Subexponential ultracontractivity and Lp-Lq functional calculus

We prove that if a symmetric submarkovian semigroup (Tt) t>0 satisfies an estimate of the form where theta is an increasing C 1 -diffeomorphism of [0,+infinity) 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<+infinity, and that a weak converse holds true if p=2. In the special case where theta(t)=Ct mi for small t and theta(t)=C' exp(ct v) for large t, mi>0, c>0, 0<v<1, one obtains a sharp and explicit result, which applies for instance to sublaplacians on solvable unimodular Lie groups with exponential growth.