arXiv subjects

Bingyang Hu

Publications and source records attributed to Bingyang Hu.

At least 19 recordsLinked to original sources

arXiv2026

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

In this paper, we obtain a full characterization of the finite positive Borel measures $\mu$ on $\mathbb D$ for which the embedding $$ \operatorname{id}:\mathcal D_{p-1}^p\longrightarrow L^p(\mu),\qquad p>2, $$ is bounded. More precisely, for any dyadic system $\mathcal D$ on $\mathbb T$, we prove that this embedding is bounded if and only if $$ \mathcal C_{p,\mathcal D}(\mu)+\mathcal H_{p,\mathcal D}(\mu)<\infty, $$ where $\mathcal C_{p,\mathcal D}(\mu)$ and $\mathcal H_{p,\mathcal D}(\mu)$ denote the packing energy and the Haar energy of $\mu$, 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 Pel\'aez in 2006 and Galanopoulos, Girela, and Pel\'aez 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 $\mu$ on the unit disc $\mathbb D$ such that the canonical embedding $$ \operatorname{id}:\mathcal Q_p \longrightarrow \mathcal T_{p,2}^2(\mu) $$ 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, Pel\'aez, P\'erez-Gonz\'alez and R\"atty\"a. Our proof is based on a new Szeg\H{o}-type recursion involving $\mathbb C^2$-valued polynomials and their reciprocal polynomials.

math.CV↗

arXiv2026

On the parabolic $H^p$ theory generated by $(p,\infty)$-atoms for $0<p<1$

We study the parabolic maximal operator $M_{\textrm{par}}$ along the moment curve $(t,t^2)$. In 1988, Christ proved that $M_{\textrm{par}}$ maps the parabolic Hardy space $H_{\textrm{par}}^1(\mathbb R^2)$, formulated using $(1,\infty)$-atoms, into $L^{1,\infty}(\mathbb R^2)$. Working directly with parabolic $(p,\infty)$-atoms, we show that this result is sharp at $p=1$: for every $0<p<1$, the natural extension from $H_{\textrm{par}}^p(\mathbb R^2)$ to $L^{p,\infty}(\mathbb R^2)$ fails even for the corresponding single-scale operator. We then introduce a curvature-adapted modified Hardy space $H_{\textrm{par}}^{p,*}(\mathbb R^2)$ and a weak tendril space $\mathcal T^{p,\infty}(\mathbb R^2)$, and prove that $$ M_{\textrm{par}}: H_{\textrm{par}}^{p,*}(\mathbb R^2) \longrightarrow \mathcal T^{p,\infty}(\mathbb R^2), \qquad 0<p<1, $$ is bounded. At $p=1$, these spaces recover those in Christ's theorem: $H_{\textrm{par}}^{1,*}(\mathbb R^2)=H_{\textrm{par}}^1(\mathbb R^2)$ and $\mathcal T^{1,\infty}(\mathbb R^2)=L^{1,\infty}(\mathbb R^2)$. Thus, our result provides a natural extension of Christ's work to the range $0<p<1$.

math.CA↗

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 $\mu$ on $\mathbb D$, we give a complete characterization of the boundedness and compactness of the embedding $$ \operatorname{id}:\mathcal B \longrightarrow L^2(\mu) $$ in terms of the Bloch capacity $\mathfrak B_{\mathcal R}(\mu)$ 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 $$ \operatorname{id}:\mathcal Q_p \longrightarrow L^2(\mu), \qquad 0<p\le1. $$ 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↗

arXiv2025

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 $\eta$ and a new structural parameter (the degree) $K_{\mathcal S}$. We characterize the corresponding sharp strong- and weak-type regimes in terms of $(n,t,\eta,K_{\mathcal S})$. 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↗

arXiv2025

On the boundedness of the curved trilinear Hilbert transform and the curved $n-$linear maximal operator in the quasi-Banach regime

