Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz

N. Merentes; J. L. Sánchez Hernández

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Displaying similar documents to “Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz”

Remark on the inequality of F. Riesz

W. Łenski (2005)

Banach Center Publications

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We prove F. Riesz’ inequality assuming the boundedness of the norm of the first arithmetic mean of the functions ${| φ ₙ |}^{p}$ with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.

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...

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 $T$ .

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...

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...

$L^{p}$ 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.

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...

Triebel-Lizorkin spaces with non-doubling measures

Yongsheng Han, Dachun Yang (2004)

Studia Mathematica

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Suppose that μ is a Radon measure on $ℝ^{d}$ , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀ > 0 such that for all x ∈ supp(μ) and r > 0, μ(B(x,r)) ≤ C₀rⁿ, where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spaces $Ḟ_{p q}^{s} (μ)$ for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure μ, C₀, n and d. The method does not use the vector-valued maximal function...

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...

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...

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, $l i m_{R \to \infty} 1 / l o g R \int_{0}^{R} (| | S ̃_{u}^{α} (f) - {f | |}_{H ¹ (ⁿ)}) / (u + 1) d u = 0$ .

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...