Exponential-square integrability, weighted inequalities for the square functions associated to operators, and applications

Peng Chen, Xuan Thinh Duong*, Liangchuan Wu, Lixin Yan

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    1 Citation (Scopus)

    Abstract

    Let X be a metric space with a doubling measure. Let L be a nonnegative self-adjoint operator acting on L2(X)⁠, hence L generates an analytic semigroup e−tL. Assume that the kernels pt(x,y) of e−tL satisfy Gaussian upper bounds and Hölder continuity in x⁠, but we do not require the semigroup to satisfy the preservation condition e−tL1=1⁠. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator L is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces Rn⁠. We then apply this result to obtain: (1) estimates of the norm on Lp as p becomes large for operators such as the square functions or spectral multipliers; (2) weighted norm inequalities for the square functions; and (3) eigenvalue estimates for Schrödinger operators on Rn or Lipschitz domains of Rn⁠.
    Original languageEnglish
    Pages (from-to)18057–18117
    Number of pages61
    JournalInternational Mathematics Research Notices
    Volume2021
    Issue number23
    DOIs
    Publication statusPublished - Dec 2021

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