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Let $T\colon H^1({\mathbb R})\to H^1({\mathbb R})$ be a bounded Fouriermultiplier on the analytic Hardy space $H^1({\mathbb R})\subset L^1({\mathbbR})$ and let $m\in L^\infty({\mathbb R}_+)$ be its symbol, that is,$\widehat{T(h)}=m\widehat{h}$ for all $h\in H^1({\mathbb R})$.Let $S^1$ be theBanach space of all trace class operators on $\ell^2$. We show that $T$ admitsa bounded tensor extension $T\overline{\otimes} I_{S_1}\colon H^1({\mathbbR};S^1) \to H^1({\mathbb R};S^1)$ if and only if there exist a Hilbert space$\mathcal H$ and two functions $\alpha, \beta \in L^\infty({\mathbbR}_+;{\mathcal H})$ such that $m(s+t) =\langle\alpha(t),\beta(s)\rangle_{\mathcal H}$ for almost every$(s,t)\in{\mathbb R}_+^2$. Such Fourier multipliers arecalled $S^1$-bounded andwe let ${\mathcal M}_{S^1}(H^1({\mathbb R}))$ denote the Banach space of all$S^1$-bounded Fourier multipliers. Next we apply this result to functionalcalculus estimates, in two steps. First we introduce a new Banach algebra${\mathcal A}_{0,S^1}({\mathbb C}_+)$ of bounded analytic functions on${\mathbb C}_+ =\bigl\{z\in{\mathbb C}\, :\, {\rm Re}(z)>0\bigr\}$ and showthat its dual space coincides with ${\mathcal M}_{S^1}(H^1({\mathbb R}))$.Second, given any bounded $C_0$-semigroup $(T_t)_{t\geq 0}$ on Hilbert space,and any $b\in L^1({\mathbb R}_+)$, we establish an estimate$\bigl\Vert\int_0^\infty b(t) T_t\, dt\bigr\Vert\lesssim \VertL_b\Vert_{{\mathcal A}_{0,S^1}({\mathbb R})}$, where $L_b$ denotes the Laplacetransform of $b$. This improves previous functional calculus estimates recentlyobtained by the first two authors.

$S^1$-bounded Fourier multipliers on $H^1({\mathbb R})$ and functional calculus for semigroups