Kavian, Otared; Kerkyacharian, Gérard; Roynette, Bernard Some remarks on ultracontractivity. (Quelques remarques sur l’ultracontractivité.) (French) Zbl 0807.47027 J. Funct. Anal. 111, No. 1, 155-196 (1993). The authors study the semigroup \(T^ \mu_ t\) generated by the operator \({1\over 2}(\Delta f- \nabla u\cdot\nabla f)\) on the Lebesgue space \(L^ 2(\mathbb{R}^ d;\mu)\) with the measure \(\mu:= e^{-u}\). They prove, via different methods using probabilistic techniques or PDE arguments, that \(T^ \mu_ t\) is ultracontractive, i.e., for \(t>0\) it maps \(L^ 1(\mu)\) into \(L^ \infty\), 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, they consider the analogous properties of the semigroup generated by the fractional powers of the above operator. Cited in 20 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations 47F05 General theory of partial differential operators 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47A60 Functional calculus for linear operators Keywords:semigroup; probabilistic techniques; PDE arguments; ultracontractive; fractional powers PDFBibTeX XMLCite \textit{O. Kavian} et al., J. Funct. Anal. 111, No. 1, 155--196 (1993; Zbl 0807.47027) Full Text: DOI