Expansive Dilation Research Articles

Operator-valued anisotropic Hardy and BMO spaces

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ, and A be an expansive dilation on Rn. In this article, the author introduces the operator-valued anisotropic BMO space BMOA(Rn,M) and obtains a predual space of BMOA(Rn,M), which is the operator-valued anisotropic Hardy space HA1(Rn,M). Moreover, as the Lq-spaces analogues of the BMO spaces, the author introduces the operator-valued anisotropic BMO-type space LqMOA(Rn,M)(2<q<∞), and establishes the duality between LqMOA(Rn,M) and the operator-valued anisotropic Hardy space HAp(Rn,M) with p=(q−1)/q. In addition, the author also obtains some interpolation results and the equivalence of HAp(Rn,M) and Lp(Rn,M) with p∈(1,∞). As an application, the boundedness of anisotropic Calderón-Zygmund operators on BMOA(Rn,M) is established. When A:=2In×n, part of results coincides with those of T. Mei [33], where In×n denotes the n×n unit matrix.

The Fourier transform of anisotropic Hardy spaces with variable exponents and their applications

Let $A$ be an expansive dilation on $\mathbb{R}^n$, and $p(\cdot):\mathbb{R}^n\rightarrow(0,\,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition. Let $\mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors obtain that the Fourier transform of $f\in \mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. As applications, the authors further conclude a higher order convergence of the continuous function $F$ at the origin and then give a variant of the Hardy-Littlewood inequality in the setting of anisotropic Hardy spaces with variable exponents.

Molecular Decomposition of Anisotropic Hardy Spaces With Variable Exponents

Let A be an expansive dilation on ℝn, and p(·): ℝn → (0, ∞) be a variable exponent function satisfying the globally log-Holder continuous condition. Let \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors establish its molecular decomposition, which is still new even in the classical isotropic setting (in the case A:= 2In×n). As applications, the authors obtain the boundedness of anisotropic Calderon-Zygmund operators from \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) to Lp(·)(ℝn) or from \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) to itself.

Molecular characterization of anisotropic variable Hardy-Lorentz spaces

Let $A$ be an expansive dilation on $\mathbb{R}^n$, $q\in(0, \infty]$ and $p(\cdot):\mathbb{R}^n\rightarrow(0, \infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. Let $H^{p(\cdot), q}_A({\mathbb{R}}^n)$ be the anisotropic variable Hardy-Lorentz space defined via the radial grand maximal function. In this paper, the authors first establish its molecular characterization via the atomic characterization of $H^{p(\cdot), q}_A(\mathbb{R}^n)$. Then, as applications, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from $H^{p(\cdot), q}_{A}(\mathbb{R}^n)$ to $L^{p(\cdot), q}(\mathbb{R}^n)$ or from $H^{p(\cdot), q}_{A}(\mathbb{R}^n)$ to itself. All these results are still new even in the classical isotropic setting.

Approximation by crystal-refinable functions

Let $$\varGamma $$ be a crystal group in $$\mathbb {R}^d$$ . A function $$\varphi :\mathbb {R}^d\longrightarrow \mathbb {C}$$ is said to be crystal-refinable (or $$\varGamma $$ -refinable) if it is a linear combination of finitely many of the rescaled and translated functions $$\varphi (\gamma ^{-1}(ax))$$ , where the translations $$\gamma $$ are taken on a crystal group $$\varGamma $$ , and a is an expansive dilation matrix such that $$a\varGamma a^{-1}\subset \varGamma .$$ A $$\varGamma $$ -refinable function $$\varphi : \mathbb {R}^d \rightarrow \mathbb {C}$$ satisfies a refinement equation $$\varphi (x)=\sum _{\gamma \in \varGamma }d_\gamma \varphi (\gamma ^{-1}(ax))$$ with $$d_\gamma \in \mathbb {C}$$ . Let $$\mathcal S(\varphi )$$ be the linear span of $$\{\varphi (\gamma ^{-1}(x)): \gamma \in \varGamma \}$$ and $$\mathcal {S}^h=\{f(x/h):f\in \mathcal {S(\varphi )}\}$$ . One important property of $$\mathcal S(\varphi )$$ is, how well it approximates functions in $$L^2(\mathbb {R}^d)$$ . This property is very closely related to the crystal-accuracy of $$\mathcal S(\varphi )$$ , which is the highest degree p such that all multivariate polynomials q(x) of $$\mathrm{degree}(q)<p$$ are exactly reproduced from elements in $$\mathcal S(\varphi )$$ . In this paper, we determine the accuracy p from the coefficients $$d_\gamma $$ . Moreover, we obtain from our conditions, a characterization of accuracy for a particular lattice refinable vector function, which simplifies the classical conditions.

