arXiv subjects

Kunyu Guo

Publications and source records attributed to Kunyu Guo.

At least 19 recordsLinked to original sources

arXiv2025

The one-weight inequality for $\mathcal{H}$-harmonic Bergman projection

Let $n\geqslant 3$ be an integer. For the Bekoll\'e-Bonami weight $\omega$ on the real unit ball $\mathbb{B}_n$, we obtain the following sharp one-weight estimate for the $\mathcal{H}$-harmonic Bergman projection: for $1<p<\infty$ and $-1<\alpha<\infty$, \[||P_\alpha||_{ L^p(\omega d\nu_\alpha)\longrightarrow L^p(\omega d\nu_\alpha)}\leqslant C [\omega]_{p,\alpha}^{\max\left\{1,\frac{1}{p-1}\right\}}, \] where $[\omega]_{p,\alpha}$ is the Bekoll\'e-Bonami constant. Our proof is inspired by the dyadic harmonic analysis, and the key ingredient involves the discretization of the Bergman kernel for the $\mathcal{H}$-harmonic Bergman spaces.

math.FA↗

arXiv2025

Contractive projections, conditional expectations, and idempotent coefficient multipliers on $H^p$ spaces $(0<p<1)$

In this paper, we investigate contractive projections, conditional expectations, and idempotent coefficient multipliers on the Hardy spaces $H^p(\mathbb{T})$ for $0<p<1$. For such values of $p$, we first establish a general extension theorem for contractive projections in a probability $L^p$-space. Combining this theorem with the study of conditional expectations on $H^p(\mathbb{T})$, we characterize a broad class of contractive projections on $H^p(\mathbb{T})$ that are of particular interest. Furthermore, we apply these results to give a complete characterization of contractive idempotent coefficient multipliers for the Hardy spaces $H^p(\mathbb{T}^d)$ on the $d$-dimensional torus for $0<p<1$ and $1\leq d\leq \infty$. This complements a remarkable result of Brevig, Ortega-Cerd\`{a}, and Seip characterizing such multipliers on $H^p(\mathbb{T}^d)$ for $1\leq p \leq \infty$.

math.FA↗

arXiv2024

Extension of contractive projections

Through the establishment of several extension theorems, we provide explicit expressions for all contractive projections and 1-complemented subspaces in the Hardy space $H^p(\mathbb{T})$ for $1\leq p<\infty$, $p\neq 2$. Our characterization leads to two corollaries: first, all nontrivial 1-complemented subspaces of $H^p(\mathbb{T})$ are isometric to $H^p(\mathbb{T})$; second, all contractive projections on $H^p(\mathbb{T})$ are restrictions of contractive projections on $L^p(\mathbb{T})$ that leave $H^p(\mathbb{T})$ invariant. The first corollary provides examples of prime Banach spaces \emph{in the isometric sense}, while the second answers a question posed by P. Wojtaszczyk in 2003.

math.FA↗

arXiv2024

Hankel matrices acting on the Dirichlet space

The characterization of the boundedness of operators induced by Hankel matrices on analytic function spaces can be traced back to the work of Z. Nehari and H. Widom on the Hardy space, and has been extensively studied on many other analytic function spaces recently. However, this question remains open in the context of the Dirichlet space [20]. By Carleson measures, the Widom type condition and the reproducing kernel thesis, this paper provides a comprehensive solution to this question. As a beneficial product, characterizations of the boundedness and compactness of operators induced by Cesàro type matrices on the Dirichlet space are given. In addition, we also show that a random Dirichlet function almost surely induces a compact Hankel type operator on the Dirichlet space.

math.CV↗

arXiv2023

The first Szegő limit theorem on multi-dimensional torus

In this paper, we consider the first Szegő limit theorems on $d$-torus $\mathbb{T}^d$ for $1\leq d\leq +\infty$. It is shown that for any Følner sequence $\{σ_N\}$ of $\mathbb{Z}^d$ and $φ\in L^1_+(\mathbb{T}^d)$, it holds that $$ \lim_{N\rightarrow \infty}\left(\det T_{σ_N}φ\right)^{\frac{1}{|σ_N|}}=\exp\left(\int_{\mathbb{T}^d} \logφ~dm_{d}\right). $$ In the case $d=+\infty$, we are associated with multiplicative Toeplitz matrix $T φ=\{\widehatφ(j/i)\}_{i,j\in\mathbb{N}}$ and the most concerned non-Følner truncation, that is, $T_N φ=\{\widehatφ(j/i)\}_{1\leq i,j\leq N}$, where $σ_N=\{1,\dots,N\}$. It is shown that for each $φ\in L^\infty_{\mathbb{R}}(\mathbb{T^{\infty}})$ and $f\in C[\text{ess-inf} ~φ,~\text{ess-sup}~φ]$, the limit $\lim_{N\rightarrow \infty} \frac{1}{N}\mathrm{Tr} f \big(T_N φ\big)$ exsits. Moreover, it is proven that the limit $\lim_{N\rightarrow \infty}\left(\det T_N φ\right)^{\frac{1}{N}}$ exists for any $φ\in L^1_+(\mathbb{T}^\infty)$ with strictly positive essential infimum. These results are directly related to two problems posed by Nikolski and Pushnitski.

