Two-weight norm inequalities for the Cesàro means of generalized Hermite expansions

Journal of Approximation Theory

2013

Self Cite

In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel expansions. For this operator, we study weighted and unweighted inequalities in the spaces L p ((0, 1), x 2ν+1 dx). Moreover, weak and restricted weak type inequalities are obtained for the critical values of p. As a consequence, we deduce the almost everywhere pointwise convergence of these means.

Section: Introduction and Main Resultssupporting

confidence: 57%

The Bochner–Riesz means for Fourier–Bessel expansions: Norm inequalities for the maximal operator and almost everywhere convergence

Ciaurri

Journal of Approximation Theory

2013

Self Cite

“…Interesting results, considering Cesàro means with potential weights, have been obtained by Muckenhoupt and Webb in [9,10] in the context of Fourier-Laguerre and Fourier-Hermite expansions; see also the previous work in [11]. Recently, the first author has obtained results related to Fourier series of generalized Hermite functions in [6] extending some of the results of Muckenhoupt and Webb [10]. It is well known that the generalized Hermite functions play a prominent role in Dunkl analysis.…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

“…For each p, let us take a value for , such that (4) and (5) are satisfied. So, using (32), Minkowski's integral inequality and (1), (31) follows from conditions (2), (3) and (6). The necessity of the conditions is obtained as in Theorem 1.…”

Section: Lemmamentioning

confidence: 89%

See 1 more Smart Citation

The Bochner–Riesz means for Fourier–Bessel expansions

Ciaurri

Journal of Functional Analysis

2005

Self Cite

“…In the case of κ ≡ 0, where the above series reduces to standard Hermite expansions, it is also well known that the Hermite expansions fail to converge in L p (R d , dx) for p = 2, unless d = 1. Even when d = 1, the series converges in L p (R, dx) (or, equivalently, the corresponding partial sum operators are uniformly bounded) if and only if 4 3 < p < 4 according to a theorem by R. Askey and S. Wainger [3].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Mixed norm estimates for the Cesàro means associated with Dunkl–Hermite expansions

Boggarapu

Trans. Amer. Math. Soc.

Thangavelu

2017

Abstract. Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with Dunkl-Hermite expansions on R d . These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator) Hκ := −∆κ + |x| 2 , where ∆κ stands for the Dunkl-Laplacian. It is shown that the desired mixed norm estimates are equivalent to vectorvalued inequalities for a sequence of Cesàro means for Laguerre expansions with shifted parameter. In order to obtain such vector-valued inequalities, we develop an argument to extend these Laguerre operators for complex values of the parameters involved and apply a version of three lines lemma.