arXiv subjects

Bingyang Hu

Publications and source records attributed to Bingyang Hu.

At least 37 records · Page 2Linked to original sources

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

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↗

arXiv2023

A Structure Theorem on Intersections of General Doubling Measures and Its Applications

We unite two themes in dyadic analysis and number theory by studying an analogue of the failure of the Hasse principle in harmonic analysis. Explicitly, we construct an explicit family of measures on the real line that are $p$-adic and $q$-adic doubling for any distinct primes $p$ and $q$, yet not doubling, and we apply these results to show analogous statements about the reverse Hölder and Muckenhoupt $A_p$ classes of weights. The proofs involve a delicate interplay among several geometric and number theoretic properties.

math.CA↗

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 Hölder 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

Dissipation enhancement for a degenerated parabolic equation

In this paper, we quantitatively consider the enhanced-dissipation effect of the advection term to the parabolic $p$-Laplacian equations. More precisely, we show the mixing property of flow for the passive scalar enhances the dissipation process of the $p$-Laplacian in the sense of $L^2$ decay, that is, the $L^2$ decay can be arbitrarily fast. The main ingredient of our argument is to understand the underlying iteration structure inherited from the parabolic $p$-Laplacian equations. This extends the dissipation enhancement result of the advection diffusion equation by Yuanyuan Feng and Gautam Iyer into a non-linear setting.

math.AP↗

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

Suppression of epitaxial thin film growth by mixing

We consider following fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus with and without advection of an incompressible vector field in the case $2<p<3$: \begin{equation*} \partial_t u + (-Δ)^2 u = -\nabla\cdot(|\nabla u|^{p-2}\nabla u). \end{equation*} The study of this form of equations arises from mathematical models that simulate the epitaxial growth of the thin film. We prove the local existence of mild solutions for any initial data lies in $L^2$ in both cases. Our main result is: in the advective case, if the imposed advection is sufficiently mixing, then the global existence of solution can be proved, and the solution will converge exponentially to a homogeneous mixed state. While in the absence of advection, there exist initial data in $H^2\cap W^{1,\infty}$ such that the solution will blow up in finite time.

math.AP↗

arXiv2021

Dyadic Carleson embedding and sparse domination of weighted composition operators on strictly pseudoconvex domains

In this paper, we study the behavior of the weighted composition operators acting on Bergman spaces defined on strictly pseudoconvex domains via the sparse domination technique from harmonic analysis. As a byproduct, we also prove a weighted type estimate for the weighted composition operators which is adapted to Sawyer-testing conditions. Our results extend the work by the first author, Li, Shi and Wick under a much more general setting.

math.CV↗

arXiv2021

Rendering Discrete Participating Media with Geometrical Optics Approximation

We consider the scattering of light in participating media composed of sparsely and randomly distributed discrete particles. The particle size is expected to range from the scale of the wavelength to the scale several orders of magnitude greater than the wavelength, and the appearance shows distinct graininess as opposed to the smooth appearance of continuous media. One fundamental issue in physically-based synthesizing this appearance is to determine necessary optical properties in every local region. Since these optical properties vary spatially, we resort to geometrical optics approximation (GOA), a highly efficient alternative to rigorous Lorenz-Mie theory, to quantitatively represent the scattering of a single particle. This enables us to quickly compute bulk optical properties according to any particle size distribution. Then, we propose a practical Monte Carlo rendering solution to solve the transfer of energy in discrete participating media. Results show that for the first time our proposed framework can simulate a wide range of discrete participating media with different levels of graininess and converges to continuous media as the particle concentration increases.

cs.GR↗

arXiv2021

A structure theorem on doubling measures with different bases

In this paper, we prove a structure theorem for the infinite union of $n$-adic doubling measures via techniques which involve far numbers. Our approach extends the results of Wu in 1998, and as a by product, we also prove a classification result related to normal numbers.

math.CA↗

arXiv2020

Sparse bounds for discrete singular Radon transforms

We show that discrete singular Radon transforms along a certain class of polynomial mappings $P:\mathbb{Z}^d\to \mathbb{Z}^n$ satisfy sparse bounds. For $n=d=1$ we can handle all polynomials. In higher dimensions, we pose restrictions on the admissible polynomial mappings stemming from a combination of interacting geometric, analytic and number-theoretic obstacles.

