Displaying similar documents to “Polar wavelets and associated Littlewood-Paley theory”

Embeddings of Besov-Morrey spaces on bounded domains

Dorothee D. Haroske, Leszek Skrzypczak (2013)

Studia Mathematica

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We study embeddings of spaces of Besov-Morrey type, i d Ω : p , u , q s ( Ω ) p , u , q s ( Ω ) , where Ω d is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of i d Ω . This continues our earlier studies relating to the case of d . Moreover, we also characterise embeddings into the scale of L p spaces or into the space of bounded continuous functions.

Multifractal analysis of the divergence of Fourier series

Frédéric Bayart, Yanick Heurteaux (2012)

Annales scientifiques de l'École Normale Supérieure

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A famous theorem of Carleson says that, given any function f L p ( 𝕋 ) , p ( 1 , + ) , its Fourier series ( S n f ( x ) ) converges for almost every x 𝕋 . Beside this property, the series may diverge at some point, without exceeding O ( n 1 / p ) . We define the divergence index at  x as the infimum of the positive real numbers β such that S n f ( x ) = O ( n β ) and we are interested in the size of the exceptional sets E β , namely the sets of  x 𝕋 with divergence index equal to  β . We show that quasi-all functions in  L p ( 𝕋 ) have a multifractal behavior with respect to...

The harmonic Cesáro and Copson operators on the spaces L p ( ) , 1 ≤ p ≤ 2

Ferenc Móricz (2002)

Studia Mathematica

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The harmonic Cesàro operator is defined for a function f in L p ( ) for some 1 ≤ p < ∞ by setting ( f ) ( x ) : = x ( f ( u ) / u ) d u for x > 0 and ( f ) ( x ) : = - - x ( f ( u ) / u ) d u for x < 0; the harmonic Copson operator ℂ* is defined for a function f in L ¹ l o c ( ) by setting * ( f ) ( x ) : = ( 1 / x ) x f ( u ) d u for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If f L p ( ) for some 1 ≤ p ≤ 2, then ( ( f ) ) ( t ) = * ( f ̂ ) ( t ) a.e., where f̂ denotes the Fourier transform of f. (ii) If f L p ( ) for some 1 < p ≤ 2, then...

The Lebesgue constants for the Franklin orthogonal system

Z. Ciesielski, A. Kamont (2004)

Studia Mathematica

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To each set of knots t i = i / 2 n for i = 0,...,2ν and t i = ( i - ν ) / n for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space ν , n of all piecewise linear and continuous functions on I = [0,1] with knots t i and the orthogonal projection P ν , n of L²(I) onto ν , n . The main result is l i m ( n - ν ) ν | | P ν , n | | = s u p ν , n : 1 ν n | | P ν , n | | = 2 + ( 2 - 3 ) ² . This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².

Theoretical analysis for 1 - 2 minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

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We investigate the recovery of k -sparse signals using the 1 - 2 minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume k -sparse signals 𝐱 with the prior support T which is composed of g true indices and b wrong...

An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner

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We prove an interpolatory estimate linking the directional Haar projection P ( ε ) to the Riesz transform in the context of Bochner-Lebesgue spaces L p ( ; X ) , 1 < p < ∞, provided X is a UMD-space. If ε i = 1 , the result is the inequality | | P ( ε ) u | | L p ( ; X ) C | | u | | L p ( ; X ) 1 / | | R i u | | L p ( ; X ) 1 - 1 / , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of L p ( ; X ) . In order to obtain the interpolatory result (1) we analyze stripe operators S λ , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

The boundedness of two classes of integral operators

Xin Wang, Ming-Sheng Liu (2021)

Czechoslovak Mathematical Journal

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The aim of this paper is to characterize the L p - L q boundedness of two classes of integral operators from L p ( 𝒰 , d V α ) to L q ( 𝒰 , d V β ) in terms of the parameters a , b , c , p , q and α , β , where 𝒰 is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

The generalized Toeplitz operators on the Fock space F α 2

Chunxu Xu, Tao Yu (2024)

Czechoslovak Mathematical Journal

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Let μ be a positive Borel measure on the complex plane n and let j = ( j 1 , , j n ) with j i . We study the generalized Toeplitz operators T μ ( j ) on the Fock space F α 2 . We prove that T μ ( j ) is bounded (or compact) on F α 2 if and only if μ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for T μ ( j ) to be in the Schatten p -class for 1 p < .

Perron-Frobenius operators and the Klein-Gordon equation

Francisco Canto-Martín, Håkan Hedenmalm, Alfonso Montes-Rodríguez (2014)

Journal of the European Mathematical Society

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For a smooth curve Γ and a set Λ in the plane 2 , let A C ( Γ ; Λ ) be the space of finite Borel measures in the plane supported on Γ , absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ . Following [12], we say that ( Γ , Λ ) is a Heisenberg uniqueness pair if A C ( Γ ; Λ ) = { 0 } . In the context of a hyperbola Γ , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly...

Complex series and connected sets

B. Jasek

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CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO Σ 1 , Σ 2 , Σ 3 , Σ 4 INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

Linear maps preserving A -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

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Let be a complex Hilbert space, A a positive operator with closed range in ( ) and A ( ) the sub-algebra of ( ) of all A -self-adjoint operators. Assume φ : A ( ) onto itself is a linear continuous map. This paper shows that if φ preserves A -unitary operators such that φ ( I ) = P then ψ defined by ψ ( T ) = P φ ( P T ) is a homomorphism or an anti-homomorphism and ψ ( T ) = ψ ( T ) for all T A ( ) , where P = A + A and A + is the Moore-Penrose inverse of A . A similar result is also true if φ preserves A -quasi-unitary operators in both directions such that there...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

Functionally countable subalgebras and some properties of the Banaschewski compactification

A. R. Olfati (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let X be a zero-dimensional space and C c ( X ) be the set of all continuous real valued functions on X with countable image. In this article we denote by C c K ( X ) (resp., C c ψ ( X ) ) the set of all functions in C c ( X ) with compact (resp., pseudocompact) support. First, we observe that C c K ( X ) = O c β 0 X X (resp., C c ψ ( X ) = M c β 0 X υ 0 X ), where β 0 X is the Banaschewski compactification of X and υ 0 X is the -compactification of X . This implies that for an -compact space X , the intersection of all free maximal ideals in C c ( X ) is equal to C c K ( X ) , i.e., M c β 0 X X = C c K ( X ) . By applying...