Weighted bounds for variational Fourier series

Yen Do; Michael Lacey

Displaying similar documents to “Weighted bounds for variational Fourier series”

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.

Integrability and continuity of functions represented by trigonometric series: coefficients criteria

Mikhail Dyachenko, Sergey Tikhonov (2009)

Studia Mathematica

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We study weighted $L_{p}$ -integrability (1 ≤ p < ∞) of trigonometric series. It is shown how the integrability of a function with weight $x^{- α}$ depends on some regularity conditions on Fourier coefficients. Criteria for the uniform convergence of trigonometric series in terms of their coefficients are also studied.

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...

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.

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...

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.

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 $A_{q}$ .

The spectral decomposition of weighted shifts and the $A^{p}$ condition

Earl Berkson, T. A. Gillespie (1990)

Colloquium Mathematicae

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Boundedness of Riesz transforms on weighted Carleson measure spaces

Ming-Yi Lee (2012)

Studia Mathematica

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Let w be in the Muckenhoupt $A_{\infty}$ weight class. We show that the Riesz transforms are bounded on the weighted Carleson measure space $C M O_{w}^{p}$ , the dual of the weighted Hardy space $H_{w}^{p}$ , 0 < p ≤ 1.

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...

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.

Weighted variable $L^{p}$ integral inequalities for the maximal operator on non-increasing functions

C. J. Neugebauer (2009)

Studia Mathematica

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Let $B_{p}$ be the Ariõ-Muckenhoupt weight class which controls the weighted $L^{p}$ -norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated $L^{p (x)}$ -norm inequalities of the Hardy operator.

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.

On the diametral dimension of weighted spaces of analytic germs

Michael Langenbruch (2016)

Studia Mathematica

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We prove precise estimates for the diametral dimension of certain weighted spaces of germs of holomorphic functions defined on strips near ℝ. This implies a full isomorphic classification for these spaces including the Gelfand-Shilov spaces $S ¹_{α}$ and $S ₁^{α}$ for α > 0. Moreover we show that the classical spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions are not isomorphic.

Existence of solutions to the (rot,div)-system in L₂-weighted spaces

Wojciech M. Zajączkowski (2009)

Applicationes Mathematicae

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The existence of solutions to the elliptic problem rot v = w, div v = 0 in Ω ⊂ ℝ³, ${v \cdot n ̅ |}_{S} = 0$ , S = ∂Ω, in weighted Hilbert spaces is proved. It is assumed that Ω contains an axis L and the weight is a negative power of the distance to the axis. The main part of the proof is devoted to examining solutions in a neighbourhood of L. Their existence in Ω follows by regularization.

Weighted norm inequalities for maximal singular integrals with nondoubling measures

Guoen Hu, Dachun Yang (2008)

Studia Mathematica

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Let μ be a nonnegative Radon measure on $ℝ^{d}$ which satisfies μ(B(x,r)) ≤ Crⁿ for any $x \in ℝ^{d}$ and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with $A_{p}^{ϱ} (μ)$ weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with $A_{\infty}^{ϱ} (μ)$ weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).

Essential norms of weighted composition operators between Hardy spaces $H^{p}$ and $H^{q}$ for 1 ≤ p,q ≤ ∞

R. Demazeux (2011)

Studia Mathematica

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We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^{p}$ and $H^{q}$ for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.

On mean value properties involving a logarithm-type weight

Nikolai G. Kuznecov (2024)

Mathematica Bohemica

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Two new assertions characterizing analytically disks in the Euclidean plane $ℝ^{2}$ are proved. Weighted mean value property of positive solutions to the Helmholtz and modified Helmholtz equations are used for this purpose; the weight has a logarithmic singularity. The obtained results are compared with those without weight that were found earlier.

Weighted composition operators from Zygmund spaces to Bloch spaces on the unit ball

Yu-Xia Liang, Chang-Jin Wang, Ze-Hua Zhou (2015)

Annales Polonici Mathematici

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Let H() denote the space of all holomorphic functions on the unit ball ⊂ ℂⁿ. Let φ be a holomorphic self-map of and u∈ H(). The weighted composition operator $u C_{φ}$ on H() is defined by $u C_{φ} f (z) = u (z) f (φ (z))$ . We investigate the boundedness and compactness of $u C_{φ}$ induced by u and φ acting from Zygmund spaces to Bloch (or little Bloch) spaces in the unit ball.

Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces

George Kyriazis, Pencho Petrushev, Yuan Xu (2008)

Studia Mathematica

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The Littlewood-Paley theory is extended to weighted spaces of distributions on [-1,1] with Jacobi weights $w (t) = {(1 - t)}^{α} {(1 + t)}^{β}$ . Almost exponentially localized polynomial elements (needlets) $φ_{ξ}$ , $ψ_{ξ}$ are constructed and, in complete analogy with the classical case on ℝⁿ, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $⟨ f, φ_{ξ} ⟩$ in respective sequence spaces.

Koecher-Maass series of a certain half-integral weight modular form related to the Duke-Imamoḡlu-Ikeda lift

Hidenori Katsurada, Hisa-aki Kawamura (2014)

Acta Arithmetica

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Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k - n/2 + 1/2 for Γ₀(4), let f be the corresponding primitive form of weight 2k-n for SL₂(ℤ) under the Shimura correspondence, and Iₙ(h) the Duke-Imamoḡlu-Ikeda lift of h to the space of cusp forms of weight k for Spₙ(ℤ). Moreover, let $ϕ_{I ₙ (h), 1}$ be the first Fourier-Jacobi coefficient of Iₙ(h), and $σ_{n - 1} (ϕ_{I ₙ (h), 1})$ be the cusp form in the generalized Kohnen plus space of weight k - 1/2 corresponding to...

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

Wojciech M. Zajączkowski

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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....

A variation norm Carleson theorem

Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, James Wright (2012)

Journal of the European Mathematical Society

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We strengthen the Carleson-Hunt theorem by proving $L^{p}$ estimates for the $r$ -variation of the partial sum operators for Fourier series and integrals, for $r > 𝚖𝚊𝚡 {p^{'}, 2}$ . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

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 $q$ .

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...

New extension of the variational McShane integral of vector-valued functions

Sokol Bush Kaliaj (2019)

Mathematica Bohemica

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We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset $G$ of $m$ -dimensional Euclidean space $ℝ^{m}$ . It is a “natural” extension of the variational McShane integral (the strong McShane integral) from $m$ -dimensional closed non-degenerate intervals to open and bounded subsets of $ℝ^{m}$ . We will show a theorem that characterizes the Hake-variational McShane integral in terms of the variational McShane integral. This theorem reduces the study...