Improved refined bilinear estimates and well-posedness for generalized KdV type equations on $\mathbb{R}$

We study the Cauchy problem for one-dimensional dispersive equations posed on $\mathbb{R}$, under the hypotheses that the dispersive operator behaves, for high frequencies, as a Fourier multiplier by $i |\xi|^\alpha \xi$ with $1 \le \alpha\le 2$, and that the nonlinear term is of the form $\partial_x f(u)$ where $f$ is a real analytic function satisfying certain conditions. We prove the unconditional local well-posedness of the Cauchy problem in $H^s(\mathbb{R})$ for $s\ge \frac{5-2\alpha}{4}$ whenever $1\le \alpha<\frac{3}{2}$, and for $s>\frac{1}{2}$ whenever $\alpha\in [\frac{3}{2},2]$. This result is optimal in the case $\alpha\ge \frac{3}{2}$ in view of the restriction $s>\frac{1}{2}$ required for the continuous embedding $H^s(\mathbb{R}) \hookrightarrow L^\infty(\mathbb{R})$. The main novelty of this work, compared to our previous studies, is an improvement of the refined linear and bilinear estimates on $\mathbb{R}$. Our local well-posedness results enable us to derive global existence of solutions for $\alpha \in [\frac{5}{4},2]$.