On the optimal local well-posedness of the wave kinetic equation in $L^r$

In this paper, we give a unified treatment of the local well-posedness for the wave kinetic equation in almost critical weighted $L^r$ spaces with $2\le r\le \mathrm{\infty }\mathrm{.}$ The proof builds on ideas from our earlier works \cite{AmLe24, AmLemain25}. Our approach is based solely on kinetic tools, with no appeal to Fourier theory.