Hermite Function Expansions Versus Hermite Polynomial Expansions

Abu-Falahah, I.; Torrea, José L.

doi:10.1017/s0017089506003004

Cited by 28 publications

(58 citation statements)

References 10 publications

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“…In the present case it is already known, see [2,Theorem 2.3], (i) ⇔ (ii) ⇔ (iii). The Riesz transform R is a Calderón-Zygmund operator on R, see [26], that can be described as a principal value operator, see [2]. Hence R loc ↓ can be defined and is a local Calderón-Zygmund operator on (0, ∞).…”

Section: Definition 36supporting

confidence: 54%

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Transference between Laguerre and Hermite settings

Betancor

Fariña

Rodríguez-Mesa

et al. 2008

Journal of Functional Analysis

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In this paper we present a new method in order to transfer boundedness results for operators associated with Hermite functions to boundedness results for operators associated with Laguerre functions. The technique relies on an exact point-wise identity relating the heat kernels of both systems. The method that we present here has the novelty that can be used backwards, that is, boundedness results for Laguerre systems can be also transfered to boundedness results for Hermite systems. We apply our method in order to get new properties of some operators in the Laguerre setting. Among others, we mention the description of Riesz transforms as principal value operators. As an application of the reversibility of the method we characterize the class of Banach spaces B for which the Riesz transforms (in the Laguerre setting) are bounded from L p B into itself. It is shown that this class coincides with the UMD class.

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Section: Definition 36supporting

confidence: 54%

“…For a Banach space B the UMD property is equivalent to the fact that the Hilbert transform admits a B-valued extension to L p B (R) for some (any) 1 < p < ∞ [4,5]. Recently, see [2], UMD property for a Banach space B has been characterized for the L p B -boundedness of the Riesz transforms defined in (2.3).…”

Section: Definition 36mentioning

confidence: 99%

Transference between Laguerre and Hermite settings

Betancor

Fariña

Rodríguez-Mesa

et al. 2008

Journal of Functional Analysis

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“…. , n; is bounded in L p (R, |x| τ dx) for −1−βp/2 < τ −βp 2 +β < p−1+βp/2 (see Theorem 3.5 of [11]) and W β is an isometry. Therefore ≤ C p f L p (y δ dy) .…”

Section: Holdmentioning

confidence: 99%

Transferring strong boundedness among Laguerre orthogonal systems

Abu-Falahah

Macías

Segovia³

et al. 2009

Proc Math Sci

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Given the family of Laguerre polynomials, it is known that several orthonormal systems of Laguerre functions can be considered. In this paper we prove that an exhaustive knowledge of the boundedness in weighted L p of the heat and Poisson semigroups, Riesz transforms and g-functions associated to a particular Laguerre orthonormal system of functions, implies a complete knowledge of the boundedness of the corresponding operators on the other Laguerre orthonormal system of functions. As a byproduct, new weighted L p boundedness are obtained. The method also allows us to get new weighted estimates for operators related with Laguerre polynomials.

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“…−1/2 H k (s)e −s 2 /2 , s ∈ R, are the eigenfunctions of the operator H , satisfying H h α = (2|α| + n)h α . The relation between the eigenfunctions can transported to the operators associated to H and L. The following Proposition can be found in [5]. dx).…”

Section: Then the Operators O(t ) And V ρ (T ) Are Bounded In L Pmentioning

confidence: 96%

The ρ-variation as an operator between maximal operators and singular integrals

et al. 2009

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The ρ-variation and the oscillation of the heat and Poisson semigroups of the Laplacian and Hermite operators (i.e. and − + |x| 2 ) are proved to be bounded fromin the case p = 1) for 1 ≤ p < ∞ and w being a weight in the Muckenhoupt's A p class. In the case p = ∞ it is proved that these operators do not map L ∞ into itself. Even more, they map L ∞ into B M O but the range of the image is strictly smaller that the range of a general singular integral operator.

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Hermite Function Expansions Versus Hermite Polynomial Expansions

Cited by 28 publications

References 10 publications

Transference between Laguerre and Hermite settings

Transference between Laguerre and Hermite settings

Transferring strong boundedness among Laguerre orthogonal systems

The ρ-variation as an operator between maximal operators and singular integrals

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