Metric spaces admitting only trivial weak contractions

Richárd Balka

Displaying similar documents to “Metric spaces admitting only trivial weak contractions”

Weak Type Inequality for the Square Function of a Nonnegative Submartingale

Adam Osękowski (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, $λ ℙ (S (f) \geq λ) \leq π / 2 ∥ f ∥ ₁$ , and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.

A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti, Stefano Bianchini, Gianluca Crippa (2014)

Journal of the European Mathematical Society

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We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_{t} u + \frac{.}{\dot{}} (b u) = 0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$ . As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

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We prove a regularity result for weak minima of integral functionals of the form $\int_{Ω} F (x, D u) d x$ where F(x,ξ) is a Carathéodory function which grows as ${| ξ |}^{p}$ with some p > 1.

A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $∥ Y ∥_{W} \leq (2 (α + 1) ²) / (2 α + 1) ∥ X ∥_{L^{\infty}}$ . Here W is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

Commutators of Littlewood-Paley [...] g κ ∗ $g_{κ}^{*}$ -functions on non-homogeneous metric measure spaces

Guanghui Lu, Shuangping Tao (2017)

Open Mathematics

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The main purpose of this paper is to prove that the boundedness of the commutator [...] Mκ,b∗ $ℳ_{κ, b}^{*}$ generated by the Littlewood-Paley operator [...] Mκ∗ $ℳ_{κ}^{*}$ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of [...] Mκ∗ $ℳ_{κ}^{*}$ satisfies a certain Hörmander-type condition, the authors prove that [...] Mκ,b∗ $ℳ_{κ, b}^{*}$ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from...

On ultra-weak convergence in $L^{p}$

Donald E. Myers (1976)

Annales Polonici Mathematici

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Relaxation of the incompressible porous media equation

László Székelyhidi Jr (2012)

Annales scientifiques de l'École Normale Supérieure

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It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T 4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T 4$ configurations. We then use this to construct weak solutions to the unstable interface...

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

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Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k} (X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_{k} (X)$ is Ascoli iff $C_{k} (X)$ is a $k_{ℝ}$ -space iff X is locally compact. Moreover, $C_{k} (X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

Exact $^{\infty}$ covering maps of the circle without (weak) limit measure

Roland Zweimüller (2002)

Colloquium Mathematicae

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We construct $^{\infty}$ maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ${(n^{- 1} \sum_{k = 0}^{n - 1} ν \circ T^{- k})}_{n \geq 1}$ of arithmetical averages of image measures does not converge weakly.

Compactness and extreme points of the set of quasi-measure extensions of a quasi-measure

Zbigniew Lipecki

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The memoir is based on a series of six papers by the author published over the years 1995-2007. It continues the work of D. Plachky (1970, 1976). It also owes some inspiration, among others, to papers by J. Łoś and E. Marczewski (1949), D. Bierlein and W. J. A. Stich (1989), D. Bogner and R. Denk (1994), and A. Ülger (1996). Let and ℜ be algebras of subsets of a set Ω with ⊂ ℜ. Given a quasi-measure μ on , i.e., μ ∈ ba₊(), we denote by E(μ) the convex set of all quasi-measure extensions...

On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

Agnieszka Kałamajska (2001)

Colloquium Mathematicae

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We study the functional $I_{f} (u) = \int_{Ω} f (u (x)) d x$ , where u=(u₁, ..., uₘ) and each $u_{j}$ is constant along some subspace $W_{j}$ of ℝⁿ. We show that if intersections of the $W_{j}$ ’s satisfy a certain condition then $I_{f}$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_{j}}_{j = 1, . . ., m}$ to have the equivalence: $I_{f}$ is weakly continuous if and only if f is Λ-affine.

Diagonal points having dense orbit

T. K. Subrahmonian Moothathu (2010)

Colloquium Mathematicae

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Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f \times f ² \times \dots \times f^{m} : X^{m} \to X^{m}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^{m}$ under the action of $f \times f ² \times \dots \times f^{m}$ . We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying...

