Ap weights and quantitative estimates in the Schrödinger setting

Ji Li, Robert Rahm*, Brett D. Wick

*Corresponding author for this work

    Research output: Contribution to journalArticle

    2 Citations (Scopus)

    Abstract

    Suppose L= -Δ+V is a Schrödinger operator on ℝn with a potential V belonging to certain reverse Hölder class RHσ with σ ≥ n/ 2. The aim of this paper is to study the Ap weights associated to L, denoted by ApL, which is a larger class than the classical Muckenhoupt Ap weights. We first prove the quantitative ApL bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative Ap,qL bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical Ap,q constant. However, since Ap,q Ap,qL, the Ap,qL constants are smaller than Ap,q constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L. Next, we prove two–weight inequalities for the fractional integral operator; these have been unknown up to this point. Finally we also have a study on the "exp–log" link between ApL and B M OL (the BMO space associated with L), and show that for w ∈ ApL, log w is in B M OL, and that the reverse is not true in general.
    Original languageEnglish
    Pages (from-to)259-283
    Number of pages25
    JournalMathematische Zeitschrift
    Volume293
    Issue number1-2
    Early online date14 Nov 2018
    DOIs
    Publication statusPublished - Oct 2019

      Fingerprint

    Keywords

    • Schrödinger operator
    • Weighted inequalities
    • Fractional integral operator

    Cite this