$$H^{\infty }$$ calculus for submarkovian semigroups on weighted $$L^2$$ spaces

Abstract

Let \((T_t)_{t \geqslant 0}\) be a markovian (resp. submarkovian) semigroup on some \(\sigma \)-finite measure space \((\Omega ,\mu )\). We prove that its negative generator A has a bounded \(H^{\infty }(\Sigma _\theta )\) calculus on the weighted space \(L^2(\Omega ,wd\mu )\) as long as the weight \(w : \Omega \rightarrow (0,\infty )\) has finite characteristic defined by \(Q^A_2(w) = \sup _{t > 0} \left\| T_t(w) T_t \left( w^{-1} \right) \right\| _{L^\infty (\Omega )}\) (resp. by a variant for submarkovian semigroups). Some additional technical conditions on the semigroup have to be imposed and their validity in examples is discussed. Any angle \(\theta > \frac{\pi }{2}\) is admissible in the above \(H^{\infty }\) calculus, and for some semigroups also certain \(\theta = \theta _w < \frac{\pi }{2}\) depending on the size of \(Q^A_2(w)\). The norm of the \(H^{\infty }(\Sigma _\theta )\) calculus is linear in the \(Q^A_2\) characteristic for \(\theta > \frac{\pi }{2}\). We also discuss negative results on angles \(\theta < \frac{\pi }{2}\). Namely we show that there is a markovian semigroup on a probability space and a \(Q^A_2\) weight w without Hörmander functional calculus on \(L^2(\Omega ,w d\mu )\).