Weighted convolution algebras on subsemigroups of the real line

H. G. Dales; H. V. Dedania

Displaying similar documents to “Weighted convolution algebras on subsemigroups of the real line”

The Properties of the Weighted Space $H_{2, α}^{k} (Ω)$ and Weighted Set $W_{2, α}^{k} (Ω, δ)$

V.A. Rukavishnikov, E.V. Matveeva, E.I. Rukavishnikova (2018)

Communications in Mathematics

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We study the properties of the weighted space $H_{2, α}^{k} (Ω)$ and weighted set $W_{2, α}^{k} (Ω, δ)$ for boundary value problem with singularity.

A class of weighted convolution Fréchet algebras

Thomas Vils Pedersen (2010)

Banach Center Publications

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For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that $ω_{m} (t) / ω ₙ (t) \to \infty$ as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists...

One-parameter semigroups in the convolution algebra of rapidly decreasing distributions

(2012)

Colloquium Mathematicae

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The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra $_{C}^{'} (ℝ ⁿ; M_{m \times m})$ of matrix valued rapidly decreasing distributions on ℝⁿ. It is proved that $G \in_{C}^{'} (ℝ ⁿ; M_{m \times m})$ is the generating distribution of an i.d.c.s. if and only if the operator $\partial_{t} \otimes_{m \times m} - G *$ on $ℝ^{1 + n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

Beurling algebra analogues of theorems of Wiener-Lévy-Żelazko and Żelazko

S. J. Bhatt, P. A. Dabhi, H. V. Dedania (2009)

Studia Mathematica

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Let 0 < p ≤ 1, let ω: ℤ → [1,∞) be a weight on ℤ and let f be a nowhere vanishing continuous function on the unit circle Γ whose Fourier series satisfies $\sum_{n \in ℤ} {| f ̂ (n) |}^{p} ω (n) < \infty$ . Then there exists a weight ν on ℤ such that $\sum_{n \in ℤ} {| \hat{(1 / f)} (n) |}^{p} ν (n) < \infty$ . Further, ν is non-constant if and only if ω is non-constant; and ν = ω if ω is non-quasianalytic. This includes the classical Wiener theorem (p = 1, ω = 1), Domar theorem (p = 1, ω is non-quasianalytic), Żelazko theorem (ω = 1) and a recent result of Bhatt and Dedania (p = 1). An...

Disjoint hypercyclic powers of weighted translations on groups

Liang Zhang, Hui-Qiang Lu, Xiao-Mei Fu, Ze-Hua Zhou (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a locally compact group and let $1 \leq p < \infty .$ Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space $L^{p} (G)$ in terms of the weights. Sufficient...

The representation of multi-hypergraphs by set intersections

Stanisław Bylka, Jan Komar (2007)

Discussiones Mathematicae Graph Theory

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This paper deals with weighted set systems (V,,q), where V is a set of indices, $\subset 2^{V}$ and the weight q is a nonnegative integer function on . The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V,,q) is defined to be an indexed family $R = {(R_{v})}_{v \in V}$ of subsets of a set S such that $| ⋂_{v \in E} R_{v} | = q (E)$ for each E ∈ . A necessary condition...

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

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For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^{p} ()$ to $L^{q} ()$ . We also give a condition on p which is necessary if this operator maps $L^{p} ()$ into L²().

Composition in ultradifferentiable classes

Armin Rainer, Gerhard Schindl (2014)

Studia Mathematica

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We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space $^{(ω)}$ and the Roumieu space $^{ω}$ as intersection and union of spaces $^{(M)}$ and $^{M}$ for associated weight sequences, respectively.

Wave front set for positive operators and for positive elements in non-commutative convolution algebras

Joachim Toft (2007)

Studia Mathematica

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Let WF⁎ be the wave front set with respect to $C^{\infty}$ , quasi analyticity or analyticity, and let K be the kernel of a positive operator from $C ₀^{\infty}$ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution $u *_{B} φ (x) = \int u (x - y) φ (y) B (x, y) d y$ , where $B \in C^{\infty}$ is appropriate, and prove that if $(u *_{B} φ, φ) \geq 0$ for every $φ \in C ₀^{\infty}$ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

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We study continuity envelopes of function spaces $B_{p, q}^{s} (ℝ ⁿ, w)$ and $F_{p, q}^{s} (ℝ ⁿ, w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

Solutions to the equation div u = f in weighted Sobolev spaces

Katrin Schumacher (2008)

Banach Center Publications

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We consider the problem div u = f in a bounded Lipschitz domain Ω, where f with $\int_{Ω} f = 0$ is given. It is shown that the solution u, constructed as in Bogovski’s approach in [1], fulfills estimates in the weighted Sobolev spaces $W_{w}^{k, q} (Ω)$ , where the weight function w is in the class of Muckenhoupt weights $A_{q}$ .

