A Faber-Krahn inequality for wavelet transforms

For some special window functions $ψ_β \in H^2(\mathbb{C}^+),$ we prove that, over all sets $Δ\subset \mathbb{C}^+$ of fixed hyperbolic measure $ν(Δ),$ the ones over which the Wavelet transform $W_{\overline{ψ_β}}$ with window $\overline{ψ_β}$ 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.

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