Stefano Meda, Błażej Wróbel
In this paper we prove a Marcinkiewicz-type multiplier result for the spherical Fourier transform on products of rank one noncompact symmetric spaces.
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Maximal noncompactness of embeddings into Marcinkiewicz spaces
本文提出了一种新的泛函分析方法，用于研究嵌入马尔钦凯维奇空间中的算子的最大非紧性特性。
This article is concerned with the question of whether Marcinkiewicz multipliers on $\mathbb R^{2n}$ give rise to bilinear multipliers on $\mathbb R^n\times \mathbb R^n$. We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.
This paper was removed by arXiv admin because it plagiarizes "Ding, Yong(PRC-BJN); Fan, Dashan(PRC-ANH); Pan, Yibiao(1-PITT) Weighted boundedness for a class of rough Marcinkiewicz integrals. Indiana Univ. Math. J. 48 (1999), no. 3, 1037--1055."
This paper was removed by arXiv admin because it plagiarizes "Ding, Yong(PRC-BJN); Fan, Dashan(PRC-ANH); Pan, Yibiao(1-PITT) Weighted boundedness for a class of rough Marcinkiewicz integrals. Indiana Univ. Math. J. 48 (1999), no. 3, 1037--1055."
A Marcinkiewicz Testing Criterion for Schur Multipliers
本文证明了Schur乘子在Schatten p-类上有界性的Marcinkiewicz检验条件，推广了Bourgain关于Toeplitz型乘子的结果，并由此得到Schatten p-类的新无条件分解。
The Marcinkiewicz multipliers are $L^{p}$ bounded for $1<p<\infty $ on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (Müller, Ricci and Stein \cite{MRS}). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$, while there is \emph{no} two parameter group of \emph{automorphic} dilations on $\mathbb{H} ^{n}$. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ in the sen…
This 1995 paper contains a sharp version of the classical Marcinkiewicz multiplier theorem for the class of homogeneous Fourier multipliers in two dimensions; here a one-dimensional Marcinkiewicz condition is sufficient. Examples are given to show that the straightforward extension of this statement to higher dimensions does not hold. We provide appropriate versions in higher dimensions which rely on a Fefferman-Stein weighted norm inequality for a relevant Córdoba type square function. The proof of this inequality is given by an early version of an induction on scales argument. For $p<1$ we p…
Improved multiplier theorems on rank one noncompact symmetric spaces
本文在秩一非紧对称空间上改进乘子定理，通过放宽Mikhlin-Hörmander条件，仅要求临界$p$条带上的乘子函数部分产生有界傅里叶乘子，从而获得更一般的结果。
We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators $L=(L_1,...,L_d);$ where some of the operators in $L$ have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair $(\mathcal{L},A)$ are of certain weak type $(1,1).$ Here $\mathcal{L}$ is the Ornstein-Uhlenbeck operator while $A$ is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms $A(\mathcal{…
Let $X$ be a Riemannian symmetric space of noncompact type and $T$ be a linear translation-invariant operator which is bounded on $L^p(X)$. We shall show that if $T$ is not a constant multiple of identity then there exist complex constants $z$ such that $zT$ is chaotic on $L^p(X)$ when $p$ is in the sharp range $2<p<\infty$. This vastly generalizes the result that dynamics of the (perturbed) heat semigroup is chaotic on $X$ proved in [15, 17].
Endpoint estimates for multiparameter multipliers of Marcinkiewicz type
本文研究多参数马尔钦维奇乘法子的端点估计，证明其最佳性并扩展了一维结果到任意多参数。
In this paper, Mikhlin and Marcinkiewicz--Lizorkin type operator-valued multiplier theorems in weighted Lebesgue-Bochner spaces are studied. By using this results embedding theorems in Sobolev-Lions type spaces is obtained. Moreover, maximal regularity properties of abstract elliptic and parabolic equations are derived
In this paper we have studied Fourier multipliers and Littlewood-Paley square functions in the context of modulation spaces. We have also proved that any bounded linear operator from modulation space $\mathcal{M}_{p,q}(\R^n), 1\leq p,q\leq \infty,$ into itself possesses an $l_2-$valued extension. This is an analogue of a well known result due to Marcinkiewicz and Zygmund on classical $L^p-$spaces.
