# $$A_{p}$$ A p weights and strong convergence of operator-valued Fourier series for Stieltjes convolutions of Marcinkiewicz functions

$$A_{p}$$ A p weights and strong convergence of operator-valued Fourier series for... Suppose that $$1<p<\infty$$ 1 < p < ∞ and let w be a bilateral weight sequence satisfying the discrete Muckenhoupt $$A_{p}$$ A p weight condition (in symbols, $$w\in A_{p}\left( \mathbb {Z}\right)$$ w ∈ A p Z ). In this setting, the left bilateral shift $$\mathcal {L}$$ L acting on the sequence space $$\ell ^{p}\left( w\right)$$ ℓ p w is known to be a trigonometrically well-bounded operator (that is, $$\mathcal {L}$$ L has a “unitary-like” spectral decomposition $$E:\mathbb {R}\rightarrow \mathfrak {B}\left( \ell ^{p}\left( w\right) \right)$$ E : R → B ℓ p w that consists of idempotent operators and has additional properties reminiscent of, but weaker than, those that would be inherited from a countably additive Borel spectral measure on $$\mathbb {R}$$ R ). In this framework we show that for any Marcinkiewicz multiplier $$\psi :\mathbb {T} \rightarrow \mathbb {C}$$ ψ : T → C , the operator-valued Fourier series of the Stieltjes convolution $$\psi \,\,\mathfrak {\circledast }$$ ψ ⊛ dE (which can be regarded as an operator-theoretic analogue of the Fourier series of $$\psi$$ ψ ) converges to $$\psi \,\,\mathfrak {\circledast }$$ ψ ⊛ dE, relative to the strong operator topology of $$\mathfrak {B}\left( \ell ^{p}\left( w\right) \right)$$ B ℓ p w , at each point of $$\mathbb {T}$$ T . In particular, the partial sums of the Fourier series of $$\psi$$ ψ are uniformly bounded in the Banach algebra of Fourier multipliers for $$\ell ^{p}\left( w\right)$$ ℓ p w . This outcome is made possible by the mutually advantageous interplay between spectral theory and real variable methods in harmonic analysis—the fundamental links being the operator ergodic discrete Hilbert transform and exrapolation of weights. These results are transferred to the framework of invertible, disjoint, modulus mean-bounded operators acting on $$L^{p}$$ L p spaces of sigma-finite measures. We note that in a still wider framework than that dealt with here—specifically the class of all super-reflexive Banach spaces—trigonometrically well-bounded operators and their spectral families of projections have recently been shown (Berkson, Studia Math 200:221–246, 2010) to constitute a disguised generalization of the dual conjugates of the operator ergodic discrete Hilbert transform generated by not necessarily power-bounded operators, in particular providing new transference vistas. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# $$A_{p}$$ A p weights and strong convergence of operator-valued Fourier series for Stieltjes convolutions of Marcinkiewicz functions

, Volume 21 (1) – May 23, 2016
27 pages

/lp/springer_journal/a-p-a-p-weights-and-strong-convergence-of-operator-valued-fourier-3xNxGo4REj
Publisher
Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0426-0
Publisher site
See Article on Publisher Site

### Abstract

Suppose that $$1<p<\infty$$ 1 < p < ∞ and let w be a bilateral weight sequence satisfying the discrete Muckenhoupt $$A_{p}$$ A p weight condition (in symbols, $$w\in A_{p}\left( \mathbb {Z}\right)$$ w ∈ A p Z ). In this setting, the left bilateral shift $$\mathcal {L}$$ L acting on the sequence space $$\ell ^{p}\left( w\right)$$ ℓ p w is known to be a trigonometrically well-bounded operator (that is, $$\mathcal {L}$$ L has a “unitary-like” spectral decomposition $$E:\mathbb {R}\rightarrow \mathfrak {B}\left( \ell ^{p}\left( w\right) \right)$$ E : R → B ℓ p w that consists of idempotent operators and has additional properties reminiscent of, but weaker than, those that would be inherited from a countably additive Borel spectral measure on $$\mathbb {R}$$ R ). In this framework we show that for any Marcinkiewicz multiplier $$\psi :\mathbb {T} \rightarrow \mathbb {C}$$ ψ : T → C , the operator-valued Fourier series of the Stieltjes convolution $$\psi \,\,\mathfrak {\circledast }$$ ψ ⊛ dE (which can be regarded as an operator-theoretic analogue of the Fourier series of $$\psi$$ ψ ) converges to $$\psi \,\,\mathfrak {\circledast }$$ ψ ⊛ dE, relative to the strong operator topology of $$\mathfrak {B}\left( \ell ^{p}\left( w\right) \right)$$ B ℓ p w , at each point of $$\mathbb {T}$$ T . In particular, the partial sums of the Fourier series of $$\psi$$ ψ are uniformly bounded in the Banach algebra of Fourier multipliers for $$\ell ^{p}\left( w\right)$$ ℓ p w . This outcome is made possible by the mutually advantageous interplay between spectral theory and real variable methods in harmonic analysis—the fundamental links being the operator ergodic discrete Hilbert transform and exrapolation of weights. These results are transferred to the framework of invertible, disjoint, modulus mean-bounded operators acting on $$L^{p}$$ L p spaces of sigma-finite measures. We note that in a still wider framework than that dealt with here—specifically the class of all super-reflexive Banach spaces—trigonometrically well-bounded operators and their spectral families of projections have recently been shown (Berkson, Studia Math 200:221–246, 2010) to constitute a disguised generalization of the dual conjugates of the operator ergodic discrete Hilbert transform generated by not necessarily power-bounded operators, in particular providing new transference vistas.

### Journal

PositivitySpringer Journals

Published: May 23, 2016

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