Sharp decay thresholds in weighted $L^{\mathrm{\infty }}$ for wave kinetic equations with power-law dispersion

We study four-wave kinetic equations in space dimension three with power-law dispersion $\omega (p)=\mathrm{\mid }p{\mathrm{\mid }}^a$ and collision kernels with high-frequency growth measured by $\beta$. In weighted $L^{\mathrm{\infty }}$ spaces, we identify the sharp decay threshold $$s_c=4\beta +3-\frac{a}{2}\mathrm{.}$$ For $s>s_c$, we prove local well-posedness by establishing trilinear bounds for the full gain-loss collision operator. For $s<s_c$, we prove ill-posedness by constructing data concentrated near a high-low-low-high resonant configuration. This threshold captures the balance between the high-frequency strength of the kernel and the geometry of the resonant manifold. The proof also shows that gain-loss cancellations are essential in the most delicate regimes.