A classification of anisotropic Besov spaces

We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices A∈GL(d,R), with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative description as decomposition spaces. This result allows to relate properties of function spaces to combinatorial properties of the underlying coverings. This principle is applied to the question of classifying dilation matrices. It turns out that the scales of homogeneous and inhomogeneous Besov spaces differ in the way they depend on the dilation matrix: Two matrices A,B that induce the same scale of homogeneous Besov spaces also induce the same scale of inhomogeneous spaces, but the converse of this statement is generally false. We give a complete characterization of the different types of equivalence in terms of the Jordan normal forms of A,B.

New real-variable characterizations of anisotropic weak Hardy spaces of Musielak-Orlicz type

A real $n\times n$ matrix $A$ is called an expansive dilation if all of its eigenvalues $\lambda $ satisfy $|\lambda |\!\!>\!\!1$. Let $\varphi : \mathbb {R}^n\times [0,\infty )\to [0,\infty )$ be a Musielak-Orlicz function. The aim of this article is to find an appropriate general space which includes the weak Hardy space of Fefferman and Soria, the weighted weak Hardy space of Quek and Yang}, the anisotropic weak Hardy space of Ding and Lan, the Musielak-Orlicz Hardy space of Ky and the anisotropic Hardy space of Musielak-Orlicz type of Li, Yang and Yuan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $m\in \mathbb {N}$ and obtain some new real-variable characterizations of $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ in terms of the radial, the non-tangential and the tangential maximal functions via a new monotone convergence theorem adapted to the weak anisotropic Musielak-Orlicz space $L^{\varphi , \infty }({\mathbb {R}}^n)$. These maximal function characterizations generalize the known results on the anisotropic weak Hardy space $H^{p, \infty }_A({\mathbb {R}}^n)$ with $p\in (0, 1]$ and are new even for their weighted variants or weak Orlicz-Hardy variants. As an application, the authors show the boundedness of a class of multilinear operators formed by the anisotropic Calderon-Zygmund operators from product weighted Lebesgue space to $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $\varphi (x,t):=t^p\omega (x)$ and $\omega \in \mathbb {A}_1(A)$, which is a weighted and non-isotropic extension of Grafakos.

LITTLEWOOD–PALEY CHARACTERIZATIONS OF ANISOTROPIC WEAK MUSIELAK–ORLICZ HARDY SPACES

Let $A$ be an expansive dilation on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D711}:\mathbb{R}^{n}\times [0,\infty )\rightarrow [0,\infty )$ an anisotropic growth function. In this article, the authors introduce the anisotropic weak Musielak–Orlicz Hardy space $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ via the nontangential grand maximal function and then obtain its Littlewood–Paley characterizations in terms of the anisotropic Lusin-area function, $g$-function or $g_{\unicode[STIX]{x1D706}}^{\ast }$-function, respectively. All these characterizations for anisotropic weak Hardy spaces $\mathit{WH}_{A}^{p}(\mathbb{R}^{n})$ (namely, $\unicode[STIX]{x1D711}(x,t):=t^{p}$ for all $t\in [0,\infty )$ and $x\in \mathbb{R}^{n}$ with $p\in (0,1]$) are new. Moreover, the range of $\unicode[STIX]{x1D706}$ in the anisotropic $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ coincides with the best known range of the $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of classical Hardy space $H^{p}(\mathbb{R}^{n})$ or its weighted variants, where $p\in (0,1]$.

