  We study the local well-posedness in the Sobolev space H^s for the modified
Korteweg-de Vries (mKdV) equation on the real line. Kenig-Ponce-Vega
\cite{KPV2} and Christ-Colliander-Tao established that the data-to-solution map
fails to be uniformly continuous on a fixed ball in H^s when s<1/4. In spite of
this, we establish that for -1/8 < s < 1/4, the solution satisfies global in
time H^s(R) bounds which depend only on the time and on the H^s(R) norm of the
initial data. This result is weaker than global well-posedness, as we have no
control on differences of solutions. Our proof is modeled on recent work by
Christ-Colliander-Tao and Koch-Tataru employing a version of Bourgain's Fourier
restriction spaces adapted to time intervals whose length depends on the
spatial frequency.
