Harmonic analysis operators associated with Laguerre polynomial expansions on variable Lebesgue spaces

Abstract

In this paper we give sufficient conditions on a measurable function p : (0, ∞) n → [1, ∞) in order that harmonic analysis operators (maximal operators, Riesz transforms, Littlewood-Paley functions and multipliers) associated with α-Laguerre polynomial expansions are bounded on the variable Lebesgue space L p(•) ((0, ∞) n , µα), where dµα(x) = 2 n n j=1 x 2α j +1 j e −x 2 j Γ(α j +1) dx, being α = (α 1 , . . . , αn) ∈ [0, ∞) n and x = (x 1 , . . . , xn) ∈ (0, ∞) n .

Maximal function characterization of Hardy spaces related to Laguerre polynomial expansions

“…We consider, for α > − 1 2 , the probability measure dγ α (x) = 2x 2α+1 Γ(α+1) e −x 2 dx on (0, ∞). The measure γ α is not doubling (not even locally doubling) with respect to the usual metric in (0, ∞).…”

“…Theorem 1.1. Let α > − 1 2 , a > 0 and 1 < q ≤ ∞. Then, H 1,q a ((0, ∞), γ α ) and H 1,∞ 1 ((0, ∞), γ α ) coincide, algebraic and topologically.…”

“…Throughout this paper, by C and c we will always denote positive constants that may change in each occurrence. We will use many times the following properties (see, for instance, [25, Section 1.2]) without mentioning them: for every a > 0 there exists C a > 1 such that 1 Ca m(x) ≤ m(y) ≤ C a m(x) and 1 Ca γ(x) ≤ γ(y) ≤ C a γ(x) provided that |x − y| ≤ am(x), where γ(x) = e −|x| 2 for x ∈ (0, ∞).…”

BLO spaces associated with Laguerre polynomials expansions

“…However, RBMO((0, ∞), γ α ) is not suitable to study harmonic analysis operators associated with Laguerre polynomial expansions because these operators are not defined by standard Calderón-Zygmund kernels ( [19], [45] and [46]). Motivated by the results in [37] in the Gaussian setting, the authors and R. Scotto ( [5]) defined a local BMO-type space related to the measure γ α as follows.…”

“…• * ,α,a , respectively. This space can be identified with the dual space of the Hardy space H 1 ((0, ∞), γ α ) studied in [5] (see [5,Theorem 1.1]).…”