Precompact Subsets Research Articles

Riesz–Kolmogorov Type Compactness Criteria in Function Spaces with Applications

We present forms of the classical Riesz–Kolmogorov theorem for compactness that are applicable in a wide variety of settings. In particular, our theorems apply to classify the precompact subsets of the Lebesgue space $$L^2$$ , Paley–Wiener spaces, weighted Bargmann–Fock spaces, and a scale of weighted Besov–Sobolev spaces of holomorphic functions that includes weighted Bergman spaces of general domains as well as the Hardy space and the Dirichlet space. We apply the compactness criteria to characterize the compact Toeplitz operators on the Bergman space, deduce the compactness of Hankel operators on the Hardy space, and obtain general umbrella theorems.

Preorders on Subharmonic Functions and Measures with Applications to the Distribution of Zeros of Holomorphic Functions

Let $$X$$ be a class of extended numerical functions on a domain $$D$$ of $$d$$ -dimensional Euclidean space $$\mathbb{R}^{d}$$ , $$H\subset X$$ . Given $$u\in X$$ and $$M\in X$$ , we write $$u\prec_{H}M$$ if there is a function $$h\in H$$ such that $$u+h\leq M$$ on $$D$$ . We consider this special preorder $$\prec_{H}$$ for a pair of subharmonic functions $$u$$ and $$M$$ on $$D$$ in cases when $$H$$ is the space of all harmonic functions on $$D$$ or $$H$$ is the convex cone of all subharmonic functions $$h\not\equiv-\infty$$ on $$D$$ . Main results are dual equivalent forms for this preorder $$\prec_{H}$$ in terms of balayage processes for Riesz measures of subharmonic functions $$u$$ and $$M$$ , for Jensen and Arens–Singer (representing) measures, for potentials of these measures, and for special test functions generated by subharmonic functions on complements $$D\backslash S$$ of non-empty precompact subsets $$S\Subset D$$ . Applications to holomorphic functions $$f$$ on a domain $$D\subset\mathbb{C}^{n}$$ relate to the distribution of zero sets of functions $$f$$ under upper restrictions $$|f|\leq\exp M$$ on $$D$$ . If a domain $$D\subset\mathbb{C}$$ is a finitely connected domain with non-empty exterior or a simply connected domain with two different points on the boundary of $$D$$ , then our conditions for the distribution of zeros of $$f\neq 0$$ with $$|f|\leq\exp M$$ on $$D$$ are both necessary and sufficient.

$P$-bases and topological groups

A topological space $X$ is defined to have a neighborhood $P$-base at any $x\in X$ from some partially ordered set (poset) $P$ if there exists a neighborhood base $(U_p[x])_{p\in P}$ at $x$ such that $U_p[x]\subseteq U_{p’}[x]$ for all $p\geq p’$ in $P$. We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a $\mathcal {K}(M)$-base for some separable metric space $M$. Banakh [Dissertationes Math. 538 (2019), p. 141] gives a positive answer to Problem 8.6.8. Let $A(X)$ be the free Abelian topological group on $X$. It is shown that if $Y$ is a retract of $X$ such that the free Abelian topological group $A(Y)$ has a $P$-base and $A(X/Y)$ has a $Q$-base, then $A(X)$ has a $P\times Q$-base. Also if $Y$ is a closed subspace of $X$ and $A(X)$ has a $P$-base, then $A(X/Y)$ has a $P$-base. It is shown that any Fréchet-Urysohn topological group with a $\mathcal {K}(M)$-base for some separable metric space $M$ is first-countable, hence metrizable. And if $P$ is a poset with calibre $(\omega _1, \omega )$ and $G$ is a topological group with a $P$-base, then any precompact subset in G is metrizable, hence $G$ is strictly angelic. Applications in function spaces $C_p(X)$ and $C_k(X)$ are discussed. We also give an example of a topological Boolean group of character $\leq \mathfrak {d}$ such that the precompact subsets are metrizable but $G$ doesn’t have an $\omega ^\omega$-base if $\omega _1<\mathfrak {d}$. Gabriyelyan, Kakol, and Liederman [Fund. Math. 229 (2015), pp. 129–158] give a consistent negative answer to Problem 6.5.

