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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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