Harmonic maps of bounded symmetric domains

Research supported in part by NNSFC, SFECC and ICTP     

Harmonic measure on simply connected domains of fixed inradius

LetD?C be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. ForR>0, letω _D (R) denote the harmonic measure at 0 of the set {z:|z|?R}??D. Then it is shown thatthere exist β>0and C>0such that for each such D,ω _D (R)≤Ce ^?βR ,for every R>0. Thus a natural question is: What is the supremum of all β′s , call it β₀, for which the above inequality holds for every suchD? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β₀ is found. Upper bounds for β₀ can be obtained by constructing examples of domainsD. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Harmonic analysis on hyperboloids

The regular representation of O(n, N) acting on $\text{L}{}^2\text{(}\text{O(n, N)}\text{O(n, N ? 1)}\text{)}$ is decomposed into a direct integral of irreducible representations. The homogeneous space $\text{O(n, N)}\text{O(n, N ? 1)}$ is realized as the Hyperboloid $\text{H = {(x, t) ? R}{}^{\mathrm{n\; +\; N}}\text{: ¦ t ¦}{}^2\text{? ¦ x ¦}{}^2\text{= 1}}$. The problem is essentially equivalent to finding the spectral resolution of a certain self-adjoint invariant differential operator □_h on H, which is the tangential part of the operator □ = Δ_x ? Δ_t on R^{n + N}. The spectrum of □_h contains a discrete part (except when N = 1) with eigenfunctions generated by restricting to H solutions of □u = 0 which vanish in the region $\text{¦ t ¦ < ¦ x ¦}$, and a continuous part $\text{H}$_?. As a representation of O(n, N), $\text{H}$_? ⊕ $\text{H}$_? is unitarily equivalent to the regular representation on L² of the cone $\text{{(x, t) : ¦ x ¦}{}^2\text{= ¦ t ¦}{}^2\text{}}$, and the intertwining operator is obtained by solving the equation □u = 0 with given boundary values on the cone. Explicit formulas are given for the spectral decomposition. The special case n = N = 2 gives the Plancherel formula for SL(2, R).     

Harmonic renewal measures and bivariate domains of attraction in fluctuation theory

Summary Each probability measure C on a first orthant is associated with a harmonic renewal measure G. Specifically we consider (N, S _N ) the ladder (time, place) of a random walk S _n. Using bivariate G we show that when S ₁ is in a domain of attraction so is (N, S _N). This unifies and generalizes results of Sinai, Rogosin.     

Harmonic analysis on unitary groups

A theory of harmonic analysis on a metric group (G, d) is developed with the model of U$\text{U}$, the unitary group of a C¹-algebra $\text{U}$, in mind. Essential in this development is the set $\text{G}\text{?}{}_{\mathrm{d}}$ of contractive, irreducible representations of G, and its concomitant set P_d(G) of positive-definite functions. It is shown that $\text{G}\text{?}{}_{\mathrm{d}}$ is compact and closed in $\text{G}\text{?}$. The set $\text{G}\text{?}{}_{\mathrm{d}}$ is determined in a number of cases, in particular when G = U($\text{U}$) with $\text{U}$ abelian. If $\text{U}$ is an AW¹-algebra, it is shown that $\text{G}\text{?}$_d is essentially the same as $\text{U}\text{?}$. Unitary groups are characterised in terms of a certain Lie algebra $\text{g}$_u and several characterisations of G = U($\text{U}$) when $\text{U}$ is abelian are given.     

Harmonic analysis of fractal measures

We consider affine systems inR ⁿ constructed from a given integral invertible and expansive matrixR, and a finite setB of translates,σ _bx:=R^–1x+b; the corresponding measure μ onR ⁿ is a probability measure fixed by the self-similarity $\mu = \left| B \right|^{ - 1} \sum\nolimits_{b \in B} {\mu o\sigma _b^{ - 1} } $ . There are twoa priori candidates for an associated orthogonal harmonic analysis: (i) the existence of some subset Λ inR ⁿ such that the exponentials {e^iλ·x}Λ form anorthogonal basis forL ²(μ); and (ii) the existence of a certaindual pair of representations of theC ^*-algebraO _N wheren is the cardinality of the setB. (For eachN, theC ^*-algebraO _N is known to be simple; it is also called the Cuntz algebra.) We show that, in the “typical” fractal case, the naive version (i) must be rejected; typically the orthogonal exponentials inL ²(μ) fail to span a dense subspace. Instead we show that theC ^*-algebraic version of an orthogonal harmonic analysis, namely (ii), is a natural substitute. It turns out that this version is still based on exponentialse ^iλ·x, but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of affine systems, based onR andB, such that μ has theC ^*-algebra property (ii). Specifically, we show that μ has an orthogonal harmonic analysis (in the sense (ii)) if the system (R, B) satisfies some specific symmetry conditions (which are geometric in nature). Our conditions for (ii) are stated in terms of two pieces of data: (a) aunitary generalized Hadamard matrix, and (b) a certainsystem of lattices which must exist and, at the same time, be compatible with the Hadamard matrix. A partial converse to this result is also given. Several examples are calculated, and a new maximality condition for exponentials is identified.     