math.FA↗

arXiv2023

Essentially normal quotient weighted Bergman modules over the bidisk and distinguished varieties

We introduce a Grassmannian structure for a class of quotient Hilbert modules and attack the polydisc version of Arveson-Douglas conjecture associated to distinguished varieties. More interestingly, we obtain an operator-theoretic characterization of distinguished varieties in the bidisk in terms of essential normality of the quotient modules. As an application, we study the K-homology of the boundary of distinguished variety.

math.OA↗

arXiv2022

Nevanlinna class, Dirichlet series and Szegö's problem

This paper is associated with Nevanlinna class, Dirichlet series and Szegö's problem in infinitely many variables. As we will see, there is a natural connection between these topics. The paper first introduces the Nevanlinna class and the Smirnov class in this context, and generalizes the classical theory in finitely many variables to the infinite-variable setting. These results applied to Szegö's problem on Hardy spaces in infinitely many variables. Moreover, this paper is also devoted to the study of the correspondence between the Nevanlinna functions and Dirichlet series.

math.CV↗

arXiv2022

On the $p$-essential normality of principal submodules of the Bergman module on strongly pseudoconvex domains

In this paper, we show that under a mild condition, a principal submodule of the Bergman module on a bounded strongly pseudoconvex domain with smooth boundary in $\mathbb{C}^n$ is $p$-essentially normal for all $p>n$. This improves a previous result by the first author and K. Wang, in which it was shown that any polynomial-generated principal submodule of the Bergman module on the unit ball $\mathbb{B}_n$ is $p$-essentially normal for all $p>n$. As a consequence, we show that the submodule of $L_a^2(\mathbb{B}_n)$ consisting of functions vanishing on an analytic subset of pure codimension $1$ is $p$-essentially normal for all $p>n$.

math.FA↗

arXiv2021

Toeplitz operators on weighted Bergman spaces induced by a class of radial weights

Suppose that $ω$ is a radial weight on the unit disk that satisfies both forward and reverse doubling conditions. Using Carleson measures and $T1$-type conditions, we obtain necessary and sufficient conditions of the positive Borel measure $μ$ such that the Toeplitz operator $T_{μ,ω}:L^p_a(ω)\to L_a^1(ω)$ is bounded and compact for $0<p\leq 1$. In addition, we obtain a bump condition for the bounded Toeplitz operators with $L^1(ω)$ symbol on $L^1_a(ω)$. This generalizes a result of Zhu in \cite{zhu1989}.

math.FA↗

arXiv2021

A Gaussian version of Littlewood's theorem on random power series

We prove a Littlewood-type theorem on random analytic functions for not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space $H^2(\dd)$ by a Gaussian process whose covariance matrix $K$ induces a bounded operator on $l^2$, then the resulting random function is almost surely in $H^p(\dd)$ for any $p>0$. The case $K=\text{Id}$, the identity operator, recovers Littlewood's theorem. A new ingredient in our proof is to recast the membership problem as the boundedness of an operator. This reformulation enables us to use tools in functional analysis and is applicable to other situations. The sharpness of the new condition and several ramifications are discussed.

math.FA↗

arXiv2021

Invariant subspaces of weighted Bergman spaces in infinitely many variables

This paper is concerned with polynomially generated multiplier invariant subspaces of the weighted Bergman space $A_{\boldsymbolβ}^2$ in infinitely many variables. We completely classify these invariant subspaces under the unitary equivalence. Our results not only cover cases of both the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$ and the Bergman space $A^{2}(\mathbb{D}_{2}^{\infty})$ in infinitely many variables, but also apply in finite-variable setting.

math.FA↗

arXiv2020

Projections in Toeplitz algebra

Motivated by Barr{\'ı}a-Halmos's \cite[Question 19]{barria1982asymptotic} and Halmos's \cite[Problem 237]{Halmos1978A}, we explore projections in Toeplitz algebra on the Hardy space. We show that the product of two Toeplitz (Hankel) operators is a projection if and only if it is the projection onto one of the invariant subspaces of the shift (backward shift) operator. As a consequence one obtains new proofs of criterion for Toeplitz operators and Hankel operators to be partial isometries. Furthermore, we completely characterize when the self-commutator of a Toeplitz operator is a projection. This provides a class of nontrivial projections in Toeplitz algebra.

math.FA↗

arXiv2020

The spectral picture of Bergman Toeplitz operators with harmonic polynomial symbols

