An extrapolation theorem for the \(H^{\infty }\) calculus on \(L^p(\Omega ;X)\). (English) Zbl 1212.47009

Summary: Let \((A_p)_{1<p<\infty }\) be a consistent family of sectorial operators on \(L^p(\Omega ;X)\), where \(\Omega \) is a homogeneous space with doubling property and \(X\) is a Banach space having the Radon-Nykodým property. If \(A_{p_0}\) has a bounded \(H^{\infty }\) calculus for some \(1<p_0<\infty \) and the resolvent or the semigroup generated by \(A_{p_0}\) satisfies a Poisson estimate, then it is proved that \(A_p\) has a bounded \(H^{\infty }\) calculus for all \(1<p\leq p_0\), and even for \(1<p<\infty \) if \(X\) is reflexive. In order to do so, the Calderón-Zygmund decomposition is generalized to the vector-valued setting.


47A60 Functional calculus for linear operators
43A85 Harmonic analysis on homogeneous spaces