In this paper we will prove the logarithmic Sobolev inequality on free loop groups for various heat kernel measures which P. Malliavin (1989, 1991, in "Diffusion Process and Related Problems in Analysis (M. A. Pinsley, Ed.), Vol. I, Birkhäuser, Basel) constructed. Those measures are associated with the Sobolev spaces of order s (s>1/2) of the free loops in the Lie algebra. We will equipp the free loop groups with those metrics and will show that a formula of Weitzenböck type holds, which enables us to apply the method of Driver and Lohrenz (1996, J. Funct. Anal.146, 381-448).
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