Abstract
Let X be a metric space with a doubling measure, and let L be a nonnegative self-adjoint operator in \(L^2(X)\) which generates a semigroup \(e^{-t L}\) whose kernels \(p_t (x, y), t > 0\), satisfy the Gaussian upper bound. Inspired by Fefferman’s paper (Fefferman in Acta Math 124:9–36), in this note, we give sufficient conditions for which the square function \(g_{L,\psi ,\alpha }^{*}\) is unbounded from \(L^p(X)\) to \(L^p(X)\). As an application, we discuss the sharpness of the exponent of aperture \(\alpha \) in the (Bui and Duong in J Geom Anal 30:874–900, Theorem 1.6).
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References
Bui, T.A., Duong, X.T.: Sharp weighted estimates for square functions associated to operators on spaces of homogeneous type. J. Geom. Anal. 30, 874–900 (2020)
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M. H. is supported by a grant from IPM.
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Hormozi, M., Yabuta, K. Comments on: “Sharp Weighted Estimates for Square Functions Associated to Operators on Spaces of Homogeneous Type ”. J Geom Anal 32, 1 (2022). https://doi.org/10.1007/s12220-021-00783-1
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DOI: https://doi.org/10.1007/s12220-021-00783-1