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Dachun Yang

Publications and source records attributed to Dachun Yang.

At least 109 records · Page 6Linked to original sources
arXiv2022

Pointwise Characterization of Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

In this article, the authors establish the pointwise characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type via clarifying the relationship among Hajłasz-Sobolev spaces, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces, grand Besov and Triebel-Lizorkin spaces, and Besov and Triebel-Lizorkin spaces. A major novelty of this article is that all results presented in this article get rid of both the dependence on the reverse doubling condition of the measure and the metric condition of the quasi-metric under consideration. Moreover, the pointwise characterization of the inhomogeneous version is new even when the underlying space is an RD-space.

math.FA↗
arXiv2022

Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón--Zygmund Operators

Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors use the Aoki--Rolewicz theorem and both the dyadic system and the exponential decay of approximations of the identity on $\mathbb{X}$, which closely connect with the geometrical properties of $\mathbb{X}$, to overcome the difficulties caused by the absence of both the triangle inequality of $\|\cdot\|_{X(\mathbb{X})}$ and the reverse doubling assumption of the measure $μ$ under consideration, and also use the relation between the convexification of $X(\mathbb{X})$ and the weak space $WX(\mathbb{X})$ associated with $X(\mathbb{X})$ to prove that the infinite summation of atoms converges in the space of distributions on $\mathbb{X}$. Moreover, all these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space, the obtained results are also new and, actually, some of them are new even on RD-spaces (namely, spaces of homogeneous type satisfying the additional reverse doubling condition).

math.FA↗
arXiv2022

Heat Kernels and Hardy Spaces on Non-Tangentially Accessible Domains with Applications to Global Regularity of Inhomogeneous Dirichlet Problems

Let $n\ge2$ and $Ω$ be a bounded non-tangentially accessible domain (for short, NTA domain) of $\mathbb{R}^n$. Assume that $L_D$ is a second-order divergence form elliptic operator having real-valued, bounded, measurable coefficients on $L^2(Ω)$ with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors prove that the heat kernels $\{K_t^{L_D}\}_{t>0}$ generated by $L_D$ are Hölder continuous. Second, for any $p\in(0,1]$, the authors introduce the `geometrical' Hardy space $H^p_r(Ω)$ by restricting any element of the Hardy space $H^p(\mathbb{R}^n)$ to $Ω$, and show that, when $p\in(\frac{n}{n+δ_0},1]$, $H^p_r(Ω)=H^p(Ω)=H^p_{L_D}(Ω)$ with equivalent quasi-norms, where $H^p(Ω)$ and $H^p_{L_D}(Ω)$ respectively denote the Hardy space on $Ω$ and the Hardy space associated with $L_D$, and $δ_0\in(0,1]$ is the critical index of the Hölder continuity for the kernels $\{K_t^{L_D}\}_{t>0}$. Third, as applications, the authors obtain the global gradient estimates in both $L^p(Ω)$, with $p\in(1,p_0)$, and $H^p_z(Ω)$, with $p\in(\frac{n}{n+1},1]$, for the inhomogeneous Dirichlet problem of second-order divergence form elliptic equations on bounded NTA domains, where $p_0\in(2,\infty)$ is a constant depending only on $n$, $Ω$, and the coefficient matrix of $L_D$. It is worth pointing out that the range $p\in(1,p_0)$ for the global gradient estimate in the scale of Lebesgue spaces $L^p(Ω)$ is sharp and the above results are established without any additional assumptions on both the coefficient matrix of $L_D$, and the domain $Ω$.

math.AP↗
arXiv2022

Global Gradient Estimates for Dirichlet Problems of Elliptic Operators with a BMO Anti-Symmetric Part

Let $n\ge2$ and $Ω\subset\mathbb{R}^n$ be a bounded NTA domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second order elliptic equations of divergence form with an elliptic symmetric part and a BMO anti-symmetric part in $Ω$. More precisely, for any given $p\in(2,\infty)$, the authors prove that a weak reverse Hölder inequality with exponent $p$ implies the global $W^{1,p}$ estimate and the global weighted $W^{1,q}$ estimate, with $q\in[2,p]$ and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second order elliptic equations of divergence form with small $\mathrm{BMO}$ symmetric part and small $\mathrm{BMO}$ anti-symmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, $C^1$ domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz--)Morrey spaces, (Musielak--)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.

math.AP↗
arXiv2021

Poincaré Inequality Meets Brezis--Van Schaftingen--Yung Formula on Metric Measure Spaces

