arXiv subjects

Xiaojing Zhou

Publications and source records attributed to Xiaojing Zhou.

11 recordsLinked to original sources

arXiv2026

Cancellation of complex kernels and sharp critical lines and endpoint theory for the Forelli--Rudin operators I: the purely hypersingular case

For $a,b,c\in\mathbb R$, we consider the Forelli--Rudin operators $$ T_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{(1-z\overline w)^c}f(w)\,dA(w) $$ and their positive counterparts $$ S_{a,b,c}f(z):=(1-|z|^2)^a\int_{\mathbb D}\frac{(1-|w|^2)^b}{|1-z\overline w|^c}f(w)\,dA(w). $$ We obtain a complete and sharp classification of their weak- and restricted weak-type mapping properties in the hypersingular regime $$ Ω_{\mathcal H}:=\{(p,q):1\leq p,q\leq\infty,\ p>q\}, $$ thereby substantially extending the recent work of the first and fourth authors on hypersingular Bergman projections. One of the main discoveries of this work is an intrinsic cancellation phenomenon associated with the complex Forelli--Rudin kernel: at certain critical endpoints, cancellation creates a sharp separation between the two operators, with $T_{a,b,c}$ remaining bounded while its positive counterpart $S_{a,b,c}$ fails to be bounded. Perhaps surprisingly, this cancellation is invisible in the strong $L^p$--$L^q$ theory established by Zhao and Zhou in 2022, where the two operators have the same boundedness range, and emerges only at the weak- and restricted weak-type levels. Allowing the parameters $a,b,c$ to vary, we show that the collection of all weak-type Forelli--Rudin pairs in $Ω_{\mathcal{H}}$ is precisely $$ \mathcal{FR}_w=\{(p,q)\inΩ_{\mathcal H}:1<p\leq2\}, $$ whereas the collection of all restricted weak-type Forelli--Rudin pairs is $$ \mathcal{FR}_{rw}=\{(p,q)\inΩ_{\mathcal H}:p\neq\infty\}. $$ These ranges, together with all corresponding endpoint estimates and failures, are sharp. Our approach combines dyadic decompositions, probabilistic constructions, and weak-type Hardy estimates.

math.FA↗

arXiv2026

A Full Characterization of the Dirichlet Carleson embedding $\operatorname{id}: \mathcal D_{p-1}^p \to L^p(μ)$ for $p>2$

In this paper, we obtain a full characterization of the finite positive Borel measures $μ$ on $\mathbb D$ for which the embedding $$ \operatorname{id}:\mathcal D_{p-1}^p\longrightarrow L^p(μ),\qquad p>2, $$ is bounded. More precisely, for any dyadic system $\mathcal D$ on $\mathbb T$, we prove that this embedding is bounded if and only if $$ \mathcal C_{p,\mathcal D}(μ)+\mathcal H_{p,\mathcal D}(μ)<\infty, $$ where $\mathcal C_{p,\mathcal D}(μ)$ and $\mathcal H_{p,\mathcal D}(μ)$ denote the packing energy and the Haar energy of $μ$, respectively. This resolves a longstanding characterization problem in the theory of Dirichlet-type spaces that arose from Wu's 1999 conjecture, corresponds to the endpoint case not covered by the work of Arcozzi, Rochberg, and Sawyer in 2002, and remained open after the works of Girela and Peláez in 2006 and Galanopoulos, Girela, and Peláez in 2011. We also construct finite measures showing that the two energy conditions are genuinely distinct. The key ingredient in the proof is a reduction of the Dirichlet embedding to a dyadic Whitney embedding, which allows us to combine weighted Hardy inequalities on trees with probabilistic arguments.

math.CV↗

arXiv2026

A Counterexample to the Liu--Lou--Zhu $\mathcal Q_p$--Carleson Embedding Conjecture

