Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem: 1. Rate of convergence

Let f(., t) be the probability density function which represents the solution of Kac's equation at time t, with initial data f_0, and let g_? be the Gaussian density with zero mean and variance ?^2, ?^22 being the value of the second moment of f_0. This is the first study which proves that the total variation distance between f(?, t) and g? goes to zero, as t->+?, with an exponential rate equal to -1/4. In the present paper, this fact is proved on the sole assumption that f_0 has finite fourth moment and its Fourier transform ?_0 satisfies |?_0(?)|=o(|?|-p) as |?|->+?, for some p>0. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.