Subexponential ultracontractivity and L^p - L^q functional calculus

We prove that if a symmetric submarkovian semigroup (T_t)_t>0 satisfies an estimate of the form ∥T_tf∥_∞ ≤ φ(t)^-1 ∥f∥₁, ∀t > 0, ∀f ∈ L¹(M), where φ is an increasing C¹-diffeomorphism of [0, +∞) with subexponential growth, then a suitable function of its infinitesimal generator is bounded from L^p(M) to L^q(M) for 1 < p < q < + ∞, and that a weak converse holds true if p = 2. In the special case where φ(t) = Ct^μ for small t and φ (t) = C′ exp(ct^ν) for large t, μ > 0, c > 0, 0 < ν < 1, one obtains a sharp and explicit result, which applies for instance to sublaplacians on solvable unimodular Lie groups with exponential growth.