\(L^p\)-gradient estimates of symmetric Markov semigroups for \(1<p\leq2\)
The authors consider a strongly continuous, positive Markov semigroup \((T_t)_{t>0}\) on \(L^2\). A Dirichlet form is assumed to be associated with this semigroup and \(L\) denotes the Dirichlet operator. Moreover, it is supposed that the semigroup admits a carré du champ operator \(\Gamma \). The main result asserts the following \(L^p\)-gradient estimates (for \(1 < p < 2\), \(q > 2\) and \(\frac 12 < \alpha < 1\)): \[ \| \Gamma^{1/2} (T_t f)\| _p \leq \frac{C_p}{\sqrt{t}}\| f\| _p,\quad \| \Gamma^{1/2}((I - L)^{-\alpha}f)\| _p \leq C_{p,\alpha} \| f\| _p, \] \[ c_{q,\alpha} \| f\| _q \leq \| \Gamma^{1/2}((I - L)^{-(1-\alpha )}f)\| _q + \| ( I - L)^{-(1-\alpha )}f\| _q. \]
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