Almost isometric embedding between metric spaces

We investigate the relations ofalmost isometric embedding and ofalmost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of ∝ form exactly one almostisometry class, and similarly with countable dense subsets of Uryson's universal separable metric spaceU. We investigate geometric, set-theoretic and model-theoretic aspects of almost isometry and of almost isometric embedding. The main results show that almost isometric embeddability behaves in the category ofseparable metric spaces differently than in the category of general metric spaces. While in the category of general metric spaces the behavior of universality resembles that in the category of linear orderings —namely, no universal structure can exist on a regular λ > ℵ₁ below the continuum—in the category of separable metric spaces universality behaves more like that in the category of graphs, that is, a small number of metric separable metric spaces on an uncountable regular λ<2^ℵ ₀ may consistently almost isometrically embed all separable metric spaces on λ. Research of the first author was supported by an Israeli Science foundation grant no. 177/01. Research of the second author was supported by the United States-Israel Binational Science Foundation. Publication 827.     

The weighted Herz-type Hardy spaces hK q α,p (ω1,ω2)

Let 1<q<∞, n(1−1/q)≤α<∞, 0<p<∞ and ω₁,ω₂ ɛA ₁(R ⁿ ) (the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces hk _q ^α,p (gw₁,ω₂) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show thatD(R ⁿ ), the class of C^∞(Rⁿ)-functions with compactly support, is dense inhK _q ^α,p (ω₁,ω₂) and there is a subsequence, which converges in distrbutional sense to some distribution ofhK _q ^α,p (ω₁,ω₂), of any bounded sequence inhK _q ^α,p (ω₁,ω₂). In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness. Supported by the NECF and the NECF and the NNSF of China.     

On the weight-spectrum of a compact space

The weight-spectrumSp(w, X) of a spaceX is the set of weights of all infinite closed subspaces ofX. We prove that ifκ>ω is regular andX is compactT ₂ withω(X)≥κ then some λ withκ≤λ≤2^<κ is inSp(ω, X). Under CH this implies that the weight spectrum of a compact space can not omitω ₁, and thus solves problem 22 of [M]. Also, it is consistent with 2^ω=c being anything it can be that every countable closed setT of cardinals less thanc withω ∈ T satisfiesSp(w, X)=T for some separable compact LOTSX. This shows the independence from ZFC of a conjecture made in [AT]. Research supported by OTKA grant no. 1908.     

On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent： （1） b = ω1; （2） t（Sω×Sω1） 〉 ω; （3） For any pair （X, Y）, which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t（X×Y）≤ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.     

On the monadic theory ofω 1 without A.C.

Buchi inLecture Notes in Mathematics, Decidable Theories II (1973) by using A.C. characterized the theoriesMT[β, <] forβ<ω ₁ and showed thatMT[ω ₁, <] is decidable. We extend Buchi’s results to a larger class of models of ZF (without A.C.) by proving the following under ZF only: (1) There is a choice function which chooses a “good” run of an automaton on countable input (Lemma 5.1). It follows that Buchi’s results cocerning countable ordinals are provable within ZF. (2) Let U.D. be the assertion that there exists a uniform denumeration ofω ₁ (i.e. a functionf: ω ₁ → ω ₁ ^ω such that for everyα<ω ₁,f(α) is a function fromω ontoα). We show that U.D. can be stated as a monadic sentence, and thereforeω ₁ is characterizable by a sentence. (3) LetF be the filter of the cofinal closed subsets ofω ₁. We show that if U.D. holds thenMT[ω ₁, <] is recursive in the first order theory of the boolean algebraP (ω ₁)/F. (We can effectively translate each monadic sentence Σ to a boolean sentenceσ such that [ω ₁, <] ⊨ Σ iffP(ω ₁)/F⊨σ). (4) As every complete boolean algebra theory is recursive we have that in every model of ZF+U.D.,MT[ω ₁, <] is recursive. All our proofs are within ZF. Buchi’s work is often referred to. Following Buchi, the main tool is finite automata. We don’t deal withMT[ω ₁, <] forω ₁ which doesn’t satisfy U.D. The results in this paper appeared in the author’s M.Sc. thesis, which was prepared at the Hebrew University under the supervision of Professor M. Rabin.     

