Some Remarks on Ultracontractivity

We study the semi-group Tμt generated by the operator 12(Δƒ − ∇u·∇ƒ) on the Lebesgue space L2(Rd; μ) with the measure μ ≔ e−u. We prove, via different methods using probabilistic techniques or PDE arguments, that Tμt is ultracontractive, i.e., for t > 0 it maps L1(μ) into L∞ when the function u satisfies a growth condition at infinity, which is essentially (for instance when the dimension d = 1) the integrability of 1/u′ at infinity. Also, we consider the analogous properties of the semi-group generated by the fractional powers of the above operator.