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Dachun Yang

Publications and source records attributed to Dachun Yang.

At least 163 records · Page 9Linked to original sources
arXiv2016

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let $p\in(1,\infty)$ and $q\in[1,\infty)$. In this article, the authors characterize the Triebel-Lizorkin space ${F}^α_{p,q}(\mathbb{R}^n)$ with smoothness order $α\in(0,2)$ via the Lusin-area function and the $g_λ^*$-function in terms of difference between $f(x)$ and its average $B_tf(x):=\frac1{|B(x,t)|}\int_{B(x,t)}f(y)\,dy$ over a ball $B(x,t)$ centered at $x\in\mathbb{R}^n$ with radius $t\in(0,1)$. As an application, the authors obtain a series of characterizations of $F^α_{p,\infty}(\mathbb{R}^n)$ via pointwise inequalities, involving ball averages, in spirit close to Hajłasz gradients, here an interesting phenomena naturally appears that, in the end-point case when $α=2$, these pointwise inequalities characterize the Triebel-Lizorkin spaces $F^2_{p,2}(\mathbb{R}^n)$, while not $F^2_{p,\infty}(\mathbb{R}^n)$. In particular, some new pointwise characterizations of Hajłasz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than $1$ on spaces of homogeneous type.

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arXiv2015

Molecular Characterizations and Dualities of Variable Exponent Hardy Spaces Associated with Operators

Let $L$ be a linear operator on $L^2(\mathbb R^n)$ generating an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ with kernels having pointwise upper bounds and $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(\cdot)}(\mathbb R^n)$, and the BMO-type space ${\mathrm{BMO}}_{p(\cdot),L}(\mathbb R^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of $H_L^{p(\cdot)}(\mathbb R^n)$ and a duality theorem between such a Hardy space and a BMO-type space. As applications, the authors study the boundedness of the fractional integral on these Hardy spaces and the coincidence between $H_L^{p(\cdot)}(\mathbb R^n)$ and the variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$.

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arXiv2015

Anisotropic Hardy-Lorentz Spaces and Their Applications

Let $p\in(0,1]$, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. The authors introduce the anisotropic Hardy-Lorentz space $H^{p,q}_A(\mathbb{R}^n)$ associated with $A$ via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic or the molecular decompositions, the radial or the non-tangential maximal functions, or the finite atomic decompositions. All these characterizations except the $\infty$-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on $\mathbb{R}^n$. As applications, the authors first prove that $H^{p,q}_A(\mathbb{R}^n)$ is an intermediate space between $H^{p_1,q_1}_A(\mathbb{R}^n)$ and $H^{p_2,q_2}_A(\mathbb{R}^n)$ with $0<p_1<p<p_2<\infty$ and $q_1,\,q,\,q_2\in(0,\infty]$, and also between $H^{p,q_1}_A(\mathbb{R}^n)$ and $H^{p,q_2}_A(\mathbb{R}^n)$ with $p\in(0,\infty)$ and $0<q_1<q<q_2\leq\infty$ in the real method of interpolation. The authors then establish a criterion on the boundedness of sublinear operators from $H^{p,q}_A(\mathbb{R}^n)$ into a quasi-Banach space; moreover, the authors obtain the boundedness of $δ$-type Calderón-Zygmund operators from $H^p_A(\mathbb{R}^n)$ to the weak Lebesgue space $L^{p,\infty}(\mathbb{R}^n)$ (or $H^{p,\infty}_A(\mathbb{R}^n)$) in the critical case, from $H_A^{p,q}(\mathbb{R}^n)$ to $L^{p,q}(\mathbb{R}^n)$ (or $H_A^{p,q}(\mathbb{R}^n)$) with $δ\in(0,\frac{\lnλ_-}{\ln b}]$, $p\in(\frac1{1+δ},1]$ and $q\in(0,\infty]$, as well as the boundedness of some Calderón-Zygmund operators from $H_A^{p,q}(\mathbb{R}^n)$ to $L^{p,\infty}(\mathbb{R}^n)$, where $b:=|\det A|$, $λ_-:=\min\{|λ|:\ λ\inσ(A)\}$ and $σ(A)$ denotes the set of all eigenvalues of $A$.

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arXiv2015

Littlewood-Paley Characterizations of Fractional Sobolev Spaces via Averages on Balls

In this paper, the authors characterize Sobolev spaces $W^{α,p}({\mathbb R}^n)$ with the smoothness order $α\in(0,2]$ and $p\in(\max\{1, \frac{2n}{2α+n}\},\infty)$, via the Lusin area function and the Littlewood-Paley $g_λ^\ast$-function in terms of centered ball averages. The authors also show that the condition $p\in(\max\{1, \frac{2n}{2α+n}\},\infty)$ is nearly sharp in the sense that these characterizations are no longer true when $p\in (1,\max\{1, \frac{2n}{2α+n}\})$. These characterizations provide a new possible way to introduce fractional Sobolev spaces with smoothness order in $(1,2]$ on metric measure spaces.

