arXiv subjects

Wen Yuan

Publications and source records attributed to Wen Yuan.

At least 109 records · Page 6Linked to original sources

arXiv2018

Real-Variable Characterizations of Orlicz-Slice Hardy Spaces

In this article, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by P. Auscher et al. Based on these Orlicz-slice spaces, the authors introduce a new kind of Hardy type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz-Hardy space of A. Bonami and J. Feuto as well as the Hardy-amalgam space of Z. V. de P. Ablé and J. Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood-Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of $δ$-type Calderón-Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood-Paley function characterizations, the dual spaces and the boundedness of $δ$-type Calderón-Zygmund operators on these Hardy-type spaces are also new.

math.CA↗

arXiv2018

Atomic and Littlewood-Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Let $\vec{a}:=(a_1,\ldots,a_n)\in[1,\infty)^n$, $\vec{p}:=(p_1,\ldots,p_n)\in(0,\infty)^n$ and $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ be the anisotropic mixed-norm Hardy space associated with $\vec{a}$ defined via the non-tangential grand maximal function. In this article, via first establishing a Calderón-Zygmund decomposition and a discrete Calderón reproducing formula, the authors then characterize $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_λ^\ast$-function. The obtained Littlewood-Paley $g$-function characterization of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ coincidentally confirms a conjecture proposed by Hart et al. [Trans. Amer. Math. Soc. (2017), DOI: 10.1090/tran/7312]. Applying the aforementioned Calderón-Zygmund decomposition as well as the atomic characterization of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, the authors establish a finite atomic characterization of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, which further induces a criterion on the boundedness of sublinear operators from $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ to itself [or to $L^{\vec{p}}(\mathbb{R}^n)$]. The obtained atomic characterizations of $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ and boundedness of anisotropic Calderón-Zygmund operators on these Hardy-type spaces positively answer two questions mentioned by Cleanthous et al. in [J. Geom. Anal. 27 (2017), 2758-2787]. All these results are new even for the isotropic mixed-norm Hardy spaces on $\mathbb{R}^n$.

math.CA↗

arXiv2017

Intrinsic Structures of Certain Musielak-Orlicz Hardy Spaces

For any $p\in(0,\,1]$, let $H^{Φ_p}(\mathbb{R}^n)$ be the Musielak-Orlicz Hardy space associated with the Musielak-Orlicz growth function $Φ_p$, defined by setting, for any $x\in\mathbb{R}^n$ and $t\in[0,\,\infty)$, $$ Φ_{p}(x,\,t):= \begin{cases} \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}} & \qquad \text{when } n(1/p-1)\notin \mathbb{N} \cup \{0\}; \\ \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}[\log(e+|x|)]^p} & \qquad \text{when } n(1/p-1)\in \mathbb{N}\cup\{0\},\\ \end{cases} $$ which is the sharp target space of the bilinear decomposition of the product of the Hardy space $H^p(\mathbb{R}^n)$ and its dual. Moreover, $H^{Φ_1}(\mathbb{R}^n)$ is the prototype appearing in the real-variable theory of general Musielak-Orlicz Hardy spaces. In this article, the authors find a new structure of the space $H^{Φ_p}(\mathbb{R}^n)$ by showing that, for any $p\in(0,\,1]$, $H^{Φ_p}(\mathbb{R}^n)=H^{ϕ_0}(\mathbb{R}^n) +H_{W_p}^p(\mathbb{R}^n)$ and, for any $p\in(0,\,1)$, $H^{Φ_p}(\mathbb{R}^n)=H^{1}(\mathbb{R}^n) +H_{W_p}^p(\mathbb{R}^n)$, where $H^1(\mathbb{R}^n)$ denotes the classical real Hardy space, $H^{ϕ_0}(\mathbb{R}^n)$ the Orlicz-Hardy space associated with the Orlicz function $ϕ_0(t):=t/\log(e+t)$ for any $t\in [0,\infty)$ and $H_{W_p}^p(\mathbb{R}^n)$ the weighted Hardy space associated with certain weight function $W_p(x)$ that is comparable to $Φ_p(x,1)$ for any $x\in\mathbb{R}^n$. As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.

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arXiv2017

Anisotropic Variable Hardy-Lorentz Spaces and Their Real Interpolation

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the anisotropic variable Hardy-Lorentz space $H_A^{p(\cdot),q}(\mathbb R^n)$ associated with $A$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain characterizations of $H_A^{p(\cdot),q}(\mathbb R^n)$, respectively, in terms of the atom and the Lusin area function. As an application, the authors prove that the anisotropic variable Hardy-Lorentz space $H_A^{p(\cdot),q}(\mathbb R^n)$ severs as the intermediate space between the anisotropic variable Hardy space $H_A^{p(\cdot)}(\mathbb R^n)$ and the space $L^\infty(\mathbb R^n)$ via the real interpolation. This, together with a special case of the real interpolation theorem of H. Kempka and J. Vybíral on the variable Lorentz space, further implies the coincidence between $H_A^{p(\cdot),q}(\mathbb R^n)$ and the variable Lorentz space $L^{p(\cdot),q}(\mathbb R^n)$ when $\mathop\mathrm{essinf}_{x\in\mathbb{R}^n}p(x)\in (1,\infty)$.