More on countably compact,locally countable spaces

Following [5], aT ₃ spaceX is called good (splendid) if it is countably compact, locally countable (andω-fair).G(κ) (resp.S(κ)) denotes the statement that a good (resp. splendid) spaceX with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2 ^ω is big+S(κ) holds unlessω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>_ω+₁+┐G(ω_ω+₁); (iii) the “Chang conjecture” (ω_ω+₁),→(ω ₁,ω) implies ┐S(κ) for allκ≧k≧ω_ω; (iv) ifP addsω ₁ dominating reals toV iteratively then, in , we haveG(λ^ω) for allλ. Research supported by Hungarian National Foundation for Scientific Research grant no. 1805.     

An extension theorem for separable Banach spaces

We construct a totally disconnected ω^*, norming subsetF of the unit ballB ^* of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B^*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ω^ω), is either isomorphic toC(ω^ω) or toc _u.     

On point to point reflection of harmonic functions across real-analytic hypersurfaces in ℝ n

Let Γ be a non-singular real-analytic hypersurface in some domainU ⊂ ℝ ⁿ and let Har₀(U, Γ) denote the linear space of harmonic functions inU that vanish on Γ. We seek a condition onx ⁰,x ¹ ∈U/Γ such that the reflection law (RL)u(x ⁰)+Ku(x ¹)=0, ∀u∈Har₀(U, Γ) holds for some constantK. This is equivalent to the class Har₀ (U, Γ) not separating the pointsx ⁰,x ¹. We find that in odd-dimensional spaces (RL)never holds unless Γ is a sphere or a hyperplane, in which case there is a well known reflection generalizing the celebrated Schwarz reflection principle in two variables. In even-dimensional spaces the situation is different. We find a necessary and sufficient condition (denoted the SSR—strong Study reflection—condition), which we described both analytically and geometrically, for (RL) to hold. This extends and complements previous work by e.g. P.R. Garabedian, H. Lewy, D. Khavinson and H. S. Shapiro.     

Universal WCG banach spaces and universal Eberlein Compacts

We prove that for cardinalsτ satisfying τ^ω=τ and forτ=ω ₁, there do not exist universal Eberlein Compacts of weightτ, or universal WCG spaces of density characterτ. Ifτ is a strong limit cardinal of countable cofinality such universal spaces do exist. Thus under GCH universal spaces exist forτ iff cof(τ)=ω. The research of the second author was supported by a grant from the United States-Israel Binational Science Foundation and the Fund for the Promotion of Research at the Technion.     

Instances of the Conjecture of Chang

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω₁, and (ρ⁺⁺,ρ⁺↠(ρ⁺, ρ), then 0⁺ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get: Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K ⁺ ≧ω₂ and (λ⁺, λ)↠(K ⁺,K. Then 0⁺ exists. Theorem B.Assume that there exists a singularcardinal κ such that(K ⁺,K↠(ω₁, ω₀. Then 0⁺ exists. Theorem C.Assume that (λ ⁺⁺, λ). Then 0⁺ exists (also ifK=ω ₀. Remark. Here, as in the paper of Donder and Koepke, “O⁺ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model. Theorem D.Assume that (λ ⁺⁺, λ)↠(K ⁺, K) and thatK≧ω ₁. Then 0⁺ exists. Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω ₂, or that κ is inaccessible. We shall recall this in due place. Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K ⁺, K↠(ω_n +1, ω_n), for κ singular andn≧0. 0266 0152 V 3 The consistency of several statements of this sort—a prototype of which is (N _ω+1,N _ω)↠(ω₁, ω₀) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.     

The Helmholtz decomposition of weightedLq-spaces for Muckenhoupt weights

We consider weights of Muckenhoupt classA _q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝⁿ we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve the weak Neumann problem for the Laplace equation in weighted spaces on ℝⁿ, ℝⁿ ₊, on bounded and on exterior domains Ω with boundary of classC ¹, which will yield the Helmholtz decomposition ofL _ω ^q(Ω)ⁿ for general ω∈A _q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of Farwig and Sohr [2] where the Helmholtz decomposition ofL _ω ^p(Ω)ⁿ is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood of ϖΩ.

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Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA _q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝⁿ, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝⁿ, ℝⁿ ₊ in domini limitati ed in domini esterni con frontiera di classeC ¹, che conduce alla decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ per un qualsiasi ω∈A _q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.

     

Imbedding theorems for the spaces H p ω,k and H p s,ω,k

Imbedding theorems for the spaces H _p ^ω,k and H _p ^s,ω,k , 1<p<+∞ (with arbitrary smoothness function ω) are studied. In particular, symmetric hulls of these spaces are described, necessary and sufficient conditions of imbedding in Orlicz and Lorentz spaces are derived, and unimprovable bounds of the moduli of continuity in L_q, p<q≤∞, are given. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 83–102, 1987.     

