  We prove local well-posedness of the Benjamin-Ono equation for a class of
bounded initial data including periodic and bore-like functions. As a
consequence, we obtain local well-posedness in
$H^s(\mathbb{R})+H^\sigma(\mathbb{T})$ for $s>\frac{1}{2}$ and
$\sigma>\frac{7}{2}$. These results follow by studying a generalized forced
Benjamin-Ono equation.
