序列位差空间的拓扑性质

利用拓扑学的一般理论,研究了序列位差空间的连通性,给出了序列位差空间为可分空间、紧致空间的充分必要条件.     

局部凸空间的正规性

本文建立局部凸拓扑向量空间的正规性、完全正规性和单调正规性的等价条件．作为这些结果的应用，研究了函数空间的单调正规性和可度量性．     

超积空间及其性质

引进了和内积空间没有相互包含关系的一类新空间:超积空间,进而研究了超积空间的性质.这类新的线性空间具有许多内积空间的重要性质,从而得到了线性空间一类新的度量刻画.     

单调空间与度量化定理

本文回答了关于MCM空间遗传性的一个问题,讨论了k-MCM空间是k半层空间的条件,得到了一些用g函数刻划的度量化定理.主要结论有:MCM空间是关于Fσ子空间遗传的;在正规空间类中,q空间(ωN空间,k-MCM空间)是关于开Fσ子空间遗传的;如果X是具有Gδ对角线的正则次中紧 k-MCM空间,则X是k半层空间;X是可度量化空间的充要条件是存在X上的g函数满足对X中任意不相交的闭集F与紧集C,都有某个n∈ω,使得(∪x∈F g(n,x))∩(∪y∈C g(n,y))=(?).     

INTERPOLATION SPACES BETWEEN H^1 AND L^∞ ON SPACES OF HOMOGENEOUS TYPE

Using the maximal function characterization of Hardy spaces,we study the interpolation spaces between H^1 and L^∞on spaces of homogeneous type.     

关于强J-空间与JHB-空间

引入了两个很有联系的空间类JHB-空间与强J HB-空间，分别推广了J-空间与强J-空间.讨论了J-空间、强J-空间、J HB-空间及强JHB-空间类间的相包含关系及此四空间类逆包含的条件，还得到了JHB-空间的内部刻画，并证明了若对每个α∈S，Xα.都是非紧的连通空间，则积空间∏α∈S Xα是强J-空间。     

核局部凸空间的一个新特征

Ａ．Ｐｉｅｔｓｃｈ＾［１］在讨论核局部凸空间时给出了两类矢值序列空间ｌ１［Ｘ］和ｌ１｛Ｘ｝。本文建立了矢值序列空间ｌ１［Ｘ］及ｌ１｛Ｘ｝和连续线性算子空间Ｌ（ｃ０，Ｘ）及绝对可和算子空间ＡＳ（ｃ０，Ｘ）之间的拓扑同胚关系。通过ｃ０上的矢值算子类Ｌ（ｃ０，Ｘ）和ＡＳ（ｃ０，Ｘ）及其上的拓扑等价关系，对局部凸空间Ｘ是核空间给出了一个新的特征刻划。     

关于弱MCP空间与弱层空间

本文引入了与已知的一些空间类非常类似的两个概念,我们分别称它们为弱MCP空间与弱层空间.本文主要讨论了这两空间类的等价命题与一些基本性质,及它们与原有空间类的关系.     

序列型空间的一个特征刻划

本文证明了拓扑向量空间E是序列型空间的一个特征为：（1）E的每个序列开集都是开集；（2）取值于E中的任意无穷矩阵（xij）i,j,若对每个j均有limxij=xj，并且limxj=x,则一定存在严格递增序列（ik）和（jk）使得limxikjk=x.作为应用证明了序列型A-空间必是k-空间．     

Summability Kernels for <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Multipliers

In this article we study the problem of extending Fourier Multipliers on L ^p (T) to those on L ^p (R) by taking convolution with a kernel, called a summability kernel. We characterize the space of such kernels for the cases p = 1 and p = 2. For other values of p we give a necessary condition for a function to be a summability kernel. For the case p = 1, we present properties of measures which are transferred from M(T) to M(R) by summability kernels. Furthermore it is shown that every l _p sequence can be extended to some L ^q (R) multipliers for certain values of p and q.     

A Noncommutative Version of <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and Characterizations of Subdiagonal Algebras

Let M{\mathcal M} be a σ-finite von Neumann algebra and \mathfrak A{\mathfrak A} a maximal subdiagonal algebra of M{\mathcal M} with respect to a faithful normal conditional expectation F{\Phi} . Based on Haagerup’s noncommutative L ^p space L^p(M){L^p(\mathcal M)} associated with M{\mathcal M} , we give a noncommutative version of H ^p space relative to \mathfrak A{\mathfrak A} . If h ₀ is the image of a faithful normal state j{\varphi} in L¹(M){L^1(\mathcal M)} such that j°F = j{\varphi\circ \Phi=\varphi} , then it is shown that the closure of {\mathfrak Ah₀^\frac1p}{\{\mathfrak Ah_0^{\frac1p}\}} in L^p(M){L^p(\mathcal M)} for 1 ≤ p < ∞ is independent of the choice of the state preserving F{\Phi} . Moreover, several characterizations for a subalgebra of the von Neumann algebra M{\mathcal M} to be a maximal subdiagonal algebra are given.     

Semiclassical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Estimates of Quasimodes on Curved Hypersurfaces

Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L ² normalized family of functions such that P(h)u(h) is O(h) in L ²(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L ^p norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch ^−δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Cyclic Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Bold">R</Emphasis>), 1 \le <Emphasis Type="Italic">p</Emphasis> < \infty

Functions whose translates span L ^p (R) are called L ^p-cyclic functions. For a fixed p \memb [1, \infty], we construct Schwartz-class functions which are L ^r -cyclic for r > p and not L ^r - cyclic for r \le p. We then construct Schwartz-class functions which are L ^r -cyclic for r \ge p and not L ^r -cyclic for r < p. The constructions differ for p \memb (1, 2) and p > 2.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞. Here, we give a sufficient condition for that the weighted L ^p -space L ^p (G, ω) is a Banach algebra. Also, we get some necessary conditions on G and the weight function ω for L ^p (G, ω) to be a Banach algebra. As a consequence, we show that if G is abelian and L ^p (G, ω) is a Banach algebra, then G is σ-compact.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to L^p(G) for all f, g L^p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g L^p(G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2. Received: 23 April 2006