Algebras of differentiable functions in the plane

Karel De Leeuw; H. Mirkil

Displaying similar documents to “Algebras of differentiable functions in the plane”

An $L^{p}$ -version of a theorem of D.A. Raikov

Gero Fendler (1985)

Annales de l'institut Fourier

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Let $G$ be a locally compact group, for $p \in (1, \infty)$ let $P f_{p} (G)$ denote the closure of $L^{1} (G)$ in the convolution operators on $L^{p} (G)$ . Denote $W_{p} (G)$ the dual of $P f_{p} (G)$ which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space $A_{p} (G)$ . It is shown that on the unit sphere of $W_{p} (G)$ the $σ (W_{p}, P f_{p})$ topology and the strong $A_{p}$ -multiplier topology coincide.

Unique continuation for Schrödinger operators in dimension three or less

Eric T. Sawyer (1984)

Annales de l'institut Fourier

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We show that the differential inequality $| Δ u | \leq v | u |$ has the unique continuation property relative to the Sobolev space $H_{l o c}^{2, 1} (Ω)$ , $Ω \subset R^{n}$ , $n \leq 3$ , if $v$ satisfies the condition $(K_{n}^{loc}) lim_{r \to 0} sup_{x \in K} \int_{| x - y | < r} | x - y |^{2 - n} v (y) d y = 0$ for all compact $K \subset Ω$ , where if $n = 2$ , we replace $| x - y |^{2 - n}$ by $- log | x - y |$ . This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H = - Δ + v$ , in the case $n \leq 3$ . The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N = 0, 1, 2, ...$ and any $u \in H_{c}^{2, 1} (R^{3} - {0}),$ $\frac{| u (x) |}{| x |^{N}} \leq C \int_{R^{3}} | x - y |^{- 1} \frac{| Δ u (y) |}{| y |^{N}} d y for a.e. x in R^{3} .$

Regularity properties of the equilibrium distribution

Hans Wallin (1965)

Annales de l'institut Fourier

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Soit $F$ un sous-ensemble compact de $R^{m}$ ayant des points intérieurs et soit $μ_{α}^{F}$ la distribution d’équilibre sur $F$ de masse totale 1 par rapport au noyau $r^{α - m}$ avec $0 < α < 2$ pour $m \geq 2$ , et $0 < α < 1$ pour $m = 1$ . La restriction de $μ_{α}^{F}$ à l’intérieur de $F$ est absolument continue et a pour densité $f_{α}^{F}$ . On donne une formule explicite pour $f_{α}^{F}$ et, pour une classe générale d’ensembles $F$ , on démontre que $f_{α}^{F}$ , définie en réalité sur un ensemble de mesure de Lebesgue nulle, croît comme la distance à la frontière $\partial F$ de $F$ élevée à la puissance...

A representation theorem for tense $n \times m$ -valued Łukasiewicz-Moisil algebras

Aldo Victorio Figallo, Gustavo Pelaitay (2015)

Mathematica Bohemica

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In 2000, Figallo and Sanza introduced $n \times m$ -valued Łukasiewicz-Moisil algebras which are both particular cases of matrix Łukasiewicz algebras and a generalization of $n$ -valued Łukasiewicz-Moisil algebras. Here we initiate an investigation into the class $_{n \times m}$ of tense $n \times m$ -valued Łukasiewicz-Moisil algebras (or tense LM $_{n \times m}$ -algebras), namely $n \times m$ -valued Łukasiewicz-Moisil algebras endowed with two unary operations called tense operators. These algebras constitute a generalization of tense...

Generalization of the topological algebra $(C_{b} (X), β)$

Jorma Arhippainen, Jukka Kauppi (2009)

Studia Mathematica

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We study subalgebras of $C_{b} (X)$ equipped with topologies that generalize both the uniform and the strict topology. In particular, we study the Stone-Weierstrass property and describe the ideal structure of these algebras.

On the range of the derivative of a real-valued function with bounded support

T. Gaspari (2002)

Studia Mathematica

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We study the set f’(X) = f’(x): x ∈ X when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a $C^{p}$ -smooth bump b:X → ℝ such that $| | b^{(p)} {| |}_{\infty}$ is finite, then any connected open subset of X* containing 0 is the range of the derivative of a $C^{p}$ -smooth bump. We also study the finite-dimensional case which is quite different....

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

Explicit cogenerators for the homotopy category of projective modules over a ring

Amnon Neeman (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $R$ be a ring. In two previous articles [12, 14] we studied the homotopy category $𝐊 (R - Proj)$ of projective $R$ -modules. We produced a set of generators for this category, proved that the category is $ℵ_{1}$ -compactly generated for any ring $R$ , and showed that it need not always be compactly generated, but is for sufficiently nice $R$ . We furthermore analyzed the inclusion $j_{!}^{} : 𝐊 (R - Proj) \to 𝐊 (R - Flat)$ and the orthogonal subcategory $𝒮 = {𝐊 (R - Proj)}^{⊥}$ . And we even showed that the inclusion $𝒮 \to 𝐊 (R - Flat)$ has a right adjoint; this forces some natural map to be...

Automatic continuity of operators commuting with translations

J. Alaminos, J. Extremera, A. R. Villena (2006)

Studia Mathematica

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Let $τ_{X}$ and $τ_{Y}$ be representations of a topological group G on Banach spaces X and Y, respectively. We investigate the continuity of the linear operators Φ: X → Y with the property that $Φ \circ τ_{X} (t) = τ_{Y} (t) \circ Φ$ for each t ∈ G in terms of the invariant vectors in Y and the automatic continuity of the invariant linear functionals on X.

Approximate amenability for Banach sequence algebras

H. G. Dales, R. J. Loy, Y. Zhang (2006)

Studia Mathematica

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We consider when certain Banach sequence algebras A on the set ℕ are approximately amenable. Some general results are obtained, and we resolve the special cases where $A = ℓ^{p}$ for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras $ℓ^{p} (ω)$ .

The jacobian map, the jacobian group and the group of automorphisms of the Grassmann algebra

Vladimir V. Bavula (2010)

Bulletin de la Société Mathématique de France

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There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic algebras). This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the Jacobian set– the set of all algebra endomorphisms of a polynomial algebra with the Jacobian...

Necessary and sufficient conditions for the chain rule in $W_{loc}^{1, 1} (ℝ^{N}; ℝ^{d})$ and $B V_{loc} (ℝ^{N}; ℝ^{d})$

Giovanni Leoni, Massimiliano Morini (2007)

Journal of the European Mathematical Society

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We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space $W_{loc}^{1, 1} (ℝ^{N}; ℝ^{d})$ and in the space $B V_{loc} (ℝ^{N}; ℝ^{d})$ of functions of bounded variation.

On the number of non-isomorphic subspaces of a Banach space

Valentin Ferenczi, Christian Rosendal (2005)

Studia Mathematica

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We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ${(e_{i})}_{i \in ℕ}$ ; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, ${[e_{i}]}_{i \in A} ≇ {[e_{i}]}_{i \in B}$ , or for a residual set of infinite subsets A of ℕ, ${[e_{i}]}_{i \in A}$ is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to $\oplus {[e_{i}]}_{i \in D}$ for any D ⊂ ℕ, and isomorphic to a denumerable Schauder...