The Neumann Problem for the Kohn-Laplacian on the Heisenberg Group ℍn$\mathbb H_{n}$

The existence and uniqueness of the solution of the Neumann problem for the Kohn-Laplacian relative to the Korányi ball on the Heisenberg group \(\mathbb {H}_{n}\) are discussed. Explicit representation for a Green’s type function (Neumann function) for the Korányi ball in \(\mathbb {H}_{n}\) for circular functions has been obtained. This function is then used on the above region in \(\mathbb {H}_{n}\) to solve the inhomogeneous Neumann boundary value problem for certain circular data.     

Green functions and related boundary value problems on the Heisenberg group

In this article, a modified Kelvin transform on ? _n using inversion with respect to a ball of arbitrary radius is defined, which gives explicit expressions for Green's function and Poisson's kernel for the Korányi ball of arbitrary radius and annular domain. The solution of the Dirichlet problem for the union of two balls is discussed using the Schwarz's alternating method.     

Möbius rigidity of invariant metrics in boundaries of symmetric spaces of rank-1

Let \({\mathbf{H}}^n_{{\mathbb K}}\) denote the symmetric space of rank-1 and of non-compact type and let \(d_{{\mathfrak H}}\) be the Korányi metric defined on its boundary. We prove that if d is a metric on \(\partial {\mathbf{H}}^n_{{\mathbb K}}\) such that all Heisenberg similarities are d-Möbius maps, then under a topological condition d is a constant multiple of a power of \(d_{{\mathfrak H}}\).     

A New Class of Graphs That Satisfies the Chen‐Chvátal Conjecture

A well‐known combinatorial theorem says that a set of n non‐collinear points in the plane determines at least n distinct lines. Chen and Chvátal conjectured that this theorem extends to metric spaces, with an appropriated definition of line. In this work, we prove a slightly stronger version of Chen and Chvátal conjecture for a family of graphs containing chordal graphs and distance‐hereditary graphs.     

Estimates for the Poisson kernel and Hardy spaces on compact manifolds

We study Hardy spaces on the boundary of a smooth open subset or Rn and prove that they can be defined either through the intrinsic maximal function or through Poisson integrals, yielding identical spaces. This extends to any smooth open subset of Rn results already known for the unit ball. As an application, a characterization of the weak boundary values of functions that belong to holomorphic Hardy spaces is given, which implies an F. and M. Riesz type theorem.     

Remarks on Fourier multipliers and applications to the wave equation

Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [A. Bényi, K.A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, preprint, April 2007 (available at ArXiv:0704.0833v1)]. The same arguments may apply to obtain local wellposedness for the NLKG equation.     

A note on specializations of Grothendieck polynomials

Buch and Rimányi proved a formula for a specialization of double Grothendieck polynomials based on the Yang–Baxter equation related to the degenerate Hecke algebra. A geometric proof was found by Yong and Woo by constructing a Gröbner basis for the Kazhdan–Lusztig ideals. In this note, we give an elementary proof for this formula by using only divided difference operators.     

Sharpness of the Korányi approach region

We prove a Littlewood-type theorem which shows the sharpness of the Korányi approach region for the boundary behavior of Poisson-Szegö integrals on the unit ball of . Our result is stronger than Hakim and Sibony (1983).

     

Asymptotic boundary value problems in Banach spaces

A continuation principle is given for solving boundary value problems on arbitrary (possibly infinite) intervals to Carathéodory differential inclusions in Banach spaces. For this aim, the appropriate fixed point index is defined to condensing decomposable multivalued operators in Fréchet spaces. This index extends and unifies the one for compact maps in Andres et al. [Trans. Amer. Math. Soc. 351 (1999) 4861-4903] as well as the one for operators in Banach spaces in Bader [Ph.D. Thesis, University of Munich, 1995]. As an application, we prove the existence of an entirely bounded solution of a semilinear evolution inclusion.     

A Lie–Rinehart Algebra with No Antipode

The aim of this note is to communicate a simple example of a Lie–Rinehart algebra whose enveloping algebra is not a Hopf algebroid, neither in the sense of Böhm and Szlachányi, nor in the sense of Lu.     

Mean-Value Theorems in Arithmetic Semigroups

This paper reports on recent progress in the theory of multiplicative arithmetic semigroups, which has been initiated by John Knopfmacher's work on abstract analytic number theory. In particular, it deals with abstract versions of the mean-value theorems of Delange, of Wirsing, and of Halász for multiplicative functions on arithmetic semigroups G with Axiom A . The Turán Kubilius inequality is transferred to G , and methods developed by Rényi, Daboussi and Indlekofer, Lucht and Reifenrath are utilized. As byproduct a new proof of the abstract prime number theorem is obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

Weak Bialgebras and Monoidal Categories

We study monoidal structures on the category of (co)modules over a weak bialgebra. Results due to Nill and Szlachányi are unified and extended to infinite algebras. We discuss the coalgebra structure on the source and target space of a weak bialgebra.     

