A function with prescribed initial derivatives in different Banach spaces

Let X₀ ? X₁ ? ··· ? X_p be Banach spaces with continuous injection of X_k into X_{k + 1} for 0 ? k ? p ? 1, and with X₀ dense in X_p. We seek a function u: [0, 1] → X₀ such that its kth derivative u^(k), k = 0, 1,…, p, is continuous from [0, 1] into x_k, and satisfies the initial condition u^(k)(0) = a_k?X_k. It is shown that such a function exists if and only if the initial values a₀, a₁, …, a_p satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ? 2, the spaces X_k (k = 0, 1, …, p) may or may not be such that the desired function exists for any given initial values a_k?X_k.     

Carleson measures and the generalized Campanato spaces of vector-valued martingales

In this paper, the so-called(p, φ)-Carleson measure is introduced and the relationship between vector-valued martingales in the general Campanato spaces Lp,φ(X) and the(p, φ)-Carleson measures is investigated. Specifically, it is proved that for q ∈ [2, ∞), the measure dμ := ||dfk||~qdP ? dm is a(q, φ)-Carleson measure on ? × N for every f ∈ L_q,φ(X)if and only if X has an equivalent norm which is q-uniformly convex; while for p ∈(1, 2], the measure dμ :=||dfk||~pdP ? dm is a(p, φ)-Carleson measure on ? × N implies that f ∈ L_p,φ(X)if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.     

Remote points in large products

A point p ∈ βX\X is a remote point of X if p? cl_βXD for any nowhere dense D ? X. Van Douwen, and independently Chae and Smith, have shown that each non-pseudocompact space of countable π-weight has a remote point. Van Mill showed that many spaces of π-weight ω₁, such as ω×2^ω₁ also have remote points.We show that arbitrarily large products of spaces with countable π-weight which are not pseudocompact have remote points. In particular, ω×2^? for any infinite cardinal ?.     

Fractional Covers for Convolution Products

A non-zero vector-valued sequence u ∈ ?^q(X′) is a cover for a subset M of ?_P(X) if, for some 0 < α ≤ 1, ∥u * h∥ _∞ ≥ α ∥u∥_q ∥h∥_p for all h ∈ M. Covers of ?¹ = ?¹(R) are important in worst case system identification in ?¹ and in the reconstruction of elements in a normed space from corrupted functional values. We investigate the existence of covers for certain naturally occurring subspaces of ?^p(X). We show that there exist finitely supported covers for some subspaces, and obtain lower bounds for their ’lengths’. We also obtain similar results for covers associated with convolution products for spaces of measurable vector-valued functions defined on the positive real axis.     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

Varieties with low dimensional dual variety

In this note we classify the complex projectiven-foldsX ? ? ^N with dimension equal to the dimension of its dualX* minus one,n ≤ 2/3N and positive defect. We show that suchn-folds are hyperplane sections of then + 1-folds withdim X=dim X* (classified by Ein) and some scrolls over curves. The natural generalization is to consider the setS _p of positive defectn-folds withdim(X)=dim(X*) ? (p ? 1) (p a fixed positive integer). We show that the set of possible values of the pair (N,n) withn ≤ 2/3N corresponding to nondegeneraten-folds inS _p which are not scrolls is finite.     

Fractional non-Archimedean calculus in one variable

Let ?? be a natural number. A function f: ? _p ?? K into a non-Archimedeanly valued complete field K ? ? _p is ??-times continuously differentiable if and only if its Mahler coefficients (a _n ) _n??? obey |a _n |n ^?? ?? 0 as n ?? ??. For a real number r ?? 0, this suggests the ad hoc definition by [1] of a C ^r -function f: ? _p ?? K by asking its Mahler coefficients (a _n ) _n??? to satisfy |a _n |n ^r ?? 0 as n?? ??. We will present for functions f: X ?? K on subsets X ? K without isolated points a general pointwise notion of r-fold differentiability through iterated difference quotients, subsequently shown on the domain X = ? _p to coincide with the one given above. For functions on open domains, we prove this notion to admit a handier characterization by its Taylor polynomial up to degree ?r?.     

Criteria for embedding of spaces constructed by the method of means with arbitrary quasi-concave functional parameters

We consider a generalization ?(X₀,X₁)_p0,p1 of the method of means to arbitrary non-degenerate functional parameter. In this case non-trivial embedding ?(X₀,X₁)_p0,p1⊂ψ(X₀,X₁)_q0,q1 take place. We find necessary and sufficient condition for such embedding if 1?q₀?p₀?∞ and 1?q₁?p₁?∞ or 1?p₀?q₀?∞ and 1?p₁?q₁?∞.     

Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.     

