L p-gradient Estimates of Symmetric Markov Semigroups for 1 < p . 2

For 1 < p . 2, an L p-gradient estimate for a symmetric Markov semigroup is derived in a general framework, i. e. ...1/2(Ttf)..p.Cpt..f.p, where . is a carré du champ operator. As a simple application we prove that .1/2((I-L)-.) is a bounded operator from L p to L p provided that 1 < p < 2 and 12<.<1. For any 1 < p < 2, q > 2 and 12<.<1, there exist two positive constants c q,.,C p,. such that .Df. p . C p,..(I - L). f. p , c q,..(I - L)1-. f. q . .Df. q + .f.q, where D is the Malliavin gradient ([2]) and L the Ornstein.Uhlenbeck operator