On potential spaces related to Jacobi expansions

Langowski, Bartosz

doi:10.1016/j.jmaa.2015.06.058

Cited by 5 publications

(27 citation statements)

References 21 publications

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“…The study of g-functions involving fractional derivatives goes back to Segovia and Wheeden [39]. More recently square functions defined via fractional derivatives were investigated in [9,10,23] in connection with potential spaces in various settings. On the other hand, extensions relying on taking any r ≥ 2 rather than the standard r = 2 are quite natural and well known in the literature; see, for instance, [8,11].…”

Section: Further Results and Commentsmentioning

confidence: 99%

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

Nowak¹,

Stempak²,

Szarek³

2016

SIGMA

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Abstract. We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to Z d 2 . Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, g-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calderón-Zygmund theory we prove that these operators are bounded on weighted L p spaces, 1 < p < ∞, and from weighted L 1 to weighted weak L 1 . We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type.

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Section: Further Results and Commentsmentioning

confidence: 99%

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

Nowak¹,

Stempak²,

Szarek³

2016

SIGMA

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“…Unweighted L p -boundedness of the interlaced Riesz-Jacobi transforms (b) and the maximal operator (d) was shown in [4,Proposition 4.2] and [4, Proposition 2.2], respectively. 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: 96%

“…We now pass to the less explored non-symmetrized Jacobi function setting. Let {H α,β t } t>0 be the Jacobi-Poisson semigroup in this context, see Sections 2 in [4,5]. We have…”

Section: 3mentioning

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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“…This article is motivated by the recent results of Nowak and Stempak [25] and the author's papers [16,17]. In [16] we investigated Sobolev and potential spaces related to discrete Jacobi function expansions.…”

Section: Introductionmentioning

confidence: 99%

Potential and Sobolev Spaces Related to Symmetrized Jacobi Expansions

Langowski¹

2015

SIGMA

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Abstract. We apply a symmetrization procedure to the setting of Jacobi expansions and study potential spaces in the resulting situation. We prove that the potential spaces of integer orders are isomorphic to suitably defined Sobolev spaces. Among further results, we obtain a fractional square function characterization, structural theorems and Sobolev type embedding theorems for these potential spaces.

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On potential spaces related to Jacobi expansions

Cited by 5 publications

References 21 publications

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

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

Potential and Sobolev Spaces Related to Symmetrized Jacobi Expansions

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