A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier–Lebesgue Spaces

Abstract We study the well-posedness of complex-valued modified Korteweg-de Vries equation (mKdV) on circle at low regularity. In our previous work (2021), we introduced second renormalized mKdV equation, based conservation momentum, which proposed as correct model to outside $$H^\frac{1}{2}({\mathbb {T}})$$ H 1 2 ( T ) . Here, employ method by Deng et al. (Commun Math Phys 384(1):1061–1107, 2021) prove local in Fourier–Lebesgue spaces $${\mathcal {F}}L^{s,p}({\mathbb F L s , p for $$s\ge \frac{1}{2}$$ ≥ and $$1\le p <\infty $$ ≤ < ∞ As a byproduct this result, show ill-posedness without renormalization initial data these with infinite momentum.