Calderón-Zygmund operators in the Bessel setting for all possible type indices

Castro, Alejandro J.; Szarek, Tomasz

doi:10.1007/s10114-014-2326-1

Cited by 15 publications

(28 citation statements)

References 23 publications

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“…In [5] the authors considered various harmonic analysis operators in the Bessel context associated with the (modified) Hankel transform. The present paper is a continuation and extension of that research.…”

Section: Introductionmentioning

confidence: 99%

On fundamental harmonic analysis operators in certain Dunkl and Bessel settings

Castro

Szarek

2014

Journal of Mathematical Analysis and Applications

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Abstract. We consider several harmonic analysis operators in the multi-dimensional context of the Dunkl Laplacian with the underlying group of reflections isomorphic to Z n 2 (also negative values of the multiplicity function are admitted). Our investigations include maximal operators, g-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and LaplaceStieltjes transform type. Using the general Calderón-Zygmund theory we prove that these objects are bounded in weighted L p spaces, 1 < p < ∞, and from L 1 into weak L 1 .

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Section: Introductionmentioning

confidence: 99%

On fundamental harmonic analysis operators in certain Dunkl and Bessel settings

Castro

Szarek

2014

Journal of Mathematical Analysis and Applications

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“…Our technique is well suited to a wider variety of operators, including more general forms of g-functions and Lusin area type integrals. In a similar spirit, analogous problems concerning analysis for "low" values of type parameters were recently investigated in the Laguerre [30], Bessel [8], and certain Dunkl [9] settings.…”

Section: Introductionmentioning

confidence: 99%

Analysis Related to All Admissible Type Parameters in the Jacobi Setting

2015

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We derive an integral representation for the Jacobi-Poisson kernel valid for all admissible type parameters α, β in the context of Jacobi expansions. This enables us to develop a technique for proving standard estimates in the Jacobi setting that works for all possible α and β. As a consequence, we can prove that several fundamental operators in the harmonic analysis of Jacobi expansions are (vector-valued) Calderón-Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory. The new JacobiPoisson kernel representation also leads to sharp estimates of this kernel. The paper generalizes methods and results existing in the literature but valid or justified only for a restricted range of α and β.

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“…Observe that for E = ∅ we recover the classical multi-dimensional Bessel context considered in [4,5,6,7,15], among many other papers. Otherwise, that is when E = ∅, the exotic situation occurs.…”

Section: 2mentioning

confidence: 70%

“…Note that the double-prime part may be void here and in what follows. Taking into account (15) and the considerations and notation from Section 2.2, we see that…”

Section: 2mentioning

confidence: 99%

“…that correspond to the (modified) Hankel transform on R + and the classical Jacobi polynomials on (−1, 1), respectively. Harmonic analysis of self-adjoint 'Laplacians' emerging from B ν and J α,β in the classical ranges of the parameters ν, α, β > −1 has been widely investigated, see for instance [3,4,5,6,7,8,9,10,11,13,14,15,16,19,38,41,44] and also references therein, which is only a small part of the related literature. In both contexts, we consider the classical and exotic self-adjoint operators and introduce the resulting general multi-dimensional Jacobi and Bessel 'Laplacians'.…”

Section: Introductionmentioning

confidence: 99%

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Maximal operators of exotic and non-exotic Laguerre and other semigroups associated with classical orthogonal expansions

Nowak

Sjögren

Szarek

2017

Advances in Mathematics

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Abstract. Classical settings of discrete and continuous orthogonal expansions, like Laguerre, Bessel and Jacobi, are associated with second order differential operators playing the role of the Laplacian. These depend on certain parameters of type that are usually restricted to a half-line, or a product of half-lines if higher dimensions are considered. Following earlier research done by Hajmirzaahmad, we deal in this paper with Laplacians in the above-mentioned contexts with no restrictions on the type parameters and bring to attention naturally associated orthogonal systems that in fact involve the classical ones, but are different. This reveals new frameworks related to classical orthogonal expansions and thus new potentially rich research areas, at least from the harmonic analysis perspective. To support the last claim we focus on maximal operators of multi-dimensional Laguerre, Bessel and Jacobi semigroups, with unrestricted type parameters, and prove that they satisfy weak type (1, 1) estimates with respect to the appropriate measures. Generally, these measures are not (locally) finite, which makes a contrast with the classical situations and generates new difficulties. An important partial result of the paper is a new proof of the weak type (1, 1) estimate for the classical multi-dimensional Laguerre semigroup maximal operator.

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Calderón-Zygmund operators in the Bessel setting for all possible type indices

Cited by 15 publications

References 23 publications

On fundamental harmonic analysis operators in certain Dunkl and Bessel settings

On fundamental harmonic analysis operators in certain Dunkl and Bessel settings

Analysis Related to All Admissible Type Parameters in the Jacobi Setting

Maximal operators of exotic and non-exotic Laguerre and other semigroups associated with classical orthogonal expansions

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