Let

$(X,d,\mu )$ be a metric measure space with the geometric doubling property and the upper doubling condition for the measure

$\mu $. In this setting, the regularized BMO space

$\text{RBMO}\left(\mu \right)$ and the Hardy space

${H}^{1}\left(\mu \right)$ have been defined and studied in a number of recent papers. Here, the authors prove that any sublinear operator

$T$ that is bounded from

${H}^{1}\left(\mu \right)$ to

${L}^{1,\infty}\left(\mu \right)$ and from

${L}^{\infty}\left(\mu \right)$ to

$\text{RBMO}\left(\mu \right)$, is also bounded on

${L}^{p}\left(\mu \right)$ for all

$p\in (1,\infty )$. This improves a result of

*B. T. Anh* and

*X. T. Duong* [“Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces”,

arXiv:1009.1274, to appear in J. Geom. Anal.] who proved it for ‘linear’ instead of ‘sublinear’ and

${L}^{1}\left(\mu \right)$ instead of

${L}^{1,\infty}\left(\mu \right)$. The proof again uses the Calderón–Zygmund decomposition of Anh and Duong [op. cit.] in this setting, but also needs some new ideas.