A Faber-Krahn inequality for wavelet transforms

Abstract

For some special window functions ${\psi }_{\beta }\in H^2({\mathbb{C}}^{+}),$ we prove that, over all sets $\Delta \subset {\mathbb{C}}^{+}$ of fixed hyperbolic measure $\nu (\Delta ),$ the ones over which the Wavelet transform $W_{\overline{{\psi }_{\beta }}}$ with window $\overline{{\psi }_{\beta }}$ concentrates optimally are exactly the discs with respect to the pseudohyperbolic metric of the upper half space. This answers a question raised by Abreu and Dörfler. Our techniques make use of a framework recently developed in a previous work by F. Nicola and the second author, but in the hyperbolic context induced by the dilation symmetry of the Wavelet transform. This leads us naturally to use a hyperbolic rearrangement function, as well as the hyperbolic isoperimetric inequality, in our analysis.

16 pages