伴随Herz空间的Hardy空间上的向量值Calderón-Zygmund算子(英文)

本文证明了一类具有向量值核的Calderon－Zygmund算子是Herz型Hard,空间HKp到向量值Herz空间KE,p有界的,应用这一结果,得到了粗糙核Calderon－Zygmund算子,极大型Calderon－Zygmund算子,极大算子等是HKp到Kp有界的．     

齐型空间上的双线性Calderon-Zygmund奇异积分算子

文在齐型空间上引入双线性Calderon-Zygmund奇异积分算子的基本概念, 研究了其基本性质以及在L^1× L¹上的弱有界性.     

Nested Commutators of Calderon-Zygmund Operators on Product Spaces

Commutators of Calderon-Zygmund operators on product spaces to be study in the paper is the commutators of Sarah H Ferguson and Michael T Lacey. The L~P boundedness of the nested commutators is proved on product spaces, where 1＜p＜∞.     

广义Calderón-Zygmund算子在加权Hardy空间的有界性

本文讨论了广义Calderon－Zygmund算子在加权Hardy空间上的性质，证明了θ（t）型Calderon－Zygmund算子是H_w~(1,q,0)到L1w及H_w~(1,q,0)到自身的有界算子．     

Calderon-Zygmund奇异积分算子交换子在Herz型Hardy空间中的有界性

本文证明了交换子[6,T]在一类Herz型Hardy空间中的强型与弱型有界性估计,其中6∈BMO(Rn),T为Calderon-zygmund奇异积分算子。     

Singular Integral Operators with Fixed Singularities on Weighted Lebesgue Spaces

The paper is devoted to study of singular integral operators with fixed singularities at endpoints of contours on weighted Lebesgue spaces with general Muckenhoupt weights. Compactness of certain integral operators with fixed singularities is established. The membership of singular integral operators with fixed singularities to Banach algebras of singular integral operators on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights is proved on the basis of Balakrishnans formula from the theory of strongly continuous semi-groups of closed linear operators. Symbol calculus for such operators, Fredholm criteria and index formulas are obtained.     

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L^p(·)(\mathbbR ₊,dm){L^{p(\cdot )}(\mathbb{R} _{+},d\mu)} where dμ is an invariant measure on multiplicative group ${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L ^p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on \mathbbR₊{\mathbb{R}_{+}} and local invertibility of singular integral operators on \mathbbR{\mathbb{R}}. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.     

从混合模空间到Zygmund空间的Volterra型复合算子

本文利用混合模空间H(p,q,φ)中函数的高阶导数的估计,通过构造一些新的检验函数,运用解析函数的性质与算子理论,给出了从混合模空间H(p,q,φ)到Zygmund空间的Volterra型复合算子的有界性和紧性的特征,获得了若干个充要条件.     

Wiener-Hopf Operators with Semi-almost Periodic Matrix Symbols on Weighted Lebesgue Spaces

Fredholm criteria and index formulas are established for Wiener-Hopf operators W(a) with semi-almost periodic matrix symbols a on weighted Lebesgue spaces where 1 < p < ∞, w belongs to a subclass of Muckenhoupt weights and . We also study the invertibility of Wiener-Hopf operators with almost periodic matrix symbols on . In the case N = 1 we also obtain a semi-Fredholm criterion for Wiener-Hopf operators with semi-almost periodic symbols and, for another subclass of weights, a Fredholm criterion for Wiener-Hopf operators with semi-periodic symbols. Work was supported by the SEP-CONACYT Project No. 25564 (México). The second author was also sponsored by the CONACYT scholarship No. 163480.     

Parameterized Littlewood-Paley Operators and Their Commutators on Lebesgue Spaces with Variable Exponent

In this paper, by applying the technique of the sharp maximal function and the equivalent representation of the norm in the Lebesgue spaces with variable exponent, the boundedness of the parameterized Littlewood-Paley operators, including the parameterized Lusin area integrals and the parameterized Littlewood-Paley g*λ-functions, is established on the Lebesgue spaces with variable exponent. Furthermore,the boundedness of their commutators generated respectively by BMO functions and Lipschitz functions are also obtained.     

Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Fredholm conditions and an index formula are obtained for Wiener-Hopf operators W(a) with slowly oscillating matrix symbols a on weighted Lebesgue spaces where 1 < p < ∞, w is a Muckenhoupt weight on and . The entries of matrix symbols belong to a Banach subalgebra of Fourier multipliers on that are continuous on and have, in general, different slowly oscillating asymptotics at ±∞. To define the Banach algebra SO_{p, w} of corresponding slowly oscillating functions, we apply the theory of pseudodifferential and Calderón-Zygmund operators. Established sufficient conditions become a Fredholm criterion in the case of Muckenhoupt weights with equal indices of powerlikeness, and also for Muckenhoupt weights with different indices of powerlikeness under some additional condition on p, w and a. Work was supported by the SEP-CONACYT Project No. 25564 (México). The second author was also sponsored by the CONACYT scholarship No. 163480.     

Ounded Operators on Weak Hardy Spaces and Applications

The atomic decomposition of weak Hardy spaces consisting of Vilenkin martingales is formulated. Some sufficient conditions for a sublinear operator T to be bounded from the weak Hardy space wHp to the weak wLp space are given. As applications a weak version of the Hardy-Littlewood inequality is obtained and it is shown that the maximal operator of the Cesàro means of a Vilenkin-Fourier series is bounded from wHp to wLp and is of weak type (1, 1). This yields that the Cesàro means of a function f L1 converge a.e. to the function in question, provided that the Vilenkin system is bounded.     

Pseudodifference Operators on Weighted Spaces, and Applications to Discrete Schrödinger Operators

We study pseudodifference operators on Z ^N with symbols which are bounded on Z ^N ×T ^N together with their derivatives with respect to the second variable. In the same way as partial differential operators on R ^N are included in an algebra of pseudodifferential operators, difference operators on Z ^N are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces l _w ^p (Z ^N ) and to Phragmen–Lindelöf type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrödinger operators and the decay of their eigenfunctions at infinity.     

关于从混合模空间到小Zygmund空间的Volterra型复合算子

利用混合模空间H(p,q,φ)中函数的高阶导数的估计、解析函数的性质与算子理论,给出了从混合模空间H(p,q,φ)到小Zygmund空间的Volterra型复合算子的有界性和紧性的特征,获得了几个充要条件.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Weighted Grand Lebesgue Space with a Mixed Norm and Integral Operators

Doklady Mathematics - Weighted grand Lebesgue spaces with mixed norms are introduced, and criteria for the boundedness of strong maximal functions and Riesz transforms in these spaces are given.     

齐型空间上的Morrey空间的算子有界性及其应用

作者建立了一大类次线性算子和交换子在齐型空间上的Morrey空间中的有界性．应用这些结果,作者研究了具有不连续系数的超抛物方程解的局部正则性．     

Algebras of Convolution-Type Operators with Piecewise Slowly Oscillating Data on Weighted Lebesgue Spaces

Let \({\mathcal B}_{p,w}\) be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space \(L^p(\mathbb {R},w)\), where \(p\in (1,\infty )\) and w is a Muckenhoupt weight. We study the Banach subalgebra \(\mathfrak {A}_{p,w}\) of \({\mathcal B}_{p,w}\) generated by all multiplication operators aI (\(a\in \mathrm{PSO}^\diamond \)) and all convolution operators \(W^0(b)\) (\(b\in \mathrm{PSO}_{p,w}^\diamond \)), where \(\mathrm{PSO}^\diamond \subset L^\infty (\mathbb {R})\) and \(\mathrm{PSO}_{p,w}^\diamond \subset M_{p,w}\) are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \(\mathbb {R}\cup \{\infty \}\), and \(M_{p,w}\) is the Banach algebra of Fourier multipliers on \(L^p(\mathbb {R},w)\). For any Muckenhoupt weight w, we study the Fredholmness in the Banach algebra \({\mathcal Z}_{p,w}\subset \mathfrak {A}_{p,w}\) generated by the operators \(aW^0(b)\) with slowly oscillating data \(a\in \mathrm{SO}^\diamond \) and \(b\in \mathrm{SO}^\diamond _{p,w}\). Then, under some condition on the weight w, we complete constructing a Fredholm symbol calculus for the Banach algebra \(\mathfrak {A}_{p,w}\) in comparison with Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 74:377–415, 2012) and Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 75:49–86, 2013) and establish a Fredholm criterion for the operators \(A\in \mathfrak {A}_{p,w}\) in terms of their symbols. A new approach to determine local spectra is found.     

Conformal Operators on Weighted Forms; Their Decomposition and Null Space on Einstein Manifolds

There is a class of Laplacian like conformally invariant differential operators on differential forms ${L^\ell_k}$ which may be considered as the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as factored polynomials in second-order differential operators. In the case that the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the ${L^\ell_k}$ in terms of the null spaces of mutually commuting second-order factors.