Variable exponent Sobolev spaces associated with Jacobi expansions

Almeida, Víctor; Castro, Alejandro J.; Sanabria, A.; Scotto, Roberto

doi:10.7153/jmi-2017-11-71

Cited by 4 publications

(11 citation statements)

References 38 publications

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“…To a large extent, this motivated our paper [3] where the symmetrization was applied in the framework of Jacobi trigonometric polynomial expansions. More precisely, we proved that fundamental harmonic analysis operators in the Jacobi symmetrized setting, including Riesz transforms, Littlewood-Paley-Stein type square functions, Jacobi-Poisson semigroup maximal operator and certain spectral multipliers, are bounded on weighted L p spaces and are of weighted weak type (1,1). Analogous results in the original non-symmetrized Jacobi context were obtained earlier by Nowak and Sjögren [8], by means of the Calderón-Zygmund theory.…”

Section: Introductionsupporting

confidence: 68%

“…We also mention that [4] contains L p results for variants of Riesz-Jacobi transforms that are different from the operators in (a) and (b), and in [5] one can find L p results for variants of square functions different from those in (e) and (f). Weighted L p boundedness results for the variants just mentioned were obtained recently in [1], though under the restriction α, β ≥ −1/2.…”

Section: 3mentioning

confidence: 88%

“…where φ α,β n are even functions on (−π, π) given by (3) and implicitly by (1). The corresponding eigenvalues are λ α,β n , L α,β Θ α,β n = λ α,β n Θ α,β n , n ≥ 0.…”

Section: 2mentioning

confidence: 99%

“…We point out that analysis in Jacobi settings received a considerable attention in recent years, see for instance [1,2,3,4,5,6,7,8,9,10,13] and numerous other references given in these articles. In particular, mapping properties of harmonic analysis operators in the classical (non-symmetrized) discrete Jacobi polynomial and function contexts were extensively studied.…”

Section: Introductionmentioning

confidence: 99%

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Harmonic analysis operators related to symmetrized Jacobi expansions for all admissible parameters

Langowski

2016

Acta Math. Hungar.

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This is an ultimate completion of our earlier paper [Acta. Math. Hungar. 140 (2013), 248-292] where mapping properties of several fundamental harmonic analysis operators in the setting of symmetrized Jacobi trigonometric expansions were investigated under certain restrictions on the underlying parameters of type. In the present article we take advantage of very recent results due to Nowak, Sjögren and Szarek to fully release those restrictions, and also to provide shorter and more transparent proofs of the previous restricted results. Moreover, we also study mapping properties of analogous operators in the parallel context of symmetrized Jacobi function expansions. Furthermore, as a consequence of our main results we conclude some new results related to the classical non-symmetrized Jacobi polynomial and function expansions.

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Section: Introductionsupporting

confidence: 68%

Section: 3mentioning

confidence: 88%

“…where φ α,β n are even functions on (−π, π) given by (3) and implicitly by (1). The corresponding eigenvalues are λ α,β n , L α,β Θ α,β n = λ α,β n Θ α,β n , n ≥ 0.…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Harmonic analysis operators related to symmetrized Jacobi expansions for all admissible parameters

Langowski

2016

Acta Math. Hungar.

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“…Assume that λ > 0. For every n ∈ N we consider the n-th ultraspherical polynomial of order λ defined by ([51, §4.7]) (2) P λ n (x) := (−1) n 2 n (λ + 1/2) n (1 − x 2 ) 1/2−λ d n dx n (1 − x 2 ) n+λ−1/2 , x ∈ [−1, 1].…”

Section: Introductionmentioning

confidence: 99%

Discrete Harmonic Analysis Associated with Ultraspherical Expansions

2019

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In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted p -boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by the difference operator ∆ λ f (n) := a λ n f (n + 1) − 2f (n) + a λ n−1 f (n − 1), n ∈ N, λ > 0, where a λ n := {(2λ+n)(n+1)/[(n+λ)(n+1+λ)]} 1/2 , n ∈ N, and a λ −1 := 0. We also prove weighted p -boundedness properties of transplantation operators associated with the system {ϕ λ n } n∈N of ultraspherical functions, a family of eigenfunctions of ∆ λ . In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.

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A weighted transplantation theorem for Jacobi coefficients

Arenas

Ciaurri

Labarga

2019

Journal of Approximation Theory

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We present a transplantation theorem for Jacobi coefficients in weighted spaces. In fact, by using a discrete vector-valued local Calderón-Zygmund theory, which has recently been furnished, we prove the boundedness of transplantation operators from ℓ p (N, w) into itself, where w is a weight in the discrete Muckenhoupt class Ap(N). Moreover, we obtain weighted weak (1, 1) estimates for those operators.2010 Mathematics Subject Classification. Primary: 42C10. Secondary: 44A20.

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Variable exponent Sobolev spaces associated with Jacobi expansions

Cited by 4 publications

References 38 publications

Harmonic analysis operators related to symmetrized Jacobi expansions for all admissible parameters

Harmonic analysis operators related to symmetrized Jacobi expansions for all admissible parameters

Discrete Harmonic Analysis Associated with Ultraspherical Expansions

A weighted transplantation theorem for Jacobi coefficients

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