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Paper   IPM / M / 16743
School of Mathematics
  Title:   Comments on ''Sharp weighted estimates for square functions associated to operators on spaces of homogeneous type
  Author(s):  Mahdi Hormozi (Joint with K. Yabuta)
  Status:   Published
  Journal: J. Geom. Anal.
  Vol.:  32
  Year:  2022
  Pages:   DOI: 10.1007/s12220-021-00783-1
  Supported by:  IPM
Let $X$ be a metric space with a doubling measure and let $L$ be a nonnegative selfadjoint 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 [2], 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 [1, Theorem 1.6].

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