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A general multiplier theorem. (English) Zbl 0760.42007
Let \({\mathcal L}\) be the generator of a semigroup \((T_ t)_{t>0}\), of class \(C_ 0\) on all the spaces \(L^ p(M)\), such that each \(T_ t\) is a contraction on every \(L^ p(M)\) and is self-adjoint on \(L^ 2(M)\). If \(m:\mathbb{R}^ +\to\mathbb{C}\) is a bounded Borel measurable function, \(m({\mathcal L})\) can be defined as an operator on \(L^ 2(M)\) by spectral theory. This paper is concerned with finding conditions on \(m\) which ensure that \(m({\mathcal L})\) extends to a bounded operator on \(L^ p(M)\), where \(p\neq 2\). The results obtained depend on the behaviour of the operator norm \(\||{\mathcal L}^{iu}\|_ p\) of the operator \({\mathcal L}^{iu}\) on \(L^ p(M)\), as \(u\) varies over \(\mathbb{R}\).

MSC:
42B15 Multipliers for harmonic analysis in several variables
47A60 Functional calculus for linear operators
47D03 Groups and semigroups of linear operators
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