ) + G ( [ f ′ ( z 1 ) ] β z ˆ ) , [ f ′ ( z 1 ) ] β z ˆ ) , z ˆ = ( z 2 , … , z n ) , where β ∈ [ 0 , 1 / 2 ] and G : C n − 1 → C is holomorphic, finding that, for Φ G , β to preserve Loewner chains, the maximum degree of terms appearing in the expansion of G is a function of β. Further applications involving Bloch mappings and radius of starlikeness are given, as are elementary results concerning extreme points and support points.

On $L_p$-affine surface areas

Let K be a convex body in R n with centroid at 0 and B be the Euclidean unit ball in R n centered at 0. We show that limt→0|K|-|K t | |B|-|B t | = O p (K) O p (B), where |K| respectively |B| denotes the volume of K respectively B, Op(K) respectively Op(B) is the p-affine surface area of K respectively B and {K t } t≥0 , {B t } t≥0 are general families of convex bodies constructed from K, B satisfying certain conditions. As a corollary we get results obtained in [23,25,26,31].

On CNC commuting contractive tuples

The characteristic function has been an important tool for studying completely non-unitary contractions on Hilbert spaces. In this note, we consider completely non-coisometric contractive tuples of commuting operators on a Hilbert space H. We show that the characteristic function, which is now an operator-valued analytic function on the open Euclidean unit ball in ℂn, is a complete unitary invariant for such a tuple. We prove that the characteristic function satisfies a natural transformation law under biholomorphic mappings of the unit ball. We also characterize all operator-valued analytic functions which arise as characteristic functions of pure commuting contractive tuples.

The minimality of the map $$\frac{x}{\|{x}\|}$$ for weighted energy

In this paper, we investigate the minimality of the map \(\frac{x}{\|{x}\|}\) from the Euclidean unit ball B n to its boundary 핊n−1 for weighted energy functionals of the type Ep,f = ∫ B n f(r)‖∇ u‖ p dx, where f is a non-negative function. We prove that in each of the two following cases: i) p = 1 and f is non-decreasing, ii) p is integer, p ≤ n−1 and f = rα with α ≥ 0, the map \(\frac{x}{\|{x}\|}\) minimizes Ep,f among the maps in W1,p(B n , 핊n−1) which coincide with \(\frac{x}{\|{x}\|}\) on ∂ B n . We also study the case where f(r) = rα with −n+2 < α < 0 and prove that \(\frac{x}{\|{x}\|}\) does not minimize Ep,f for α close to −n+2 and when n ≥ 6, for α close to 4−n.

Euclidean embeddings in spaces of finite volume ratio via random matrices

Let (ℝ n , || ⋅ ||) be the space ℝ N equipped with a norm || ⋅ || whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N × n matrix with N &gt; n , whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n -dimensional Euclidean space (ℝ N , | ⋅ |) onto its image in (ℝ N , || ⋅ ||) , i.e. there exist α , β &gt; 0 such that for all x ∈ ℝ N , . This solves a conjecture of Schechtman on random embeddings of ℓ 2 n into ℓ 1 N .

Characteristic function for polynomially contractive commuting tuples

In this note, we develop the theory of characteristic function as an invariant for n-tuples of operators. The operator tuple has a certain contractivity condition put on it. This condition and the class of domains in C n that we consider are intimately related. A typical example of such a domain is the open Euclidean unit ball. Given a polynomial P in C [ z 1 , z 2 , … , z n ] whose constant term is zero, all the coefficients are nonnegative and the coefficients of the linear terms are nonzero, one can naturally associate a Reinhardt domain with it, which we call the P-ball (Definition 1.1). Using the reproducing kernel Hilbert space H P ( C ) associated with this Reinhardt domain in C n , S. Pott constructed the dilation for a polynomially contractive commuting tuple (Definition 1.2) [S. Pott, Standard models under polynomial positivity conditions, J. Operator Theory 41 (1999) 365–389. MR 2000j:47019]. Given any polynomially contractive commuting tuple T we define its characteristic function θ T which is a multiplier. We construct a functional model using the characteristic function. Exploiting the model, we show that the characteristic function is a complete unitary invariant when the tuple is pure. The characteristic function gives newer and simpler proofs of a couple of known results: one of them is the invariance of the curvature invariant and the other is a Beurling theorem for the canonical operator tuple on H P ( C ) . It is natural to study the boundary behaviour of θ T in the case when the domain is the Euclidean unit ball. We do that and here essential differences with the single operator situation are brought out.

A Modification of the Roper-Suffridge Extension Operator

The Roper-Suffridge extension operator extends a locally univalent mapping defined on the unit disk of ℂ to a locally biholomorphic mapping defined on the Euclidean unit ball of ℂn. Furthermore, the extension of a one variable mapping that is either convex or starlike has the analogous property in several variables. Motivated by recent results concerning the extreme points of the family K n of normalized convex mappings of the Euclidean ball in ℂn, we introduce a new extension operator that, under precise conditions, takes the extreme points of K 1 to extreme points of K n. In general, we examine the conditions under which this new extension operator will take a convex or starlike mapping of the unit disk to a mapping of the same type defined on the unit ball.

