arXiv subjects

Guangbin Ren

Publications and source records attributed to Guangbin Ren.

44 records · Page 3Linked to original sources

arXiv2016

Boundary Schwarz lemma for holomorphic self-mappings of strongly pseudoconvex domains

In this paper, we generalize a recent work of Liu et al. from the open unit ball $\mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Carathéodory metric near the boundary of $C^2$ domains and the well-known Graham's estimate on the boundary behavior of the Carathéodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.

math.CV↗

arXiv2015

Algebraic approach to slice monogenic functions

In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this paper, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. Subsequently, an inner product is defined corresponding to this slice Dirac operator and its polynomial null-solutions are determined. Finally, analogues of the Hermite polynomials and Hermite functions are constructed in this context and their properties are studied.

math.CV↗

arXiv2015

Extremal functions of boundary Schwarz lemma

In this paper, we present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Carathéodory theorem for univalent holomorphic self-mappings of the open unit disk $\mathbb D\subset \mathbb C$. Our approach has its extra advantage to get the extremal functions of the inequality in the boundary Schwarz lemma.

math.CV↗

arXiv2011

(Discrete) Almansi Type Decompositions: An umbral calculus framework based on $\mathfrak{osp}(1|2)$ symmetries

We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multivariate polynomials $\BR[\underline{x}]$ shall be described in terms of the generators of the Weyl-Heisenberg algebra. The extension of $\BR[\underline{x}]$ to the algebra of Clifford-valued polynomials $\mathcal{P}$ gives rise to an algebra of Clifford-valued operators whose canonical generators are isomorphic to the orthosymplectic Lie algebra $\mathfrak{osp}(1|2)$. This extension provides an effective framework in continuity and discreteness that allow us to establish an alternative formulation of Almansi decomposition in Clifford analysis (c.f. \cite{Ryan90,MR02,MAGU}) that corresponds to a meaningful generalization of Fischer decomposition for the subspaces $\ker (D')^k$. We will discuss afterwards how the symmetries of $\mathfrak{sl}_2(\BR)$ (even part of $\mathfrak{osp}(1|2)$) are ubiquitous on the recent approach of \textsc{Render} (c.f. \cite{Render08}), showing that they can be interpreted in terms of the method of separation of variables for the Hamiltonian operator in quantum mechanics.

math.CV↗

arXiv2009

Complex Dunkl operators

The real theory of the Dunkl operators has been developed very extensively, while there still lacks the corresponding complex theory. In this paper we introduce the complex Dunkl operators for certain Coxeter groups. These complex Dunkl operators have the commutative property, which makes it possible to establish a corresponding complex analysis of Dunkl operators.

math.CV↗

arXiv2009

Almansi Theorems in Umbral Clifford Analysis and the Quantum Harmonic Oscillator

We introduce the Umbral calculus into Clifford analysis starting from the abstract of the Heisenberg commutation relation $[\frac{d}{dx}, x] = {\bf id}$. The Umbral Clifford analysis provides an effective framework in continuity and discreteness. In this paper we consider functions defined in a star-like domain $Ω\subset \BR^n$ with values in the Umbral Clifford algebra $C\ell_{0,n}'$ which are Umbral polymonogenic with respect to the (left) Umbral Dirac operator $D'$, i.e. they belong to the kernel of $(D')^k$. We prove that any polymonogenic function $f$ has a decomposition of the form $$f=f_1+ x'f_2 + ... + (x')^{k-1}f_k,$$ where $x'=x'_1e_1 + ... + x'_ne_n$ and $f_j, j=1,..., k,$ are Umbral monogenic functions. As examples, this result recoveries the continuous version of the classical Almansi theorem for derivatives and establishes the discrete version of Almansi theorem for difference operator. The approach also provides a similar result in quantum field about Almansi decomposition related to Hamilton operators. Some concrete examples will presented for the discrete analog version of Almansi Decomposition and for the quantum harmonic oscillator.

math.CA↗

arXiv2008

Liouville Theorem for Dunkl Polyharmonic Functions

Assume that $f$ is Dunkl polyharmonic in $\mathbb{R}^n$ (i.e. $(Δ_h)^p f=0$ for some integer $p$, where $Δ_h$ is the Dunkl Laplacian associated to a root system $R$ and to a multiplicity function $κ$, defined on $R$ and invariant with respect to the finite Coxeter group). Necessary and successful condition that $f$ is a polynomial of degree $\le s$ for $s\ge 2p-2$ is proved. As a direct corollary, a Dunkl harmonic function bounded above or below is constant.

math.CA↗

arXiv1996

Gleason's problem in weighted Bergman space on egg domains

In the paper, we discuss on the egg domains: $$ Ω_a=\left\{ξ=(z,w)\in\bold C^{n+m}: \ z\in\bold C^n, \ w\in\bold C^m, |z|^2+|w|^{2/a}<1\right\}, \qquad 0<a\le 2. $$ We show that Gleason's problem can be solved in the weight Bergman space on theegg domains. The proof will need the help of the recent work of the second named author on the weighted Bergman projections on this kind of domain. As an application, we obtain a multiplier theorem on the egg domains.

math.CV↗