Sequences in the range of a vector measure and Banach spaces satisfying Π1(X,l 1) = Π2(X,l 1)

If X is a Banach space such that Π₁(X,l ₁) = Π₂ (X, l ₁), we prove the following results: 1) A bounded sequence (x _n ) lies inside the range of some X-valued measure if and only if the operator (α _n ) ? l ₁ → Σ _n α _n x _n ? X is 1-summing, and 2) If A is a bounded subset of X lying in the range of some X ^??-valued measure, then A is necessarily contained in the range of some X-valued measure.     

On (N,ϵ)-pseudospectra of operators on Banach spaces

In this paper we extend the concept of the $(N,?)$-pseudospectra of Hansen to the case of bounded linear operators on Banach spaces and prove several relations to the usual spectrum. We particularly discuss the approximation by rectangular finite sections and the impact of the fundamental result of Shargorodsky on “jumping” pseudospectra.     

On Polarized Surfaces (X, L) with h0(L) > 0, κ(X) ≥ 0 and g(L) = q(X) + 1

Let (X,L) be a polarized surface. If h⁰(L)>0, then g(L) q(X). In our previous papers, we classified polarized surfaces (X,L) with g(L)=q(X) and h⁰(L)>0. In this paper, we classify polarized surfaces (X,L) with g(L)=q(X)+1, h⁰(L)>0, and (X) 0.     

On property (β) in Banach lattices, Calderón-Lozanowskii and Orlicz-Lorentz spaces

The geometry of Calderón-Lozanowskii spaces, which are strongly connected with the interpolation theory, was essentially developing during the last few years (see [4, 9, 10, 12, 13, 17]). On the other hand many authors investigated property (β) in Banach spaces (see [7, 19, 20, 21, 25, 26]). The first aim of this paper is to study property (β) in Banach function lattices. Namely a criterion for property (β) in Banach function lattice is presented. In particular we get that in Banach function lattice property (β) implies uniform monotonicity. Moreover, property (β) in generalized Calderón-Lozanowskii function spaces is studied. Finally, it is shown that in Orlicz-Lorentz function spaces property (β) and uniform convexity coincide.     

Product (α1, α2)-modulation spaces

We study fundamental properties of product (α₁, α₂)-modulation spaces built by (α₁, α₂)-coverings of ℝⁿ¹ × ℝⁿ². Precisely we prove embedding theorems between these spaces with different parameters and other classical spaces. Furthermore, we specify their duals. The characterization of product modulation spaces via the short time Fourier transform is also obtained. Families of tight frames are constructed and discrete representations in terms of corresponding sequence spaces are derived. Fourier multipliers are studied and as applications we extract lifting properties and the identification of our spaces with (fractional) Sobolev spaces with mixed smoothness.

     

Variational-type inequalities for (η, θ, δ)-pseudomonotone-type set-valued mappings in nonreflexive Banach spaces

In this paper, we introduce (η, θ, δ)-pseudomonotone-type set-valued mappings and consider the existence of solutions to variational-type inequality problems for (η, θ, δ)-pseudomonotone-type set-valued mappings in nonreflexive Banach spaces.     

On the integersn for which Ω(n)=k (II)

Let (n) be the number of prime divisors ofn, counted with multiplicity. We denote byS(x, k) the set of thenx for which (n)=k, and byV _p(n) the exponent of the primep in the factorization ofn. In a previous paper we proved a result which implies that, ify=x/2 ^k tends to infinity withk>2loglogx where >1, then the distribution of the numbers on the setS(x, k) converges to the normal distribution of Gauss. Here, besides a slight improvement of that result, we give, for the moment of orderq of the above mentioned distribution, a formula which holds uniformly for 2loglogxklog (x/3)/log2 where 1<<3/2.     

