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In this paper we study some applications of the L\'evy logarithmic Sobolevinequality to the study of the regularity of the solution of the fractal heatequation, i. e. the heat equation where the Laplacian is replaced with thefractional Laplacian. It is also used to the study of the asymptotic behaviourof the L\'evy-Ornstein-Uhlenbeck process.

Logarithmic Sobolev inequalities: regularizing effect of L\'evy operators and asymptotic convergence in the L\'evy-Fokker-Planck equation