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Guanlong Bao

Publications and source records attributed to Guanlong Bao.

14 recordsLinked to original sources
arXiv2025

New characterizations for Fock spaces

We show that the maximal Fock space $F^\infty_α$ on $C^n$ is a Lipschitz space, that is, there exists a distance $d_α$ on $C^n$ such that an entire function $f$ on $C^n$ belongs to $F^\infty_α$ if and only if $$|f(z)-f(w)|\le Cd_α(z,w)$$ for some constant $C$ and all $z,w\in C^n$. This can be considered the Fock space version of the following classical result in complex analysis: a holomorphic function $f$ on the unit ball $B_n$ in $C^n$ belongs to the Bloch space if and only if there exists a positive constant $C$ such that $|f(z)-f(w)|\le Cβ(z,w)$ for all $z,w\in B_n$, where $β(z,w)$ is the distance on $B_n$ in the Bergman metric. We also present a new approach to Hardy-Littlewood type characterizations for $F^p_α$.

math.CV↗
arXiv2025

Embedding and compact embedding between Bergman and Hardy spaces

For Hardy spaces and weighted Bergman spaces on the open unit ball in ${\mathbb C}^n$, we determine exactly when $A^p_α\subset H^q$ or $H^p\subset A^q_α$, where $0<q<\infty$, $0<p<\infty$, and $-\infty<α<\infty$. For each such inclusion we also determine exactly when it is a compact embedding. Although some special cases were known before, we are able to completely cover all possible cases here. We also introduce a new notion called {\it tight fitting} and formulate a conjecture in terms of it, which places several prominent known results about contractive embeddings in the same framework.

math.CV↗
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

Duality for $α$-Möbius invariant Besov spaces

For $1\leq p\leq \infty$ and $α>0$, Besov spaces $B^p_α$ play a key role in the theory of $α$-Möbius invariant function spaces. In some sense, $B^1_α$ is the minimal $α$-Möbius invariant function space, $B^2_α$ is the unique $α$-Möbius invariant Hilbert space, and $B^\infty_α$ is the maximal $α$-Möbius invariant function space. In this paper, under the $α$-Möbius invariant pairing and by the space $B^\infty_α$, we identify the predual and dual spaces of $B^1_α$. In particular, the corresponding identifications are isometric isomorphisms. The duality theorem via the $α$-Möbius invariant pairing for $B^p_α$ with $p>1$ is also given.

math.CV↗
arXiv2022

A Carleson type measure and a family of Möbius invariant function spaces

For $0<s<1$, let $\{z_n\}$ be a sequence in the open unit disk such that $\sum_n (1-|z_n|^2)^s δ_{z_n}$ is an $s$-Carleson measure. In this paper, we consider the connections between this $s$-Carleson measure and the theory of Möbius invariant $F(p, p-2, s)$ spaces by the Volterra type operator, the reciprocal of a Blaschke product, and second order complex differential equations having a prescribed zero sequence.

math.CV↗
arXiv2022

Double integral estimates for Besov type spaces and their applications

For $0<p<\infty$, we give a complete description of nonnegative radial weight functions $ω$ on the open unit disk $\mathbb{D}$ such that $$ \int_{\mathbb{D}} |f'(z)|^p (1-|z|^2)^{p-2}ω(z)dA(z)<\infty $$ if and only if $$ \int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(ζ)|^p}{|1-\overlineζz|^{4+τ+σ}}(1-|z|^2)^τ(1-|ζ|^2)^σω(ζ)dA(z)A(ζ)<\infty $$ for all analytic functions $f$ in $\mathbb{D}$, where $τ$ and $σ$ are some real numbers. As applications, we give some geometric descriptions of functions in Besove type spaces $B_p(ω)$ with doubling weights, and characterize the boundedness and compactness of Hankel type operators related to Besov type spaces with radial Békollé-Bonami weights. Some special cases of our results are new even for some standard weighted Besov spaces.

math.CV↗
arXiv2017

The Bloch space and the dual space of a Luecking-type subspace of $A^1$

Let $X$ be the dual space of a Luecking-type subspace of the Bergman space $A^1$. It is known that the Bloch space $\mathcal{B}$ is a subset of $X$. In 1990, Ghatage and Sun asked whether $\mathcal{B}$ is dense in $X$. They also asked whether the little version of $X$ is a subset of $\mathcal{B}$. In this note, based on results and methods of Girela, Peláez, Pérez-González and Rättyä in 2008, we answer the two questions in the negative.

math.CV↗
arXiv2016

On absolute values of QK functions

In this paper, the effect of absolute values on the behavior of functions $f$ in the spaces $\mathcal{Q}_K$ is investigated. It is clear that $f\in \mathcal{Q}_K(\partial {\mathbb{D}}) \Rightarrow |f|\in \mathcal{Q}_K(\partial {\mathbb{D}})$, but the converse is not always true. For $f$ in the Hardy space $H^2$, we give a condition involving the modulus of the function only, such that this condition together with $|f|\in \mathcal{Q}_K(\partial {\mathbb{D}})$ is equivalent to $f\in \mathcal{Q}_K$. As an application, a new criterion for inner-outer factorisation of $\mathcal{Q}_K$ spaces is given. These results are also new for $\mathcal{Q}_p$ spaces.

math.CV↗
arXiv2015

Boundary multipliers of a family of Möbius invariant function spaces

For $1<p<\infty$ and $0<s<1$, let $\mathcal{Q}^p_ s (\mathbb{T})$ be the space of those functions $f$ which belong to $ L^p(\mathbb{T})$ and satisfy \[ \sup_{I\subset \mathbb{T}}\frac{1}{|I|^s}\int_I\int_I\frac{|f(ζ)-f(η)|^p}{|ζ-η|^{2-s}}|dζ||dη|<\infty, \] where $|I|$ is the length of an arc $I$ of the unit circle $\mathbb{T}$ . In this paper, we give a complete description of multipliers between $\mathcal{Q}^p_ s (\mathbb{T})$ spaces. The spectra of multiplication operators on $\mathcal{Q}^p_ s (\mathbb{T})$ are also obtained.

math.CV↗
arXiv2015

The pseudoanalytic extensions for some spaces of analytic functions

Using the Cauchy-Riemann operator, we characterize $Q_K$ spaces, Besov spaces and analytic Morrey spaces in terms of pseudoanalytic extensions of primitive functions. Our results are also true on some classical Banach spaces, such as the Bloch space, $BMOA$ and the Dirichlet space.

math.CV↗