From Toeplitz Eigenvalues through Green's Kernels to Higher-Order Wirtinger-Sobolev Inequalities

Toeplitz operators on Fock-Sobolev type spaces

In this paper, we characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on weighted Fock-Sobolev spaces of fractional order.

Commutants of Toeplitz operators with radial symbols on the Fock-Sobolev space

In the setting of the Fock space over the complex plane, Bauer and Lee have recently characterized commutants of Toeplitz operators with radial symbols, under the assumption that symbols have at most polynomial growth at infinity. Their characterization states: If one of the symbols of two commuting Toeplitz operators is nonconstant and radial, then the other must be also radial. We extend this result to the Fock-Sobolev spaces.

经典环面与量子环面上的尖锐完全修正对数索伯列夫不等式

Sharp Complete Modified Log-Sobolev Inequalities on Classical and Quantum Tori

该研究证明圆周热半群的最优完全修正对数索伯列夫常数为1，借助矩阵值Wirtinger不等式得到Fisher信息收缩，进而通过张量化与转移原理推广到经典、量子环面热半群，确定其尖锐常数。

Toeplitz operators and weighted Bergman kernels

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. Finally, we also exhibit a holomorphic continuation of the kernels with respect to the Sobolev parameter to the entire complex plane. Our main tool are the generalized Toeplitz operators of Bout…

正交多项式的零点与若干矩阵不等式

Zeros of orthogonal polynomials and some matrix inequalities

本研究利用正交多项式的矩阵方法、有界点赋值概念与Wirtinger型不等式，证明了多类向量测度下Sobolev正交多项式的零点有界性，突破了序列控制的前提限制。

Loops in SU(2) and Factorization, II

In the prequel to this paper, we proved that for a $SU(2,\mathbb C)$ valued loop having the critical degree of smoothness (one half of a derivative in the $L^2$ Sobolev sense), the following statements are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. This hinges on some Plancherel-esque formulas for determinants of Toeplitz operators. The main point of this report is is to outline a generalization of this result to loops of vanishi…

Polyanalytic Toeplitz operators: isomorphisms, symbolic calculus and approximation of Weyl operators

We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.

Fokker-Planck equations and one-dimensional functional inequalities for heavy tailed densities

We study one-dimensional functional inequalities of the type of Poincaré, logarithmic Sobolev and Wirtinger, with weight, for probability densities with polynomial tails. As main examples, we obtain sharp inequalities satisfied by inverse Gamma densities, taking values on $R_+$, and Cauchy-type densities, taking values on $R$. In this last case, we improve the result obtained by Bobkov and Ledoux in 2009 by introducing a better weight function in the logarithmic Sobolev inequality. The results are obtained by resorting to Fokker-Planck type equations which possess these densities as steady sta…