LOGARITHMIC SOBOLEV INEQUALITIES ON LOOP-GROUPS

Let G be a compact Lie group. Denote by m the Brownian bridge measure on the loop group Y ≡ {g ε{lunate} C([0, 1]; G): g(0) = g(1) = e }. The finite energy subgroup of Y determines in a natural way a gradient operation for functions on Y. The following logarithmic Sobolev inequality is proven, ∝ f2, log |f|dm ≤ ∝ {|gradf(y)|2 + V(y) f (y)2} dm + ∥f∥2log∥f∥wherein ∥f∥ denotes the L2(m) norm and V is a potential which is quadratic in the associated Lie algebra valued Brownian motion. The inequality is derived by a method of inheritance from the known inequality for the G valued Brownian motion. © 1991.