The Bergman kernel: Explicit formulas,deflation, Lu Qi-Keng problem and Jacobi polynomials

We investigate the Bergman kernel function for the intersection of two complex ellipsoids {(z,w ₁,w ₂) ∈ C ⁿ⁺²: |z ₁|²+...+|z _n |²+|w ₁| ^q < 1, |z ₁|²+...+|z _n |²+|w ₂| ^r < 1}. We also compute the kernel function for {(z ₁,w ₁,w ₂) ∈ C³: |z ₁|^2/n + |w ₁| ^q < 1, |z ₁|^2/n + |w ₂| ^r < 1} and show deflation type identity between these two domains. Moreover in the case that q = r = 2 we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.     

Torsional rigidity of submanifolds with controlled geometry

We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P ^m with controlled radial mean curvature in ambient Riemannian manifolds N ⁿ with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz-symmetrization of the domains in warped product model spaces. Our main results are obtained using methods from previously established isoperimetric inequalities, as found in, e.g., Markvorsen and Palmer (Proc Lond Math Soc 93:253--272, 2006; Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below, p. 39, preprint, 2007). As in that paper we also characterize the geometry of those situations in which the bounds for the torsional rigidity are actually attained and study the behavior at infinity of the so-called geometric average of the mean exit time for Brownian motion.     

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

We prove estimates of a p-harmonic measure, p∈(n?m,∞], for sets in Rⁿ which are close to an m-dimensional hyperplane Λ?Rⁿ, m∈[0,n?1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω?Rⁿ?Λ for p-subharmonic functions. Moreover, we give local and global growth estimates for p-harmonic functions, vanishing on sets in Rⁿ, which are close to an m-dimensional hyperplane.     

Über das Schwarzsche Lemma und Verwandte Sätze

Upper bounds for the Jacobian determinant by holomorphic mappings of bounded domainsD into itself were given first more then thirty years ago by Stefan Bergman by means of his theory of the kernel function ofD. In this paper a different method shall be developed and distortion theorems for holomorphic mappings of bounded domains of a Kähler manifoldM ⁿ into a Kähler manifoldM ₀ ⁿ shall be proved. The special casesM ⁿ =C _n (unit sphere of C ⁿ ) andM ⁿ =M ₀ ⁿ =|C _n shall also be considered. The proof depends essentially on the two Hermitian quadratic forms corresponding to the metric and to the Ricci tensor. The manifolds must be of negative Ricci curvature and fulfil two conditions given in section 4.     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Bergman type operators on mixed norm spaces over the ball in ℂ<Superscript>n</Superscript>

The paper considers Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions. We prove that there exist values of parameter ß for which these operators are bounded on mixed norm spaces L(p, q, ß) on the unit ball in Cⁿ.     

Estimates on conjectures of Minkowski and woods II

Let ? ⁿ be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < Rwhich contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ? ⁿ of radius \(\sqrt n /2\) contains a point of ∧. This is known to be true for n ≤ 8. Recently we gave estimates on a more general conjecture of Woods for n ≥ 9. This lead to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms. Here we shall refine our method to obtain improved estimates for Woods Conjecture. These give improved estimates of Minkowski’s conjecture for 9 ≤ n ≤ 31.     

Estimation of the depth of reversible circuits consisting of NOT,CNOT and 2-CNOT gates

The paper discusses the asymptotic depth of a reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The reversible circuit depth function D(n, q) is introduced for a circuit implementing a mapping f: Z₂ⁿ → Z₂ⁿ as a function of n and the number q of additional inputs. It is proved that for the case of implementation of a permutation from A(Z₂ⁿ) with a reversible circuit having no additional inputs the depth is bounded as D(n, 0) ? 2ⁿ/(3log₂n). It is also proved that for the case of transformation f: Z₂ⁿ → Z₂ⁿ with a reversible circuit having q₀ ~ 2ⁿ additional inputs the depth is bounded as D(n,q₀) ? 3n.     

On homogenization for non-self-adjoint locally periodic elliptic operators

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R ^d of the form A ^ε = ?divA(x, x/ε)?. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A ^ε ? μ)^?1, including one with a corrector, and for (?Δ) ^s/2(A ^ε ? μ)^?1 in the operator norm on L ₂(R ^d ) ⁿ . For s ≠ 0, we also give estimates of the rates of approximation.     

