Strong convergence in the weighted setting of operator-valued Fourier series defined by the Marcinkiewicz multipliers

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• 作者：Earl Berkson ^{berkson@math.uiuc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Marcinkiewicz class ; Fourier multiplier ; Fourier series ; Modulus mean-bounded operator ; Shift operator ; ApAp weight sequence
• 刊名：Comptes Rendus Mathematique
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：354
• 期：2
• 页码：181-184
• 全文大小：221 K

文摘

Suppose that 1<p<∞$1<p<\infty$ and let w   be a bilateral weight sequence satisfying the discrete Muckenhoupt A_p$A_p$ weight condition. We show that every Marcinkiewicz multiplier ψ:T→C$\psi :\mathbb{T}\to \mathbb{C}$ has an associated operator-valued Fourier series which serves as an analogue in B(ℓ^p(w))$\mathfrak{B}\left({\ell }^p\left(w\right)\right)$ of the usual Fourier series of ψ, and this operator-valued Fourier series is everywhere convergent in the strong operator topology. In particular, we deduce that the partial sums of the usual Fourier series of ψ   are uniformly bounded in the Banach algebra of Fourier multipliers for ℓ^p(w)${\ell }^p\left(w\right)$. These results transfer to the framework of invertible, modulus mean-bounded operators acting on L^p$L^p$ spaces of sigma-finite measures.