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Operator means and spectral integration of Fourier multipliers. (English) Zbl 1056.42008
A bounded operator $$U$$ on $$L^p(\mu)$$ ($$U\in B(L^p(\mu))$$) is said to be trigonometrically well-bounded if $$U$$ has a spectral representation $$U=\int_{0-}^{2\pi} e^{it} \,dE(t)$$, where $$E:{\mathbb R}\rightarrow B(L^p(\mu))$$ is an idempotent-valued function called (after suitable normalization) the spectral decomposition of $$U$$. Now, if $$\Psi\in BV({\mathbb T})$$, the integral $$\int_{0-}^{2\pi} \Psi(e^{it}) dE(t)$$ exists strongly as a Riemann-Stieltjes integral and the mapping $\Psi \longrightarrow \Psi(U)=\Psi(1) E(0)+ \int_{0-}^{2\pi} \Psi(e^{it}) dE(t)\tag{1}$ is a norm continuous representation of the Banach algebra $$BV({\mathbb T})$$ in $$B(L^p(\mu))$$. The goal of this paper, which is a continuation of a previous paper of the authors [Ill. J. Math. 43, No. 3, 500–519 (1999; Zbl 0930.42004)] is to find sufficient conditions on $$T\in B(L^p(\mu))$$ to ensure that $$T$$ will be trigonometrically well-bounded with a spectral decomposition that extends the spectral integration in (1) from $$BV({\mathbb T})$$ to the Marcinkiewicz $$q$$-class of Fourier multipliers $$M_q({\mathbb T})$$. The main result of this paper asserts that if $$T$$ is a bounded, invertible and separation-preserving mapping on $$L^p(\mu)$$ such that $$T$$ is mean$$_2$$-bounded, that is, for every $$f\in L^p(\mu)$$ and every $$N\in\mathbb N$$, ${1\over (2N+1)^{1/2}} \bigg\| \bigg\{ \sum_{k=-N}^N | T^k f| ^2\bigg\}^{1/2}\bigg\|_{L^p(\mu)}\leq A | | f| | _{L^p(\mu)}$ the result follows.
##### MSC:
 42A45 Multipliers in one variable harmonic analysis 42B15 Multipliers for harmonic analysis in several variables 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B40 Spectral operators, decomposable operators, well-bounded operators, etc.