Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series
Volume 222 / 2014
Abstract
Let $f\in V_{r}( \mathbb {T}) \cup \mathfrak {M}_{r}( \mathbb {T}) $, where, for $1\leq r<\infty $, $V_{r}( \mathbb {T}) $ (resp., $\mathfrak {M}_{r}( \mathbb {T}) $) denotes the class of functions (resp., bounded functions) $g:{}\mathbb {T\rightarrow C}$ such that $g$ has bounded $r$-variation (resp., uniformly bounded $r$-variations) on $\mathbb {T}$ (resp., on the dyadic arcs of $\mathbb {T}$). In the author's recent article [New York J. Math. 17 (2011)] it was shown that if $\mathfrak {X}$ is a super-reflexive space, and $E( \cdot ) :\mathbb {R}\rightarrow \mathfrak {B}( \mathfrak {X}) $ is the spectral decomposition of a trigonometrically well-bounded operator $U\in \mathfrak {B}( \mathfrak {X}) $, then over a suitable non-void open interval of $r$-values, the condition $f\in V_{r}( \mathbb {T}) $ implies that the Fourier series $\sum _{k=-\infty }^{\infty }\widehat{f}( k) z^{k}U^{k}$ ($z\in \mathbb {T}$) of the operator ergodic “Stieltjes convolution” $\mathfrak {S}_{U}:\mathbb {T}\rightarrow \mathfrak {B}( \mathfrak {X}) $ expressed by $\int _{[ 0,2\pi ] }^{\oplus }f( ze^{it}) \,dE( t) $ converges at each $z\in \mathbb {T}$ 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 $\mathfrak {M}_{r}( \mathbb {T}) $.
Since it is known that there are a trigonometrically well-bounded operator $U_{0}$ acting on the Hilbert sequence space $\mathfrak {X}=\ell ^{2}( \mathbb {N}) $ and a function $f_{0}\in \mathfrak {M}_{1}( \mathbb {T}) $ which cannot be integrated against the spectral decomposition of $U_{0}$, 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}( \mu ) $, where $\mu $ is an arbitrary sigma-finite measure, and $1< p< \infty $). 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 $\mathfrak {M}_{r}( \mathbb {T}) $-functions in the setting of $A_{p}$-weighted sequence spaces.
