Fractional very fast diffusion equations in Lebesgue spaces: uniqueness and smoothing effects

We investigate forward and backward smoothing effects in Lebesgue spaces $L^p$ and $\mathcal{M}^p:=L^{p,\infty}$ for the Cauchy problem associated to the nonlinear and nonlocal fractional diffusion equation $\partial_t u+(-Δ)^{\frac\sigma2}|u|^{m-1}u=0$ in $\mathbb{R}^N$, $0<σ<2$, in the very fast r