Orbit structure of non-compact hermitian symmetric spaces

SiaX ₀ uno spazio simmetrico hermitiano di tipo non compatto,x ₀ un punto diX ₀ eK ₀ il sottogruppo d'isotropia dix ₀. In questo articolo viene studiata approfonditamente la struttura dello spazio delle orbite dell'azione diK ₀ suX ₀. Ad ogni puntox∈X ₀ vengono associatir=rankX ₀ numeri reali non negative dettimoduli, che parametrizzano lo spazio delle orbite. Usando questi moduli, è possibile stabilire effettivamente quando due orbite sono isomorfe e, più in generale, quando esiste una applicazione fra due orbite che commuta con l'azione diK ₀. Inoltre è anche possibile determinare la dimensione di ogni orbita, e descrivere la struttura del bordo di ?ilov diX ₀, generalizzando alcuni risultati di Wolf, Korányi e Harris. Infine, viene data la realizzazione concreta delle definizioni introdotte e dei teoremi dimostrati nel caso dei quattro domini classici nella realizzazione di Cartan.     

Subspaces of Hilbert-generated Banach spaces and the quantification of super weak compactness

We introduce a measure of super weak noncompactness Γ defined for bounded subsets and bounded linear operators in Banach spaces that allows to state and prove a characterization of the Banach spaces which are subspaces of a Hilbert-generated space. The use of super weak compactness and Γ casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, Hájek, Montesinos, Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits connections with Banach-Saks type properties.     

Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces H p σ and B p σ

In a bounded Lipschitz domain, we consider a strongly elliptic second-order equation with spectral parameter without assuming that the principal part is Hermitian. For the Dirichlet and Neumann problems in a weak setting, we prove the optimal resolvent estimates in the spaces of Bessel potentials and the Besov spaces. We do not use surface potentials. In these spaces, we derive a representation of the resolvent as a ratio of entire analytic functions with sharp estimates of their growth and prove theorems on the completeness of the root functions and on the summability of Fourier series with respect to them by the Abel-Lidskii method. Preliminarily, such questions for abstract operators in Banach spaces are discussed. For the Steklov problem with spectral parameter in the boundary condition, we obtain similar results. We indicate applications of the resolvent estimates to parabolic problems in a Lipschitz cylinder. We also indicate generalizations to systems of equations. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 4, pp. 2–23, 2008 Original Russian Text Copyright ? by M. S. Agranovich To dear Israel Moiseevich Gelfand in connection with his 95th birthday Supported by RFBR grant no. 07-01-00287.     

On the average‐case complexity of Shellsort

We prove a lower bound expressed in the increment sequence on the average‐case complexity of the number of inversions of Shellsort. This lower bound is sharp in every case where it could be checked. A special case of this lower bound yields the general Jiang‐Li‐Vitányi lower bound. We obtain new results, for example, determining the average‐case complexity precisely in the Yao‐Janson‐Knuth 3‐pass case.     

对称随机变量序列的Hájek-Rényi型不等式和强大数律

得到了对称随机变量序列的Hájek-Rényi型不等式,并利用它研究了对称随机变量序列的强大数律.     

Fourier multipliers of classical modulation spaces

Based on the observation that translation invariant operators on modulation spaces are convolution operators we use techniques concerning pointwise multipliers for generalized Wiener amalgam spaces in order to give a complete characterization of the Fourier multipliers of modulation spaces. We deduce various applications, among them certain convolution relations between modulation spaces, as well as a short proof for a generalization of the main result of a recent paper by Bènyi et al., see [À. Bènyi, L. Grafakos, K. Gröchenig, K.A. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal. 19 (1) (2005) 131–139]. Finally, we show that any function with ([d/2]+1)-times bounded derivatives is a Fourier multiplier for all modulation spaces with p(1,∞) and q[1,∞].     

Besov spaces on surfaces with singularities

The trace to the boundary of a domain Ω of functions in Besov spaces and Sobolev spaces defined in Ω is characterized, in the case when the boundary has singularities of a certain type.     

Locally Scalar Graph Representations in the Category of Hilbert Spaces

The condition of being locally scalar is imposed on graph (or quiver) representations in the category of Hilbert spaces. Under this condition, reflection and Coxeter functors are constructed in categories of such representations and used to prove an analog of the Gabriel theorem.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 13–30, 2005Original Russian Text Copyright © by S. A. Kruglyak and A. V. Roiter