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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
##### Keywords:
multiplier operator; Mellin transform; $$g$$-function; semigroup
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##### References:
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