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arXiv2015

Weighted Endpoint Estimates for Commutators of Calderón-Zygmund Operators

Let $δ\in(0,1]$ and $T$ be a $δ$-Calderón-Zygmund operator. Let $w$ be in the Muckenhoupt class $A_{1+δ/n}({\mathbb R}^n)$ satisfying $\int_{{\mathbb R}^n}\frac {w(x)}{1+|x|^n}\,dx<\infty$. When $b\in{\rm BMO}(\mathbb R^n)$, it is well known that the commutator $[b, T]$ is not bounded from $H^1(\mathbb R^n)$ to $L^1(\mathbb R^n)$ if $b$ is not a constant function. In this article, the authors find out a proper subspace ${\mathop\mathcal{BMO}_w({\mathbb R}^n)}$ of $\mathop\mathrm{BMO}(\mathbb R^n)$ such that, if $b\in {\mathop\mathcal{BMO}_w({\mathbb R}^n)}$, then $[b,T]$ is bounded from the weighted Hardy space $H_w^1(\mathbb R^n)$ to the weighted Lebesgue space $L_w^1(\mathbb R^n)$. Conversely, if $b\in{\rm BMO}({\mathbb R}^n)$ and the commutators of the classical Riesz transforms $\{[b,R_j]\}_{j=1}^n$ are bounded from $H^1_w({\mathbb R}^n)$ into $L^1_w({\mathbb R}^n)$, then $b\in {\mathop\mathcal{BMO}_w({\mathbb R}^n)}$.

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arXiv2015

Weighted $L^p$ Estimates of Kato Square Roots Associated to Degenerate Elliptic Operators

Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\geq 2$. In this article, the authors establish some weighted $L^p$ estimates of Kato square roots associated to the degenerate elliptic operators $L_w$. More precisely, the authors prove that, for $w\in A_{p}(\mathbb{R}^n)$, $p\in(\frac{2n}{n+1},\,2]$ and any $f\in C^\infty_c(\mathbb{R}^n)$, $\|L_w^{1/2}(f)\|_{L^p(w,\,\mathbb{R}^n)}\sim \|\nabla f\|_{L^p(w,\,\mathbb{R}^n)}$, where $C_c^\infty(\mathbb{R}^n)$ denotes the set of all infinitely differential functions with compact supports.

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arXiv2015

Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$. In this article, the authors establish the Riesz transform characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$, for $w\in A_{q}(\mathbb{R}^n)$ and $w^{-1}\in A_{2-\frac{2}{n}}(\mathbb{R}^n)$ with $n\geq 3$, $q\in[1,2]$ and $p\in(q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]$ if, for some $r\in[1,\,2)$, $\{tL_w e^{-tL_w}\}_{t\geq 0}$ satisfies the weighted $L^r-L^2$ full off-diagonal estimate.

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arXiv2015

Boundedness of Commutators on Hardy Spaces over Metric Measure Spaces of Non-homogeneous Type

Let $(\mathcal{X},d,μ)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and $b\in\rm{\widetilde{RBMO}(μ)}$ (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $T_b:=bT-Tb$ generated by $T$ and $b$ from the atomic Hardy space $\widetilde{H}^1(μ)$ with the discrete coefficient into the weak Lebesgue space $L^{1,\,\infty}(μ)$. The boundedness of the commutator generated by the generalized fractional integral $T_α\,(α\in(0,1))$ and the $\rm{\widetilde{RBMO}(μ)}$ function from $\widetilde{H}^1(μ)$ into $L^{1/{(1-α)},\,\fz}(μ)$ is also presented. Moreover, by an interpolation theorem for sublinear operators, the authors show that the commutator $T_b$ is bounded on $L^p(μ)$ for all $p\in(1,\infty)$.

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arXiv2015

Products of Functions in ${\mathop\mathrm{BMO}}({\mathcal X})$ and $H^1_{\rm at}({\mathcal X})$ via Wavelets over Spaces of Homogeneous Type

Let $({\mathcal X},d,μ)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, the authors prove that the product $f\times g$ of $f\in H^1_{\rm at}({\mathcal X})$ and $g\in\mathop\mathrm{BMO}({\mathcal X})$, viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $H^1_{\rm at}({\mathcal X})\times\mathop\mathrm{BMO}({\mathcal X})$ into $L^1({\mathcal X})$ and from $H^1_{\rm at}({\mathcal X})\times\mathop\mathrm{BMO}({\mathcal X})$ into $H^{\log}({\mathcal X})$, which affirmatively confirms the conjecture suggested by A. Bonami and F. Bernicot (This conjecture was presented by L. D. Ky in [J. Math. Anal. Appl. 425 (2015), 807-817]).

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arXiv2015

Wavelet Characterizations of the Atomic Hardy Space $H^1$ on Spaces of Homogeneous Type

Let $({\mathcal X},d,μ)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, together with obtaining some crucial lower bounds for regular wavelets, the authors give an unconditional basis of $H^1_{\rm at}({\mathcal X})$ and several equivalent characterizations of $H^1_{\rm at}({\mathcal X})$ in terms of wavelets, which are proved useful.

