SOME MULTIPLIER THEOREMS FOR ANISOTROPIC HARDY SPACES——In Memory of Professor Yongsheng Sun

Let A be a symmetric expansive matrix and H^p（R^n） be the anisotropic Hardy space associated with A. For a function m in L∞（R^n）, an appropriately chosen function η in Cc^∞（R^n） and j ∈ Z define mj（ξ） = m（A^jξ）η（ξ）. The authors show that if 0 〈 p 〈 1 and mj belongs to the anisotropic nonhomogeneous Herz space K1^1/P^-1,p（R^n）, then m is a Fourier multiplier from H^p（R^n） to L^V（R^n）. For p = 1, a similar result is obtained if the space K1^0.1（R^n） is replaced by a slightly smaller space K（w）. Moreover, the authors show that if 0 〈 p 〈 1 and if the sequence {（mj）^v} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from H^p（R^n） to L^v（R^n）.

Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators

Let a:=(a_(1),...,a_(n))2[1,∞)^(n),p∈(0,1),andα:=1/p-1.For any x∈R^(n)and t∈[0,∞),letΦ_(p)(x,t):={t/1+(t[x]_(a)^(ν))^(1-p)if να■N,t/1+(t[x]_(a)^(ν))^(1-p)[log(e+|x|a)]^(p)if να∈N,let where [·]a:=1+|·|a,|·|a denotes the anisotropic quasi-homogeneous norm with respect to a,and ν:=a_(1)+…+a_(n).Let H_(a)^(p)(R^(n)),L_(a)^(a)(R^(n)),and H_(a)^(Φ_(p))(R^(n))be,respectively,the anisotropic Hardy space,the anisotropic Campanato space,and the anisotropic Musielak-Orlicz Hardy space associated with Φ_(p) on R^(n).In this article,via first establishing the wavelet characterization of anisotropic Campanato spaces,we prove that for any f∈H_(a)^(p)(R^(n))and g∈L_(a)^(a)(R^(n)),the product of f and g can be decomposed into S(f,g)+T(f,g) in the sense of tempered distributions,where S is a bilinear operator bounded from H_(a)^(p)(R^(n))*L_(a)^(a)(R^(Φ_(p))) to L^(1)(R^(n)) and T is a bilinear operator bounded from H_(a)^(p)(R^(n))*L_(a)^(a)(R^(n)) to H_(a)^(Φ_(p))(R^(n)) .Moreover,this bilinear decomposition is sharp in the dual sense that any y■H_(a)^(Φ_(p))(R^(n)) that fits into the above bilinear decomposition should satisfy(L^(1)(R^(n))+y)*=(L^(1)(R^(n)+H_(a)^(Φ_(p))(R^(n))*.As applications,for any non-constant b∈L_(a)^(a)(R^(n)) and any sublinear operator T satisfying some mild bounded assumptions,we find the largest subspace of H_(a)^(p)(R^(n)),denoted by H_(a,b)^(p)(R^(n)),such that the commutator [b,T] is bounded from H_(a,b)^(p)(R^(n))to L^(1)(R^(n)).In addition,when T is an anisotropic CalderónZygmund operator,the boundedness of [b,T] from H_(a,b)^(p)(R^(n))to L^(1)(R^(n))(or to H_(a)^(1)(R^(n)) is also presented.The key of their proofs is the wavelet characterization of function spaces under consideration.

各向异性加权Herz型Hardy空间(英文)

对于矩阵伸缩A和A1权函数,给出了各向异性加权Herz型Hardy空间的概念,并证明了一类极大函数的加权Lp有界性,得到了各向异性加权Herz型Hardy空间原子分解定理.

各向异性Heisenberg群上一类强Hardy型不等式及其应用

通过推广、改进欧氏空间中的思想,对各向异性Heisenberg群上的Hardy型不等式给出了一个新证明.注意到原有许多结果中,由于使用方法的原因把原点排出在外,首先构造了一类C1向量场,结合逼近的思想不仅改进了这个缺陷而且得到常数cpQ,p的最佳性.作为应用,讨论了一类p次非线性算子的正定性与下无界性.

各向异性Hardy空间上一类奇异积分算子

设A为一个n×n阶实扩张矩阵,引入了一类奇异积分核,并得到由此导出的算子在伴随于伸缩A的各向异性Hardy空间的有界性.

各向异性Hardy空间上的一类卷积型算子

引入了各向异性Hardy空间上的一类卷积型算子,即带(α,r)型核的算子,0≤α<1,r为正整数.研究了这类算子作用在某些原子上的性质,并得到了一个各向异性非齐性Herz型Hardy空间到各向异性Hardy空间有界性的定理.

各向异性Heisenberg群上带余项的Hardy型不等式

利用一些非常精细的估计技巧,证明了各向异性Heisenberg群上的一类带余项的Hardy型不等式,推广了最近文献中关于Heisenberg群上的带余项的Hardy型不等式的结果.

非凸区域上的一些Hardy型不等式

研究非凸区域上的Hardy型不等式.通过选取特殊的向量值函数以及仔细的分析与计算,得到了各向异性Heisenberg群上一类非凸区域上的Hardy不等式,更进一步得到了非凸区域上与Greiner型向量场相关的几类Hardy型不等式.

