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. $$ {\left\| {\Gamma ^{{1/2}} {\left( {T_{t} f} \right)}} \right\|}_{p} \leqslant \frac{{C_{p} }} {{{\sqrt t }}}{\left\| f \right\|}_{p} $$ , where Γ is a carré du champ operator. As a simple application we prove that Γ1/2(( I- L)-α) is a bounded operator from L p to L p provided that 1 < p < 2 and $$ \frac{1} {2} < \alpha < 1 $$ . For any 1 < p < 2, q > 2 and $$ \frac{1} {2} < \alpha < 1 $$ , there exist two positive constants c q,α, C p,α such that ∥ Df∥ p ≤ C p,α∥( I - L)α f∥ p , c q,α∥( I - L)1-α f∥ q ≤ ∥ Df∥ q + ∥ f∥ q, where D is the Malliavin gradient ([2]) and L the Ornstein–Uhlenbeck operator. [ABSTRACT FROM AUTHOR]