Ultracontractivity and embedding into \(L^\infty\) (Q996008)

This paper provides a generalization to the Galiardo--Nirenberg inequality for selfadjoint operators with ultracontractive semigroups. More precisely, let \((X,\mu)\) be a \(\sigma\)-finite measure space and \(A\) a self-adjoint operator on \(L^2(\mu)\) with \[ \| e^{-tA}\| _{2\to \infty}\leq e^{m(t)},\;t>0 \] for some positive function \(m\) on \((0,\infty)\), where \(\|\cdot\|_{2\to \infty}\) is the operator norm from \(L^2(\mu)\) to \(L^\infty(\mu)\). An explicit correspondence is provided for \(m\) on the one hand and positive sequence \(\phi_n\) on the other hand to ensure that any \(g\in D(A^n)\) for all \(n\geq 1\) with \[ \limsup_{n\to\infty} \frac{\| A^n g\| _2^{1/2}}{\phi_n}\leq 1 \] implies that \(g\in L^\infty(\mu)\).