Explicit formula for the Benjamin--Ono equation with square integrable and real valued initial data and applications to the zero dispersion limit

In this paper, we extend G{\'e}rard's formula for the solution of the Benjamin--Ono equation on the line to square integrable and real valued initial data. Combined with this formula, we also extend the G{\'e}rard's formula for the zero dispersion limit of the Benjamin--Ono equation on the line to more singular initial data. In the derivation of the extension of the formula for the zero dispersion limit, we also find an interesting integral equality, which might be useful in other contexts.