On the spectrum of the hierarchical Laplacian

Let (X, d) be a locally compact separable ultrametric space. We assume that (X, d) is proper, that is, any closed ball B?X is a compact set. Given a measure m on X and a function C(B) defined on the set of balls (the choice function) we define the hierarchical Laplacian L _C which is closely related to the concept of the hierarchical lattice of F.J. Dyson. L _C is a non-negative definite self-adjoint operator in L ²(X, m). In this paper we address the following question: How general can be the spectrum \(\mathsf {Spec}(L_{C})\subseteq \mathbb {R}_{+}?\) When (X, d) is compact, S p e c(L _C ) is an increasing sequence of eigenvalues of finite multiplicity which contains 0. Assuming that (X, d) is not compact we show that under some natural conditions concerning the structure of the hierarchical lattice (≡ the tree of d-balls) any given closed subset S ? ?₊, which contains 0 as an accumulation point and is unbounded if X is non-discrete, may appear as S p e c(L _C ) for some appropriately chosen function C(B). The operator ?L _C extends to L ^q (X, m), 1 ≦ q < ∞, as Markov generator and its spectrum does not depend on q. As an example, we consider the operator ?? ^α of fractional derivative defined on the field ? _p of p-adic numbers.     

On interpolation of interpolating sequences

Let A be a uniform algebra on the compact space X and σ a probability measure on X. We define the Hardy spaces H^P(σ) and the H^P(σ) interpolating sequences S in the p-spectrum M_p of σ. Under some structural hypotheses on (A, σ), we prove that if a sequence S ⊂ M_p is H^P(σ) interpolating, then it is H^s(σ) interpolating for s < p. In the special case of the unit ball B of ?ⁿ this answers a natural question asked in [8].     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.     

A spectral sequence for string cohomology

Let X be a 1-connected space with free-loop space ΛX. We introduce two spectral sequences converging towards H^*(ΛX;Z/p) and H^*((ΛX)_hT;Z/p). The E₂-terms are certain non-Abelian-derived functors applied to H^*(X;Z/p). When H^*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.     

Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on ?² (namely operators bounded on both ?¹ and ?^∞) is not inverse-closed. When ?²=?²(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A∈Aω(X) satisfies ‖Afp_‖?‖fp_‖, for some 1?p?∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in ?². The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X=Zd, under additional conditions on the decay.     

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Let θ ∈ (0, 1), λ ∈ [0, 1) and p, p ₀, p ₁ ∈ (1,∞] be such that (1 ? θ)/p ₀ + θ/p ₁ = 1/p, and let φ, φ₀, φ₁ be some admissible functions such that φ, φ₀ ^p/p0 and φ₁ ^p/p1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L _φ0 ^p0),λ (X), L _φ1 ^{p1), λ} (X), θ> of two generalized grand Morrey spaces on a quasi-metric measure space X is the generalized grand Morrey space L _φ ^p),λ (X). Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.     

Dunford-Pettis like properties on tensor products

In this paper we study equivalent formulations of the DP^? P_p (1 < p < ∞). We show that X has the DP^? P_p if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give su?cient conditions on Banach spaces X and Y so that the projective tensor product X ?_π Y, the dual (X ?_? Y)^? of their injective tensor product, and the bidual (X ?_π Y)^?? of their projective tensor product, do not have the DP P_p, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP^? P_p, 1 < p < ∞.     

On the number of permutatively inequivalent basic sequences in a Banach space

Let X be a Banach space with a Schauder basis (en)_n∈N. The relation E₀ is Borel reducible to permutative equivalence between normalized block-sequences of (en)_n∈N or X is c₀ or ?p saturated for some 1?p<+∞. If (en)_n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c₀ or ?p, 1<p<+∞, or the relation E₀ is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X^∗. If (en)_n∈N is unconditional, then either X is isomorphic to ?₂, or X contains ω² subspaces or ω² quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases.     

Three-space problems for the bounded compact approximation property

In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the BCAP, then L∞/X has the BCAP. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y∈X. Let M be a closed subspace of X such that M⊥ is complemented in X*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p.     

Convergence and left-K-sequential completeness in asymmetrically normed lattices

If (X,∥.∥) is a real normed lattice, then p(x)=∥x ⁺∥ defines an asymmetric norm on X. We study the convergence of sequences in the asymmetrically normed lattice (X,p) and give a characterization of the set of limit points of a convergent sequence in the case X=? ^m . These results enable us to prove the left-K-sequential completeness of the asymmetrically normed lattices ? ^m , C(Ω), c ₀, ? _∞ and ? _p (1≦p<∞).     

Some generalizations of the concept of a p-space

In this article, some generalizations of the concept of a p-space are introduced and studied. The notion of a source of a space in a larger space and the concepts of partial plumage, s-embedding, p-embedding, p^?-embedding, s-space, and p^?-space are defined and studied in depth (see Theorems 2.6, 2.7, 3.2, 4.3, 4.4, 4.10 and their corollaries). An example of a hereditarily p^?-space which is not a p-space and is a perfect image of a hereditarily p-space is indicated (Example 2.9). Among the main results, we establish that if a paracompact space X is p-embedded in a pseudocompact space as a dense subspace, then X is a p-space (Corollary 4.8), and that if X has a countable network and is p^?-embedded in a pseudocompact space, then X is metrizable (Corollary 4.11). The following problem is posed: is every paracompact Gδ-subspace of a pseudocompact space ?ech-complete?