Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

Earl Berkson

Studia Mathematica, Tome 223 (2014), p. 123-155 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $f ∈ V_{r}() ∪ _{r}()$, where, for 1 ≤ r < ∞, $V_{r}()$ (resp., $_{r}()$) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition $f ∈ V_{r}()$ implies that the Fourier series $∑_{k=-∞}^{∞} f̂(k)z^{k}U^{k}$ (z ∈ ) of the operator ergodic “Stieltjes convolution” $_{U}: → ()$ expressed by $∫t_{[0,2π]}^{⊕} f(ze^{it})dE(t)$ converges at each z ∈ 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 $_{r}()$. Since it is known that there are a trigonometrically well-bounded operator U₀ acting on the Hilbert sequence space = ℓ²(ℕ) and a function f₀ ∈ ₁() 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 $_{r}()$-functions in the setting of $A_{p}$-weighted sequence spaces.

Détail

Publié le : 2014-01-01

Zbl 1322.47035

EUDML-ID : urn:eudml:doc:286621