Some remarks on ultracontractivity
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.
scientific article; zbMATH DE number 220690