Let $(\mathcal{X}, ρ, μ)$ be a metric measure space of homogeneous type which supports a certain Poincaré inequality. Denote by the symbol $\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$ the space of all continuous functions $f$ with compact support satisfying that $\operatorname{Lip} f:=\limsup _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ is also a continuous function with compact support and $\operatorname{Lip} f=\lim _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ converges uniformly. Let $p \in[1,\infty)$. In this article, the authors prove that, for any $f\in\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$, \begin{align*} &\sup_{λ\in(0,\infty)}λ^p\int_{\mathcal{X}} μ\left(\left\{y\in \mathcal{X}:\ |f(x)-f(y)|>λρ(x,y) [V(x,y)]^{\frac 1p}\right\}\right)\, dμ(x)\\ &\quad\sim \int_{\mathcal{X}} [\operatorname{Lip}f(x)]^p \,dμ(x) \end{align*} with the positive equivalence constants independent of $f$, where $V(x,y):=μ(B(x,ρ(x,y)))$. This generalizes a recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung from the $n$-dimensional Euclidean space ${\mathbb R}^n$ to $\mathcal{X}$. Applying this generalization, the authors establish new fractional Sobolev and Gagliardo--Nirenberg inequalities in $\mathcal{X}$. All these results have a wide range of applications. Particularly, when applied to two concrete examples, namely, ${\mathbb R}^n$ with weighted Lebesgue measure and the complete $n$-dimensional Riemannian manifold with non-negative Ricci curvature, all these results are new. The proofs of these results strongly depend on the geometrical relation of differences and derivatives in the metric measure space and the Poincaré inequality.

math.FA↗
arXiv2021

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Characterizations of Maximal Functions, Decompositions, and Dual Spaces

Let $({\mathcal X},ρ,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximal inequality, and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}^*({\mathcal X})$, associated with $Y({\mathcal X})$, via the grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of $H_{Y}^*({\mathcal X})$, which proves to be a ball Campanato-type function space associated with $Y({\mathcal X})$. All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this article exist in that, to escape the reverse doubling condition of $μ$ and the triangle inequality of $ρ$, the authors cleverly construct admissible sequences of balls, and fully use the geometrical properties of ${\mathcal X}$ expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of $H_{Y}^*({\mathcal X})$, the authors further prove that $Y({\mathcal X})$ can be embedded into the weighted Lebesgue space with certain special weight, and then can fully use the known results of the weighted Lebesgue space.

math.FA↗
arXiv2021

Boundedness of Calderón--Zygmund Operators on Special John--Nirenberg--Campanato and Hardy-Type Spaces via Congruent Cubes

Let $p\in[1,\infty]$, $q\in(1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $α\in\mathbb{R}$. In this article, the authors introduce a reasonable version $\widetilde T$ of the Calderón--Zygmund operator $T$ on $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, the special John--Nirenberg--Campanato space via congruent cubes, which coincides with the Campanato space $\mathcal{C}_{α,q,s}(\mathbb{R}^n)$ when $p=\infty$. Then the authors prove that $\widetilde T$ is bounded on $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$ if and only if, for any $γ\in\mathbb{Z}_+^n$ with $|γ|\leq s$, $T^*(x^γ)=0$, which is a well-known assumption. To this end, the authors find an equivalent version of this assumption. Moreover, the authors show that $T$ can be extended to a unique continuous linear operator on the Hardy-kind space $HK_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, the predual space of $JN_{(p',q',s)_α}^{\mathrm{con}}(\mathbb{R}^n)$ with $\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}$, if and only if, for any $γ\in\mathbb{Z}_+^n$ with $|γ|\leq s$, $T^*(x^γ)=0$.

math.CA↗
arXiv2021

A Survey on Function Spaces of John--Nirenberg Type

In this article, the authors give a survey on the recent developments of both the John--Nirenberg space $JN_p$ and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO, $VJN_p$, and $CJN_p$ on $\mathbb{R}^n$ or a given cube $Q_0\subset\mathbb{R}^n$ with finite side length. In addition, some related open questions are also presented.

