The Continuity of Commutators on Triebel-Lizorkin Spaces

The purpose of this paper is to study the continuity in the context of Triebel-Lizorkin spaces for some commutators related to certain convolution operators. The operators include Littlewood-Paley operator, Marcinkiewicz integral and Bochner-Riesz operator.     

Littlewood-Paley Functions and Triebel-Lizorkin Spaces,Besov Spaces

We establish Littlewood-Paley characterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average, the ball average, the Bochner-Riesz means and some other well-known operators. We provide a simple proof so that we are able to extend and improve many results published in recent papers.     

Atomic Decompositions of Triebel-Lizorkin Spaces with Local Weights and Applications

In this paper,the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fp,q s,w(Rn)with local weight w by using the Lusin-area functions for the full ranges of the indices,and then establish their atomic decompositions for s ∈ R,p ∈(0,1] and q ∈ [p,∞).The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in(0,1].Finite atomic decompositions for smooth functions in Fp,q s,w(Rn)are also obtained,which further implies that a(sub)linear operator that maps smooth atoms of Fp,q s,w(Rn)uniformly into a bounded set of a(quasi-)Banach space is extended to a bounded operator on the whole Fp,q s,w(Rn).As an application,the boundedness of the local Riesz operator on the space Fp,q s,w(Rn)is obtained.     

New characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces over spaces of homogeneous type

In this paper we use the T1 theorem to prove a new characterization with minimum regularity and cancellation conditions for inhomogeneous Besov and Triebel-Lizorkin spaces over spaces of homogeneous type. These results are new even for R^n.     

THE BOUNDEDNESS OF MAXIMAL BOCHNER-RIESZ OPERATOR AND MAXIMAL COMMUTATOR ON MORREY TYPE SPACES

The boundedness of maximal Bochner-Riesz operator B^δ＊ and that of maximal commutator B^bδ＊, generated by this operator and Lipschitz function on the classical Morrey space and generalized Morrey space are established.     

Weighted Lipschitz Estimate for Commutator of Bochner-Riesz Operators on Weighted Morrey Spaces

In this paper, we will use a method of sharp maximal function approach to show the boundedness of commutator [b,TδR] by Bochner-Riesz operators and the function b on weighted Morrey spaces Lp,κ(ω) under appropriate conditions on the weight ω, where b belongs to Lipschitz space or weighted Lipschitz space.     

极大多线性Bochner-Riesz算子Triebel-Lizorkin空间估计

设BδA,*是由Bochner-Riesz算子生成的极大多线性Bochner-Riesz算子,其中DγA∈Λ.β(γ=m).我们得到了在一定条件下,极大多线性Bochner-Riesz算子在Triebel-Lizorkin空间中的有界性.     

A New Estimate for Bochner-Riesz Operators at the Critical Index on Weighted Hardy Spaces

Let w be a Muckenhoupt weight and Hwp （JRn） be the weighted Hardy space. In this paper, by using the atomic decomposition of Hwp（Rn）, we will show that the Bochner-Riesz operators TRδ are bounded from Hwp（Rn） to the weighted weak Hardy spaces WHwp （Rn） for 0 〈 p 〈 1 and δ = n/p- （n ＋ 1）/2. This result is new even in the unweighted case.     

Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderón's identity. This is inspired by the work of discrete Littlewood-Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe's covering lemma in multiparameter setting.     

Singular integral operators on product Triebel-Lizorkin spaces

In this paper, we consider the rough singular integral operators on product Triebel-Lizorkin spaces and prove certain boundedness properties on the Triebel-Lizorkin spaces. We also use the same method to study the fractional integral operator and the Littlewood-Paley functions. The results extend some known results.     

Triebel-lizorkin space estimates for multilinear operators of sublinear operators

In this paper, we obtain the continuity for some multilinear operators related to certain non-convolution operators on the Triebel-Lizorkin space. The operators include Littlewood-Paley operator and Marcinkiewicz operator.     

Lipschitz estimates for multilinear commutator of Littlewood-Paley operator

In this paper, we will study the continuity of multilinear commutator generated by Littlewood-Paley operator and Lipschitz functions on Triebel-Lizorkin space, Hardy space and Herz-Hardy space.      

Boundedness of Fractional Integrals with a Rough Kernel on the Pro duct Triebel-Lizorkin Spaces

By using the Littlewood-Paley decomposition and the interpolation the-ory, we prove the boundedness of fractional integral on the product Triebel-Lizorkin spaces with a rough kernel related to the product block spaces.     

双线性Fourier乘子在Triebel-Lizorkin和Besov空间的有界性

本文利用Littlewood-Paley分解,Fourier变换和逆变换等方法,研究了双线性Fourier乘子在非齐次正光滑性Triebel-Lizorkin空间和Besov空间的有界性.     

A characterization of Haj?asz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions

In this paper, we establish the equivalence between the Haj?asz-Sobolev spaces or classical Triebel-Lizorkin spaces and a class of grand Triebel-Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when p∈(n/(n+1),∞), we give a new characterization of the Haj?asz-Sobolev spaces via a grand Littlewood-Paley function.     

A note on convolution-type Calderón-Zygmund operators

For convolution-type Calderon-Zygmund operators, by the boundedness on Besov spaces and Hardy spaces, applying interpolation theory and duality, it is known that H5rmander condition can ensure the boundedness on Triebel-Lizorkin spaces Fp^0,q （1 〈 p,q 〈 ∞） and on a party of endpoint spaces FO,q （1 ≤ q ≤ 2）, hut this idea is invalid for endpoint Triebel-Lizorkin spaces F1^0,q （2 〈 q ≤ ∞）. In this article, the authors apply wavelets and interpolation theory to establish the boundedness on F1^0,q （2 〈 q ≤ ∞） under an integrable condition which approaches HSrmander condition infinitely.     

LITTLEWOOD-PALEY THEOREM FOR SCHROEDINGER OPERATORS

Let H be a Schroedinger operator on R^n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

WELL-POSEDNESS IN CRITICAL SPACES FOR THE FULL COMPRESSIBLE MHD EQUATIONS

In this paper we prove local well-posedness in critical Besov spaces for the full compressible MHD equations in R^N, N≥ 2, under the assumptions that the initialdensity is bounded away from zero. The proof relies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term.     

LITTLEWOOD-PALEY THEOREM FOR SCHR DINGER OPERATORS

Let H be a Schrodinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

Riesz transforms and multipliers for the Grushin operator

We show that Riesz transforms associated to the Grushin operator G = ?Δ ? |x|²～ _t ² are bounded on L ^p (? ⁿ⁺¹). We also establish an analogue of the Hörmander-Mihlin Multiplier Theorem and study Bochner-Riesz means associated to the Grushin operator. The main tools used are Littlewood-Paley theory and an operator-valued Fourier multiplier theorem due to L. Weis.