Two-weight norm inequalities for Cesàro means of Laguerre expansions

Abstract

Abstract. Two-weight L p norm inequalities are proved for Cesàro means of Laguerre polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and "unweighted" cases, by including all values of p ≥ 1 for all positive orders of the Cesàro summation and all values of the Laguerre parameter α > −1. Almost everywhere convergence results are obtained as a corollary. For the Cesàro means the hypothesized conditions are shown to be necessary for the norm inequalities. Necessity results are also obtained for the norm inequalities with the supremum of the Cesàro means; in particular, for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth.

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

“…They also obtain similar estimates for Cesàro means of Hermite polynomial series and for the supremum of those means in [15]. An almost everywhere convergence result is obtained as a corollary in [14] and [15]. The result about Laguerre polynomials is an extension of a previous result in [18].…”

“…Cesàro means are other of the most usual summation methods. B. Muckenhoupt and D. W. Webb [14] give inequalities for Cesàro means of Laguerre polynomial series and for the supremum of these means with certain parameters and 1 < p ≤ ∞. For p = 1, they prove a weak type result.…”

“…These are stated below. They are small modifications of the six lemmas contained in Section 3 of [14] where a sketch of their proofs can be found.…”

Norm inequalities for the conjugate operator in two-weighted Lebesgue spaces

“…In succession of classical weighted-norm inequalities, starting from Muckenhoupt's result in 1975 [6], there have been extensive studies on two-weighted norm inequalities (for textbooks [7][8][9][10] and for related topics [11][12][13][14][15][16][17]). In [6], Muckenhoupt derives a necessary and sufficient condition on two-weighted norm inequalities for the Poisson integral operator.…”

Laguerre expansions on conic domains

“…If f (x, t) = f 0 (t) with f 0 : R + → R, then the orthogonal expansions of f on the conic domain becomes the classical Laguerre expansions on R + , which has been studied extensively in the literature; see [1,7,8,9,10,14,17,18] and, for some more recent works, see [2,11,12,15] and the references therein. As in the study of classical Laguerre expansions, our study relies on intrinsic properties of orthogonal polynomials that hold only for particular weight functions and special domains.…”