$L^p$-$L^q$ Fourier multipliers and Hausdorff-Young-Paley inequalities on Riemannian symmetric spaces of noncompact type
本文在任意秩非紧型黎曼对称空间上建立Hörmander型Lp-Lq Fourier乘子定理，推导谱乘子定理，证明Laplace-Beltrami算子函数的有界性，并给出Hausdorff-Young和Paley不等式的混合范数版本，结果适用于Damek-Ricci空间。
Essential norms of pointwise multipliers in the non-algebraic setting
本文计算了在不同Köthe空间之间作用的乘法算子的本质范数和弱本质范数，探讨了Banach序列空间和单位圆上解析函数空间的相关应用。
Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator $M$ is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. Suppose $a$ is a Fourier multiplier on the space $X(\mathbb{R})$. We show that the Fourier convolution operator $W^0(a)$ with symbol $a$ is compact on the space $X(\mathbb{R})$ if and only if $a=0$. This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.
We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.
The well-known factorization theorem of Lozanovski{\u ı} may be written in the form $L^{1}\equiv E\odot E^{\prime}$, where $\odot $ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize $F$ through $E$, i.e., when $F\equiv E\odot M(E, F) \,$, where $M(E, F) $ is the space of pointwise multipliers from $E$ to $F$. Properties of $M(E, F) $ were investigated in our earlier paper [KLM12] and here we collect and prove some properties of the construction $E\odot F$. The formulas for pointwise product of Calderón-Loza…
We investigate nontrigonometric forms of Riesz transforms in the context of Schur multipliers. This refines Grothendieck-Haagerup's endpoint criterion with a new condition for the Schatten p-boundedness of Schur multipliers and strengthens Potapov/Sukochev's solution of Arazy's conjecture. We recover as well dimension-free estimates for trigonometric Riesz transforms. Our discrete approach is much simpler than previous harmonic analysis and probabilistic approaches. As an application, we find a very simple proof of recent criteria for Schur multipliers of Hörmander-Mikhlin and Marcinkiewicz ty…
Singular multipliers on multiscale Zygmund sets
本文刻画了使 Hörmander-Mihlin 或 Marcinkiewicz 乘子满足弱型端点模不等式的闭零集，引入多尺度 Zygmund 性质，并给出稀疏加权估计，解决了 Lerner 猜想。
In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X^1(M), X^2(M), ... of new Hardy spaces on M, the sequence Y^1(M/, Y^2(M), ... of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for purely imaginary powers of the Laplace-Beltrami operator and for more general spectral multipliers associated to the Laplace--Beltrami operator L on M. Under the additional condition that the volume of the geodesic balls of radius r i…
The invariant subspaces of periodic Fourier multipliers with application to abstract evolution equations
本文用调和分析识别周期傅里叶乘子不变的大类Banach空间，并应用于抽象二阶积分微分方程，刻画其适定性与最大正则性条件，推广了现有理论。
In the paper compact multiplier operators on Banach spaces of analytic functions on the unit disk with the range in Banach sequence lattices are studied. If the domain space $X$ is such that $H_\infty\hookrightarrow X\hookrightarrow H_1$, necessary and sufficient conditions for compactness are presented. Moreover, the calculation of the Hausdorff measure of noncompactness for diagonal operators between Banach sequence lattices is applied to obtaining the characterization of compact multipliers in case the domain space $X$ satisfies $H_\infty\hookrightarrow X\hookrightarrow H_2$.
Lower estimates for the norm and the Kuratowski measure of noncompactness of Wiener-Hopf type operators
本文研究了Wiener-Hopf型算子在Banach函数空间中的范数及非紧性测度的下界估计，引入了弱倍增和分离倍增性质，得出了算子范数与非紧性测度的明确下界关系。
In this paper we define difference operators and homogeneous Sobolev-type spaces on the dual of a compact Lie group. As an application and to show that this defines a relevant differential structure, we state and prove multiplier theorems of Hörmander, Mihlin and Marcinkiewicz types together with the sharpness in the Sobolev exponent for the one of Hörmander type.
In this paper we prove new inequalities describing the relationship between the "size" of a function on a compact homogeneous manifold and the "size" of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in $L^p$ on SU(2) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on gener…
Weighted Bourgain-Morrey-Besov type and Triebel-Lizorkin type spaces associated with operators
本文在齐次型空间上，针对非负自伴算子引入加权Bourgain-Morrey-Besov型与Triebel-Lizorkin型空间，给出连续刻画、原子分子分解，并证明分数幂与谱乘子的有界性。
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