On proximal fineness of topological groups in their right uniformity

&lt;p&gt;A uniform space X is said to be proximally fine if every proximally continuous function defined on X into an arbitrary uniform pace Y is uniformly continuous. We supply a proof that every topological group which is functionally generated by its precompact subsets is proximally fine with respect to its right uniformity. On the other hand, we show that there are various permutation groups G on the integers N that are not proximally fine with respect to the topology generated by the sets {g ∈ G : g(A) ⊂ B}, A, B ⊂ N.&lt;/p&gt;

On Nonperiodic Euler Flows with Hölder Regularity

In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager’s conjecture on the failure of energy conservation for incompressible Euler flows with Holder regularity not exceeding \({1/3}\). This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space \({L_t^\infty B_{3,\infty}^{1/3}}\) due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Holder exponent less than \({1/5}\). The main result of the present paper shows that any given smooth Euler flow can be perturbed in \({C^{1/5-\epsilon}_{t,x}}\) on any pre-compact subset of \({\mathbb{R}\times \mathbb{R}^3}\) to violate energy conservation. Furthermore, the perturbed solution is no smoother than \({C^{1/5-\epsilon}_{t,x}}\). As a corollary of this theorem, we show the existence of nonzero \({C^{1/5-\epsilon}_{t,x}}\) solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Holder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.

Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation

For a Banach spaceXand a measure space(Ω,Σ), letM(Ω,X)be the space of allX-valued countably additive measures on(Ω,Σ)of bounded variation, with the total variation norm. In this paper we give a characterization of weakly precompact subsets ofM(Ω,X).

Sobolev Inequalities and the $$\overline{\partial }$$ ∂ ¯ -Neumann Operator

We study a complex-valued version of the Sobolev inequalities and its relationship to compactness of the \(\overline{\partial }\)-Neumann operator. For this purpose we use an abstract characterization of compactness derived from a general description of precompact subsets in \(L^2\)-spaces. Finally we remark that the \(\overline{\partial }\)-Neumann operator can be continuously extended provided a subelliptic estimate holds.

Prüfer domains of integer-valued polynomials on a subset

Let D be an integral domain with quotient field K and let E be an additive subgroup of D. We investigate the question of when Int(E,D)={f∈K[X]|f(E)⊆D} is a Prüfer domain. We show that if D is a Krull-type Prüfer domain, an SFT Prüfer domain, or the ring of entire functions, then Int(E,D) is a Prüfer domain if and only if sup{|E/IDM||M∈Max(D)}<∞ for each nonzero fractional ideal I of D. In particular, for a valuation domain D, we also show that: Int(E,D) is a Prüfer domain if and only if Int(E,D) has the strong 2-generator property if and only if Int(E,D) is dense in C(Eˆ,Dˆ) for the uniform convergence topology if and only if E is a precompact subset of K. This gives a partial answer to the question posed by Cahen, Chabert, and Loper.

Critically large entropy of absolutely convex hulls in Banach spaces of type [formula omitted

Given a precompact subset A of a Banach space of type p, we investigate and quantify in which way the rate of decay of the entropy numbers of A affects the rate of decay of the entropy numbers of the real absolutely convex hull of A. This problem has been studied by various researchers in different settings during the last 25 years. We get new and best possible results both in the case of logarithmic decay of entropy numbers of A and in the case that the entropy numbers of A belong to some Lorentz sequence space making a major step to completing the picture.

PRECOMPACTNESS OF THE SET OF TRAJECTORIES OF THE CONTROLLABLE SYSTEM DESCRIBED BY A NONLINEAR VOLTERRA INTEGRAL EQUATION

In this paper the controllable system whose behaviour is described by a nonlinear Volterra integral equation, is studied. The set of admissible control functions is the closed ball of the space L p (p &gt; 1) with radius µ 0 and centered at the origin. It is shown that the set of trajectories of the system is a bounded and precompact subset of the space of continuous functions.

Kuelbs–Li Inequalities and Metric Entropy of Convex Hulls

Let T be a precompact subset of a Hilbert space. We make use of a precise link between the absolutely convex hull \(\operatorname{aco}(T)\) and the reproducing kernel Hilbert space of a Gaussian random variable constructed from T. Firstly, we avail ourselves of it for optimality considerations concerning the well-known Kuelbs–Li inequalities. Secondly, this enables us to apply small deviation results to the problem of estimating the metric entropy of \(\operatorname{aco}(T)\) in dependence of the metric entropy of T.