In this paper, we disprove a conjecture of Liu, Lou, and Zhu concerning Carleson embeddings of $\mathcal Q_p$ spaces into tent spaces for $0<p<1$. More precisely, we construct a finite positive $p$-Carleson measure $μ$ on the unit disc $\mathbb D$ such that the canonical embedding $$ \operatorname{id}:\mathcal Q_p \longrightarrow \mathcal T_{p,2}^2(μ) $$ is not bounded. The main ingredient is a new family of $\mathcal Q_p$ test functions that encodes Cantor-type structures on the unit circle $\mathbb T$ into the analytic behavior of functions in $\mathcal Q_p$. This construction is inspired by ideas developed in a recent work of the first and third authors on composition operators on $\mathcal Q_p$ spaces.

math.CV↗

arXiv2026

On the holomorphic differential operator $\frac{d}{dz}: Q_K\to L^q(WdA)$

In this paper, we obtain non-testing characterizations, in terms of dyadic capacity gauges, of the boundedness and compactness of the differentiation operator $$ \frac{d}{dz}:Q_K\longrightarrow L^q(W\,dA), \qquad 0<q<\infty. $$ We also characterize the limiting case as $q\to0^+$, formulated in terms of a logarithmic geometric mean, while the endpoint $q=\infty$ is treated separately using a standard testing argument. These results greatly extend the previous work on ${\mathcal Q}_p$-spaces to the general setting of $Q_K$-spaces. As applications, we characterize composition operators and Volterra-type integral operators between different $Q_K$-spaces. In particular, the off-diagonal characterization established here, together with the previously established diagonal case, completely resolves Zhao's 2009 open question on composition operators between ${\mathcal Q}_p$-spaces.

math.CV↗

arXiv2026

On the growth of Bloch functions

We prove that there exist two Bloch functions $f_1$ and $f_2$ on $\mathbb D$ such that $$ |f_1(z)|+|f_2(z)| \geq \left(\log\frac{1}{1-|z|}\right)^{1/2}, \qquad z\in\mathbb D, $$ thereby resolving an open problem posed in 2008 by Girela, Peláez, Pérez-González and Rättyä. Our proof is based on a new Szegő-type recursion involving $\mathbb C^2$-valued polynomials and their reciprocal polynomials.

math.CV↗

arXiv2026

On the Bourgain--Brezis--Mironescu spaces over Carleson tents

We introduce Carleson analogs of the Bourgain--Brezis--Mironescu spaces $B$ and $B_0$ by measuring mean oscillation over upper Carleson tents. For these spaces, denoted by $B_{\mathcal C}^p$ and $B_{\mathcal C,0}^p$, we prove two types of structural results. First, we show that they contain several natural classes of functions, including BMO/VMO--Carleson spaces, tent-space potential classes, and fractional Sobolev classes. Second, motivated by Zhu's structural theorem for BMO spaces induced by the Bergman metric, we establish decompositions of $B_{\mathcal C}^p$ and $B_{\mathcal C,0}^p$ into bounded-oscillation and bounded-average components. We then revisit the Bourgain--Brezis--Mironescu rigidity phenomenon in the Carleson setting. Although the direct rigidity statement fails for $B_{\mathcal C,0}^p$, we introduce a natural $B_{\mathcal C}^p$-trace and prove that the rigidity theorem survives at the level of traces.

math.CA↗

arXiv2026

On the Bloch and $\mathcal Q_p$--Carleson measure problems

In this paper, we study the Bloch and $\mathcal Q_p$--Carleson measure problems on the unit disc $\mathbb D$. In the Bloch case, for a positive Borel measure $μ$ on $\mathbb D$, we give a complete characterization of the boundedness and compactness of the embedding $$ \operatorname{id}:\mathcal B \longrightarrow L^2(μ) $$ in terms of the Bloch capacity $\mathfrak B_{\mathcal R}(μ)$ associated with an admissible dyadic resolution $\mathcal R$ of $\mathbb D$. The proof is based on the Bergman projection representation of Bloch functions, conditional expectations on admissible dyadic resolutions, and a finite-dimensional semidefinite programming argument. We also adapt this dyadic framework to the more general $\mathcal Q_p$--Carleson measure problem and obtain a corresponding complete boundedness and compactness characterization for $$ \operatorname{id}:\mathcal Q_p \longrightarrow L^2(μ), \qquad 0<p\le1. $$ This work further develops the dyadic approach introduced in our recent work on composition operators on $\mathcal Q_p$ spaces, but in a different setting where the embedding involves recovering function values from derivative information.