Let $n\in\mathbb{N}$, $\vec{\alpha}=(\alpha_1,\ldots,\alpha_n)\in (0,\infty)^n$, $\vec{\beta}=(\beta_1,\ldots,\beta_n)\in (\mathbb{R}\setminus\{0\})^n$, $\vec{f}:=(f_1,\ldots, f_n)\in \mathcal{S}^n(\mathbb{R})$ and set $$H_{n,\vec{\alpha},\vec{\beta}}(\vec{f})(x):=p.v. \int_{\mathbb{R}} f_1(x+\beta_1 t^{\alpha_1})\ldots f_n(x+\beta_n t^{\alpha_n}) \frac{dt}{t}, \quad x \in \mathbb{R}\,,$$ with $\mathcal{M}_{n,\vec{\alpha},\vec{\beta}}$ being its maximal operator counterpart. Assume that $1<p_j<\infty$ and $\frac{1}{2}<r<\infty$ satisfy $\sum_{j=1}^n \frac{1}{p_j}=\frac{1}{r}$. Under the \emph{non-resonant} assumption that $\{\alpha_j\}_{j=1}^n$ are pairwise distinct we show that $$\|H_{3,\vec{\alpha},\vec{\beta}}(\vec{f})\|_{L^r}\lesssim_{\vec{\alpha},\vec{\beta}} \prod_{j=1}^3\|f_j\|_{L^{p_j}}\qquad \textrm{and} \qquad \|\mathcal{M}_{n,\vec{\alpha},\vec{\beta}}\|_{L^r}\lesssim_{n,\vec{\alpha},\vec{\beta}} \prod_{j=1}^n\|f_j\|_{L^{p_j}} \qquad \forall\:n\geq 2\,.$$ (For the maximal operator the boundedness range can be extended to include the endpoint $\infty$ for both $p_j$ and $r$.)

math.CA↗

arXiv2025

The Bilinear Hilbert-Carleson operator along curves. The purely non-zero curvature case

In this paper, we provide the maximal boundedness range (up to end-points) for the Bilinear Hilbert-Carleson operator along curves in the (purely) non-zero curvature setting. More precisely, we show that the operator $$ BHC_{[\vec{a},\vec{\alpha}]}(f_1,f_2)(x) := \sup_{\lambda\in\mathbb{R}} \left|\,p.v.\, \int_{\mathbb{R}} f_1(x - a_1 t^{\alpha_1}) \,f_2(x - a_2 t^{\alpha_2}) \,e^{i\,\lambda\,a_3 \,t^{\alpha_3}} \,\frac{dt}{t}\right|$$ obeys the bounds $$\|BHC_{[\vec{a},\vec{\alpha}]} (f_1,f_2)\|_{L^r} \lesssim_{\vec{a} \,\vec{\alpha},r,p_1,p_2} \|f_1\|_{L^{p_1}}\,\|f_2\|_{L^{p_2}}$$ whenever $\vec{a}=(a_1,a_2,a_3),\,\vec{\alpha}=(\alpha_1,\alpha_2,\alpha_3)\in (\mathbb{R}\setminus\{0\})^3$ with $\vec{\alpha}$ having pairwise distinct coordinates and for any H\"older range $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{r}$ with $1<p_1,p_2<\infty$ and $\frac{1}{2}<r<\infty$. This result is achieved via the Rank II LGC method introduced in arXiv:2308.10706.

math.CA↗

arXiv2025

Boundedness and compactness of Bergman projection commutators in two-weight setting

The goal of this paper is to study the boundedness and compactness of the Bergman projection commutators in two weighted settings via the weighted BMO and VMO spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. In the analytic setting, we completely characterize boundedness and compactness, while in the non-analytic setting, we provide a sufficient condition which generalizes the Euclidean case and is also necessary in many cases of interest. Our work initiates a study of the commutators acting on complex function spaces with different symbols.

