Optimal Weighted Smoothing and Asymptotics of Ancient Solutions for Fast Diffusion Equations

We establish sharp weighted smoothing estimates for limit solutions to the Cauchy-Dirichlet problem for the fast diffusion equation on smooth bounded domains. We demonstrate that the critical exponent governing these estimates coincides with the classical Brezis–Turner exponent known in the theory of semilinear elliptic equations. As a primary application, we derive improved global Harnack inequalities and describe asymptotic behavior of positive ancient solutions.