On the Cauchy problem for convolution equations

(2013)

Colloquium Mathematicae

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We consider one-parameter (C₀)-semigroups of operators in the space $^{'} (ℝ ⁿ; ℂ^{m})$ with infinitesimal generator of the form ${(G *) |}_{^{'} (ℝ ⁿ; ℂ^{m})}$ where G is an $M_{m \times m}$ -valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $(ℝ ⁿ; ℂ^{m})$ , $_{L^{p}} (ℝ ⁿ; ℂ^{m})$ , p ∈ [1,∞], $(_{a}) (ℝ ⁿ; ℂ^{m})$ , a ∈ ]0,∞[, or the spaces $_{L^{q}}^{'} (ℝ ⁿ; ℂ^{m})$ , q ∈ ]1,∞], of bounded distributions.

Existence of solutions to the (rot,div)-system in $L_{p}$ -weighted spaces

Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

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The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , S = ∂Ω in weighted $L_{p}$ -Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.

The $V_{a}$ -deformation of the classical convolution

Anna Dorota Krystek (2007)

Banach Center Publications

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We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $μ *_{T} ν = T^{- 1} (T μ * T ν)$ . We deal with the $V_{a}$ -deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_{a}$ -deformed classical convolution and give the orthogonal...

Note on duality of weighted multi-parameter Triebel-Lizorkin spaces

Wei Ding, Jiao Chen, Yaoming Niu (2019)

Czechoslovak Mathematical Journal

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We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces ${\dot{F}}_{p}^{α, q} (ω; ℝ^{n_{1}} \times ℝ^{n_{2}})$ . This space has been introduced and the result ${({\dot{F}}_{p}^{α, q} (ω; ℝ^{n_{1}} \times ℝ^{n_{2}}))}^{*} = {CMO}_{p}^{- α, q^{'}} (ω; ℝ^{n_{1}} \times ℝ^{n_{2}})$ for $0 < p \leq 1$ has been proved in Ding, Zhu (2017). In this paper, for $1 < p < \infty$ , $0 < q < \infty$ we establish its dual space ${\dot{H}}_{p}^{α, q} (ω; ℝ^{n_{1}} \times ℝ^{n_{2}})$ .

On a theorem of Saeki concerning convolution squares of singular measures

Thomas Körner (2008)

Bulletin de la Société Mathématique de France

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If $1 > α > 1 / 2$ , then there exists a probability measure $μ$ such that the Hausdorff dimension of the support of $μ$ is $α$ and $μ * μ$ is a Lipschitz function of class $α - \frac{1}{2}$ .

Weighted Frobenius-Perron operators and their spectra

Mohammad Reza Jabbarzadeh, Rana Hajipouri (2017)

Mathematica Bohemica

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First, some classic properties of a weighted Frobenius-Perron operator $𝒫_{ϕ}^{u}$ on $L^{1} (Σ)$ as a predual of weighted Koopman operator $W = u U_{ϕ}$ on $L^{\infty} (Σ)$ will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of $𝒫_{ϕ}^{u}$ under certain conditions.

Boundedness of commutators of strongly singular convolution operators on Herz-type spaces

Zongguang Liu (2003)

Studia Mathematica

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The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, $T_{b}^{m}$ , on Herz spaces $K ̇_{q}^{α, p} (ℝ ⁿ)$ and $K_{q}^{α, p} (ℝ ⁿ)$ , and on a new class of Herz-type Hardy spaces $H K ̇_{q, b, m}^{α, p, 0} (ℝ ⁿ)$ and $H K_{q, b, m}^{α, p, 0} (ℝ ⁿ)$ , where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

Polyanalytic Besov spaces and approximation by dilatations

Ali Abkar (2024)

Czechoslovak Mathematical Journal

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Using partial derivatives $\partial f / \partial z$ and $\partial f / \partial \bar{z}$ , we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree $q$ can be approximated in norm by polyanalytic polynomials of degree at most $q$ .

A remark on the asymmetry of convolution operators

Saverio Giulini (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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A convolution operator, bounded on $L^{q}(\mathbb{R}^{n})$ , is bounded on $L^{p}(\mathbb{R}^{n})$ , with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace $\mathbb{R}^{n}$ with a general non-commutative locally compact group $G$ . In this paper we give a simple construction of a convolution operator on a suitable compact group $G$ , wich is bounded on $L^{q}(G)$ for every $q\in[2,\infty)$ and is unbounded on $L^{p}(G)$ if $p\in[1,2)$ .