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↗

arXiv2024

Polynomial Convergence Rate for Quasi-Periodic Homogenization of Hamilton-Jacobi Equations and Application to Ergodic Estimates

In this paper, we demonstrate a polynomial convergence rate for homogenization of Hamilton-Jacobi equations with quasi-periodic potentials. We establish a connection between the convergence rate of homogenization and the regularity of the effective Hamiltonian, by using a new quantitative ergodic estimate for bounded quasi-periodic functions with Diophantine frequencies. As an application, we also study the convergent rate for Birkhoff average of unbounded quasi-periodic functions.

math.AP↗

arXiv2023

On the curved Trilinear Hilbert transform

Building on the (Rank I) LGC-methodology introduced by the second author and on the novel perspective employed in the time-frequency discretization of the non-resonant bilinear Hilbert--Carleson operator, we develop a new, versatile method -- referred to as Rank II LGC -- that has as a consequence the resolution of the $L^p$ boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator \begin{equation*} H_{C}(f_1, f_2, f_3)(x):= \textrm{p.v.}\,\int_{\mathbb{R}} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb{R}\,, \end{equation*} is bounded from $L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\times L^{p_3}(\mathbb{R})$ into $L^{r}(\mathbb{R})$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1<p_1,p_3<\infty$, $1<p_2\leq \infty$ and $1\leq r <\infty$. A crucial difficulty in approaching this problem is the lack of absolute summability for the linearized discretized model (derived via Rank I LGC method) of the quadrilinear form associated to $H_{C}$. In order to overcome this, we develope a so-called correlative time-frequency model whose control is achieved via the following interdependent elements: (1) a sparse-unform decomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space, (2) a structural analysis of suitable maximal ``joint Fourier coefficients", and (3) a level set analysis with respect to the time-frequency correlation set.

math.CA↗

arXiv2022

The Commutator of the Bergman Projection on Strongly Pseudoconvex Domains with Minimal Smoothness

Consider a bounded, strongly pseudoconvex domain $D\subset \mathbb C^n$ with minimal smoothness (namely, the class $C^2$) and let $b$ be a locally integrable function on $D$. We characterize boundedness (resp., compactness) in $L^p(D), p > 1$, of the commutator $[b, P]$ of the Bergman projection $P$ in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on $b$. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.

math.CV↗

arXiv2021

Global existence of a non-local semilinear parabolic equation with advection and applications to shear flow

In this paper, we consider the following non-local semi-linear parabolic equation with advection: for $1 \le p<1+\frac{2}{N}$, \begin{equation*} \begin{cases} u_t+v \cdot \nabla u-Δu=|u|^p-\int_{\mathbb T^N} |u|^p \quad & \textrm{on} \quad \mathbb T^N, \\ \\ u \ \textrm{periodic} \quad & \textrm{on} \quad \partial \mathbb T^N \end{cases} \end{equation*} with initial data $u_0$ defined on $\mathbb T^N$. Here $v$ is an incompressible flow, and $\mathbb T^N=[0, 1]^N$ is the $N$-torus with $N$ being the dimension. We first prove the local existence of mild solutions to the above equation for arbitrary data in $L^2$. We then study the global existence of the solutions under the following two scenarios: (1). when $v$ is a mixing flow; (2). when $v$ is a shear flow. More precisely, we show that under these assumptions, there exists a global solution to the above equation in the sense of $L^2$.

math.AP↗

arXiv2021

Global Existence for the unstable Cahn-Hilliard equation in 2D with a Shear Flow

In this paper, we study the advective unstable Cahn--Hilliard equation on $\mathbb T^2$ with shear flow: \begin{equation*} \begin{cases} u_t+Av_1(y) \partial_x u+\varepsilon \Delta^2 u= \Delta(a u^3+ b u^2) \quad & \quad \textrm{on} \quad \mathbb T^2; \\ \\ u \ \textrm{periodic} \quad & \quad \textrm{on} \quad \partial \mathbb T^2, \end{cases} \end{equation*} where $u_0\in H_0^2(\mathbb T^2)$, $A,\varepsilon>0$, $a<0$, and $b\in\mathbb R$. The condition $a<0$ puts the model in an unstable phase-field regime: the nonlinear chemical potential may amplify, rather than restore, concentration fluctuations, as in spinodal decomposition. The shear term $Av_1(y)\partial_xu$ models imposed stirring along the shear direction; through mixing, it enhances dissipation and counteracts the growth driven by the unstable cubic term $\Delta(au^3)$. Assuming that the shear profile has finitely many critical points and that linearly growing modes occur only in the shear direction, we prove that the $L^2$-energy converges exponentially to zero, provided $|a|$ and $\|\int_{\mathbb T} u_0(x,\cdot)\,dx\|_{L_y^2}$ are sufficiently small.

math.AP↗