math.FA↗
arXiv2021

Nontriviality of Riesz--Morrey Spaces

In this article, the authors completely answer an open question, presented in [Banach J. Math. Anal. 15 (2021), no. 1, 20], via showing that the Riesz--Morrey space is truly a new space larger than a particular Lebesgue space with critical index. Indeed, this Lebesgue space is just the real interpolation space of the Riesz--Morrey space for suitable indices. Moreover, the authors further show the aforementioned inclusion is also proper, namely, this embedding is sharp in some sense, via constructing two nontrivial spare functions, respectively, on $\mathbb{R}^n$ and any given cube $Q_0$ of $\mathbb{R}^n$ with finite side length. The latter constructed function is inspired by the striking function constructed by Dafni et al. [J. Funct. Anal. 275 (2018), 577--603]. All the proofs of these results strongly depend on some exquisite geometrical analysis on cubes of $\mathbb{R}^n$. As an application, the relationship between Riesz--Morrey spaces and Lebesgue spaces is completely clarified on all indices.

math.FA↗
arXiv2021

Wavelet Characterization of Besov and Triebel--Lizorkin Spaces on Spaces of Homogeneous Type and Its Applications

In this article, the authors establish the wavelet characterization of Besov and Triebel--Lizorkin spaces on a given space $(X,d,μ)$ of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel--Lizorkin sequence spaces on $X$, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel--Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood--Paley characterizations of Triebel--Lizorkin spaces on $X$. The main novelty of this article is that all these results get rid of their dependence on the reverse doubling property of $μ$ and also the triangle inequality of $d$, by fully using the geometrical property of $X$ expressed via its equipped quasi-metric $d$, dyadic reference points, dyadic cubes, and wavelets

math.FA↗
arXiv2021

Difference Characterization of Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

In this article, the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss, and discuss their relations with Besov and Triebel-Lizorkin spaces. As an application, the authors establish the difference characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. A major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space ${\mathcal X}$ via using the geometrical property of ${\mathcal X}$ expressed by its dyadic reference points, dyadic cubes, and the (local) lower bound. Moreover, some results when $p\le 1$ but near to $1$ are new even when ${\mathcal X}$ is an RD-space.

math.FA↗
arXiv2021

Functional Calculus on BMO-type Spaces of Bourgain, Brezis and Mironescu

A nonlinear superposition operator $T_g$ related to a Borel measurable function $g:\ {\mathbb C}\to {\mathbb C}$ is defined via $T_g(f):=g\circ f$ for any complex-valued function $f$ on ${\mathbb R^n}$. This article is devoted to investigating the mapping properties of $T_g$ on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion result and the continuity property of $T_g$ on these spaces are obtained.

math.CA↗
arXiv2021

Compactness Characterizations of Commutators on Ball Banach Function Spaces

Let $X$ be a ball Banach function space on ${\mathbb R}^n$. Let $Ω$ be a Lipschitz function on the unit sphere of ${\mathbb R}^n$,which is homogeneous of degree zero and has mean value zero, and let $T_Ω$ be the convolutional singular integral operator with kernel $Ω(\cdot)/|\cdot|^n$. In this article, under the assumption that the Hardy--Littlewood maximal operator $\mathcal{M}$ is bounded on both $X$ and its associated space, the authors prove that the commutator $[b,T_Ω]$ is compact on $X$ if and only if $b\in{\rm CMO}({\mathbb R}^n)$. To achieve this, the authors mainly employ three key tools: some elaborate estimates, given in this article, on the norm in $X$ of the commutators and the characteristic functions of some measurable subset,which are implied by the assumed boundedness of ${\mathcal M}$ on $X$ and its associated space as well as the geometry of $\mathbb R^n$; the complete John--Nirenberg inequality in $X$ obtained by Y. Sawano et al.; the generalized Fréchet--Kolmogorov theorem on $X$ also established in this article. All these results have a wide range of applications. Particularly, even when $X:=L^{p(\cdot)}({\mathbb R}^n)$ (the variable Lebesgue space), $X:=L^{\vec{p}}({\mathbb R}^n)$ (the mixed-norm Lebesgue space), $X:=L^Φ({\mathbb R}^n)$ (the Orlicz space), and $X:=(E_Φ^q)_t({\mathbb R}^n)$ (the Orlicz-slice space or the generalized amalgam space), all these results are new.

math.FA↗
arXiv2020

Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type with Applications to Boundedness of Calderón-Zygmund Operators