math.CA↗

arXiv2020

Sharp Mei's lemma with different bases

In this paper, we prove a sharp Mei's Lemma with assuming the bases of the underlying general dyadic grids are different. As a byproduct, we specify all the possible cases of adjacent general dyadic systems with different bases. The proofs have connections with certain number-theoretic properties.

math.CA↗

arXiv2020

On the general dyadic grids in $\mathbb{R}^d$

Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson and Wei) we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here we extend this work to all dimensions, which turns out to have many surprising difficulties due to the fact that $d+1$, not $2^d$, grids is the optimal number in an adjacent dyadic system in $\mathbb{R}^d$. As a byproduct, we show that a collection of $d+1$ dyadic systems in $\mathbb{R}^d$ is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on $\mathbb{R}$. The underlying geometric structures that arise in this higher dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and $n$-adic, for any $n$) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific example.

math.CA↗

arXiv2020

Dyadic decomposition of convex domains of finite type and applications

In this paper, we introduce a dyadic structure on convex domains of finite type via the so-called dyadic flow tents. This dyadic structure allows us to establish weighted norm estimates for the Bergman projection $P$ on such domains with respect to Muckenhoupt weights. In particular, this result gives an alternative proof of the $L^p$ boundedness of $P$. Moreover, using extrapolation, we are also able to derive weighted vector-valued estimates and weighted modular inequalities for the Bergman projection.

math.CA↗

arXiv2020

Sparse domination of weighted composition operators on weighted Bergman spaces in the upper half-plane

The purpose of this paper is to study sparse domination estimates of composition operators in the setting of complex function theory. The method originates from proofs of the $A_2$ theorem for Calderón-Zygmund operators in harmonic analysis. Using this tool from harmonic analysis, some new characterizations are given for the boundedness and compactness of weighted composition operators acting between weighted Bergman spaces in the upper half plane. Moreover, we establish a new weighted type estimate for the holomorphic Bergman-class functions, for a new class of weights, which is adapted to Sawyer--testing conditions. We also extend our results to the unit ball $\mathbb B$ in $\mathbb C^n$.

math.CV↗

arXiv2019

On the translates of general dyadic systems on $\mathbb{R}$

Many techniques in harmonic analysis use the fact that a continuous object can be written as a sum (or an intersection) of dyadic counterparts, as long as those counterparts belong to an adjacent dyadic system. Here we generalize the notion of adjacent dyadic system and explore when it occurs, leading to some new and perhaps surprising classifications. In particular, we show that every dyadic grid is determined by two parameters, the \emph{shift} and the \emph{location}; moreover two dyadic grids form an adjacent dyadic system if and only if their shifts and locations satisfy readily verifiable conditions.

math.CA↗

arXiv2019

A unified method for maximal truncated Calderón-Zygmund operators in general function spaces by sparse domination

In this note we give simple proofs of several results involving maximal truncated Caldeón-Zygmund operators in the general setting of rearrangement invariant quasi-Banach function spaces by sparse domination. Our techniques allow us to track the dependence of the constants in weighted norm inequalities; additionally, our results hold in $\mathbb{R}^n$ as well as in many spaces of homogeneous type.

math.CA↗

arXiv2019

Sparse domination of singular Radon transform

The purpose of this paper is to study the sparse bound of the operator of the form $f \mapsto ψ(x) \int f(γ_t(x))K(t)dt$, where $γ_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(x, t) \in \mathbb R^n \times \mathbb R^k$, satisfying $γ_0(x) \equiv x$, $ψ$ is a $C^\infty$ cut-off function supported on a small neighborhood of $0 \in \mathbb R^n$ and $K$ is a Calderón-Zygmund kernel suppported on a small neighborhood of $0 \in \mathbb R^k$. Christ, Nagel, Stein and Wainger gave conditions on $γ$ under which $T: L^p \mapsto L^p (1<p<\infty)$ is bounded. Under the these same conditions, we prove sparse bounds for $T$, which strengthens their result. As a corollary, we derive weighted norm estimates for such operators.

math.CA↗