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arXiv2015

Characterizations of Variable Exponent Hardy Spaces via Riesz Transforms

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying that there exists a constant $p_0\in(0,p_-)$, where $p_-:=\mathop{\mathrm {ess\,inf}}_{x\in \mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(\cdot)/p_0}(\mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $H^{p(\cdot)}(\mathbb R^n)$ via the first order Riesz transforms when $p_-\in (\frac{n-1}n,\infty)$, and via compositions of all the first order Riesz transforms when $p_-\in(0,\frac{n-1}n)$.

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arXiv2015

Characterizations of Besov and Triebel-Lizorkin Spaces via Averages on Balls

Let $\ell\in\mathbb{N}$ and $p\in(1,\infty]$. In this article, the authors prove that the sequence $\{f-B_{\ell,2^{-k}}f\}_{k\in\mathbb{Z}}$ consisting of the differences between $f$ and the ball average $B_{\ell,2^{-k}}f$ characterizes the Besov space $\dot B^α_{p,q}(\rn)$ with $q\in (0, \infty]$ and the Triebel-Lizorkin space $\dot F^α_{p,q}(\rn)$ with $q\in (1,\infty]$ when the smoothness order $α\in(0,2\ell)$. More precisely, it is proved that $f-B_{\ell,2^{-k}}f$ plays the same role as the approximation to the identity $φ_{2^{-k}}\ast f$ appearing in the definitions of $\dot B^α_{p,q}(\rn)$ and $\dot F^α_{p,q}(\rn)$. The corresponding results for inhomogeneous Besov and Triebel-Lizorkin spaces are also obtained. These results, for the first time, give a way to introduce Besov and Triebel-Lizorkin spaces with any smoothness order in $(0, 2\ell)$ on spaces of homogeneous type, where $\ell\in{\mathbb N}$.

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arXiv2015

Interpolation of Morrey-Campanato and Related Smoothness Spaces

In this article, the authors study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e.\,g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the $\pm$-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasize is given to the quasi-Banach case and to the interpolation property.

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arXiv2015

Besov-Type and Triebel--Lizorkin-Type Spaces Associated with Heat Kernels

Let $(M, ρ,μ)$ be an RD-space satisfying the non-collapsing condition. In this paper, the authors introduce Besov-type spaces $B_{p,q}^{s,τ}(M)$ and Triebel--Lizorkin-type spaces $F_{p,q}^{s,τ}(M)$ associated to a non-negative self-adjoint operator $L$ whose heat kernels satisfy some Gaussian upper bound estimate, Hölder continuity, and the stochastic completeness property. Characterizations of these spaces via Peetre maximal functions and heat kernels are established for full range of indices. Also, frame characterizations of these spaces are given. When $L$ is the Laplacian operator on $\mathbb R^n$, these spaces coincide with the Besov-type and Triebel-Lizorkin-type spaces on $\mathbb R^n$ studied in [Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In the case $τ=0$ and the smoothness index $s$ is around zero, comparisons of these spaces with the Besov and Triebel--Lizorkin spaces studied in [Abstr. Appl. Anal. 2008, Art. ID 893409, 250 pp] are also presented.

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arXiv2015

Maximal Function Characterizations of Hardy Spaces Associated to Homogeneous Higher Order Elliptic Operators

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and $(p_-(L),\, p_+(L))$ be the maximal interval of exponents $q\in[1,\,\infty]$ such that the semigroup $\{e^{-tL}\}_{t>0}$ is bounded on $L^q(\mathbb{R}^n)$. In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces $H_L^p(\mathbb{R}^n)$ for all $p\in(0,\,p_+(L))$, which, when $p=1$, answers a question asked by Deng et al. in [J. Funct. Anal. 263 (2012), 604-674]. Moreover, the authors characterize $H_L^p(\mathbb{R}^n)$ via various versions of square functions and Lusin-area functions associated to the operator $L$.

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arXiv2015

Triebel-Lizorkin-Type Spaces with Variable Exponents

In this article, the authors first introduce the Triebel-Lizorkin-type space $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ with variable exponents, and establish its $φ$-transform characterization in the sense of Frazier and Jawerth, which further implies that this new scale of function spaces is well defined. The smooth molecular and the smooth atomic characterizations of $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ are also obtained, which are used to prove a trace theorem of $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$. The authors also characterize the space $F_{p(\cdot),q(\cdot)}^{s(\cdot),ϕ}(\mathbb R^n)$ via Peetre maximal functions.

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arXiv2015

Besov-Type Spaces with Variable Smoothness and Integrability

In this article, the authors introduce Besov-type spaces with variable smoothness and integrability. The authors then establish their characterizations, respectively, in terms of $φ$-transforms in the sense of Frazier and Jawerth, smooth atoms or Peetre maximal functions, as well as a Sobolev-type embedding. As an application of their atomic characterization, the authors obtain a trace theorem of these variable Besov-type spaces.

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