In this paper we introduce the one-sided sharp functions defined by \[ f_ +^ \# (x) = \sup_{h > 0} {1 \over h} \int^{x + h}_ x \left( f(y) - {1 \over h} \int^{x + 2h}_{x + h} f \right)^ + dy \] and \[ f_ -^ \# (x) = \sup_{h > 0} {1 \over h} \int^ x_{x - h} \left( f(y) - {1 \over h} \int^{x - h}_{x-2h} f \right)^ + dy \] where \(z^ + = \max (z,0)\). We study the BMO spaces associated to \(f^ \#_ +\) and \(f^ \#_ -\) and their relation with the good weights for the one-sided Hardy-Littlewood maximal functions. Finally, as an application of our results, we characterize the weights for one-sided fractional integrals and one-sided fractional maximal operators.