Entropy dissipation estimates for the Landau equation with Coulomb potentials

We prove a lower bound for the entropy dissipation of the Landau equation with Coulomb potentials by a weighted Lebesgue norm $L_{-5\mathrm{/}3}^3$. In particular, we enhance the weight exponent from $-5$, which was established by Desvillettes, to $-5\mathrm{/}3$. Moreover, we prove that the weighted Lebesgue norm $L_{-5\mathrm{/}3}^3$ is optimal for both exponents.