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。     

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.     

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。     

一类包含S—闭空间和紧空间的拓扑空间

本文定义了WS—闭空间的概念,它是S—闭空间和紧空间的推广.文中讨论了WS—闭空间的一些性质,推广了s—闭空间的一些结果.     

关于GB型空间的若干结果

本文对GB型空间进行了若干讨论,得到了一些结果。 设X为Banach空间,X为X的共轭空间,B(X,X)为X到X的所有有界线性算子组成的空间。如果X中的弱收敛序列为弱收敛序列,则称X为GB型空间。Grothendieck.A于1953年证明了l~∞是GB型空间。显然自反空间是GB型空间。     

关于Lasnev空间的超空间

讨论了Ｌａｓｎｅｖ空间的超空间的某些性质．文中构造一反例，证明存在可数Ｌａｓｎｅｖ空间，其紧子集超空间不是层型空间．并指出文［６］中关于上述结果的证明中有一关键性失误，故［６］中的反例尚不能说明上述结论成立．本文还对具有σ－ＣＦ拟基的ｋ′空间给出一个刻画定理     

由给定的子空间构造新的子空间

本给出并证明了若干个子空间的并以及两个子空间的基构成子空间的充要条件，从而本质地揭示了除子空间的交与和是构造新的予空间的方法外，集合的其它运算不能构造新的子空间，最后分析了子空间直和的两种不同定义的优缺点，指出了张禾瑞教材中子空间直和定义推广时应注意的一个问题。     

Quasi-Nagata空间的映射定理

本文讨论了quasi—Nagata空间与一类重要广义度量空间k—半分层空间的关系。证明了quasi—Nagata空间的一些映射性质。     

Multi-parameter singular radon transforms I: The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

The purpose of this paper is to study the L ² boundedness of operators of the form f ↦ ψ(x) ∫ f (γ _t (x))K(t)dt, where γ _t (x) is a C ^∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ ^N × ℝ ⁿ , satisfying γ ₀(x) ≡ x, ψ is a C ^∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ ⁿ , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ ^N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L ². The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L ^p boundedness, while the third paper deals with the special case when γ is real analytic.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> boundedness for multilinear fractional integral on product spaces

For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript> by the use of finite submatrices

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l ²(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J _n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].     

Generalized inverses in<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-algebras, 2

Sea é un elemento di unaC ^*-algebra tale chepa=a per qualche proiezionep, alloraa é una unitá inversa generalizzata, chiamata inversa di Moore-Penrose, e si denotata cona ⁺. Per leC ^*-algebras, si ottiene altrest un risultato sulla minimizzazione di ∥axb−c∥.      

Upper bounds for the first dirichlet eigenvalue of a tube around an algebraic complex curve of ℂ<Emphasis Type="Italic">P</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>(<Emphasis Type="Italic">λ</Emphasis>)

We give an upper bound for the first Dirichlet eigenvalue of a tube around a complex curve P of ℂP ⁿ (λ) which depends only on the radius of the tube and the degrees of the polynomials defining P. The bound is sharp on a totally geodesic ℂP ¹(λ) and gives a gap between the eigenvalue of a tube around ℂP ¹(λ) and around other complex curves.     

An Estimate of Growth Bound of Positive <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-Semigroup on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Space and its Applications

Let be a positive C ₀-semigroup on L ^p (Ω), with infinitesimal generator A. In this paper, it is proved that if there exists a such that and , where A ^* is the adjoint of A, then the growth bound of T(t) is upper bounded by b when p = 1, and by when 1 lt; p lt; α and c D(A), where . This is an operator version of a classical stability result on Z-matrix. As application examples, some new results on the asymptotic behaviours of population system and neutron transport system are obtained. Submitted: March 1, 2001?Revised: August 28, 2002     

A weighted estimate for the square function on the unit ball in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show that the Luzin area integral or the square function on the unit ball of ℂ ⁿ , regarded as an operator in the weighted space L ²(w) has a linear bound in terms of the invariant A ₂ characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted L ²(w) norm of the square function. We prove the equivalence of the classical and the invariant A ₂ classes.     

