Let \((\mathcal {X},\rho ,\mu )\) be a space of homogeneous type in the sense of Coifman and Weiss, \(Y(\mathcal {X})\) a ball quasi-Banach function space satisfying some mild assumptions, and \(q\in (0,\infty )\) . In this article, we prove that, for any given \(\alpha \in \mathbb {R}\setminus \{0\}\) and for any \(f\in Y(\mathcal {X})\) , \(\begin{aligned} \sup _{\lambda \in (0,\infty )} \lambda \left\| \left\{ \int _{\{(\cdot ,y):\ \frac{|f(\cdot )-f(y)|}{[U(\cdot ,y)]^{\frac{\alpha }{q}}}>\lambda \}} [U(\cdot ,y)]^{\alpha -1}d\mu (y) \right\} ^{\frac{1}{q}}\right\| _{{Y(\mathcal {X})}} \sim \Vert f\Vert _{Y(\mathcal {X})} \end{aligned}\) and where \(U(x,y){:}{=}\min \{\mu (B(x,\rho (x,y))),\,\mu (B(y,\rho (y,x)))\}\) for any \(x,y\in \mathcal {X}\) , \(\Lambda (\alpha )=0\) if \(\alpha \in (0,\infty )\) or \(\Lambda (\alpha )=\infty \) if \(\alpha \in (-\infty ,0)\) , and the implicit positive constants are independent of f. These results are of wide generality and applied to eight specific ball quasi-Banach function spaces. In particular, when \(Y(\mathcal {X}){:}{=}L^q(\mathbb {R}^n)\) with \(q\in [1,\infty )\) , the first formula exactly coincides with the recent formula of Gu and Yung in 2021 (for \(\alpha =1\) ) and Brezis et al. in 2022 (for all \(\alpha \in \mathbb {R}\setminus \{0\}\) ). The novelty of this article lies in that we use the method of extrapolation to overcome the difficulty caused by the deficiency of the explicit expression of the quasi-norm of \(Y(\mathcal {X})\) , apply admissible sequences of balls to circumvent the reverse doubling condition of the measure \(\mu \) under consideration, and employ the Borel-semiregular property of \(\mu \) to establish the density of the Hölder space with bounded support in \(Y(\mathcal {X})\) .
