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

Studia Mathematica (2014)

- Volume: 222, Issue: 2, page 123-155
- ISSN: 0039-3223

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topEarl Berkson. "Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series." Studia Mathematica 222.2 (2014): 123-155. <http://eudml.org/doc/286621>.

@article{EarlBerkson2014,

abstract = {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.},

author = {Earl Berkson},

journal = {Studia Mathematica},

keywords = {spectral decomposition; spectral integral; Marcinkiewicz multiplier of higher variation; trigonometrically well-bounded operator; disjoint operator; modulus mean-bounded operator; Stieltjes convolution; operator-valued Fourier series},

language = {eng},

number = {2},

pages = {123-155},

title = {Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series},

url = {http://eudml.org/doc/286621},

volume = {222},

year = {2014},

}

TY - JOUR

AU - Earl Berkson

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

JO - Studia Mathematica

PY - 2014

VL - 222

IS - 2

SP - 123

EP - 155

AB - 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.

LA - eng

KW - spectral decomposition; spectral integral; Marcinkiewicz multiplier of higher variation; trigonometrically well-bounded operator; disjoint operator; modulus mean-bounded operator; Stieltjes convolution; operator-valued Fourier series

UR - http://eudml.org/doc/286621

ER -

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