Finite generation properties for various rings of integer-valued polynomials

For any subset E of a Dedekind domain D, we show the ring Int { r } ( E , D ) of polynomials that are integer-valued on E together with all their divided differences of order up to r not to be a finitely generated D-algebra, contrary to the ring Int x ( E , D ) of integer-valued polynomials on E having a given non-zero modulus x (which is hence Noetherian, since the domain D is so). Localization properties allow us to focus on valuation domains; furthermore, the consideration of precompact subsets allows us to consider valuation domains V of arbitrary dimension.

P -adic subsets whose factorials satisfy a generalized Legendre formula

Amice studied the notion of a regular compact subset in a local field K, with valuation v and maximal ideal 𝔐. In her work, she introduced the notion of well distributed sequences and showed that every regular compact subset S admits well distributed sequences and that its factorial sequence (n!S) satisfies a generalized Legendre formula: ν ( n ! s ) = ∑ i = 1 i = ∞ [ n q i ] for every integer n and where qi denotes the number of classes of S modulo 𝔐i. In this article, in more general settings, we show the converse assertions. More precisely, we prove that, for every precompact subset of any discrete valuation domain V, the following assertions are equivalent: the topological closure of S is a regular subset, S admits a very well distributed sequence, S satisfies the generalized Legendre formula.

Necessary and sufficient conditions for precompact sets to be metrisable

This self-contained paper characterises those locally convex spaces whose (weakly) precompact (respectively, compact) subsets are metrisable. Applications and examples are provided. Our approach also applies to get Cascales-Orihuela's, Valdivia's and Robertson's metrisation theorems for (pre)compact sets.

Contraction mapping on probabilistic hyperspace and an application

We considered a class of precompact subsets of a PM space (here we called probabilistic hyperspace) and applied the fixed point principle in this space to obtain an attractor of the dynamical system defined through the iterated function systems (IFS).

Weight of precompact subsets and tightness

Pfister (1976) and Cascales and Orihuela (1986) proved that precompact sets in (DF)- and (LM)-spaces have countable weight, i.e., are metrizable. Improvements by Valdivia (1982), Cascales and Orihuela (1987), and Ka̧kol and Saxon (preprint) have varying methods of proof. For these and other improvements a refined method of upper semi-continuous compact-valued maps applied to uniform spaces will suffice. At the same time, this method allows us to dramatically improve Kaplansky's theorem, that the weak topology of metrizable spaces has countable tightness, extending it to include all (LM)-spaces and all quasi-barrelled (DF)-spaces, both in the weak and original topologies. One key is showing that for a large class G including all (DF)- and (LM)-spaces, countable tightness of the weak topology of E in G is equivalent to realcompactness of the weak∗ topology of the dual of E.

Weak compactness in dual spaces

We present in this note an extension and an elementary proof of a characterization of¶\( \sigma(E^*, E) \)-sequentially precompact subsets of E *, where E is a \( \sigma \)-Dedekind complete M-space with a unit.

Metric entropy of convex hulls

LetT be a precompact subset of a Hilbert space. The metric entropy of the convex hull ofT is estimated in terms of the metric entropy ofT, when the latter is of order εℒ2. The estimate is best possible. Thus, it answers a question left open in [CKP].

Metric entropy of convex hulls in type $p$ spaces—The critical case

Given a precompact subset A of a type p Banach space E, where p ∈ (1, 2], we prove that for every β E [0,1) and all n ∈ N sup k 1/p' (log k) β-1 e k (aco A) ≤ c sup k 1/p' (log k) β e k (A) k≤n holds, where acoA is the absolutely convex hull of A and e k (.) denotes the k th dyadic entropy number. With this inequality we show in particular that for given A and β ∈ (-∞,1) with e n (A) ≤ n -1/p' (log n) -β for all n ∈ N the inequality e n (aco A) < c n -1/p' (log n) -β+1 holds true for all n E N. We also prove that this estimate is asymptotically optimal whenever E has no better type than p. For β = 0 this answers a question raised by Carl, Kyrezi, and Pajor which has been solved up to now only for the Hilbert space case by F. Gao.

An Ascoli theorem for sequential spaces

Ascoli theorems characterize “precompact” subsets of the set of morphisms between two objects of a category in terms of “equicontinuity” and “pointwise precompactness,” with appropriate definitions of precompactness and equicontinuity in the studied category. An Ascoli theorem is presented for sets of continuous functions from a sequential space to a uniform space. In our development we make extensive use of the natural function space structure for sequential spaces induced by continuous convergence and define appropriate concepts of equicontinuity for sequential spaces. We apply our theorem in the context of C*‐algebras.