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In this paper we study the mapping properties of the one-sided fractional integrals in the Calderón-Hardy spaces ${\mathscr{H}}_{q,\alpha }^{p,+}\left(\omega \right)$ for $0<p\le 1$, $0<\alpha <\infty$ and $1<q<\infty$. Specifically, we show that, for suitable values of $p,q,\gamma ,\alpha$ and $s$, if $\omega \in A_s^{+}$ (Sawyer’s classes of weights) then the one-sided fractional integral $I_{\gamma }^{+}$ can be extended to a bounded operator from ${\mathscr{H}}_{q,\alpha }^{p,+}\left(\omega \right)$ to ${\mathscr{H}}_{q,\alpha +\gamma }^{p,+}\left(\omega \right)$. The result is a consequence of the pointwise inequality $$N_{q,\alpha +\gamma }^{+}\left(I_{\gamma }^{+}F;x\right)\le C_{\alpha ,\gamma }{N}_{q,\alpha }^{+}\left(F;x\right),$$ where $N_{q,\alpha }^{+}(F;x)$ denotes the Calderón maximal function.