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. ∥Gamma ^1/2 (T_t f)∥_p ≤ C_p/ √t∥f∥_p, where Γ is a carré du champ operator. As a simple application we prove that Γ^1/2((I-L) ^-α) is a bounded operator from L ^p to L ^p provided that 1 < p < 2 and 1/2 < α < 1. For any 1 < p < 2, q > 2 and 1/2 < α < 1, there exist two positive constants c _q,α,C _p,α such that equation presented, where D is the Malliavin gradient ([2]) and L the Ornstein-Uhlenbeck operator.