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

Earl Berkson

Studia Mathematica, Tome 222 (2014) no. 2, p. 123.

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 = - ∞ ∞ f ̂ ( k ) z k U k (z ∈ ) of the operator ergodic “Stieltjes convolution” U : → ( ) expressed by ∫ t [ 0 , 2 π ] ⊕ f ( z e i t ) d E ( 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

Classification : 47A35, 47B40, 46E30, 42A45, 40E05, 26A45
Mots-clés : 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