Remark on the inequality of F. Riesz

W. Łenski

Displaying similar documents to “Remark on the inequality of F. Riesz”

On the Fejér-F. Riesz inequality in $L^{p}$

Yoram Sagher (1977)

Studia Mathematica

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Generalized Riesz products produced from orthonormal transforms

Nikolaos Atreas, Antonis Bisbas (2012)

Colloquium Mathematicae

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Let $ℳ_{p} = {m_{k}}_{k = 0}^{p - 1}$ be a finite set of step functions or real valued trigonometric polynomials on = [0,1) satisfying a certain orthonormality condition. We study multiscale generalized Riesz product measures μ defined as weak-* limits of elements $μ_{N} \in V_{N} (N \in ℕ)$ , where $V_{N}$ are $p^{N}$ -dimensional subspaces of L₂() spanned by an orthonormal set which is produced from dilations and multiplications of elements of $ℳ_{p}$ and $\bar{⋃_{N \in ℕ} V_{N}} = L ₂ ()$ . The results involve mutual absolute continuity or singularity of such Riesz products extending previous...

Dichotomy of global density of Riesz capacity

Hiroaki Aikawa (2016)

Studia Mathematica

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Let $C_{α}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲ (C_{α}, E, r) = i n f_{x \in ℝ ⁿ} C_{α} (E \cap B (x, r)) / C_{α} (B (x, r))$ for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that $l i m_{r \to \infty} ̲ (C_{α}, E, r)$ is either 0 or 1; the first case occurs if and only if $̲ (C_{α}, E, r)$ is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also...

Riesz sequences and arithmetic progressions

Itay Londner, Alexander Olevskiĭ (2014)

Studia Mathematica

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Given a set of positive measure on the circle and a set Λ of integers, one can ask whether $E (Λ) : = e_{λ \in Λ}^{i λ t}$ is a Riesz sequence in L²(). We consider this question in connection with some arithmetic properties of the set Λ. Improving a result of Bownik and Speegle (2006), we construct a set such that E(Λ) is never a Riesz sequence if Λ contains an arithmetic progression of length N and step $ℓ = O (N^{1 - ε})$ with N arbitrarily large. On the other hand, we prove that every set admits a Riesz sequence E(Λ) such that...

$L^{p}$ boundedness of Riesz transforms for orthogonal polynomials in a general context

Liliana Forzani, Emanuela Sasso, Roberto Scotto (2015)

Studia Mathematica

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Nowak and Stempak (2006) proposed a unified approach to the theory of Riesz transforms and conjugacy in the setting of multi-dimensional orthogonal expansions, and proved their boundedness on L². Following them, we give easy to check sufficient conditions for their boundedness on $L^{p}$ , 1 < p < ∞. We also discuss the symmetrized version of these transforms.

A short proof on lifting of projection properties in Riesz spaces

Marek Wójtowicz (1999)

Commentationes Mathematicae Universitatis Carolinae

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Let $L$ be an Archimedean Riesz space with a weak order unit $u$ . A sufficient condition under which Dedekind [ $σ$ -]completeness of the principal ideal $A_{u}$ can be lifted to $L$ is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of $C (X)$ -spaces. Similar results are obtained for the Riesz spaces $B_{n} (T)$ , $n = 1, 2, \dots$ , of all functions of the $n$ th Baire class on a metric space $T$ .

On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)

Annales Polonici Mathematici

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For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^{q}, l^{p})}^{α}$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^{q}, l^{p})$ and contains the Lebesgue space $L^{α}$ . We study the closure ${(L^{q}, l^{p})}_{c, 0}^{α}$ in ${(L^{q}, l^{p})}^{α}$ of the space of test functions (infinitely differentiable and with compact support in $ℝ^{d}$ ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W ¹ ({(L^{q}, l^{p})}^{α})$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

Riesz potentials derived by one-mode interacting Fock space approach

Nobuhiro Asai (2007)

Colloquium Mathematicae

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The main aim of this short paper is to study Riesz potentials on one-mode interacting Fock spaces equipped with deformed annihilation, creation, and neutral operators with constants $c_{0, 0}, c_{1, 1} \in ℝ$ and $c_{0, 1} > 0$ , $c_{1, 2} \geq 0$ as in equations (1.4)-(1.6). First, to emphasize the importance of these constants, we summarize our previous results on the Hilbert space of analytic L² functions with respect to a probability measure on ℂ. Then we consider the Riesz kernels of order 2α, $α = c_{0, 1} / c_{1, 2}$ , on ℂ if $0 < c_{0, 1} < c_{1, 2}$ , which can be derived from...

Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case

Carlota Maria Cuesta, Xuban Diez-Izagirre (2023)

Czechoslovak Mathematical Journal

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We study the large time behaviour of the solutions of a nonlocal regularisation of a scalar conservation law. This regularisation is given by a fractional derivative of order $1 + α$ , with $α \in (0, 1)$ , which is a Riesz-Feller operator. The nonlinear flux is given by the locally Lipschitz function ${| u |}^{q - 1} u / q$ for $q > 1$ . We show that in the sub-critical case, $1 < q < 1 + α$ , the large time behaviour is governed by the unique entropy solution of the scalar conservation law. Our proof adapts the proofs of the analogous results for...

Orthosymmetric bilinear map on Riesz spaces

Elmiloud Chil, Mohamed Mokaddem, Bourokba Hassen (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $E$ be a Riesz space, $F$ a Hausdorff topological vector space (t.v.s.). We prove, under a certain separation condition, that any orthosymmetric bilinear map $T : E \times E \to F$ is automatically symmetric. This generalizes in certain way an earlier result by F. Ben Amor [On orthosymmetric bilinear maps, Positivity 14 (2010), 123–134]. As an application, we show that under a certain separation condition, any orthogonally additive homogeneous polynomial $P : E \to F$ is linearly represented. This fits in the type of...

Sharp inequalities for Riesz transforms

Adam Osękowski (2014)

Studia Mathematica

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We establish the following sharp local estimate for the family ${R_{j}}_{j = 1}^{d}$ of Riesz transforms on $ℝ^{d}$ . For any Borel subset A of $ℝ^{d}$ and any function $f : ℝ^{d} \to ℝ$ , $\int_{A} | R_{j} f (x) | d x \leq C_{p} {| | f | |}_{L^{p} (ℝ^{d})} {| A |}^{1 / q}$ , 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_{p} = {[2^{q + 2} Γ (q + 1) / π^{q + 1} \sum_{k = 0}^{\infty} {(- 1)}^{k} / {(2 k + 1)}^{q + 1}]}^{1 / q}$ , 1 < p < 2, and $C_{p} = {[4 Γ (q + 1) / π^{q} \sum_{k = 0}^{\infty} 1 / {(2 k + 1)}^{q}]}^{1 / q}$ , 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

Variations on Bochner-Riesz multipliers in the plane

Daniele Debertol (2006)

Studia Mathematica

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We consider the multiplier $m_{μ}$ defined for ξ ∈ ℝ by $m_{μ} (ξ) ≐ {((1 - ξ ₁ ² - ξ ₂ ²) / (1 - ξ ₁))}^{μ} 1_{D} (ξ)$ , D denoting the open unit disk in ℝ. Given p ∈ ]1,∞[, we show that the optimal range of μ’s for which $m_{μ}$ is a Fourier multiplier on $L^{p}$ is the same as for Bochner-Riesz means. The key ingredient is a lemma about some modifications of Bochner-Riesz means inside convex regions with smooth boundary and non-vanishing curvature, providing a more flexible version of a result by Iosevich et al. [Publ. Mat. 46 (2002)]. As an application, we show...

A strong convergence theorem for H¹(𝕋ⁿ)

Feng Dai (2006)

Studia Mathematica

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Let ⁿ denote the usual n-torus and let $S ̃_{u}^{δ} (f)$ , u > 0, denote the Bochner-Riesz means of order δ > 0 of the Fourier expansion of f ∈ L¹(ⁿ). The main result of this paper states that for f ∈ H¹(ⁿ) and the critical index α: = (n-1)/2, $l i m_{R \to \infty} 1 / l o g R \int_{0}^{R} (| | S ̃_{u}^{α} (f) - {f | |}_{H ¹ (ⁿ)}) / (u + 1) d u = 0$ .

Valiron-Titchmarsh Theorem for Subharmonic Functions in With Masses on a Half-Line

Alexander I. Kheyfits (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros is extended to subharmonic functions in $ℝ^{n}, n \geq 3$ , having the Riesz masses on a ray.

Variation for the Riesz transform and uniform rectifiability

Albert Mas, Xavier Tolsa (2014)

Journal of the European Mathematical Society

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For $1 \leq n < d$ integers and $ρ > 2$ , we prove that an $n$ -dimensional Ahlfors-David regular measure $μ$ in $ℝ^{d}$ is uniformly $n$ -rectifiable if and only if the $ρ$ -variation for the Riesz transform with respect to $μ$ is a bounded operator in $L^{2} (μ)$ . This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the $L^{2} (μ)$ boundedness of the Riesz transform to the uniform rectifiability of $μ$ .

On the Riesz means of n/ϕ(n) - III

Ayyadurai Sankaranarayanan, Saurabh Kumar Singh (2015)

Acta Arithmetica

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Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term $E_{k} (x)$ where $1 / k! \sum_{n \leq x} n / ϕ (n) {(1 - n / x)}^{k} = M_{k} (x) + E_{k} (x)$ . For instance, the upper bound for |Ek(x)| established here improves the earlier known upper bounds for all integers k satisfying $k ≫ {(l o g x)}^{1 + ϵ}$ .