Imaginary powers of Laplace operators (Q2706590)

Let \(L\) be a second-order uniformly elliptic operator in divergence form on \(\mathbb{R}^d\). The authors show that the following inequality holds NEWLINE\[NEWLINEC_1(1+ |\alpha|)^{d/2}\leq \|L^{i\alpha}\|_{L^1\to L^{1,\infty}}\leq C_2(1+ |\alpha|)^{d/2}NEWLINE\]NEWLINE for any \(\alpha\in\mathbb{R}\), where \(\|\cdot\|_{L^1\to L^{1,\infty}}\) is the weak type \((1,1)\) norm. This is an improvement of the inequality NEWLINE\[NEWLINEC_1(1+ |\alpha|)^{d/2}\leq \|(-\Delta_d)^{i\alpha} \|_{L^1\to L^{1,\infty}}\leq C_2(1+ |\alpha|)^{d/2} \log(1+ |\alpha|),NEWLINE\]NEWLINE which is obtained by applying the classical Hörmander multiplier theorem to \((-\Delta_d)^{i\alpha}\), where \(\Delta_d\) is the standard Laplace operator on \(\mathbb{R}^d\).