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1.
Let Ω be a bounded Lipschitz domain. Define B 0,1 1, r (Ω) = {fL 1 (Ω): there is an FB 0,1 1 (ℝ n ) such that F|Ω = f} and B 0,1 1 z (Ω) = {fB 0,1 1 (ℝ n ) : f = 0 on ℝ n \}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation of ℝ n +. Received June 8, 2000, Accepted October 24, 2000  相似文献   

2.
This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i.e. f(P)∃LipAα, then the corresponding Bernstein Bezier net fnLip Asec αφα, here φ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net fnLip Bα, then its elevation Bezier net EfnLip Bα; and (3) If f(P)∃Lip Aα, then the corresponding Bernstein polynomials Bn(f;P)∃Lip Asec αφα, and the constant Asecαφ is best in some sense. Supported by NSF and SF of National Educational Committee  相似文献   

3.
It is proved that, for any Lipschitz function f(t 1, ..., t n ) of n variables, the corresponding map f op: (A 1, ...,A n ) → f(A 1, ..., A n ) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal S p , p ∈ (1,∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in S p . It is also proved that the map f op is Fréchet differentiable in the norm of S p if f is continuously differentiable.  相似文献   

4.
Univalent mappings associated with the Roper-Suffridge extension operator   总被引:15,自引:0,他引:15  
The Roper-Suffridge extension operator provides a way of extending a (locally) univalent functionfεH(U) to a (locally) biholomorphic mappingFH(Bn). In this paper, we give a simplified proof of the Roper-Suffridge theorem: iff is convex, then so isF. We also show that iffS *, theF is starlike and that iff is a Bloch function inU, thenF is a Bloch mapping onB n. Finally, we investigate some open problems. Partially supported by the Natural Sciences and Engineering Research Council of Canada under grant A9221.  相似文献   

5.
In this paper we shall construct proper holomorphic mappings from strictly pseudoconvex domains in Cn into the unit ball in CN which satisfy some regularity conditions up to the boundary. If we only require continuity of the map, but not more, then there is a large class of such maps (see [2], [3], and [5]). On the other hand, if F is Ck on the closure, k > N ? n + 1, then there is a very small class of such maps. In fact such F must be holomorphic across the boundary (see [1] and [4]). We are interested in maps F that are less than CN ? n + 1, but more than continuous on the closure. Namely, we want to find out if this is a very small or a large class. Our main result is as follows. Theorem, (a) Let ga < 1/6; then there exists an N = N(α, n) such that we can find a map F: Bn → BN that is proper, holomorphic, and Lipschitz α up to the boundary, but F is not holomorphic across the boundary. (b) If D is a general strictly pseudoconvex domain with C -boundary in Cn, then we can find a map F: D → BN, N = N(α, n), that is proper, holomorphic, and Lipschitz α up to the boundary of D. To do part (a) of the theorem we only need to show that we can find a proper holomorphic map F = (f1, …, FN): Bn → BN that is Lipschitz α and fN(z) = c(1 - Z1)1/6 for some constant c > 0. With this we can in fact ensure that the map in (a) is at most Lipschitz 1/6 on the closure of Bn.  相似文献   

6.
Let fC ω (∂B n ), where B n is the unit ball of ℂ n . We prove that if a,b ? [`(B)] na,b \in {\overline B ^n}, ab, for every complex line L passing through one of a or b, the restricted function f|L ??Bnf{|_{L \cap \partial {B^n}}} has a holomorphic extention to the cross-section LB n , then f is the boundary value of a holomorphic function in B n .  相似文献   

7.
Denote by Hol(B n ) the space of all holomorphic functions in the unit ball B n of ℂ n , n ≥ 1. Given gHol(B m ) and a holomorphic mapping φ: B m B n , put C φ g f = g · (fφ) for fHol(B n ). We characterize those g and φ for which C φ g is a bounded (or compact) operator from the growth space A −log(B n ) or A β (B n ), β > 0, to the weighted Bergman space A α p (B m ), 0 < p < ∞, α > −1. We obtain some generalizations of these results and study related integral operators.  相似文献   

8.
Liczberski–Starkov gave a sharp lower bound for DΦn(f)(z) near the origin, where Φn 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 Bn in Cn. In this paper, we will give a sharp lower bound on Bn 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 Cn.  相似文献   

9.
We derive an error bound for fixed-point iterationsx n+1=f(x n ) by using monotonicity in the sense of [2]. The new bound is preferable to the classical one which bounds the error in terms of the Lipschitz constant off.  相似文献   

10.
For any symmetric function f: ? n  → ? n , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.  相似文献   

11.
We study the Bloch constant for Κ-quasiconformal holomorphic mappings of the unit ball B of C n . The final result we prove in this paper is: If f is a Κ-quasiconformal holomorphic mappig of B into C n such that det(f′(0)) = 1, then f(B) contains a schlicht ball of radius at least where C n > 1 is a constant depending on n only, and as n→∞. Received June 24, 1998, Accepted January 14, 1999  相似文献   