The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

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Operators on a Hilbert space similar to a part of the backward shift of multiplicity one

Yoichi Uetake (2001)

Studia Mathematica

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Let A: X → X be a bounded operator on a separable complex Hilbert space X with an inner product $⟨ \cdot, \cdot ⟩_{X}$ . For b, c ∈ X, a weak resolvent of A is the complex function of the form $⟨ {(I - z A)}^{- 1} b, c ⟩_{X}$ . We will discuss an equivalent condition, in terms of weak resolvents, for A to be similar to a restriction of the backward shift of multiplicity 1.

Extensions of weak type multipliers

P. Mohanty, S. Madan (2003)

Studia Mathematica

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We prove that if $Λ \in M_{p} (ℝ^{N})$ and has compact support then Λ is a weak summability kernel for 1 < p < ∞, where $M_{p} (ℝ^{N})$ is the space of multipliers of $L^{p} (ℝ^{N})$ .

The subspace of weak $P$ -points of $ℕ^{*}$

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $W$ be the subspace of $ℕ^{*}$ consisting of all weak $P$ -points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$ -pseudocompact space for all $p \in ℕ^{*}$ .

On weak sharp minima for a special class of nonsmooth functions

Marcin Studniarski (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a characterization of weak sharp local minimizers of order one for a function f: ℝⁿ → ℝ defined by $f (x) : = m a x f_{i} (x) | i = 1, . . ., p$ , where the functions $f_{i}$ are strictly differentiable. It is given in terms of the gradients of $f_{i}$ and the Mordukhovich normal cone to a given set on which f is constant. Then we apply this result to a smooth nonlinear programming problem with constraints.

Weak-bases and $D$ -spaces

Dennis K. Burke (2007)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that certain weak-base structures on a topological space give a $D$ -space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are $D$ -spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Hence, quotient mappings, with compact fibers, from metric spaces have a $D$ -space image. What about quotient $s$ -mappings? Arhangel’skii and Buzyakova have shown that spaces with a point-countable base...

The power set of ω Elementary submodels and weakenings of CH

István Juhász, Kenneth Kunen (2001)

Fundamenta Mathematicae

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We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH*, the weak Freeze-Nation property of (ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle $C ₂^{s} (ω ₂)$ , but does not follow from $C ₂^{s} (ω ₂)$ , or even $C^{s} (ω ₂)$ .

Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics

Zofia Adamowicz, Konrad Zdanowski (2011)

Fundamenta Mathematicae

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We prove that for i ≥ 1, the arithmetic $I Δ ₀ + Ω_{i}$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1 + ε) l o g^{i + 2}$ , where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in $l o g^{i + 3}$ .

A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions

Adam Osękowski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate $ℙ (s u p_{t \geq 0} | Y_{t} | \geq 1) \leq 3.375 . . . ∥ X ∥ ₁$ . As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.

The Lindelöf property in Banach spaces

B. Cascales, I. Namioka, J. Orihuela (2003)

Studia Mathematica

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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent: (i) K is fragmented by $d_{D}$ , where, for each S ⊂ D, $d_{S} (x, y) = s u p ϱ (x (t), y (t)) : t \in S$ . (ii) For each countable subset...

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set $E$ with $C^{1}$ boundary and the spherical Hausdorff measure $S^{Q - 1}$ restricted to $\partial E$ , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and $Q$ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of $E$ is less than or equal to $S^{Q - 1} (\partial E)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

Weak-type inequalities for maximal operators acting on Lorentz spaces

Adam Osękowski (2014)

Banach Center Publications

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We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space $L^{p, q}$ , 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant $C_{p, q, r}$ such that for any $ϕ \in L^{p, q}$ , ${| | ℳ ϕ | |}_{r, \infty} \leq C_{p, q, r} {| | ϕ | |}_{p, q}$ .

$L^{q}$ -approach to weak solutions of the Oseen flow around a rotating body

Stanislav Kračmar, Šárka Nečasová, Patrick Penel (2008)

Banach Center Publications

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We consider the time-periodic Oseen flow around a rotating body in ℝ³. We prove a priori estimates in $L^{q}$ -spaces of weak solutions for the whole space problem under the assumption that the right-hand side has the divergence form. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional term -(ω ∧ x)·∇u + ω ∧ u in the equation of momentum where ω denotes the angular velocity. We prove the existence of generalized weak solutions...