S​1​​-bounded Fourier multipliers on ​​H​​1​(R​​)​​ and functional calculus for semigroups​​​

Loris Arnold, Christian Le Merdy*, Safoura Zadeh

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

Let T : H1(R) → H1(R) be a bounded Fourier multiplier on the analytic Hardy space H1(R) ⊂ L1(R) and let m ∈ L∞(R+) be its symbol, that is, T(h) = mh for all h ∈ H1(R). Let S1 be the Banach space of all trace class operators on 2. We show that T admits a bounded tensor extension T⊗IS1 : H1(R; S1) → H1(R; S1) if and only if there exist a Hilbert space H and two functions α, β ∈ L∞(R+;H) such that m(s+t) = α(t), β(s)H for almost every (s,t) ∈ R2+. Such Fourier multipliers are called S1-bounded and we letMS1 (H1(R)) denote the Banach space of all S1-bounded Fourier multipliers. Next we apply this result to functional calculus estimates, in two steps. First we introduce a new Banach algebra A0,S1 (C+) of bounded analytic functions on C+ = {z ∈ C : Re(z) > 0} and show that its dual space coincides with MS1 (H1(R)). Second, given any bounded C0-semigroup (Tt)t≥0 on Hilbert space, and any b ∈ L1(R+), we establish an estimate || ∞0 b(t)Tt dt LbA0,S1 , where Lb denotes the Laplace transform of b. This improves previous functional calculus estimates recently obtained by the first two authors.
Original languageEnglish
Number of pages37
JournalJournal d'Analyse Mathématique
Early online date12 Dec 2023
DOIs
Publication statusE-pub ahead of print - 12 Dec 2023

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