S1-bounded Fourier multipliers on H1(ℝ) and functional calculus for semigroups

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, \(\widehat {T(h)} = m\hat h\) for all h ∈ H¹(ℝ). Let S¹ be the Banach space of all trace class operators on ℓ². We show that T admits a bounded tensor extension \(T\overline \otimes {I_{{S_1}}}:{H^1}(\mathbb{R};{S^1}) \to {H^1}(\mathbb{R};{S^1})\) 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) ∈ ℝ ₊ ² . Such Fourier multipliers are called S¹-bounded and we let \({{\cal M}_{{S^1}}}({H^1}(\mathbb{R}))\) 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 \({{\cal A}_{0,{S^1}}}({\mathbb{C}_ +})\) of bounded analytic functions on ℂ₊ = {z ∈ ℂ:Re(z) > 0} and show that its dual space coincides with \({{\cal M}_{{S^1}}}({H^1}(\mathbb{R}))\) . Second, given any bounded C₀-semigroup (T_t)_t≥0 on Hilbert space, and any b ∈ L¹(ℝ₊), we establish an estimate \(||\int_0^\infty {b(t)} {T_t}dt||\,\, \lesssim\,\,||{L_b}|{|_{{{\cal A}_{0,{S^1}}}}}\) , where L_b denotes the Laplace transform of b. This improves previous functional calculus estimates recently obtained by the first two authors.