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let $$\vec A: = \left( {{A_1},{A_2}} \right)$$ be a pair of expansive dilations and φ: ℝ n ×ℝ m ×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ via the anisotropic Lusin-area function and establish its atomic characterization, the $$\vec g$$ -function characterization, the $$\vec g_\lambda ^*$$ -function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet ( $$\left( {\varphi ,q,\vec s} \right)$$ ), if T is a sublinear operator and maps all ( $$\left( {\varphi ,q,\vec s} \right)$$ )-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to L φ (R n × R m ) and from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to itself, whose kernels are adapted to the action of $$\vec A$$ . The results of this article essentially extend the existing results for weighted product Hardy spaces on ℝ n × ℝ m and are new even for classical product Orlicz-Hardy spaces.

Molecular characterization of anisotropic Musielak–Orlicz Hardy spaces and their applications

Let A be an expansive dilation on ℝn and φ : ℝn × [0, ∞) → [0, ∞) an anisotropic Musielak–Orlicz function. Let HAφ(ℝn) be the anisotropic Hardy space of Musielak–Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of HAφ(ℝn). The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case (namely, A := 2In×n) coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space Hp(ℝn) with p ∈ (0, 1] (in this case, A := 2In×n, φ(x, t) := tp for all x ∈ ℝn and t ∈ [0, ∞)), this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Calderon–Zygmund operators from HAφ(ℝn) to Lφ(ℝn) or from HAφ(ℝn) to itself.

LITTLEWOOD-PALEY CHARACTERIZATIONS OF ANISOTROPIC HARDY SPACES OF MUSIELAK-ORLICZ TYPE

Let $\varphi : \mathbb{R}^n\times [0,\,\infty)\to[0,\infty)$ be a Musielak-Orlicz function and $A$ an expansive dilation. Let $H^\varphi_A({\mathbb {R}}^n)$ be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. Its atomic characterization and some other maximal function characterizations of $H^\varphi_A({\mathbb {R}^n})$, in terms of the radial, the non-tangential and the tangential maximal functions, are known. In this article, the authors further obtain their characterizations in terms of the Lusin-area function, the $g$-function or the $g^\ast_\lambda$-function via first establishing an anisotropic Peetre's inequality of Musielak-Orlicz type. Moreover, the range of $\lambda$ in the $g^\ast_\lambda$-function characterization of $H^\varphi_A({\mathbb {R}}^n)$ coincides with the known best conclusions in the case when $H^\varphi_A({\mathbb {R}}^n)$ is the classical Hardy space $H^p({\mathbb{R}}^n)$ or the anisotropic Hardy space $H^p_A({\mathbb {R}}^n)$ or their weighted variants, where $p\in(0,\,1]$.

A Mean Characterization of Weighted Anisotropic Besov and Triebel-Lizorkin Spaces

In this article, the authors study weighted anisotropic Besov and Triebel-Lizorkin spaces associated with expansive dilations and A_\infty -weights. The authors show that elements of these spaces are locally integrable when the smoothness parameter \alpha is positive. The authors also characterize these spaces for small values of \alpha in terms of a mean square function recently introduced in the context of Sobolev spaces in [Math. Ann. 354 (2012), 589–626] and isotropic Triebel–Lizorkin spaces in [Publ. Mat. 57 (2013), 57–82].

Littlewood–Paley characterization and duality of weighted anisotropic product Hardy spaces

The authors study anisotropic product Hardy spaces Hwp(A→) associated with a pair A→:=(A1,A2) of expansive dilations and a class of product Muckenhoupt weights A∞(A→) on Rn×Rm. This article is a continuation of their earlier work in Bownik et al. (2010) [8]. The authors establish the Littlewood–Paley g-function characterization and the φ-transform characterization of Hwp(A→), 0<p⩽1. The authors also introduce the weighted anisotropic product Campanato space Lp,w(A→) and establish its φ-transform characterization. As an application, the authors identify the dual space of Hwp(A→) with Lp,w(A→). The results of this article improve the existing results for weighted product Hardy spaces on R×R and are new even in the weighted isotropic setting.

Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators

Let φ : ℝn × [0, ∞)→[0, ∞) be a Musielak-Orlicz function and A an expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, H A φ(ℝn), via the grand maximal function. The authors then obtain some real-variable characterizations of H A φ(ℝn) in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space H A p(ℝn) with p ∈ (0,1] and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization of H A φ(ℝn), and, as an application, the authors prove that, for a given admissible triplet (φ, q, s), if T is a sublinear operator and maps all (φ, q, s)-atoms with q < ∞ (or all continuous (φ, q, s)-atoms with q = ∞) into uniformly bounded elements of some quasi-Banach spaces ℬ, then T uniquely extends to a bounded sublinear operator from H A φ(ℝn) to ℬ. These results are new even for anisotropic Orlicz-Hardy spaces on ℝn.

Dimension Functions of Self-Affine Scaling Sets

Abstract.In this paper, the dimension function of a self-affine generalized scaling set associated with an n×n integral expansive dilation A is studied. More specifically, we consider the dimension function of an A-dilation generalized scaling set K assuming that K is a self-affine tile satisfying BK = (K+d1)[ (K + d2), where B = At , A is an n×n integral expansive matrix with |det A| = 2, and d1, d2 ∊ ℝn. We show that the dimension function of K must be constant if either n = 1 or 2 or one of the digits is 0, and that it is bounded by 2|K| for any n.

Closure of dilates of shift-invariant subspaces

AbstractLet V be any shift-invariant subspace of square summable functions. We prove that if for some A expansive dilation V is A-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of V, among them the origin is a point of A*-approximate continuity of the spectral function if we assume this value to be one. We present our results also in a more general setting of A-reducing spaces. We also prove that the origin is a point of A*-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero.

Homogeneous approximation property for wavelet frames with matrix dilations, II

The homogeneous approximation property (HAP) states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.

On Parseval super-frame wavelets

Suppose that η1, ..., ηn are measurable functions in L2(ℝ). We call the n-tuple (η1, ..., ηn) a Parseval super frame wavelet of length n if $$\left\{ {2^{\tfrac{k} {2}} \eta _1 \left( {2^k t - \ell } \right) \oplus \cdots \oplus 2^{\tfrac{k} {2}} \eta _n \left( {2^k t - \ell } \right):k,\ell \in \mathbb{Z}} \right\}$$ is a Parseval frame for L2(ℝ)⊕n. In high dimensional case, there exists a similar notion of Parseval super frame wavelet with some expansive dilation matrix. In this paper, we will study the Parseval super frame wavelets of length n, and will focus on the path-connectedness of the set of all s-elementary Parseval super frame wavelets in one-dimensional and high dimensional cases. We will prove the corresponding path-connectedness theorems.

Extrapolation of discrete Triebel‐Lizorkin spaces

AbstractWe show an extrapolation formula for discrete Triebel‐Lizorkin spaces which extends a formula of Cwikel and Nilsson 14 to quasi‐Banach lattices. This is done in the general setting of anisotropic Triebel‐Lizorkin spaces associated with expansive dilations and doubling measures on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb R^n}$\end{document} introduced by the author 4, 5. Our main result is new even in the standard dyadic setting.

Anisotropic classes of inhomogeneous pseudodifferential symbols

We introduce a class of pseudodifferential operators in the anisotropic setting induced by an expansive dilation A which generalizes the classical isotropic class \({S^{m}_{\gamma, \delta}}\) of inhomogeneous symbols. We extend a well-known L 2-boundedness result to the anisotropic class \({S_{\delta, \delta}^0(A)}\), 0 ≤ δ < 1. As a consequence, we deduce that operators with symbols in the anisotropic class \({S^0_{1,0}(A)}\) are bounded on L p spaces, 1 < p < ∞.