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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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