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

概要

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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 HpL(X) be the Hardy space associated with L. We prove a Hörmander-type spectral multiplier theorem for L on HpL(X) for 0 < p < ∞: the operator m(L) is bounded from HpL(X) to HpL(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 HpL(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.
2011-01-01