On the Fejér-F. Riesz inequality in
Yoram Sagher (1977)
Studia Mathematica
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Yoram Sagher (1977)
Studia Mathematica
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Nikolaos Atreas, Antonis Bisbas (2012)
Colloquium Mathematicae
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Let 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 , where are -dimensional subspaces of L₂() spanned by an orthonormal set which is produced from dilations and multiplications of elements of and . The results involve mutual absolute continuity or singularity of such Riesz products extending previous...
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 with p ≥ 2 instead of boundedness of the functions φₙ of an orthonormal system.
Hiroaki Aikawa (2016)
Studia Mathematica
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Let be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density 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 is either 0 or 1; the first case occurs if and only if 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...
Adam Osękowski (2014)
Studia Mathematica
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We establish the following sharp local estimate for the family of Riesz transforms on . For any Borel subset A of and any function , , 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, , 1 < p < 2, and , 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.
Elmiloud Chil, Mohamed Mokaddem, Bourokba Hassen (2015)
Commentationes Mathematicae Universitatis Carolinae
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Let be a Riesz space, a Hausdorff topological vector space (t.v.s.). We prove, under a certain separation condition, that any orthosymmetric bilinear map 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 is linearly represented. This fits in the type of...
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 , with , which is a Riesz-Feller operator. The nonlinear flux is given by the locally Lipschitz function for . We show that in the sub-critical case, , 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...
Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)
Annales Polonici Mathematici
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For 1 ≤ q ≤ α ≤ p ≤ ∞, is a complex Banach space which is continuously included in the Wiener amalgam space and contains the Lebesgue space . We study the closure in of the space of test functions (infinitely differentiable and with compact support in ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...
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 and , 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, , on ℂ if , which can be derived from...
Albert Mas, Xavier Tolsa (2014)
Journal of the European Mathematical Society
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For integers and , we prove that an -dimensional Ahlfors-David regular measure in is uniformly -rectifiable if and only if the -variation for the Riesz transform with respect to is a bounded operator in . 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 boundedness of the Riesz transform to the uniform rectifiability of .
Daniele Debertol (2006)
Studia Mathematica
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We consider the multiplier defined for ξ ∈ ℝ by , D denoting the open unit disk in ℝ. Given p ∈ ]1,∞[, we show that the optimal range of μ’s for which is a Fourier multiplier on 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...
Jeffrey S. Geronimo, Plamen Iliev (2014)
Journal of the European Mathematical Society
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We give a complete characterization of the positive trigonometric polynomials on the bi-circle, which can be factored as where is a polynomial nonzero for and . The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities...
Feng Dai (2006)
Studia Mathematica
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Let ⁿ denote the usual n-torus and let , 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, .
Neil A. Watson (2015)
Mathematica Bohemica
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We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set whose supports are compact polar subsets of . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set...
Bahri Turan (2006)
Studia Mathematica
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Let E be a Riesz space. By defining the spaces and of E, we prove that the center of is and show that the injectivity of the Arens homomorphism m: Z(E)” → Z(E˜) is equivalent to the equality . Finally, we also give some representation of an order continuous Banach lattice E with a weak unit and of the order dual E˜ of E in which are different from the representations appearing in the literature.
Marek Wójtowicz (1999)
Commentationes Mathematicae Universitatis Carolinae
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Let be an Archimedean Riesz space with a weak order unit . A sufficient condition under which Dedekind [-]completeness of the principal ideal can be lifted to is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of -spaces. Similar results are obtained for the Riesz spaces , , of all functions of the th Baire class on a metric space .