Parametric representation and extension operators for biholomorphic mappings on some Reinhardt domains§

In this article we obtain a generalization of the Pfaltzgraff–Suffridge extension operator, denoted by Φ n,p , on some Reinhardt domains in C n . We prove that if f∈ S 0(Bn ) and p≥ 2n/(n+1) then where . In particular, if f is starlike on the Euclidean unit ball B n then is starlike on Ω n,p . We also give examples of starlike mappings on Ω n,p . Moreover, we study certain convexity properties associated with the operator Φ n,p . Further, we prove that the Roper–Suffridge extension operator Ψn preserves starlikeness of order 1/2. In the last section we obtain certain subordination results associated with the operator Φ n,p .

Roper–Suffridge extension operator and the lower bound for the distortion

Liczberski–Starkov gave a sharp lower bound for ‖ D Φ n ( f ) ( z ) ‖ near the origin, where Φ n is the Roper–Suffridge extension operator and f is a normalized convex mapping on the unit disk in C . They gave a conjecture that the sharp lower bound holds on the Euclidean unit ball B n in C n . In this paper, we will give a sharp lower bound on B n for a more general extension operator and for normalized univalent mappings f or normalized convex mappings f. We will give a lower bound for mappings f in a linear invariant family. We will also give a similar sharp lower bound on bounded convex complete Reinhardt domains in C n .

Random Euclidean embeddings in spaces of bounded volume ratio

Let ( R N,‖·‖) be the space R N equipped with a norm ‖·‖ whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N× n matrix with N> n, whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n-dimensional Euclidean space ( R n,|·|) onto its image in ( R N,‖·‖) : there exist α, β>0 such that for all x∈ R n , α N |x|⩽‖Γx‖⩽β N |x| . This solves a conjecture of Schechtman on random embeddings of ℓ 2 n into ℓ 1 N . To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Duality of metric entropy

AbstractFor two convex bodies Kand T in R n , the covering number of KbyT, denoted N(K,T), is defined as the minimal number of translates ofT needed to cover K. Let us denote by K ◦ the polar body of K andby D the euclidean unit ball in R n . We prove that the two functionsof t, N(K,tD) and N(D,tK ◦ ), are equivalent in the appropriate sense,uniformly over symmetric convex bodies K ⊂ R n and over n ∈ N. Inparticular, this verifies the duality conjecture for entropy numbers of linearoperators, posed by Pietsch in 1972, in the central case when either thedomain or the range of the operator is a Hilbert space. 1 Introduction For two convex bodies Kand T in R n , the covering number of Kby T,denoted N(K,T), is defined as the minimal number of translates of Tneeded to cover KN(K,T) = min{N: ∃x 1 ...x N ∈ R n , K⊂[ i≤N x i +T}.We denote by Dthe euclidean unit ball in R n . In this paper we prove thefollowing duality result for covering numbers.Theorem 1 (Main theorem) There exist two universal constants

Bounce law at the corners of convex billiards

Let C be a convex subset of R n . Given any elastic shock solution x (·) of the differential inclusion x ̈ (t)+N C (x(t))∋0, t>0, the bounce of the trajectory at a regular point of the boundary of C follows the Descartes law. The aim of the paper is to exhibit the bounce law at the corners of the boundary. For that purpose, we define a sequence ( C ε ) of regular sets tending to C as ε →0, then we consider the approximate differential inclusion x ̈ ε (t)+N C ε (x ε (t))∋0 , and finally we pass to the limit when ε →0. For approximate sets defined by C ε =C+ε B (where B is the unit euclidean ball of R n ), we recover the bounce law associated with the Moreau–Yosida regularization.

Bounce law at the corners of convex billiards

Let C be a convex subset of R n . Given any elastic shock solution x(·) of the differential inclusion x ̈ (t)+N C(x(t))∋0, t>0, the bounce of the trajectory at a regular point of the boundary of C follows the Descartes law. The aim of the paper is to exhibit the bounce law at the corners of the boundary. For that purpose, we define a sequence ( C ε ) of regular sets tending to C as ε→0, then we consider the approximate differential inclusion x ̈ ε(t)+N C ε (x ε(t))∋0 , and finally we pass to the limit when ε→0. For approximate sets defined by C ε=C+ε B (where B is the unit euclidean ball of R n ), we recover the bounce law associated with the Moreau–Yosida regularization.

Almost contractive retractions in Orlicz spaces

Let Bk denote the Euclidean unit ball in ℝκ equipped with the k-dimensional Lebesgue measure and let φ: ℝ+ → ℝ+ be a convex function satisfying φ(0) = 0, φ(t) &gt; 0 for some t &gt; 0. Denote by Eφ = Eφ(Bk) the Orlicz space of finite elements (see (1.6)) generated by φ. The aim of this paper is to show that there exists a retraction of the closed unit ball in Eφ onto the unit sphere in Eφ being a (2 + ɛ)γφ;-set contraction (Theorem 3.6), which generalises [9, Corollary 6] proved for the case of Lp[−1, 1], 1 ≤ p &lt; ∞. Here γφ, denote the Hausdorff measure of noncompactness. This theorem is proved both for the Amemiya and the Luxemburg norms. Also some related results concerning the case of s-convex (0 &lt; s ≤ 1) functions are presented.

Duality of metric entropy in Euclidean space

Let K be a convex body in a Euclidean space, K° its polar body and D the Euclidean unit ball. We prove that the covering numbers N( K, tD) and N( D, tK°) are comparable in the appropriate sense, uniformly over symmetric convex bodies K, over t>0 and over the dimension of the space. In particular this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space. To cite this article: S. Artstein et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).