Modules of vector fields,differential forms and degenerations of differential systems

We deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X ₁,..., X_n−1) (codimension 1 differential systems) defined locally on ℝ ⁿ or ℂ ⁿ , and extend the standard duality(X ₁,..., X_n−1)↦(ω), ω=Ω(X₁,...,X_n−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsX _i are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension 1 differential systemsV and are equivalent if and only if so are the corresponding Pfaffian equations (ω) and provided that ω has1-division property: ωΛμ=0, μ—any 1-form ⇒ μ=fω for certain function germf. The 1-division property of ω turns out to be equivalent to the following properties ofV: (a)fX∈V, f—not a 0-divisor function germ ⇒X∈V (thedivision property); (b) (V ^⊥)^⊥=V; (c)V ^⊥=(ω); (d) (ω)^⊥=V, where ⊥ denotes the passing from a module (of vector fields or differential 1-forms) to its annihilator. Supported by Polish KBN grant N^o 2 1090 91 01. Partially supported by the fund for the promotion of research at the Technion, 100–942.     

The List-Chromatic Number of Infinite Graphs Defined on Euclidean Spaces

If D is a countable set of positive reals, 2≤n<ω, let X _n (D) be the graph with the points of R ⁿ as vertices where two vertices are joined iff their distance is in D. We determine the list-chromatic number of X _n (D) as much as possible.     

Theories of linear order

LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2^ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2^ω. (iii) IfS ₁(T) is countable then so isS _n(T) for everyn. (iv) In caseS ₁(T) is countable we find a relation between the Cantor Bendixon rank ofS ₁(T) and the Cantor Bendixon rank ofS _n(T). (v) We define a class of modelsL, and show thatS ₁(T) is finite iff the models ofT belong toL. We conclude that ifS ₁(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models. Most of the results in this paper appeared in the author’s Master of Science thesis which was prepared at the Hebrew University under the supervision of Professor H. Gaifman.     

Various Slicing Indices on Banach Spaces

We give a short and direct proof for the computation of the Szlenk index of the C(K) spaces, when K is a countable compact space and determine their Lavrientiev indices. We also compute the Szlenk index of certain C(α) spaces, where α is an uncountable ordinal. Finally, we show that if the Szlenk index of a Banach space is ω (first infinite ordinal), then its weak*-dentability index is at most ω² and that this estimate is optimal. The first author was supported by the grants: Institutional Research Plan AV0Z10190503, A100190502, GA ČR 201/04/0090.     

Subspaces of<Emphasis Type="Italic">l</Emphasis><Superscript>∞</Superscript> (Γ) without quasicomplements

J. Lindenstrauss proves in [L] thatc ₀(Γ) is not quasicomplemented inl ^∞(Γ) while H. P. Rosenthal in [R] proves that subspaces, whose dual balls are weak* sequentially compact and weak* separable, are quasicomplemented inl ^∞(Γ). In this note it is proved that weak* separability of the dual is the precise condition determining whether a subspace, without isomorphic copies ofl ₁ and whose dual balls are weak* sequentially compact, is quasicomplemented or not inl ^∞(Γ). Especially spaces isomorphic tol _p(Γ), for 1<p<∞, have no quasicomplements inl ^∞(Γ) if Γ is uncountable.     

Random ergodic theorems and real cocycles

We study mean convergence of ergodic averages associated to a measure-preserving transformation or flow τ along the random sequence of times κ _n (ω) given by the Birkhoff sums of a measurable functionF for an ergodic measure-preserving transformationT. We prove that the sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem, i.e., that, for almost every, ω, the averages (*) converge for every choice of τ, if and only if the “cocycle”F satisfies a cohomological condition, equivalent to saying that the eigenvalue group of the “associated flow” ofF is countable. We show that this condition holds in many natural situations. When no assumption is made onF, the random sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem on the class of mildly mixing transformations τ. However, for any aperiodic transformationT, we are able to construct an integrable functionF for which the sequence (k _n(ω)) is not almost surely universally good for the class of weakly mixing transformations.     

On equivalent strictly G-convex renormings of Banach spaces

We study strictly G-convex renormings and extensions of strictly G-convex norms on Banach spaces. We prove that ℓ_ω(Γ) space cannot be strictly G-convex renormed given Γ is uncountable and G is bounded and separable.