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 \(L^2(\Omega ,w d\mu )\).

令\（（T_t）_ {t \ geqslant 0} \）是某个\（\ sigma \）有限度量空间\（（\ Omega，\ mu）\）上的马尔可夫（resp。submarkovian）半群。我们证明了它的负发生器甲已经有界\（H ^ {\ infty}（\西格玛_ \ THETA）\）上的加权空间演算\（L ^ 2（\欧米茄，WD \亩）\）只要权重\（w：\ Omega \ rightarrow（0，\ infty）\）具有由\（Q ^ A_2（w）= \ sup _ {t> 0} \ left \ | T_t（w）T_t \定义的有限特征left（w ^ {-1} \ right）\ right \ | _ {L ^ \ infty（\ Omega}} \）（由亚马尔科夫半群的变体表示）。必须对半群施加一些附加的技术条件，并讨论它们在示例中的有效性。在上述\（H ^ {\ infty} \）演算中，任何角度\（\ theta> \ frac {\ pi} {2} \）都是允许的，对于某些半群，某些\（\ theta = \ theta _w < \ frac {\ pi} {2} \）取决于\（Q ^ A_2（w）\）的大小。所述的规范\（H ^ {\ infty}（\西格玛_ \ THETA）\）演算是线性的\（Q ^ A_2 \）为特征\（\ THETA> \压裂{\ PI} {2} \ ）。我们还将讨论角度\（\ theta <\ frac {\ pi} {2} \）上的负结果。也就是说，我们证明在概率空间上存在一个马尔可夫半群，在\（L ^ 2（\ Omega，wd \ mu）\）上没有Hörmander函数演算的情况下，一个\（Q ^ A_2 \）权重w。