Remarks on the critical Besov space and its embedding into weighted Besov-Orlicz spaces

Hidemitsu Wadade (2010)

Studia Mathematica

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We present several continuous embeddings of the critical Besov space $B_{p}^{n / p, ρ} (ℝ ⁿ)$ . We first establish a Gagliardo-Nirenberg type estimate ${| | u | |}_{Ḃ_{q, w_{r}}^{0, ν}} \leq C ₙ {(1 / (n - r))}^{1 / q + 1 / ν - 1 / ρ} {(q / r)}^{1 / ν - 1 / ρ} {| | u | |}_{Ḃ_{p}^{0, ρ}}^{(n - r) p / n q} {| | u | |}_{Ḃ_{p}^{n / p, ρ}}^{1 - (n - r) p / n q}$ , for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function $w_{r} (x) = 1 / {(| x |}^{r})$ with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding $B_{p}^{n / p, ρ} (ℝ ⁿ) ↪ B_{Φ ₀, w_{r}}^{0, ν} (ℝ ⁿ)$ , where the function Φ₀ of the weighted Besov-Orlicz space $B_{Φ ₀, w_{r}}^{0, ν} (ℝ ⁿ)$ is a Young function of the exponential type. Another point of interest is to embed $B_{p}^{n / p, ρ} (ℝ ⁿ)$ into the weighted Besov...

The linear bound in A₂ for Calderón-Zygmund operators: a survey

Michael Lacey (2011)

Banach Center Publications

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For an L²-bounded Calderón-Zygmund Operator T acting on $L ² (ℝ^{d})$ , and a weight w ∈ A₂, the norm of T on L²(w) is dominated by $C_{T} {| | w | |}_{A ₂}$ . The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A₂ character of the weight can...

Weighted $w$ -core inverses in rings

Liyun Wu, Huihui Zhu (2023)

Czechoslovak Mathematical Journal

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Let $R$ be a unital $*$ -ring. For any $a, s, t, v, w \in R$ we define the weighted $w$ -core inverse and the weighted dual $s$ -core inverse, extending the $w$ -core inverse and the dual $s$ -core inverse, respectively. An element $a \in R$ has a weighted $w$ -core inverse with the weight $v$ if there exists some $x \in R$ such that $a w x v x = x$ , $x v a w a = a$ and ${(a w x)}^{*} = a w x$ . Dually, an element $a \in R$ has a weighted dual $s$ -core inverse with the weight $t$ if there exists some $y \in R$ such that $y t y s a = y$ , $a s a t y = a$ and ${(y s a)}^{*} = y s a$ . Several characterizations of weighted $w$ -core invertible and weighted dual $s$ -core...

Beurling-Figà-Talamanca-Herz algebras

Serap Öztop, Volker Runde, Nico Spronk (2012)

Studia Mathematica

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For a locally compact group G and p ∈ (1,∞), we define and study the Beurling-Figà-Talamanca-Herz algebras $A_{p} (G, ω)$ . For p = 2 and abelian G, these are precisely the Beurling algebras on the dual group Ĝ. For p = 2 and compact G, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We...

A remark on the asymmetry of convolution operators

Saverio Giulini (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

Existence and regularity of solutions of some elliptic system in domains with edges

Wojciech M. Zajączkowski

Similarity:

CONTENTS1. Introduction.......................................................................52. Notation and auxiliary results............................................93. Statement of the problem (1.1)-(1.3)..............................204. The problem (3.14).........................................................225. Auxiliary results in $D_{ϑ}$ ...............................................346. Existence of solutions of (3.14) in $H_{μ}^{k} (D_{ϑ})$ ............417. Green function................................................................528....

Weighted sub-Bergman Hilbert spaces

Maria Nowak, Renata Rososzczuk (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces $A_{α}^{2}$ , $- 1 < α < \infty$ . These spaces have already been studied in [8], [7], [5] and [1]. We extend some results from these papers.

Characterization of the convolution operators on quasianalytic classes of Beurling type that admit a continuous linear right inverse

José Bonet, Reinhold Meise (2008)

Studia Mathematica

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Extending previous work by Meise and Vogt, we characterize those convolution operators, defined on the space $_{(ω)} (ℝ)$ of (ω)-quasianalytic functions of Beurling type of one variable, which admit a continuous linear right inverse. Also, we characterize those (ω)-ultradifferential operators which admit a continuous linear right inverse on $_{(ω)} [a, b]$ for each compact interval [a,b] and we show that this property is in fact weaker than the existence of a continuous linear right inverse on $_{(ω)} (ℝ)$ .

On $L_{p} - L_{q}$ boundedness for convolutions with kernels having singularities on a sphere

Alexey N. Karapetyants (2001)

Studia Mathematica

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For the convolution operators $A_{a}^{α}$ with symbols ${a (| ξ |) | ξ |}^{- α} e x p i | ξ |$ , 0 ≤ Re α < n, $a (| ξ |) \in L_{\infty}$ , we construct integral representations and give the exact description of the set of pairs (1/p,1/q) for which the operators are bounded from $L_{p}$ to $L_{q}$ .