Displaying similar documents to “An inversion formula and a note on the Riesz kernel”

Regularity properties of the equilibrium distribution

Hans Wallin (1965)

Annales de l'institut Fourier

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Soit F un sous-ensemble compact de R m ayant des points intérieurs et soit μ α F la distribution d’équilibre sur F de masse totale 1 par rapport au noyau r α - m avec 0 < α < 2 pour m 2 , et 0 < α < 1 pour m = 1 . La restriction de μ α F à l’intérieur de F est absolument continue et a pour densité f α F . On donne une formule explicite pour f α F et, pour une classe générale d’ensembles F , on démontre que f α F , définie en réalité sur un ensemble de mesure de Lebesgue nulle, croît comme la distance à la frontière F de F élevée à la puissance...

When is a Riesz distribution a complex measure?

Alan D. Sokal (2011)

Bulletin de la Société Mathématique de France

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Let α be the Riesz distribution on a simple Euclidean Jordan algebra, parametrized by α . I give an elementary proof of the necessary and sufficient condition for α to be a locally finite complex measure (= complex Radon measure).

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 C α ( E 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 ̲ ( 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 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 and c 0 , 1 > 0 , c 1 , 2 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...

Generalized Riesz products produced from orthonormal transforms

Nikolaos Atreas, Antonis Bisbas (2012)

Colloquium Mathematicae

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Let 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 V N ( 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 p and N V N ¯ = L ( ) . The results involve mutual absolute continuity or singularity of such Riesz products extending previous...

Regularity properties of commutators and B M O -Triebel-Lizorkin spaces

Abdellah Youssfi (1995)

Annales de l'institut Fourier

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In this paper we consider the regularity problem for the commutators ( [ b , R k ] ) 1 k n where b is a locally integrable function and ( R j ) 1 j n are the Riesz transforms in the n -dimensional euclidean space n . More precisely, we prove that these commutators ( [ b , R k ] ) 1 k n are bounded from L p into the Besov space B ˙ p s , p for 1 &lt; p &lt; + and 0 &lt; s &lt; 1 if and only if b is in the B M O -Triebel-Lizorkin space F ˙ s , p . The reduction of our result to the case p = 2 gives in particular that the commutators ( [ b , R k ] ) 1 k n are bounded form L 2 into the Sobolev space H ˙ s if and only if b ...

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 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...

Global approximations for the γ-order Lognormal distribution

Thomas L. Toulias (2013)

Discussiones Mathematicae Probability and Statistics

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A generalized form of the usual Lognormal distribution, denoted with γ , is introduced through the γ-order Normal distribution γ , with its p.d.f. defined into (0,+∞). The study of the c.d.f. of γ is focused on a heuristic method that provides global approximations with two anchor points, at zero and at infinity. Also evaluations are provided while certain bounds are obtained.

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 n < d integers and ρ > 2 , we prove that an n -dimensional Ahlfors-David regular measure μ in 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 .

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 ² - ξ ² ) / ( 1 - ξ ) ) 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...

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 d . For any Borel subset A of d and any function f : d , A | R j f ( x ) | d x C p | | f | | L p ( 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 k = 0 ( - 1 ) k / ( 2 k + 1 ) q + 1 ] 1 / q , 1 < p < 2, and C p = [ 4 Γ ( q + 1 ) / π q k = 0 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.

Uniformly convex spiral functions and uniformly spirallike functions associated with Pascal distribution series

Gangadharan Murugusundaramoorthy, Basem Aref Frasin, Tariq Al-Hawary (2022)

Mathematica Bohemica

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The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes 𝒮𝒫 p ( α , β ) and 𝒰𝒞𝒱 p ( α , β ) of uniformly spirallike functions. Further, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

Inequalities involving heat potentials and Green functions

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 E whose supports are compact polar subsets of E . 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...

On asymmetric distributions of copula related random variables which includes the skew-normal ones

Ayyub Sheikhi, Fereshteh Arad, Radko Mesiar (2022)

Kybernetika

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Assuming that C X , Y is the copula function of X and Y with marginal distribution functions F X ( x ) and F Y ( y ) , in this work we study the selection distribution Z = d ( X | Y T ) . We present some special cases of our proposed distribution, among them, skew-normal distribution as well as normal distribution. Some properties such as moments and moment generating function are investigated. Also, some numerical analysis is presented for illustration.