arXiv subjects

Lijia Ding

Publications and source records attributed to Lijia Ding.

5 recordsLinked to original sources

arXiv2024

Essential normality of quotient submodules over strongly pseudoconvex finite manifolds

We investigate the $p$-essential normality of Hilbert quotient submodules on a relatively compact smooth strongly pseudoconvex domain in a complex manifold satisfying Property (S). For analytic subvarieties that have compact singularities and transversely intersect the strongly pseudoconvex boundary, we prove that the corresponding Bergman-Sobolev quotient submodules are $p$-essentially normal whenever $p$ exceeds the dimension of the noncompact part of the analytic subvarieties. As a consequence, we partially confirm the geometric Arveson-Douglas Conjecture and resolve an open problem regarding the trace-class antisymmetric sum of truncated Toeplitz operators within a broader context. Moreover, we provide applications in $K$-homology and geometric invariant theory.

math.CV↗

arXiv2023

The biholomorphic invariance of essential normality on bounded symmetric domains

This paper mainly concerns the biholomorphic invariance of $p$-essential normality of Hilbert modules on bounded symmetric domains. By establishing new integral formulas concerning rational function kernels for the Taylor functional calculus, we prove a biholomorphic invariance result related to the $p$-essential normality. Furthermore, for quotient analytic Hilbert submodules determined by analytic varieties, we develop an algebraic approach to proving that the $p$-essential normality is preserved invariant if the coordinate multipliers are replaced by arbitrary automorphism multipliers. Moreover, the Taylor spectrum of the compression tuple is calculated under a mild condition, which gives a solvability result of the corona problem for quotient submodules. As applications, we extend the recent results on the equivalence between $\infty$-essential normality and hyperrigidity.

math.FA↗

arXiv2020

Schatten class Bergman-type and Szegö-type operators on bounded symmetric domains

This is our third work on Bergman-type operator over bounded domains. In the previous two articles, we systematically study the boundedness, compactness and Schatten membership of Bergman-type on the Hilbert unit ball. In the present paper, we investigate singular integral operators induced by the Bergman kernel and Szegö kernel on the irreducible bounded symmetric domain in its standard Harish-Chandra realization. We completely characterize when Bergman-type operators and Szegö-type operators belong to Schatten class operator ideals by several analytic numerical invariants of the bounded symmetric domain. These results generalize a recent result on the Hilbert unit ball due to the author and his coauthor but also cover all irreducible bounded symmetric domains. Moreover, we obtain two trace formulae and a new integral estimate related to the Forelli-Rudin estimate. The key ingredient of the proofs involves the function theory on the bounded symmetric domain and the spectrum estimate of Bergman-type and and Szegö-type operators.

math.FA↗

arXiv2020

On the compactness of Bergman-type integral operators

Bergman-type integral operators are classical operators in complex analysis and operator theory. Recently, the first author and his collaborator \cite{DiW} completely characterized the $L^p$-$L^q$ boundedness of Bergman-type integral operators $K_α,K_α^+$ and the $L^p$-$L^q$ compactness of $K_α$ on the unit ball. In this paper, we will use a substantially new method to completely characterize the $L^p$-$L^q$ compactness of $K_α^+,$ but also prove that the $L^p$-$L^q$ compactness of operators $K_α,K_α^+$ is in fact equivalent. Moreover, we completely characterize Schatten class and Macaev class Bergman-type integral operator $K_α$ on $L^2$ space and Bergman space via inequalities related to the dimension of the unit ball, and we also give an intrinsic characterization by introducing the concept of Hausdorff dimension of compact operators. The Dixmier trace of $K_α$ are also calculated in this paper.

math.FA↗

arXiv2020

The Lp-Lq problems of Bergman-type operators

Let $\mathbb{B}^d$ be the unit ball on the complex space $\mathbb{C}^d$ with normalized Lebesgue measure $dv.$ For $α\in\mathbb{R},$ denote $k_α(z,w)=\frac{1}{(1-\langle z,w\rangle)^α},$ the Bergman-type integral operator $K_α$ on $L^1(\mathbb{B}^d,dv)$ is defined by $$ K_αf(z)=\int_{\mathbb{B}^d}k_α(z,w)f(w)dv(w).$$ It is an important class of operators in the holomorphic function space theory over the unit ball. We also consider the integral operator $K_α^+$ on $L^1(\mathbb{B}^d,dv)$ which is given by $$ K_α^+ f(z)=\int_{\mathbb{B}^d}\vert k_α(z,w)\vert f(w)dv(w).$$ In this paper, we completely characterize the $L^p$-$L^q$ boundedness of $K_α,K_α^+$ and $L^p$-$L^q$ compactness of $K_α.$ The results of boundedness are in fact the Hardy-Littlewood-Sobolev theorem but also prove the conjecture of G. Cheng et al [Trans. Amer. Math. Soc. (2017), MR3710638 ] in the case of bounded domain $\mathbb{B}^d.$ Meanwhile, a trace formula and some sharp norm estimates of $K_α,K_α^+$ are given.

math.FA↗