math.CV↗

arXiv2026

A non-testing characterization of bounded and compact composition operators on $\mathcal Q_p$ spaces

In this paper, we give symbol-only, non-testing characterizations of bounded and compact composition operators on $\mathcal Q_p$, $0<p\le 1$, via a novel dyadic trace formulation over Carleson tents for generalized $p$-Nevanlinna counting functions. Our results resolve an open question raised in Xiao's 2001 book, as well as the diagonal case of Zhao's 2009 question, a longstanding problem in the theory of $\mathcal Q_p$ spaces. As an application of our main theorems, we also obtain boundedness and compactness criteria for the $\mathcal Q_p$-Carleson measure embedding $\textrm{id}:\mathcal Q_p\to L^2(WdA)$ for weights $W$ in the Littlewood--Paley class.

math.CV↗

arXiv2026

From Complex-Analytic Models to Dyadic Methods: A Real-Variable Approach to Hypersingular Operators

Motivated by the work of Cheng-Fang-Wang-Yu on the hypersingular Bergman projection, we develop a real-variable framework for hypersingular operators in regimes where strong-type bounds fail on the critical line. Our main new ingredient is the Forelli-Rudin method: a dyadic mechanism, inspired by complex-analytic Forelli-Rudin type arguments, that yields sharp critical-line and endpoint estimates. On the unit disc, for $1 q$, valid for all $t>0$. For the hypersingular Bergman projection \[ K_{2t}f(z)=\int_{\mathbb D}\frac{f(w)}{(1-z\overline w)^{2t}}\,dA(w), \] we establish sharp critical-line bounds, with emphasis on the endpoint weak-type estimate at $(p,q)=\bigl(\tfrac{1}{3-2t},1\bigr)$. In particular, this result resolves an open question on the critical-line behavior of the Bergman projection in the hypersingular regime. Finally, we introduce a class of hypersingular cousins of sparse operators in $\mathbb R^n$ associated with graded sparse families, quantified by the sparseness $η$ and a new structural parameter (the degree) $K_{\mathcal S}$. We characterize the corresponding sharp strong- and weak-type regimes in terms of $(n,t,η,K_{\mathcal S})$. This real-variable perspective addresses an inquiry of Cheng-Fang-Wang-Yu on developing effective real-analytic tools in the hypersingular regime for both $\mathcal M_t^{\mathcal D}$ and $K_{2t}$, and it also provides a new route to critical-line analysis for Forelli-Rudin type and related hypersingular operators in both real and complex settings.

math.CA↗

arXiv2024

Near-endpoints Carleson Embedding of $\mathcal Q_s$ and $F(p, q, s)$ into tent spaces

This paper aims to study the $\mathcal Q_s$ and $F(p, q, s)$ Carleson embedding problems near endpoints. We first show that for $0 1$. Our method is different from the previously known approach which involves a delicate study of Carleson measures (or logarithmic Carleson measures) on weighted Dirichlet spaces. As some byproducts, the corresponding compactness results are also achieved. Finally, we compare our approach with the existing solutions of Carleson embedding problems proposed by Xiao, Pau, Zhao, Zhu, etc. Our results assert that a "tiny-perturbed" version of a conjecture on the $\mathcal Q_s$ Carleson embedding problem due to Liu, Lou, and Zhu is true. Moreover, we answer an open question by Pau and Zhao on the $F(p, q, s)$ Carleson embedding near endpoints.

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↗