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Imaginary powers of Laplace operators. (English) Zbl 0969.42007
Let $$L$$ be a second-order uniformly elliptic operator in divergence form on $$\mathbb{R}^d$$. The authors show that the following inequality holds $C_1(1+ |\alpha|)^{d/2}\leq \|L^{i\alpha}\|_{L^1\to L^{1,\infty}}\leq C_2(1+ |\alpha|)^{d/2}$ 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 $C_1(1+ |\alpha|)^{d/2}\leq \|(-\Delta_d)^{i\alpha} \|_{L^1\to L^{1,\infty}}\leq C_2(1+ |\alpha|)^{d/2} \log(1+ |\alpha|),$ which is obtained by applying the classical Hörmander multiplier theorem to $$(-\Delta_d)^{i\alpha}$$, where $$\Delta_d$$ is the standard Laplace operator on $$\mathbb{R}^d$$.

##### MSC:
 42B15 Multipliers for harmonic analysis in several variables 35P99 Spectral theory and eigenvalue problems for partial differential equations
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##### References:
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