Gagliardo-Nirenberg inequalities for semigroups of operators and applications (Q1202899)

Let \(-A\) be the generator of a bounded holomorphic semigroup on \(L^ p\), \(1\leq p<\infty\), and let \(n\in (0,\infty)\). The author proves equivalence of (i) \(\| T(t)\|_{{\mathcal L}(L^ p,L^ \infty)}\leq \text{const.}t^{-n/2p}\) (\(t>0\)); and (ii) \(\| f\|_ \infty\leq C\| f\|_ p^{1-n/\alpha p} \| A^{\alpha/2} f\|_ p^{n/\alpha p}\) \((f\in L^ p\cap D(A^{\alpha/2}))\) where \(\alpha p>n\). In the case where \(-A\) is the Laplacian, property (ii) is close to an inequality shown by Gagliardo and Nirenberg. Another characterization of (i) in terms of an inequality of type Nash is obtained. Also exponential decay \(\| T(t)\|_{{\mathcal L}(L^ p,L^ \infty)} \leq ct^{-n/2p} e^{-\lambda t}\) (\(t>0\)) is considered. Finally, discrete versions are given; i.e. a power bounded operator \(T\) is considered which is analytical in the sense that \(k\|(I-T)T^ k\|\leq c\), \(k\in\mathbb{N}\). Condition (i) is then replaced by \[ \| T^ k\|_{{\mathcal L}(L^ p,L^ \infty)} \leq \text{const.}k^{- n/2p} \qquad (k\in\mathbb{N}).\leqno(i)' \] {}.