Convenient Vector Spaces of Smooth Functions

By a convenient vector space is meant a locally convex IR-vector space which is separated, bornological and Mackey-complete. The theory of such spaces, initiated in [Kr 82], [Fr 82], and [FGK 83], has evolved into a book [FK 88]. In the preliminaries below we outline the principal features of this theory relevant to this paper. We are concerned mainly with questions about the reflexiveness of spaces C^∞(X, ?) for various X and matters closely related to this.     

光滑Banach空间上扩张值函数的Frechet次微分

首先证明了Frechet光滑Banach空间上齐次函数的次微分的一个有用定理,然后利用下半连续函数和的次微分规则把Clarke-Ledyaev多方向中值不等式推广到多个函数的情形.     

Density of Smooth Functions in Anisotropic Weighted Sobolev Spaces with Weights that are Locally Bounded and Locally Bounded Away from Zero

Mathematical Notes - The density of smooth functions in weighted Sobolev spaces that are anisotropic with respect to the order of the derivatives and the weight functions is established.     

Implementation of the Sparr Interpolation Method in Classes of Spaces of Smooth Functions

The multidimensional Sparr interpolation method is implemented in the Besov spaces and the Lizorkin--Triebel spaces . It is shown that this results in Besov spaces of type . An interpolation theorem for Besov spaces using weak conditions of the form is formulated.     

K—very Smooth Banach Spaces

We introduce the notion of K-very smoothness which is a generalization of very smoothness in Banach spaces. A necessary and sufficient condition for a Banach space to be K-very smooth is obtained. We also consider some relations between K-very smoothness and other geometrical notions.     

某些Banach空间的有限阶光滑点和强光滑点（英）

本文给出l_∞(X)以及L(l₁(X),Y),L(X,l_∞(Y))和L(X,c₀(Y))的单位球的有限阶光滑点和强光滑点的充要条件，这里X和Y都是任意的Banach空间.特别地，本文给出这些空间的单位球的光滑点和强光滑点的充要条件.     

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.     

On Factorizations of Smooth Nonnegative Matrix-Values Functions and on Smooth Functions with Values in Polyhedra

We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to a similar problem for smooth functions with values in a polyhedron. The work was partially supported by NSF Grant DMS-0653121.     

Musielak-Orlicz空间的w*光滑点

Musielak-Orlicz空间是经典Orlicz空间的一种推广，ω^*光滑是Orlicz空间的一种重要几何性质，本讨论了Musielak-Orlicz空间中ω^*光滑点的充分必要条件。     

Random processes in Sobolev-Orlicz spaces

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L _p (Ω) and L ₂ (Ω) in the norm of the space L _q (ℝ). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1340–1356, October, 2006.     

Uniqueness of Minimal Projections in Smooth Matrix Spaces

Let X=(M(n, m), ·), where · fulfills Condition 0.3 and W=M(n, 1)+M(1, m). A formula for a minimal projection from X onto W is given in (E. W. Cheney and W. A. Light, 1985, “Approximation Theory in Tensor Product Spaces,” Lecture Notes in Mathematics, Springer-Verlag, Berlin; E. J. Halton and W. A. Light, 1985, Math. Proc. Cambridge Philos. Soc.97, 127–136; and W. A. Light, 1986, Math. Z.191, 633–643). We will show that this projection is the unique minimal projection (see Theorem 2.1).     

Dual Spaces of Weighted Spaces of Analytic Functions

Siberian Mathematical Journal -     

R-Boundedness of Smooth Operator-Valued Functions

In this paper we study R-boundedness of operator families , where X and Y are Banach spaces. Under cotype and type assumptions on X and Y we give sufficient conditions for R-boundedness. In the first part we show that certain integral operator are R-bounded. This will be used to obtain R-boundedness in the case that is the range of an operator-valued function which is in a certain Besov space . The results will be applied to obtain R-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems. Mark Veraar is supported by the Alexander von Humboldt foundation. His visit to Helsinki, which started this project, was funded by the Finnish Centre of Excellence in Analysis and Dynamics Research.     

Analytic Extension of Smooth Functions

Let F be a closed proper subset of ?ⁿ and let ?* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ? ∈ ?* (?ⁿ) there is ${\widetilde f} \in {\cal E}_{*}(\rm R ^{n})$ which is real analytic on ?ⁿF and such that ?^a ? ¦ _F = ?^a ? ¦ _F for any a ∈ ?₀ ⁿ. For bounded ultradifferentiable functions ? we can obtain ${\widetilde f}$ by means of a continuous linear operator.     

Characterizations of K-smooth Spaces and K-strongly Smooth Spaces

Some necessary and sufficient conditions for a Banach space to be K-smooth or Kstrongly smooth are obtained.     

关于一致光滑Banch空间中的Ishikawa迭代

设E为一致光滑Banach空间,K为E的非空闭凸子集,T：K→K为具有有界值域的连续φ－强伪压缩算子。使用新的分析技巧证明了在非常普遍的条件下,Ishikawa迭代序列{xn}强收敛于T的唯一不动点x^*。改进和扩展了近期许多相关的结果。     

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.     

Invariant Spaces of Entire Functions

Mathematical Notes - Subspaces of the space of analytic functions on a convex domain in the complex plane that are invariant with respect to the differentiation operator are studied. The problem of...