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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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...
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...
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...
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...
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.
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...
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.
Xuefang Yan (2015)
Czechoslovak Mathematical Journal
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Let be a metric measure space endowed with a distance and a nonnegative Borel doubling measure . Let be a non-negative self-adjoint operator of order on . Assume that the semigroup generated by satisfies the Davies-Gaffney estimate of order and satisfies the Plancherel type estimate. Let be the Hardy space associated with We show the boundedness of Stein’s square function arising from Bochner-Riesz means associated to from Hardy spaces to , and also study...
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 .
Benedetta Noris, Hugo Tavares, Susanna Terracini, Gianmaria Verzini (2012)
Journal of the European Mathematical Society
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In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system , as the interspecies scattering length goes to . For this system we consider the associated energy functionals , with -mass constraints, which limit (as ) is strongly irregular. For such functionals, we construct multiple critical points...
Steve Hofmann, Svitlana Mayboroda, Alan McIntosh (2011)
Annales scientifiques de l'École Normale Supérieure
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Let be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in , Sobolev, and some new Hardy spaces naturally associated to . First, we show...
Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang (2016)
Czechoslovak Mathematical Journal
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Let be a Schrödinger operator and let be a Schrödinger type operator on , where is a nonnegative potential belonging to certain reverse Hölder class for . The Hardy type space is defined in terms of the maximal function with respect to the semigroup and it is identical to the Hardy space established by Dziubański and Zienkiewicz. In this article, we prove the -boundedness of the commutator generated by the Riesz transform , where , which is larger...
Michael Usher (2014)
Annales de la faculté des sciences de Toulouse Mathématiques
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For a Morse function on a compact oriented manifold , we show that has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in whose components have nontrivial linking number, such that the minimal value of on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of in terms of the Betti numbers of and the behavior of with respect...
M.S. Shahrokhi-Dehkordi (2017)
Communications in Mathematics
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Let be a bounded starshaped domain and consider the -Laplacian problem where is a positive parameter, , and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.
David Swanson (2010)
Colloquium Mathematicae
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Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function possesses an derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.
Yoshihiro Mizuta, Tetsu Shimomura (2023)
Czechoslovak Mathematical Journal
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Our aim is to establish Sobolev type inequalities for fractional maximal functions and Riesz potentials in weighted Morrey spaces of variable exponent on the half space . We also obtain Sobolev type inequalities for a function on . As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents , where and satisfy log-Hölder conditions, for , and is nonnegative and Hölder continuous of order .
Richard Lechner
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We prove an interpolatory estimate linking the directional Haar projection to the Riesz transform in the context of Bochner-Lebesgue spaces , 1 < p < ∞, provided X is a UMD-space. If , the result is the inequality , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of . In order to obtain the interpolatory result (1) we analyze stripe operators , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....