Some function spaces on spaces of homogeneous type

We introduce Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete metric space which possess properties comparable to their classical analogues. In particular we show integral characterizations, the validity of the John–Nirenberg theorem, Poincarè and Sobolev inequalities, Sobolev's embedding theorem and estimates on the pointwise behavior of Sobolev-type mappings. Received: 4 December 2000 / Revised version: 5 July 2001     

A large-deviation result for the range of random walk and for the Wiener sausage

Let {S _n } be a random walk on ℤ ^d and let R _n be the number of different points among 0, S ₁,…, S _{n −1}. We prove here that if d≥ 2, then ψ(x) := lim _{n →∞}(−:1/n) logP{R _n ≥nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper. We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ ^d let Λ _t = Λ _t (A) be the d-dimensional Lebesgue measure of the `sausage' ∪_{0≤ s ≤ t} (B(s) + A). Then φ(x) := lim _t→∞: (−1/t) log P{Λ _t ≥tx exists for x≥ 0 and has similar properties as ψ. Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001     

Homogeneous nilmanifolds attached to representations of compact Lie groups

For each compact Lie algebra ? and each real representation V of ? we consider a two-step nilpotent Lie group N(?,V), endowed with a natural left-invariant riemannian metric. The homogeneous nilmanifolds so obtained are precisely those which are naturally reductive. We study some geometric aspects of these manifolds, finding many parallels with H-type groups. We also obtain, within the class of manifolds N(?,V), the first examples of non-weakly symmetric, naturally reductive spaces and new examples of non-commutative naturally reductive spaces. Received: 16 September 1998 / Revised version: 24 February 1999     

Spectral asymptotics of periodic elliptic operators

We demonstrate that the structure of complex second-order strongly elliptic operators H on with coefficients invariant under translation by can be analyzed through decomposition in terms of versions , , of H with z-periodic boundary conditions acting on where . If the s emigroup S generated by H has a H?lder continuous integral kernel satisfying Gaussian bounds then the semigroups generated by the have kernels with similar properties and extends to a function on which is analytic with respect to the trace norm. The sequence of semigroups obtained by rescaling the coefficients of by converges in trace norm to the semigroup generated by the homogenization of . These convergence properties allow asymptotic analysis of the spectrum of H. Received September 1, 1998; in final form January 14, 1999     

Strongly τ-Decomposable and Selfdecomposable Laws on Simply Connected Nilpotent Lie Groups

 Let ? be a simply connected nilpotent Lie group with Lie Algebra ? and let τ be a contraction on ?. A probability measure μ on ? is strongly τ-decomposable iff it is representable as the limit of for some probability ν on ?. We show that such a limit exists if and only if ν possesses a finite logarithmic moment with respect to a homogeneous norm on ?. This result is then generalized to the class of selfdecomposable laws on ?. We also show that selfdecomposable laws on ? correspond in a 1–1 way to operator selfdecomposable laws on the tangent space ?. Received 1 October 1998; in revised form 29 March 1999     

UV-laser ablation of sensory cells in living insects

Locusta migratoria L. ). The handling of living insects, the survival, and physiological response after treatment are described. We selectively interrupted the d-receptor of the tympanal organ, which is the receptor system responsible for the locust’s sensitivity in the high-frequency range (>10 kHz). The effects of the laser treatment were tested by determining hearing thresholds in electrophysiological recordings from the tympanal nerves. In agreement with the literature, the interruption of the d-receptors led to a significant shift towards higher values of the thresholds in the high-frequency range. Future perspectives and biological applications of UV-laser ablation are discussed. Received: 10 Dezember 1998 / Accepted: 15 December 1998 / Published online: 24 February 1999     

Hörmander Multipliers on the Heisenberg Group

 Let be the Heisenberg group of dimension . Let be the partial sub-Laplacians on and T the central element of the Lie algebra of . For any we prove that the operator is bounded on the Hardy spaces , if the function m satisfies a Hrmander-type condition on which depends on . We also obtain analogous results for the operators and , where the function m satisfies analogous H?rmander-type conditions on and on , respectively. Here is the Kohn-Laplacian on . (Received 28 July 1999; in final form 6 March 2000)     

An algebraic approach to discrete dilations. Application to discrete wavelet transforms

We investigate the connections between continuous and discrete wavelet transforms on the basis of algebraic arguments. The discrete approach is formulated abstractly in terms of the action of a semidirect product A×Γ on ℓ²(Γ), with Γ a lattice and A an abelian semigroup acting of Γ. We show that several such actions may be considered, and investigate those which may be written as deformations of the canonical one. The corresponding deformed dilations (the pseudodilations) turn out to be characterized by compatibility relations of a cohomological nature. The connection with multiresolution wavelet analysis is based on families of pseudodilations of a different type.     

The structure of singular spaces of dimension 2

In this paper, we study the structure of locally compact metric spaces of Hausdorff dimension 2. If such a space has non-positive curvautre and a local cone structure, then every simple closed curve bounds a conformal disk. On a surface (a topological manifold of dimension 2), a distance function with non-positive curvature and whose metric topology is equivalent to the surface topology gives a structure of a Riemann surface. The construction of conformal disks in these spaces uses minimal surface theory; in particular, the solution of the Plateau Problem in metric spaces of non-positive curvature. Received: 18 November 1997/ Revised versions: 15 January and 7 June 1999     

一种“裸眼”可视化检测溶液的pH值及细胞中铁离子的BODIPY类探针

以吡啶-2-甲酸(4-甲酰基苯酚)酯代替对羟基苯甲醛,将对羟基苯基引入到BODIPY染料的3-位,以高产率得到一个BODIPY类探针2。探针2易溶解到各种有机溶剂中,在甲醇溶液中的最大吸收和发射波长分别为570和585 nm,荧光量子产率达到0.9。当溶液的pH值从酸性过渡到碱性时,对羟基苯乙烯基团会慢慢失去质子,其溶液最大吸收波长发生红移,荧光发生淬灭,而且溶液颜色也发生明显的变化。因此,探针2可作为一种"裸眼"可视化的pH探针。此外,探针2对Fe~(3+)也有明显的响应,在溶液中加入Fe~(3+),探针2分子结构中乙烯基团发生断裂,其溶液最大吸收波长和最大发射峰均发生蓝移。因此,探针2也可以作为Fe~(3+)的比率型探针。同时探针2毒性较低、渗透性好、对Fe~(3+)选择性高,可以应用到细胞中检测Fe~(3+)的含量。