Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

Xuan Thinh Duong*, Lixin Yan

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    33 Citations (Scopus)

    Abstract

    Let (X,d,μ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ. Let L be a non-negative self-adjoint operator on L2(X). Assume that the semigroup e-tL generated by L satisfies the Davies-Gaffney estimates. Let HL p(X) be the Hardy space associated with L. We prove a Hörmander-type spectral multiplier theorem for L on HL p(X) for 0 < p < ∞ the operator m(L) is bounded from HLp(X) to HLp(X) if the function m possesses s derivatives with suitable bounds and s > n(1/p - 1/2) where n is the "dimension" of X. By interpolation, m(L) is bounded on H Lp(X) for all 0 < p < ∞ if m is infinitely differentiable with suitable bounds on its derivatives. We also obtain a spectral multiplier theorem on Lp spaces with appropriate weights in the reverse Hölder class.

    Original languageEnglish
    Pages (from-to)295-319
    Number of pages25
    JournalJournal of the Mathematical Society of Japan
    Volume63
    Issue number1
    DOIs
    Publication statusPublished - 2011

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