Sharp estimates for imaginary powers of discrete Laplacians

Let ${\mathrm{\Delta }}_{\mathrm{d}}$ be the nonnegative discrete Laplacian on ${\mathbb{Z}}^d$. We prove that, for $0<p\le 1$, $$\mathrm{\parallel }{\mathrm{\Delta }}_{\mathrm{d}}^{i\gamma }{\mathrm{\parallel }}_{H_{{\mathrm{\Delta }}_{\mathrm{d}}}^p\to H_{{\mathrm{\Delta }}_{\mathrm{d}}}^p}\simeq (1+\mathrm{\mid }\gamma \mathrm{\mid }{)}^{d(1\mathrm{/}p-1\mathrm{/}2)},$$ and, for $1<p<\mathrm{\infty }$, $$\mathrm{\parallel }{\mathrm{\Delta }}_{\mathrm{d}}^{i\gamma }{\mathrm{\parallel }}_{{\mathrm{\ell }}^p\to {\mathrm{\ell }}^p}\simeq (1+\mathrm{\mid }\gamma \mathrm{\mid }{)}^{d\mathrm{\mid }1\mathrm{/}p-1\mathrm{/}2\mathrm{\mid }}\mathrm{.}$$ The estimates are sharp with respect to the exponent of $1+\mathrm{\mid }\gamma \mathrm{\mid }$. We also explain why the general spectral multiplier theorem and the usual Calderón–Zygmund approach do not give the sharp power.