$$A_p$$ A p Weights and quantitative estimates in the Schrödinger setting

Abstract

Part of the intrinsic structure of singular integrals in the Bessel setting is captured by Muckenhoupt-type weights. Anderson-Kerman showed that the Bessel Riesz transform is bounded on weighted L p w if and only if w is in the class A p,λ . We introduce a new class of Muckenhoupt-type weights A p,λ in the Bessel setting, which is different from A p,λ but characterizes the weighted boundedness for the Hardy-Littlewood maximal operators. We also establish the weighted L p boundedness and compactness, as well as the endpoint weak type boundedness of Riesz commutators. The quantitative weighted bound is also established.

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

“…We prove Theorem 1.3 in Section 2 by borrowing some ideas from [1], where Bongioanni et al introduced a ρ-localized method to obtain the boundedness of operators associated to the Schrödinger operator L on weighted Lebesgue spaces. This is quite different from the method used in [12]. Besides this, we also need a quantitative version of the extrapolation theorem for A ρ, θ p (R n ) weights (see Lemma 2.6 below).…”

“…Moreover, the weighted boundedness, related to A ρ, θ p (R n ), of many operators associated to the Schrödinger operator L := −∆ + V was obtained in [1,2,16,23,25,26]. Recently, Li et al [12] introduced the fractional weight class A ρ, θ p, q (R n ) adapted to L and obtained the quantitative weighted boundedness of the fractional maximal function and the fractional integral operator associated to L.…”

“…Motivated by [1,12,15], in this article, we consider the Littlewood-Paley functions g L , S L and g * L, λ associated to the Schrödinger operator L, which are defined, respectively, by setting, for any f ∈ L 2 (R n ) and x ∈ R n ,…”

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

“…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.…”