  The unified transform method introduced by Fokas can be used to analyze
initial-boundary value problems for integrable evolution equations. The method
involves several steps, including the definition of spectral functions via
nonlinear Fourier transforms and the formulation of a Riemann-Hilbert problem.
We provide a rigorous implementation of these steps in the case of the mKdV
equation in the quarter plane under limited regularity and decay assumptions.
We give detailed estimates for the relevant nonlinear Fourier transforms. Using
the theory of $L^2$-RH problems, we consider the construction of quarter plane
solutions which are $C^1$ in time and $C^3$ in space.