Harmonic analysis in infinite dimension

We study the Hermite transform onL ²() where is a Gaussian measure on a Lusin locally convex spaceE. We are then lead to a Hilbert space () of analytic functions onE which is also a natural range for the Laplace transform. LetB be a convenient Hilbert-Schmidt operator on the Cameron-Martin spaceH of . There exists a natural sequence Cap _n of capacities onE associated toB. This implies the Kondratev-Yokoi theorem about positive linear forms on the Hida test-functions space.     

Harmonic analysis on symmetric Stein manifolds from the point of view of complex analysis

The classical theory of finite dimensional representations of compact and complex semisimple Lie groups is discussed from the perspective of multidimensional complex geometry and analysis. The key tool is the complex horospherical transform which establishes a duality between spaces of holomorphic functions on symmetric Stein manifolds and dual horospherical manifolds. Communicated by: Toshiyuki Kobayashi     

Harmonic analysis for resistance forms

In this paper, we define the Green functions for a resistance form by using effective resistance and harmonic functions. Then the Green functions and harmonic functions are shown to be uniformly Lipschitz continuous with respect to the resistance metric. Making use of this fact, we construct the Green operator and the (measure valued) Laplacian. The domain of the Laplacian is shown to be a subset of uniformly Lipschitz continuous functions while the domain of the resistance form in general consists of uniformly 1/2-Hölder continuous functions.     

A result on quasi-periodic solutions of a nonlinear beam equation with a quasi-periodic forcing term

In this paper, a quasi-periodically forced nonlinear beam equation \({u_{tt}+u_{xxxx}+\mu u+\varepsilon\phi(t)h(u)=0}\) with hinged boundary conditions is considered, where μ > 0, \({\varepsilon}\) is a small positive parameter, \({\phi}\) is a real analytic quasi-periodic function in t with a frequency vector ω = (ω ₁,ω ₂ . . . , ω _m ), and the nonlinearity h is a real analytic odd function of the form \({h(u)=\eta_1u+\eta_{2\bar{r}+1}u^{2\bar{r}+1}+\sum_{k\geq \bar{r}+1}\eta_{2k+1}u^{2k+1},\eta_1,\eta_{2\bar{r}+1} \neq0, \bar{r} \in {\mathbb {N}}.}\) The above equation admits a quasi-periodic solution.     

Harmonic analysis on the Pascal graph

In this paper, we completely determine the spectral invariants of an auto-similar planar 3-regular graph. Using the same methods, we study the spectral invariants of a natural compactification of this graph.     

Harmonic analysis on perturbed Cayley Trees

We study some spectral properties of the adjacency operator of non-homogeneous networks. The graphs under investigation are obtained by adding density zero perturbations to the homogeneous Cayley Trees. Apart from the natural mathematical meaning, such spectral properties are relevant for the Bose Einstein Condensation for the pure hopping model describing arrays of Josephson junctions on non-homogeneous networks. The resulting topological model is described by a one particle Hamiltonian which is, up to an additive constant, the opposite of the adjacency operator on the graph. It is known that the Bose Einstein condensation already occurs for unperturbed homogeneous Cayley Trees. However, the particles condensate on the perturbed graph, even in the configuration space due to non-homogeneity. Even if the graphs under consideration are exponentially growing, we show that it is enough to perturb in a negligible way the original graph in order to obtain a new network whose mathematical and physical properties dramatically change. Among the results proved in the present paper, we mention the following ones. The appearance of the Hidden Spectrum near the zero of the Hamiltonian, or equivalently below the norm of the adjacency. The latter is related to the value of the critical density and then with the appearance of the condensation phenomena. The investigation of the recurrence/transience character of the adjacency, which is connected to the possibility to construct locally normal states exhibiting the Bose Einstein condensation. Finally, the study of the volume growth of the wave function of the ground state of the Hamiltonian, which is nothing but the generalized Perron Frobenius eigenvector of the adjacency. This Perron Frobenius weight describes the spatial distribution of the condensate and its shape is connected with the possibility to construct locally normal states exhibiting the Bose Einstein condensation at a fixed density greater than the critical one.