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We study the well-posedness of the complex-valued modified Korteweg-de Vriesequation (mKdV) on the circle at low regularity. In our previous work (2019),we introduced the second renormalized mKdV equation, based on the conservationof momentum, which we proposed as the correct model to study the complex-valuedmKdV outside of $H^\frac12(\mathbb{T})$. Here, we employ the method introducedby Deng-Nahmod-Yue (2019) to prove local well-posedness of the secondrenormalized mKdV equation in the Fourier-Lebesgue spaces$\mathcal{F}L^{s,p}(\mathbb{T})$ for $s\geq \frac12$ and $1\leq p <\infty$. Asa byproduct of this well-posedness result, we show ill-posedness of thecomplex-valued mKdV without the second renormalization for initial data inthese Fourier-Lebesgue spaces with infinite momentum.

A refined well-posedness result for the modified KdV equation in the Fourier-Lebesgue spaces