Regular Article
Some Remarks on Ultracontractivity

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Abstract

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 ≔ eu. 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.

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