In this paper, it is shown that some new phenomenon related to the spectra of Toeplitz operators with bounded harmonic symbols on the Bergman space. On the one hand, we prove that the spectrum of the Toeplitz operator with symbol ${\bar{z}+p}$ is always connected for every polynomial $p$ with degree less than $3$. On the other hand, we show that for each integer $k$ greater than $2$, there exists a polynomial $p$ of degree $k$ such that the spectrum of the Toeplitz operator with symbol ${\bar{z}+p}$ has at least one isolated point but has at most finitely many isolated points. Then these results are applied to obtain a new class of non-hyponormal Toeplitz operators with bounded harmonic symbols on the Bergman space for which Weyl's theorem holds.

math.FA↗

arXiv2020

Power dilation systems $\{f(z^k)\}_{k\in\mathbb{N}}$ in Dirichlet-type spaces

In this paper, we concentrate on power dilation systems $\{f(z^k)\}_{k\in\mathbb{N}}$ in Dirichlet-type spaces $\mathcal{D}_t\ (t\in\mathbb{R})$. When $t\neq0$, we prove that $\{f(z^k)\}_{k\in\mathbb{N}}$ is orthogonal in $\mathcal{D}_t$ only if $f=cz^N$ for some constant $c$ and some positive integer $N$. We also give complete characterizations of unconditional bases and frames formed by power dilation systems for Drichlet-type spaces.

math.FA↗

arXiv2020

The Kozlov completeness problem

This paper concerns a long-standing problem raised by Kozlov on completeness of the dilation systems $\{\mathbf{1}_{(\alpha,\beta)}(kx):k=1,2,\cdots\}$ generated by odd periodic extensions on $\mathbb{R}$ of characteristic functions $\mathbf{1}_{(\alpha,\beta)}$, where $0\leq\alpha<\beta\leq1$. Up to now there has only some fragmentary results under the assumption $\alpha=0$. Focusing on the dilation completeness problem for characteristic functions $\mathbf{1}_V$ of open subsets $V\subset(0,1)$ that are finite unions of intervals with rational endpoints, we exhibit the exact forms of such $V$ in almost all interesting situations by using substantially techniques from analytic number theory. As a consequence, it yields a complete solution for the rational version of the Kozlov completeness problem. Moreover, our results also illustrate the fascinating connection among the Completeness Problem, the Twin Prime Conjecture and the Sophie Germain Prime Conjecture.

math.CA↗

arXiv2020

Dilation theory and analytic model theory for doubly commuting sequences of $C_{.0}$-contractions

Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has motivated lots of further works on model theory and generalizations to commuting tuples of $C_{\cdot0}$-contractions. In this paper, we focus on doubly commuting sequences of $C_{\cdot0}$-contractions, and establish the dilation theory and the analytic model theory for these sequences of operators. These results are applied to generalize the Beurling-Lax theorem and Jordan blocks in the multivariable operator theory to the operator theory in countably infinitely many variables.

math.FA↗

arXiv2020

The Periodic Dilation Completeness Problem: Cyclic vectors in the Hardy space over the infinite-dimensional polydisk

The classical completeness problem raised by Beurling and independently by Wintner asks for which $ψ\in L^2(0,1)$, the dilation system $\{ψ(kx):k=1,2,\cdots\}$ is complete in $L^2(0,1)$, where $ψ$ is identified with its extension to an odd $2$-periodic function on $\mathbb{R}$. This difficult problem is nowadays commonly called as the Periodic Dilation Completeness Problem (PDCP). By Beurling's idea and an application of the Bohr transform, the PDCP is translated as an equivalent problem of characterizing cyclic vectors in the Hardy space $\mathbf{H}_\infty^2$ over the infinite-dimensional polydisk for coordinate multiplication operators. In this paper, we obtain lots of new results on cyclic vectors in the Hardy space $\mathbf{H}_\infty^2$. In almost all interesting cases, we obtain sufficient and necessary criterions for characterizing cyclic vectors, and hence in these cases we completely solve the PDCP. Our results cover almost all previous known results on this subject.

math.FA↗

arXiv2019

A Sharp Inequality of Hardy-Littlewood Type Via Derivatives

In this paper we consider a generalized version of Carleman's inequality. An equivalent version of it states that $\|f\|_{A_α^{2α}}\leq\|f\|_{H^2}$, where $f$ is a holomorphic function and $α>1$. If the norms $\|f\|_{A_α^{2α}}$ are decreasing in $α$, then the inequality holds for $f$. For a dense set of functions, we calculate the derivative of the norms $\|f\|_{A_α^{2α}}$ in $α$ and give sufficient conditions for this derivative to be non-positive. As an application, we prove the inequality for linear combinations of two reproducing kernels. Some numerical evidences are also provided.

math.FA↗