A Faber–Krahn inequality for Wavelet transforms

For some special window functions (Formula presented.), we prove that, over all sets (Formula presented.) of fixed hyperbolic measure (Formula presented.), those for which the Wavelet transform (Formula presented.) with window (Formula presented.) concentrates optimally are exactly the discs with respect to the pseudo-hyperbolic metric of the upper half space. This answers a question raised by Abreu and Dörfler in Abreu and Dörfler (Inverse Problems 28 (2012) 16). Our techniques make use of a framework recently developed by Nicola and Tilli in Nicola and Tilli (Invent. Math. 230 (2022) 1–30), 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.