In this article, the authors introduce Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss, prove that these (in)homogeneous Besov and Triebel-Lizorkin spaces are independent of the choices of both exp-ATIs (or exp-IATIs) and underlying spaces of distributions, and give some basic properties of these spaces. As applications, the authors show that some known function spaces coincide with certain special cases of Besov and Triebel-Lizorkin spaces and, moreover, obtain the boundedness of Calderón-Zygmund operators on these Besov and Triebel-Lizorkin spaces. All these results strongly depend on the geometrical properties, reflected via its dyadic cubes, of the considered space of homogeneous type. Comparing with the known theory of these spaces on metric measure spaces, a major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space and hence give a final real-variable theory of these function spaces on spaces of homogeneous type.

math.FA↗
arXiv2020

Gaussian Estimates for Heat Kernels of Higher Order Schrödinger Operators with Potentials in Generalized Schechter Classes

Let $m\in\mathbb N$, $P(D):=\sum_{|α|=2m}(-1)^m a_αD^α$ be a $2m$-order homogeneous elliptic operator with real constant coefficients on $\mathbb{R}^n$, and $V$ a measurable function on $\mathbb{R}^n$. In this article, the authors introduce a new generalized Schechter class concerning $V$ and show that the higher order Schrödinger operator $\mathcal{L}:=P(D)+V$ possesses a heat kernel that satisfies the Gaussian upper bound and the Hölder regularity when $V$ belongs to this new class. The Davies--Gaffney estimates for the associated semigroup and their local versions are also given. These results pave the way for many further studies on the analysis of $\mathcal{L}$.

math.AP↗
arXiv2020

XMO and Weighted Compact Bilinear Commutators

To study the compactness of bilinear commutators of certain bilinear Calderón--Zygmund operators which include (inhomogeneous) Coifman--Meyer bilinear Fourier multipliers and bilinear pseudodifferential operators as special examples, Torres and Xue [Rev. Mat. Iberoam. 36 (2020), 939--956] introduced a new subspace of BMO$\,(\mathbb{R}^n)$, denoted by XMO$\,(\mathbb{R}^n)$, and conjectured that it is just the space VMO$\,(\mathbb{R}^n)$ introduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO$\,(\mathbb{R}^n)$, which further clarifies that XMO$\,(\mathbb{R}^n)$ is a proper subspace of VMO$\,(\mathbb{R}^n)$. This equivalent characterization of XMO$\,(\mathbb{R}^n)$ is formally similar to the corresponding one of CMO$\,(\mathbb{R}^n)$ obtained by A. Uchiyama, but its proof needs some essential new techniques on dyadic cubes as well as some exquisite geometrical observations. As an application, the authors also obtain a weighted compactness result on such bilinear commutators, which optimizes the corresponding result in the unweighted setting.

math.CA↗
arXiv2020

Bilinear Decomposition and Divergence-Curl Estimates on Products Related to Local Hardy Spaces and Their Dual Spaces

Let $p\in(0,1)$, $α:=1/p-1$ and, for any $τ\in [0,\infty)$, $Φ_{p}(τ):=τ/(1+τ^{1-p})$. Let $H^p(\mathbb R^n)$, $h^p(\mathbb R^n)$ and $Λ_{nα}(\mathbb{R}^n)$ be, respectively, the Hardy space, the local Hardy space and the inhomogeneous Lipschitz space on $\mathbb{R}^n$. In this article, applying the inhomogeneous renormalization of wavelets, the authors establish a bilinear decomposition for multiplications of elements in $h^p(\mathbb R^n)$ [or $H^p(\mathbb R^n)$] and $Λ_{nα}(\mathbb{R}^n)$, and prove that these bilinear decompositions are sharp in some sense. As applications, the authors also obtain some estimates of the product of elements in the local Hardy space $h^p(\mathbb R^n)$ with $p\in(0,1]$ and its dual space, respectively, with zero $\lfloor nα\rfloor$-inhomogeneous curl and zero divergence, where $\lfloor nα\rfloor$ denotes the largest integer not greater than $nα$. Moreover, the authors find new structures of $h^{Φ_p}(\mathbb R^n)$ and $H^{Φ_p}(\mathbb R^n)$ by showing that $h^{Φ_p}(\mathbb R^n)=h^1(\mathbb R^n)+h^p(\mathbb R^n)$ and $H^{Φ_p}(\mathbb R^n)=H^1(\mathbb R^n)+H^p(\mathbb R^n)$ with equivalent quasi-norms, and also prove that the dual spaces of both $h^{Φ_p}(\mathbb R^n)$ and $h^p(\mathbb R^n)$ coincide. These results give a complete picture on the multiplication between the local Hardy space and its dual space.

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