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2014 | 222 | 2 | 123-155
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Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

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Let $f ∈ V_{r}(𝕋) ∪ 𝔐_{r}(𝕋)$, where, for 1 ≤ r < ∞, $V_{r}(𝕋)$ (resp., $𝔐_{r}(𝕋)$) denotes the class of functions (resp., bounded functions) g: 𝕋 → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on 𝕋 (resp., on the dyadic arcs of 𝕋). In the author's recent article [New York J. Math. 17 (2011)] it was shown that if 𝔛 is a super-reflexive space, and E(·): ℝ → 𝔅(𝔛) is the spectral decomposition of a trigonometrically well-bounded operator U ∈ 𝔅(𝔛), then over a suitable non-void open interval of r-values, the condition $f ∈ V_{r}(𝕋)$ implies that the Fourier series $∑_{k=-∞}^{∞} f̂(k)z^{k}U^{k}$ (z ∈ 𝕋) of the operator ergodic "Stieltjes convolution" $𝔖_{U}: 𝕋 → 𝔅(𝔛)$ expressed by $∫t_{[0,2π]}^{⊕} f(ze^{it})dE(t)$ converges at each z ∈ 𝕋 with respect to the strong operator topology. The present article extends the scope of this result by treating the Fourier series expansions of operator ergodic Stieltjes convolutions when, for a suitable interval of r-values, f is a continuous function that is merely assumed to lie in the broader (but less tractable) class $𝔐_{r}(𝕋)$.
Since it is known that there are a trigonometrically well-bounded operator U₀ acting on the Hilbert sequence space 𝔛 = ℓ²(ℕ) and a function f₀ ∈ 𝔐₁(𝕋) which cannot be integrated against the spectral decomposition of U₀, the present treatment of Fourier series expansions for operator ergodic convolutions is confined to a special class of trigonometrically well-bounded operators (specifically, the class of disjoint, modulus mean-bounded operators acting on $L^{p}(μ)$, where μ is an arbitrary sigma-finite measure, and 1 < p < ∞). The above-sketched results for operator-valued Stieltjes convolutions can be viewed as a single-operator transference machinery that is free from the power-boundedness requirements of traditional transference, and endows modern spectral theory and operator ergodic theory with the tools of Fourier analysis in the tradition of Hardy-Littlewood, J. Marcinkiewicz, N. Wiener, the (W. H., G. C., and L. C.) Young dynasty, and others. In particular, the results show the behind-the-scenes benefits of the operator ergodic Hilbert transform and its dual conjugates, and encompass the Fourier multiplier actions of $𝔐_{r}(𝕋)$-functions in the setting of $A_{p}$-weighted sequence spaces.
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  • Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, U.S.A.
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