\(\mathrm {L}^p\) -Bounds for the \(\mathrm {H}^\infty \) -Calculus and Riesz Transform

We have seen in Corollary 14.15 that L has a bounded \(\mathrm {H}^\infty \) -calculus in \(\mathrm {L}^2\) of optimal angle. The first main result of this chapter will be \(\mathrm {L}^p\) -estimates for the \(\mathrm {H}^\infty \) -calculus in the optimal range \((p_{-}(L), p_{+}(L))\) .