A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

On the long time behaviour of solutions to dissipative wave equations in \mathbb{R}^{2}

In this paper we present a parabolic approach to studying the diffusive long time behaviour of solutions to the Cauchy problem:

Syzygies of Multi-Variable Higher Spin Dirac Operators on $$\mathbb{R}^{3}$$

This paper is a short report on the generalization of some results of our previous paper [12] to the case of spin j/2 Dirac operators in real dimension three for arbitrary odd integer j. We use an explicit formula for the local expression of such operators to study their algebraic properties, construct the compatibility conditions of the overdetermined system associated to the operator in several spatial variables, and we prove that its associated algebraic complex, dual do the BGG sequence coming from representation theory, has substantially the same pattern as the Cauchy-Fueter complex. The author is a member of the Eduard Čech Center and his research is supported by the relative grants.     

The second main theorem for moving targets

The defect relation for holomorphic maps in respect to slowly moving target hyperplanes in is proved. The sharp defect boundn+1 is obtained. Communicated by Yoram Sagher     

Estimates of holomorphic functions in zero-free domains

We study functions f(z) holomorphic in having the property f(z) ≠ 0 for 0 < Im z < 1 and we obtain lower bounds for |f(z)| for 0 < Im z < 1. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I + A(z) assuming that A(z) is a trace class operator for and I + A(z) is invertible for 0 < Im z < 1 and is unitary for . A. Borichev was partially supported by the ANR project DYNOP.     

Uniform estimates with weights for the\bar \partial - equation

This paper concernsL ^∞-variants of Hörmanders weightedL ²-estimates for the $\bar \partial - equation$ . In particular, we discuss a conjecture concerning suchL ^∞-estimates which is related to the corona problem in the ball, and show a weaker version of this conjecture. The proof uses a refinedL ²-estimate for the canonical solution to the $\bar \partial - equation$ . An alternative approach based on von Neumann’s Minimax theorem is also given.     

Nevanlinna-Pick Interpolation for {\mathbb{C}} + BH^\infty

We study the Nevanlinna-Pick problem for a class of subalgebras of H ^∞. This class includes algebras of analytic functions on embedded disks, the algebras of finite codimension in H ^∞ and the algebra of bounded analytic functions on a multiply connected domain. Our approach uses a distance formula that generalizes Sarason’s [23] work. We also investigate the difference between scalar-valued and matrix-valued interpolation through the use of C ^*-envelopes. This research was partially supported by the NSF grant DMS 0300128. This research was completed as part of my Ph.D. dissertation at the University of Houston.     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

On the nonextendability of germs of holomorphic functions

We construct explicit supporting manifolds and local holomorphic peak functions as obstructions to the extendability of holomorphic functions on a class of domains not necessarily pseudoconvex in C^N, N >2.     

Maximum boundary regularity of bounded Hua-harmonic functions on tube domains

In this article we prove that bounded Hua-harmonic functions on tube domains that satisfy some boundary regularity condition are necessarily pluriharmonic. In doing so, we show that a similar theorem is true on one-dimensional extensions of the Heisenberg group or equivalently on the Siegel upper half-plane.     

Intersection domains and the sum of two complex hyperbolic metrics

In this article, we prove that the intersection of the unit ball in ℂⁿ with an affine transformation of it is a negatively curved domain. The two-dimensional case is due to Cheung and Wu.     

Wiener-Hopf operators on L_{\omega}^{2}(\mathbb{R}^{+})

Let be a weighted space with weight . In this paper we show that for every Wiener-Hopf operator T on and for every a I, there exists a function such that

The action of S n on the cohomology of \overline{M}_{0,n}(\mathbb{R})

In recent work by Etingof, Henriques, Kamnitzer, and the author, a presentation and explicit basis was given for the rational cohomology of the real locus of the moduli space of stable genus 0 curves with n marked points. We determine the graded character of the action of S_n on this space (induced by permutations of the marked points), both in the form of a plethystic formula for the cycle index, and as an explicit product formula for the value of the character on a given cycle type.      

Unique continuation theorems for the\bar \partial - Operator and applications

We formulate a unique continuation principle for the inhomogeneous Cauchy-Riemann equations near a boundary pointz ₀ of a smooth domain in complex euclidean space. The principle implies that the Bergman projection of a function supported away fromz ₀ cannot vanish to infinite order atz ₀ unless it vanishes identically. We prove that the principle holds in planar domains and in domains where the problem is known to be analytic hypoelliptic. We also demonstrate the relevance of such questions to mapping problems in several complex variables. The last section of the paper deals with unique continuation properties of the Szegő projection and kernel in planar domains. Research supported by NSF Grant DMS-8922810.     

On estimates of biharmonic functions on Lipschitz and convex domains

Using Maz ’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in ℝⁿ. Forn ≥ 8, combinedwitharesultin[18], these estimates lead to the solvability of the L^p Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow us to show that the L^p Dirichlet problem is uniquely solvable for any 2 − ε < p < ∞ and n ≥ 4.     

On proper holomorphic self-maps of generalized complex ellipsoids

The purpose of this paper is to prove that every proper holomorphic self-mapping of a Reinhardt domain Ω in C ⁿ which is a generalization of a complex ellipsoid is biholomorphic. The main novelty of our result is that Ω is a domain in C ⁿ such that it is allowed to have a boundary point at which the Levi determinant has infinite order of vanishing.     

Nondicritical $${\mathbb {C}^*}$$-actions on two-dimensional Stein manifolds

We study and classify actions of the complex multiplicative group on a nonsingular Stein surface with an isolated nondicritical singularity. We prove that the corresponding foliation exhibits a holomorphic first integral of a type F = f ⁿ g ^m where f and g are global holomorphic functions and . Under some additional conditions on the functions f and g we prove analytic linearization for the action. Our results can be viewed as extension of the original work of Masakazu Suzuki.     

Inequalities for the Volume of the Unit Ball in \mathbb {R}^{n}, II

We present several sharp inequalities for the volume of the unit ball in ,

A boundary Morera theorem

LetD ⊂C ^N ,N ≥ 2 be a bounded open set withC ² boundary and letL be an open connected set of affine complex hyperplanes inC ^N containing a hyperplane that misses . LetE = ∪_Λ∈LΛ, Γ =E ∩bD. Suppose thatf ∈C(Γ) and assume that