Atomic hardy-type spaces between <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> on metric spaces with non-doubling measures

Let ( Y,d,dl )\left( {\mathcal{Y},d,d\lambda } \right) be (ℝ ⁿ , |·|, μ), where |·| is the Euclidean distance, μ is a nonnegative Radon measure on ℝ ⁿ satisfying the polynomial growth condition, or the Gauss measure metric space (ℝ ⁿ , |·|, d _λ ), or the space (S, d, ρ), where S ≡ ℝ ⁿ ⋉ ℝ⁺ is the (ax + b)-group, d is the left-invariant Riemannian metric and ρ is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces { X_s ( Y ) }_{0 < s \leqslant ￥}\left\{ {X_s \left( \mathcal{Y} \right)} \right\}_{0 < s \leqslant \infty } and the BMO-type spaces { BMO( Y, s ) }_{0 < s \leqslant ￥}\left\{ {BMO\left( {\mathcal{Y}, s} \right)} \right\}_{0 < s \leqslant \infty }. Let H ¹ ( Y )\left( \mathcal{Y} \right) be the known atomic Hardy space and L ₀¹ ( Y )\left( \mathcal{Y} \right) the subspace of f ∈ L ¹ ( Y )\left( \mathcal{Y} \right) with integral 0. The authors prove that the dual space of X _s ( Y )\left( \mathcal{Y} \right) is BMO( Y,s )BMO\left( {\mathcal{Y},s} \right) when s ∈ (0,∞), X _s ( Y )\left( \mathcal{Y} \right) = H ¹ ( Y )\left( \mathcal{Y} \right) when s ∈ (0, 1], and X _∞ ( Y )\left( \mathcal{Y} \right) = L ₀¹ ( Y )\left( \mathcal{Y} \right) (or L ¹ ( Y )\left( \mathcal{Y} \right)). As applications, the authors show that if T is a linear operator bounded from H ¹ ( Y )\left( \mathcal{Y} \right) to L ¹ ( Y )\left( \mathcal{Y} \right) and from L ¹ ( Y )\left( \mathcal{Y} \right) to L ^1,∞ ( Y )\left( \mathcal{Y} \right), then for all r ∈ (1,∞) and s ∈ (r,∞], T is bounded from X _r ( Y )\left( \mathcal{Y} \right) to the Lorentz space L ^1,s ( Y )\left( \mathcal{Y} \right), which applies to the Calderón-Zygmund operator on (ℝ ⁿ , |·|, μ), the imaginary powers of the Ornstein-Uhlenbeck operator on (ℝ ⁿ , |·|, d _γ ) and the spectral operator associated with the spectral multiplier on (S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.     

On Stability of Cones in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript> with Zero Scalar Curvature

In this work we generalize the case of scalar curvature zero the results of Simmons (Ann. Math. 88 (1968), 62–105) for minimal cones in Rⁿ⁺¹. If Mⁿ⁻¹ is a compact hypersurface of the sphere Sⁿ(1) we represent by C(M)ε the truncated cone based on M with center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone base on M also has zero scalar curvature. Hounie and Leite (J. Differential Geom. 41 (1995), 247–258) recently gave the conditions for the ellipticity of the partial differential equation of the scalar curvature. To show that, we have to assume n ⩾ 4 and the three-curvature of M to be different from zero. For such cones, we prove that, for n ≤slant 7 there is an ε for which the truncate cone C(M)_ε is not stable. We also show that for n ⩾ 8 there exist compact, orientable hypersurfaces Mⁿ⁻¹ of the sphere with zero scalar curvature and S₃ different from zero, for which all truncated cones based on M are stable. Mathematics Subject Classifications (2000): 53C42, 53C40, 49F10, 57R70.     

The ultrametric corona problem

Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk D: |x| < 1. Let Mult(A, ∥ · ∥) be the set of continuous multiplicative semi-norms of A, let Mult _m (A, ∥ · ∥) be the subset of the ϕ ∈ Mult(A, ∥ · ∥) whose kernel is a maximal ideal and let Mult _a (A, ∥ · ∥) be the subset of the ϕ ∈ Mult _m (A, ∥ · ∥) whose kernel is of the form (x − a)A, a ∈ D ( if ϕ ∈ Mult _m (A, ∥ · ∥) \ Mult _a (A, ∥ · ∥), the kernel of ϕ is then of infinite codimension). We examine whether Mult _a (A, ∥ · ∥) is dense inside Mult _m (A, ∥ · ∥) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult _a (A, ∥ · ∥) in the whole set Mult(A, ∥ · ∥): this is the corresponding problem to the well-known complex corona problem. We notice that if ϕ ∈ Mult _m (A, ∥ · ∥) is defined by an ultrafilter on D, then ϕ lies in the closure of Mult _a (A, ∥ · ∥). Particularly, we show that this is case when a maximal ideal is the kernel of a unique ϕ ∈ Multm(A, ∥ · ∥). Particularly, when K is strongly valued all maximal ideals enjoy this property. And we can prove this is also true when K is spherically complete, thanks to the ultrametric holomorphic functional calculus. More generally, we show that if ψ ∈ Mult(A, ∥ · ∥) does not define the Gauss norm on polynomials (∥ · ∥), then it is defined by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ ∈ Multm(A, ∥ · ∥) \ Multa(A, ∥ · ∥) or if φ does not lie in the closure of Mult _a (A, ∥ · ∥), then its restriction to polynomials is the Gauss norm. The first situation does happen. The second is unlikely. The text was submitted by the authors in English.