Inequalities of Littlewood--Paley Type for n-Harmonic Functions on the Polydisk

For n-harmonic functions on the unit polydisk in the space $\mathbb{C}^n $ we define g-functions of Littlewood--Paley type and establish L ^p-inequalities related to them. In the present paper, the main theorems deal with the extension of results of Littlewood, Paley, and Flett to the polydisk and their generalizion to fractional derivatives of arbitrary order. This gives an answer to a question posed by Littlewood.     

Gaussian Analytic Functions in the Polydisk

We study hyperbolic Gaussian analytic functions in the unit polydisk of \(\mathbb {C}^n\). Following the scheme previously used in the unit ball, we first study the asymptotics of fluctuations of linear statistics as the directional intensities \(L_j\), \(j=1,\dots ,n\) tend to \(\infty \). Then, we estimate the probability of large deviations of such linear statistics and use the estimate to prove a hole theorem. Our proofs are inspired by the methods of M. Sodin and B. Tsirelson for the one-dimensional case, and B. Shiffman and S. Zelditch for the study of the analogous problem for compact Kähler manifolds.     

Ribbon Schur Functions

We present a new determinantal expression for Schur functions. Previous expressions were due to Jacobi, Trudi, Giambelli and others (see [7]) and involved elementary symmetric functions or hook functions. We give, in Theorem 1.1, a decomposition of a Schur function into ribbon functions (also called skew hook functions, new functions by MacMahon, and MacMahon functions by others). We provide two different proofs of this result in Sections 2 and 3. In Section 2, we use Bazin's formula for the minors of a general matrix, as we already did in [6], to decompose a skew Schur function into hooks. In Section 3, we show how to pass from hooks to ribbons and conversely. In Section 4, we generalize to skew Schur functions. In Section 5, we give some applications, and show how such constructions, in the case of staircase partitions, generalize the classical continued fraction for the tangent function due to Euler.     

Equality of Schur and Skew Schur Functions

We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables. Received May 29, 2004     

Multiplicity Free Schur, Skew Schur, and Quasisymmetric Schur Functions

In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape have distinct descent sets. We conclude by providing such a classification for quasisymmetric Schur function families, giving a classification of Schur functions that are in some sense almost F-multiplicity free.     

Row-Strict Quasisymmetric Schur Functions

Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions, called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions, called the row-strict quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as quasisymmetic Schur functions are generated through fillings of composition diagrams. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships.     

The Schur Algorithm for Generalized Schur Functions II: Jordan Chains and Transformations of Characteristic Functions

 In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1–36) it was shown that for a generalized Schur function s(z), which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal s(z)⁻¹ of the characteristic function s(z) and general factorization results for characteristic functions. Received October 31, 2001; in revised form August 21, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

A Formula for the Specialization of Skew Schur Functions

We give a formula for \({s_{\lambda/\mu}(1, q, q^{2},\ldots) /s_{\lambda}(1, q, q^{2},\ldots)}\), which generalizes a result of Okounkov and Olshanski about \({f^{\lambda/\mu} / f ^{\lambda}}\).     

Lagrange Inversion and Schur Functions

Macdonald defined an involution on symmetric functions by considering the Lagrange inverse of the generating function of the complete homogeneous symmetric functions. The main result we prove in this note is that the images of skew Schur functions under this involution are either Schur positive or Schur negative symmetric functions. The proof relies on the combinatorics of Lagrange inversion. We also present a q-analogue of this result, which is related to the q-Lagrange inversion formula of Andrews, Garsia, and Gessel, as well as the operator of Bergeron and Garsia.     

On Odd Symplectic Schur Functions

Proctor defined combinatorially a family of Laurent Polynomials, called odd symplectic Schur functions, indexed by pairs (λ, c), where λ is partition andcis a column length of λ. A conjecture of Proctor (Invent. Math.92,1988, 307–332) includes the statement that the odd symplectic Schur functions are actually characters ofSp(2n + 1, C). The purpose of the present note is to prove this.     

Multiplicity-Free Products of Schur Functions

. We classify all multiplicity-free products of Schur functions and all multiplicity-free products of characters of SL(n, C).     

非可加集函数的Lebesgue分解

本文讨论一般的非可加集函数的Lebesgue分解定理,它是经典测度论中相应结果的扩充,同时,也为经典可加测度的Lebessue分解定理提供了另一证明方法.     

完全对称函数的Schur凸性

定义了一完全对称函数并研究该称函数的Schur凸性,Schur乘性凸性及Schur调和凸性,作为应用探讨了与其相关的一些不等式.     

m-Berezin Transform on the Polydisk

m-Berezin transforms are introduced for bounded operators on the Bergman space of the polydisk. We show several properties of m-Berezin transform and use them to show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the polydisk. A useful and sharp approximate identity of its m-Berezin transforms is obtained for a bounded operator.     

Multiple Integral Inequalities for Schur Convex Functions on Symmetric and Convex Bodies

In this paper, by making use of Divergence theorem for multiple integrals, we establish some integral inequalities for Schur convex functions defined on bodies $B⊂\mathbb{R}^n$ that are symmetric, convex and have nonempty interiors. Examples for three dimensional balls are also provided.     

Interpolation, Schur Functions and Moment Problems

No Abstract.  .