Let \(f\in \Ran ๐_{(-\infty ,E]}(-\Delta)\) and \(\alpha\in \mathbb{N}^d\). Then
\[ \lVert \partial^\alpha f\rVert_{\ell^2(\mathbb{Z}^d)} \le (\sqrt{E})^{\lvert \alpha\rvert}\lVert f\rVert_{\ell^2(\mathbb{Z}^d)}. \]
Proof
Let \(f\in \Ran ๐_{(-\infty ,E}(-\Delta)\) and \(\alpha\in \mathbb{N}^d\) with \(\lvert \alpha\rvert=k\). Using integration by parts on \(\mathbb{Z}^d\) , we obtain
\begin{align*} \lVert \partial^\alpha f\rVert_{\ell^2(\mathbb{Z}^d)}^2 \le \sum_{\lvert \beta\rvert=k} \binom{k}{\beta} \lVert \partial^\beta f\rVert_{\ell^2(\mathbb{Z}^d)}^2 =\langle (-\Delta)^k f, f\rangle_{\ell^2(\mathbb{Z}^d)} \le E^k \lVert f\rVert_{\ell^2(\mathbb{Z}^d)}^{2}. \end{align*}
Remarks
- This implies \(\Ran ๐_{(-\infty ,E]}(-\Delta)\subseteq \cap_{k\in \mathbb{N}} H^k(\mathbb{Z}^d)\).