On Local Invertible Operators in L2 (ℝ1, H)

We study operators of the form Lu = — G(t) u(t) in L²([t₀ — δ, t₀ + δ], H) with = L² ([t₀— δ, t₀ + δ], H ) in the neighbourhood [t₀— δ, t₀ + δ] of a point t₀ ∈ ℝ¹. Such problems arise in questions on local solvability of partial differential equations (see [6] and [7]). For these operators,one of the major questions is if they are invertible in a neighbourhood of a point t ∈ ℝ¹. To solve this problem we establish needed commutator estimates. Using the commutator estimates and factorization theorems for nonanalytic operator-functions we give additional conditions for the nonanalytic operator -function G(t) and show that the operator L (or ) with some boundary conditions is local invertible.     

On uniformly Gâteaux smooth C (n)-smooth norms on separable Banach spaces

Every separable Banach space with C ⁽ⁿ⁾-smooth norm (Lipschitz bump function) admits an equivalent norm (a Lipschitz bump function) which is both uniformly Gâteaux smooth and C ⁽ⁿ⁾-smooth. If a Banach space admits a uniformly Gâteaux smooth bump function, then it admits an equivalent uniformly Gâteaux smooth norm.     

When Does the Equality J(X~*) = J(X) Hold for a Two-dimensional Banach Space X ?

In this paper, we consider the following problem about the James constant: When does the equality J(X*) = J(X) hold for a Banach space X ? It is known that the James constant of a Banach space does not coincide with that of its dual space in general. In fact, we already have counterexamples of two-dimensional normed spaces that are equipped with either symmetric or absolute norms. However,we show that if the norm on a two-dimensional space X is both symmetric and absolute, then the equality J(X*) = J(X) holds. This provides a global answer to the problem in the two-dimensional case.     

On left (right) ρ-transitive geometric spaces

In this article, first we generalize the concept of $\mathfrak{B}$-parts in a geometric space with respect to a binary relation $\rho$ and then we study left (right) $l{\mathfrak{B}}_{\rho }$-closures ($r{\mathfrak{B}}_{\rho }$-closures, respectively). Finally we introduce and investigate the notions left $\rho$-strongly transitive geometric spaces and left $\rho$-emphatic strongly transitive geometric spaces.     

On the limit theorems for random variables with values in the spaces L p (2≦p<∞)

Summary We prove that whenever B is an infinite dimensional Banach space, there exists a B-valued random variable X failing the Central Limit Theorem (in short the CLT) and such that IEX²= but yet satisfying the Law of the Iterated Logarithm (in short the LIL). We obtain a new sufficient condition for the LIL in Hilbert space and we characterize the random variables with values in l _p or L _p with 2<p< which satisfy the CLT. As an application we show that in l _p (2<p<) the stochastic boundedness of the weighed partial sums does not imply the CLT.Research partially supported by NSF Grant MCS 75-07605 A01     

Banach代数L∞(T;X)

讨论向量值函数的Banach代数L∞（T；X）的极大理想空间的拓扑性质和代数性质，得到了若干结果；给出了Banach空间H∞（D；X）中闭单位球的端点的一条性质．     

On C~(1 ,α) Regularity for Monge- Ampre Equation

In this paper,we consider the well known problem concerning global regularity ofsolutions of the Dirichlet problems for the Monge-Ampère equations.　Theorem1　LetΩ be a uniformly convex domain in Rn with the boundary Ω∈ C3,1 ,φ∈ C2 ,1 ( Ω ) and letf be a nonnegative function in Rnsuch thatf1 / n∈ C1 ,1 (Rn) . Then thereexistsa unique convex solution u∈C1 ,1 / 4 (Ω)∩C1 ,1 (Ω) of the Dirichletproblemdet D2 u =f(x) ,　　 inΩ (1 )u =φ(x) ,　　 on Ω , (2 )where D2 u=[Diju] …     

On complemented subspaces of the spaces (l p ) N ∩l q (l q )

Summary In this paper we study the complemented subspaces of the spaces (l^p)^N∩l^q(l^q), with 1≤p<q≤∞ or q=0, thereby showing that if (l^p)^N∩l^q(l^q)=F⊕G then either F or G contains a complemented copy of the whole space. Entrata in Redazione il 19 aprile 1999. Ricevuta versione finale il 4 febbraio 2000. Research supported by the Italian MURST.