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Imaginary powers of Laplace operators

Authors:	Adam Sikora James Wright

Institution:	Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia (or University of Wroclaw, KBN 2 P03A 058 14, Poland) ; School of Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia

Abstract:	We show that if is a second-order uniformly elliptic operator in divergence form on $\mathbf{R}^d$ , then $C_1(1+\vert\alpha\vert)^{d/2} \le \Vert L^{i\alpha}\Vert _{L^1 \to L^{1,\infty}} \le C_2 (1+\vert\alpha\vert)^{d/2}$ . We also prove that the upper bounds remain true for any operator with the finite speed propagation property.

Keywords:	Spectral multiplier imaginary powers

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