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.     

级联算法在Besov和Triebel-Lizorkin空间上的有界性和收敛性(Ⅱ)(英文)

本文利用联合谱半径刻画了级联算法在Ｂｅｓｏｖ和Ｔｈｉｅｂｅｌ－Ｌｉｚｏｒｋｉｎ空间上的收敛性,给出了级联算法初值函数矩条件的新证明,并利用到细分分布的光滑性和非齐次细分方程解的存在性等方面．特别地,在某些条件下,我们证明了级联算法的有界性和收敛性相互等价．     

级联算法在Besov和Triebel—Lizorkin空间上的有界性和收敛性（Ⅱ）

本文利用联合谱半径刻画了级联算法在Besov和Triebel-Lizorkin空间上的收敛性,给出了级联算法初值函数矩条件的新证明,并利用到细分分布的光滑性和非齐次细分方程解的存在性等方面,特别地,在某些条件下,我们证明了级联算法的有界性和收敛性相互等价。     

Pointwise Multiplication of Besov and Triebel-Lizorkin Spaces

It is shown that para-multiplication applies to a certain product π(u, v) defined for appropriate u and v in ??'(IRⁿ). Boundedness of π(.,.) is investigated for the anisotropic Besov and Triebel-Lizorkin spaces - i.e., for B^M,s/_p,q and F^M,s/_p,q with s ∈ R and p and q in [0, ∞] (though p < ∞ in the F-case) - with a treatment of the generic as well as of various borderline cases. For max (s_o,s,) > 0 the spaces B^M,so/_p,qo ⊕ B^M,so/_p,qo and B^M,so/_p,qo ⊕ B^M,so/_p,qo to which π(.,.) applies arc determined. For generic B^so/_po,qo ⊕ B^s1/_p1,q1 the receiving F^s_p,q spaces are characterized. It is proved that π(f, g) = f · g holds for functions f and g when f · g ∈L_1,loc, roughly speaking. In addition, π(f, u) = fu when f ∈ ??_M and u ∈??' . Moreover, for an arbitrary open set Ω? IRⁿ, a product π_Ω(., .) is defined by lifting to IRⁿ. Boundedness of n on R′ is shown to carry over to π_Ω is general.     

Lipschitz曲面上的Besov空间和Triebel-Lizorkin空间

利用Clifford分析工具,给出了Lipschitz曲面上Besov空间与Triebel-Lizorkin空间定义,并研究其特征刻划．     

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.     

Herz型Besov和Triebel-Lizorkin空间的原子分解

该文给出了Herz型Besov和Triebel-Lizorkin空间的原子分解.     

级联算法在Besov和Triebel-Lizorkin空间上的有界性和收敛性(I)

本文通过引入级联算法的特征多项式和利用一维分解技术,完整地刻画了级联算法在Besov和Triebel-Lizorkin空间上的增量和收敛性。     

Harmonic Functions on Metric Measure Spaces: Convergence and Compactness

We investigate the properties of harmonic functions defined on a metric measure space. Especially, sequences of harmonic functions are examined, i.e. their convergence and compactness. Moreover, Harnack‘s inequality is shown.     

Triebel-Lizorkin空间和Besov空间上关于一类带Hardy核的振荡奇异积分

In this paper,the boundedness is obtained on the Triebel-Lizorkin spaces and the Besov spaces for a class of oscillatory singular integrals with Hardy kernels.     

On 2-Microlocal Herz Type Besov and Triebel-Lizorkin Spaces with Variable Exponents

In this paper, 2-microlocal Herz type Besov and Triebel-Lizorkin spaces with variable exponents are introduced for the first time. Then, we give characterizations of these spaces by so-called Peetre's maximal functions. Further, the atomic and molecular decompositions of these spaces are obtained. Finally, using the characterizations of the spaces by local means and molecular decomposition we obtain the wavelet characterizations.     

Pointwise Characterizations of Besov and Triebel-Lizorkin Spaces with Generalized Smoothness and Their Applications

In this article,the authors first establish the pointwise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness on Rn via the Haj?a...     

The Boundedness of Composition Operators on Triebel–Lizorkin and Besov Spaces with Different Homogeneities

In this paper, we introduce new Triebel–Lizorkin and Besov Spaces associated with the different homogeneities of two singular integral operators. We then establish the boundedness of composition of two Calder′on–Zygmund singular integral operators with different homogeneities on these Triebel–Lizorkin and Besov spaces.     

Spaces of Harmonic Functions

It is important and interesting to study harmonic functionson a Riemannian manifold. In an earlier work of Li and Tam [21]it was demonstrated that the dimensions of various spaces ofbounded and positive harmonic functions are closely relatedto the number of ends of a manifold. For the linear space consistingof all harmonic functions of polynomial growth of degree atmost d on a complete Riemannian manifold Mⁿ of dimension n,denoted by H_d(Mⁿ), it was proved by Li and Tam [20] that thedimension of the space H₁(M) always satisfies dimH₁(M) dimH₁(Rⁿ)when M has non-negative Ricci curvature. They went on to askas a refinement of a conjecture of Yau [32] whether in generaldim H_d(Mⁿ) dimH_d(Rⁿ)for all d. Colding and Minicozzi made animportant contribution to this question in a sequence of papers[5–11] by showing among other things that dimH_d(M) isfinite when M has non-negative Ricci curvature. On the otherhand, in a very remarkable paper [16], Li produced an elegantand powerful argument to prove the following. Recall that Msatisfies a weak volume growth condition if, for some constantA and , (1.1) for all x M and r R, where V_x(r) is the volume of the geodesicball B_x(r) in M; M has mean value property if there exists aconstant B such that, for any non-negative subharmonic functionf on M, (1.2) for all p M and r > 0.     

Tangential Boundary Behaviour of Harmonic and Holomorphic Functions

Let K be a kernel on Rⁿ, that is, K is a non-negative, unboundedL¹ function that is radially symmetric and decreasing. We definethe convolution K * F by and note from L^p-capacity theory [11, Theorem 3] that, if F L^p, p > 1, then K * F exists as a finite Lebesgue integraloutside a set A Rⁿ with C_K,p(A) = 0. For a Borel set A, where We define the Poisson kernel for = {(x, y) : x Rⁿ, y > 0} by and set Thus u is the Poisson integral of the potential f = K * F, andwe write u=P_y*(K*F)=P_y*f=P[f]. We are concerned here with the limiting behaviour of such harmonicfunctions at boundary points of , and in particular with the tangential boundary behaviour ofthese functions, outside exceptional sets of capacity zero orHausdorff content zero.     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

On Restrictions and Extensions of the Besov and Triebel-Lizorkin Spaces with Respect to Lipschitz Domains

The restrictions B^s_pq() and F^s_pq() of the Besov and Triebel–Lizorkinspaces of tempered distributions B^s_pq(Rⁿ) and F^s_pq(Rⁿ) to Lipschitzdomains Rⁿ are studied. For general values of parameters (sR,p>0, q>0) a ‘universal’ linear bounded extensionoperator from B^s_pq() and F^s_pq() into the corresponding spaceson Rⁿ is constructed. The construction is based on a new variantof the Calderón reproducing formula with kernels supportedin a fixed cone. Explicit characterizations of the elementsof B^s_pq() and F^s_pq() in terms of their values in are also obtained.