math.CA↗

arXiv2017

Interpolation between $H^{p(\cdot)}(\mathbb R^n)$ and $L^\infty(\mathbb R^n)$: Real Method

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into "good" and "bad" parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(\cdot)}(\mathbb R^n)$ and the space $L^{\infty}(\mathbb R^n)$: \begin{equation*} (H^{p(\cdot)}(\mathbb R^n),L^{\infty}(\mathbb R^n))_{θ,\infty} =W\!H^{p(\cdot)/(1-θ)}(\mathbb R^n),\quad θ\in(0,1), \end{equation*} where $W\!H^{p(\cdot)/(1-θ)}(\mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb R^n)$ with $p_-:=\mathop\mathrm{ess\,inf}_{x\in\rn}p(x)\in(1,\infty)$ is proved to coincide with the variable Lebesgue space $W\!L^{p(\cdot)}(\mathbb R^n)$.

math.CA↗

arXiv2016

Variable Weak Hardy Spaces and Their Applications

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on $\mathbb R^n$, $W\!H^{p(\cdot)}(\mathbb R^n)$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W\!H^{p(\cdot)}(\mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_λ^\ast$-function. As an application, the authors establish the boundedness of convolutional $δ$-type and non-convolutional $γ$-order Calderón-Zygmund operators from $H^{p(\cdot)}(\mathbb R^n)$ to $W\!H^{p(\cdot)}(\mathbb R^n)$ including the critical case $p_-={n}/{(n+δ)}$, where $p_-:=\mathop\mathrm{ess\,inf}_{x\in \rn}p(x).$

math.CA↗

arXiv2016

Littlewood-Paley Characterizations of Anisotropic Hardy-Lorentz Spaces

Let $p\in(0,1]$, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. Let $H^{p,q}_A(\mathbb{R}^n)$ be the anisotropic Hardy-Lorentz spaces associated with $A$ defined via the non-tangential grand maximal function. In this article, the authors characterize $H^{p,q}_A(\mathbb{R}^n)$ in terms of the Lusin-area function, the Littlewood-Paley $g$-function or the Littlewood-Paley $g_λ^*$-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space $L^{p,q}(\mathbb{R}^n)$. All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on $\mathbb{R}^n$. Moreover, the range of $λ$ in the $g_λ^*$-function characterization of $H^{p,q}_A(\mathbb{R}^n)$ coincides with the best known one in the classical Hardy space $H^p(\mathbb{R}^n)$ or in the anisotropic Hardy space $H^p_A(\mathbb{R}^n)$.

math.CA↗

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.

math.CA↗

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 $A$.

math.CA↗

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.

math.CA↗

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 $\ell\in{\mathbb N}$.

math.CA↗

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.

math.CA↗

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.

math.CA↗

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.

math.CA↗

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.

math.CA↗

arXiv2014

Hajlasz Gradients Are Upper Gradients

Let $(X, d, μ)$ be a metric measure space, with $μ$ a Borel regular measure. In this paper, we prove that, if $u\in L^1_{\mathop\mathrm{\,loc\,}}(X)$ and $g$ is a Hajłasz gradient of $u$, then there exists $\widetilde u$ such that $\widetilde u=u$ almost everywhere and $4g$ is a $p$-weak upper gradient of $\widetilde u$. This result avoids a priori assumption on the quasi-continuity of $u$ used in [Rev. Mat. Iberoamericana 16 (2000), 243-279]. As an application, an embedding of the Morrey-type function spaces based on Hajłasz-gradients into the corresponding function spaces based on upper gradients is obtained. We also introduce the notion of local Hajłasz gradient, and investigate the relations between local Hajłasz gradient and upper gradient.

math.FA↗

arXiv2014

Weighted multilinear Hardy operators and commutators

In this paper, we introduce a type of weighted multilinear Hardy operators and obtain their sharp bounds on the product of Lebesgue spaces and central Morrey spaces. In addition, we obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted multilinear Hardy operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Riemann-Liouville and Weyl.

math.FA↗

arXiv2014

M Times Photon Subtraction-Addition Coherent Superposition Operated Odd-Schrődinger-cat State: Nonclassicality and Decoherence

We introduce a new non-Gaussian state, generated by m times coherent superposition operation $a\cos θ+a^{\dagger }e^{iφ}\sin θ$ (MCSO) on odd-Schrodinger-cat state (OSCS). Its normalized constant is turned out to be related with the Hermite polynomial. We further investigate the nonclassical properties of the MCSO-OSCS through Mandel's Q-parameter, quadrature squeezing, the photocount distribution and Wigner function (WF). It is shown that the nonclassicality of the MCSO-OSCS is influenced by the number of times (m) of coherent superpositon operation, the angle $θ$ and the amplitude of the coherent state (|$α_{0}$|). Especially the volume of negative region of WF increases with the increment of parameters m, $θ$ and $α_{0}$. We also investigate the decoherence of the MCSO-OSCS in terms of the fadeaway of the negativity of WF in a thermal environment.

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