Quantitative boundedness of Littlewood–Paley functions on weighted Lebesgue spaces in the Schrödinger setting

Abstract

Let L := −∆ + V be the Schrödinger operator on R n with n ≥ 3, where V is a nonnegative potential which belongs to certain reverse Hölder class RH q (R n ) with q ∈

Quantitative Weighted Bounds for Littlewood-Paley Functions Generated by Fractional Heat Semigroups Related with Schrödinger Operators

“…We point out that the quantitative weighted boundedness of Littlewood-Paley functions generated by fe −tL α g t>0 is not a simple analogue of [19] which deals with the case α = 1. For α = 1, it is well known that the heat kernel K t ð•Þ of the Laplace operator has a good decay properties.…”

“…In recent years, the quantitative weighted bounds for operators in the Schrödinger settings have been investigated by many researchers. Li et al [18] established the quantitative weighted boundedness of maximal functions, maximal heat semigroups, and fractional integral operators related to L. In 2020, Zhang and Yang [19] showed that the quantitative weighted boundedness for Littlewood-Paley functions in the Schrödinger setting. Bui et al [20] investigated the quantitative boundedness for square functions with new class of weights.…”

“…Inspired by the results in [18,19,22], in this paper, we investigate the quantitative weighted boundedness of Littlewood-Paley functions generated by fractional heat semigroups associated with L. Let K L t ð•, • Þ denote the integral kernel of the heat semigroup fe −tL g t≥0 . For α > 0, the subordinative formula (cf.…”

“…The arbitrariness of N ensures that the square functions generated with fe −tð−ΔÞ g can be dominated by the intrinsic Lusin area function: for ϵ ∈ ð1,∞Þ (see [19], Lemma 3.1).…”

Intrinsic Square Function Characterizations of Several Hardy-Type Spaces — A Survey

“…Very recently, Yan et al [49] continued the above line of research and established the intrinsic square function characterizations of the Hardy type space H X (R n ) related to a ball quasi-Banach function space X satisfying some mild additional assumptions. For more applications of such intrinsic square functions, we refer the reader to [10,11,23,37,39,52].…”