12.
We consider a stochastic control model with linear transition law and arbitrary convex cost functions, a far-reaching generalization of the familiar linear quadratic model. Firstly conditions are given under which the continuous state version has minimizersf n at each stagen which are increasing and in addition either right continuous or continuous or Lipschitz continuous with explicitly given Lipschitz constant. For the computationally important discrete version we verify some analogous properties under stronger assumptions.  相似文献   

13.
Let h(t) = Σn ≥ 1hntn, h1 > 0, and exp(xh(t)) = Σn ≥ 0Pn(x) tn/n!. For f C[0,1], the associated Bernstein-Sheffer operator of degree n is defined by Bhnf(x) = Pn− 1 Σnk = 0f(k/n)(nk) Pk(x) Pnk(1 − x) where pn = pn(1). We characterize functions h for which Bhn is a positive operator for all n ≥ 0. Then we give a necessary and sufficient condition insuring the uniform convergence of Bhnf to f. When h is a polynomial, we give an upper bound for the error fBhnf . We also discuss the behavior of Bhnf when h is a series with a finite or infinite radius of convergence.  相似文献   

14.
In this paper, we characterize the Hardy class ofM-harmonic functions on the unit ballB in ℂ n in terms of the Berezin transform. We define and study the Besovp-spaces ofM-harmonic functions. For anM-harmonic symbolf, we give various criteria for the Hankel operatorsH f andH f to be bounded, compact or in the Schatten-von-Neumann classS p . These criteria establish a close relationship among Besovp-spaces, Berezin transform, the invariant Laplacian, and Hankel operators on the unit ballB.  相似文献   

15.
We prove that the symplectic group Sp(2n,\mathbbZ){Sp(2n,\mathbb{Z})} and the mapping class group Mod S of a compact surface S satisfy the R property. We also show that B n (S), the full braid group on n-strings of a surface S, satisfies the R property in the cases where S is either the compact disk D, or the sphere S 2. This means that for any automorphism f{\phi} of G, where G is one of the above groups, the number of twisted f{\phi}-conjugacy classes is infinite.  相似文献   

16.
In this note, the authors show the boundedness of the maximal commutators of Bocher-Riesz operator B δ and that of maximal commutator B δ1* b generated by B δ and a Lipschitz function b mapping from M p q (R n ) into BMO space and also maps from M p q (R n ) into L (βn/q).  相似文献   

17.
In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space ℂ n . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric representation, where AL(ℂ n , ℂ n ), and prove that if dt < ∞ (this integral is convergent if k +(A) < 2m(A)), then a mapping fS(B n ) is A-asymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f(z, t) such that D f(0, t) = e At , {e −At f(·, t)} t≥0 is a normal family on B n and f = f(·, 0). In particular, a spirallike mapping with respect to AL(ℂ n , ℂ n ) with dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A* = 2I n , then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings. Partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221. Partially supported by Grant-in-Aid for Scientific Research (C) no. 19540205 from Japan Society for the Promotion of Science, 2007. Partially supported by Romanian Ministry of Education and Research, CEEX Program, Project 2-CEx06-11-10/2006.  相似文献   

18.
In this paper we give an astonishingly simple proof of C1, 1 regularity in elliptic theory. Our technique yields both new simple proofs of old results as well as new optical results. The setting we'll consider is the following. Let u be a solution to where B, is the unit ball in ?n, f(x, t) is a bounded Lipschitz function in x, and ft′ is bounded from below. Then we prove that u ? C1, 1 (B1/2). Our method is a simple corollary to a recent monotonicity argument due to Caffarelli, Jerison, and Kenig. © 2002 Wiley Periodicals, Inc.  相似文献   

19.
Our purpose here is to consider on a homogeneous tree two Pompeiutype problems which classically have been studied on the plane and on other geometric manifolds. We obtain results which have remarkably the same flavor as classical theorems. Given a homogeneous tree, letd(x, y) be the distance between verticesx andy, and letf be a function on the vertices. For each vertexx and nonnegative integern let Σ n f(x) be the sum Σ d(x, y)=n f(y) and letB n f(x)=Σ d(x, y)≦n f(y). The purpose is to study to what extent Σ n f andB n f determinef. Since these operators are linear, this is really the study of their kernels. It is easy to find nonzero examples for which Σ n f orB n f vanish for one value ofn. What we do here is to study the problem for two values ofn, the 2-circle and the 2-disk problems (in the cases of Σ n andB n respectively). We show for which pairs of values there can exist non-zero examples and we classify these examples. We employ the combinatorial techniques useful for studying trees and free groups together with some number theory.  相似文献   

20.
A generalization of the Blaschke product is constructed. This product enables one to factor out the zeros of the members of certain non-Nevanlinna classes of functions analytic in the unit disc, so that the remaining (non-vanishing) functions still belong to the same class. This is done for the classesA −n (0<n<∞) andB −n (0<n<2) defined as follows:fA −n iff |f(z)|≦C f (1−|z|)n ,fB n iff |f(z)|≦exp {C f (1−|z|)n }, whereC f depends onf.  相似文献   

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