Imaginary powers of Laplace operators

Abstract

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

Locations

• Proceedings of the American Mathematical Society - View - PDF