Fourier transform of anisotropic mixed-norm Hardy spaces

Let a:=(a1,…,an)∈[1,∞)n,p:=(p1,…,pn)∈(0,1]n,Hpa(R^(n))be the anisotropic mixed-norm Hardy space associated with adefined via the radial maximal function,and let f belong to the Hardy space Hpa(R^(n)).In this article,we show that the Fourier transform fcoincides with a continuous function g onℝn in the sense of tempered distributions and,moreover,this continuous function g,multiplied by a step function associated with a,can be pointwisely controlled by a constant multiple of the Hardy space norm of f.These proofs are achieved via the known atomic characterization of Hpa(R^(n))and the establishment of two uniform estimates on anisotropic mixed-norm atoms.As applications,we also conclude a higher order convergence of the continuous function gat the origin.Finally,a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained.All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(R^(n))with p∈0,1],and are even new for isotropic mixed-norm Hardy spaces on∈n.

各向异性Herz型Hardy空间上的振荡奇异积分算子

研究了振荡奇异积分算子T在各向异性Herz型Hardy空间上的有界性问题。当相函数P(x,y)满足▽yP(0,y)=0并且p,q满足一定条件时,利用原子分解定理,证明了这类算子T是从HKq,αp到Kq,αp上的有界算子。这一结论丰富了各向异性Herz型Hardy空间上算子有界性理论。

θ(t)型奇异积分算子在各向异性Hardy空间的有界性

对于伴随于一个扩张矩阵A的各向异性Hardy空间Hp(Rn),利用此空间的原子分解和分子分解,本文讨论了伴随于A的θ(t)型奇异积分算子在各向异性Hardy空间H1(Rn)到L1(Rn)空间的有界性,以及在各向异性Hardy空间Hp(Rn)自身上的有界性。这些结果拓展了θ(t)型奇异积分算子在Hardy空间有界性的结论。

Littlewood-Paley函数在各向异性Musielak-Orlicz型弱Hardy空间上的有界性（英文）

设A是一个扩张矩阵,p∈(0,1]及?:Rn×[0,∞)→[0,∞)是一个各向异性p-增长函数.本文通过主极大函数定义了各向异性Musielak-Orlicz型弱Hardy空间H?,∞A(Rn),并用此空间上的原子分解证明了各向异性LittlewoodPaley Lusin-area函数,各向异性g-函数及各向异性g*λ-函数从H?,∞A(Rn)到弱Musielak-Orlicz-型空间上的有界性.我们指出在g*λ-函数关于空间H?,∞A(Rn)有界性的结论中,参数λ的范围与H?,∞A(Rn)被下述空间所替代时λ的最佳范围仍保持一致,即,被经典Hardy空间或其加权形式,Musielak-Orlicz Hardy空间或各向异性Musielak-Orlicz Hardy空间所替代.

Anisotropic weak Hardy spaces and interpolation theorems

In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.

New Hardy Spaces Associated with Some Anisotropic Herz Spaces and Their Applications

In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝ n are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy spaces.

Molecular Characterization of Anisotropic Musielak–Orlicz Hardy Spaces and Their Applications

Let A be an expansive dilation on Rn and φ ： Hn×[0, ∞）→[0, ∞） an anisotropic Musielak-Orlicz function. Let HAφ（R^n） 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（R^n）. 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（R^n） with p∈[（0, 1] （in this case, A ：= 2In×n,φ（x, t） ：= t^p for all x ∈ R^n and t∈[0,∞））, this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Caldeon-Zygmund operators from HA^φ（Hn） to L^φ（R^n） or from HA^φ（Hn） to itself.

Hormander Type Multipliers on Anisotropic Hardy Spaces

The main purpose of this paper is to establish, using the Littlewood–Paley–Stein theory(in particular, the Littlewood–Paley–Stein square functions), a Calderón–Torchinsky type theorem for the following Fourier multipliers on anisotropic Hardy spaces Hp(Rn;A) associated with expensive dilation A:■Our main Theorem is the following: Assume that m(ξ) is a function on Rn satisfying ■with s > ζ--1(1/p-1/2). Then Tm is bounded from Hp(Rn;A) to Hp(Rn;A) for all 0 < p ≤ 1 and ■where A* denotes the transpose of A. Here we have used the notations mj(ξ) = m(A*jξ)φ(ξ) and φ(ξ) is a suitable cut-off function on Rn, and Ws(A*) is an anisotropic Sobolev space associated with expansive dilation A* on Rn.

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

Let A :=(A_1, A_2) 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~φ_A(R^n× R^m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-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~φ_A(R^n× R^m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R^n× R^m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R^n× R^m) to L~φ(R^n× R^m)and from H~φ_A(R^n×R^m) to itself, whose kernels are adapted to the action of 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.

Weakly Strongly Singular Integral Operators on Anisotropic Hardy Spaces and Their Dual Operators

Let A be an expansive dilation. We define weakly strongly singular integral kernels and study the action of the operators induced by these kernels on anisotropic Hardy spaces associated with A.

极大算子交换子在各向异性Morrey-Herz空间上的有界性(英文)

本文引进了伴随伸缩矩阵A的各向异性齐次Morrey-Herz型空间,利用Hardy-Littlewod极大算子交换子的Lp有界性,证明了Hardy-Littlewod极大算子交换子在各向异性齐次Morrey-Herz型空间上的有界性,对于分数次Hardy-Littlewod极大算子交换子也得到了类似的结果.

Boundedness of the Anisotropic Maximal and Anisotropic Singular Integral Operators in Generalized Morrey Spaces

In this paper we give the conditions on the pair （~1, W2） which ensures the boundedness of the anisotropic maximal operator and anisotropic singular integral operators from one generalized Morrey space Mp,w1 to another Mp,w2, 1 〈 p 〈 ∞, and from the space M1,w1 to the weak space M1,w2.