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

Let $T\colon H^1({\mathbb R})\to H^1({\mathbb R})$ be a bounded Fourier multiplier on the analytic Hardy space $H^1({\mathbb R})\subset L^1({\mathbb R})$ 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 the Banach space of all trace class operators on $\ell^2$. We show that $T$ admits a bounded tensor extension $T\overline{\otimes} I_{S_1}\colon H^1({\mathbb R};S^1) \to H^1({\mathbb R};S^1)$ if and only if there exist a Hilbert space $\mathcal H$ and two functions $\alpha, \beta \in L^\infty({\mathbb R}_+;{\mathcal H})$ 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 and we 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 functional calculus 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 show that 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 $\bigl\Vert\int_0^\infty b(t) T_t\, dt\bigr\Vert\lesssim \Vert L_b\Vert_{{\mathcal A}_{0,S^1}({\mathbb R})}$, where $L_b$ denotes the Laplace transform of $b$. This improves previous functional calculus estimates recently obtained by the first two authors.