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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.
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.