Removable singularities of solutions of linear uniformly elliptic second order equations

Let L be a uniformly elliptic linear second order differential operator in divergence form with bounded measurable real coefficients in a bounded domain G ? ?ⁿ (n ? 2). We define classes of continuous functions in G that contain generalized solutions of the equation L? = 0 and have the property that the compact sets removable for such solutions in these classes can be completely described in terms of Hausdorff measures.     

A note on K<Emphasis Type="Italic">ä</Emphasis>hler manifolds with almost nonnegative bisectional curvature

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M ⁿ is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ ?ε(n, Λ), then M ⁿ is diffeomorphic to the product M ₁ × ? × M _k , where each M _i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.     

Removable Singularities of Solutions of the Minimal Surface Equation

Suppose that G is a bounded domain in ? ⁿ (n ? 2), E ≠ G is a relatively closed set in G, and 0 < α < 1. We prove that E is removable for solutions of the minimal surface equation in the class C ^1,α(G)_loc if and only if the (n ? 1 + α)-dimensional Hausdorff measure of E is zero.     

Compactness of the Hardy-Littlewood operator on some spaces of harmonic functions

We study the compactness of the Hardy-Littlewood operator on several spaces of harmonic functions on the unit ball in ? ⁿ such as: a-Bloch, weighted Hardy, weighted Bergman, Besov, BMO _p , and Dirichlet spaces.     

Division Algebras with Left Algebraic Commutators

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a ₀ + a ₁ x + ? + a _n x ⁿ (resp. right polynomial a ₀ + x a ₁ + ? + x ⁿ a _n ) over K such that a ₀ + a ₁ a + ? + a _n a ⁿ = 0 (resp. a ₀ + a a ₁ + ? + a ⁿ a _n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.     

Nonlinear Korn Inequalities on a Hypersurface

The authors establish several estimates showing that the distance in W~(1,p),1 p ∞,between two immersions from a domain of R~n into R~(n+1) is bounded by the distance in L~p between two matrix fields defined in terms of the first two fundamental forms associated with each immersion. These estimates generalize previous estimates obtained by the authors and P. G. Ciarlet and weaken the assumptions on the fundamental forms at the expense of replacing them by two different matrix fields.     

On Bounded Positive (m,p)-Circle Domains

Let D be a bounded positive (m, p)-circle domain in ?². The authors prove that if dim(Iso(D)⁰) = 2, then D is holomorphically equivalent to a Reinhardt domain; if dim(Iso(D)⁰) = 4, then D is holomorphically equivalent to the unit ball in ?². Moreover, the authors prove the Thullen’s classification on bounded Reinhardt domains in ?² by the Lie group technique.     

On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure

It is shown that, for any compact set K ? ? ⁿ (n ? 2) of positive Lebesgue measure and any bounded domain G ? K, there exists a function in the Hölder class C^1,1(G) that is a solution of the minimal surface equation in G \ K and cannot be extended from G \ K to G as a solution of this equation.     

Complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow

In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation among λ, extrinsic radius k, intrinsic diameter d, and dimension n of the complete λ-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded λ-hypersurface with some natural and general restrictions.     

Holomorphic invariant forms of a bounded domain

Given a complete ortho-normal system  = (0, 1, 2, . . .) of L2H(D), the space of holomorphic and absolutely square integrable functions in the bounded domain D of Cn, we construct a holomorphic imbedding ι : D →■(n, ∞), the complex infinite dimensional Grassmann manifold of rank n. It is known that in ■(n, ∞) there are n closed (μ, μ)-forms τμ (μ = 1, . . . , n) which are invariant under the holomorphic isometric automorphism of ■(n, ∞) and generate algebraically all the harmonic differential forms of ...     

Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion

We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so–called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self–adjoint integral operator in a suitable L ₂(μ)–space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X _H with Hurst parameter H∈(0,1), indexed by a self–similar set T?? ^N of Hausdorff dimension D. This rate turns out to be of order n ^?H/D (log?n)^1/2. In the case T=[0,1] ^N we present a concrete wavelet representation of X _H leading to an approximation of X _H with the optimal rate n ^?H/N (log?n)^1/2.