A Faber-Krahn inequality for Wavelet transforms

For some special window functions psi alpha is an element of H2(C+), we prove that, over all sets Delta subset of C+ of fixed hyperbolic measure nu(Delta), those for which the Wavelet transform W psi alpha with window psi alpha 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 Dorfler in Abreu and Dorfler (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.

Automated summary

For certain special window functions psi alpha belonging to H2(C+), we prove that the optimal concentrations of the Wavelet transform W psi alpha with window psi alpha, over all sets Delta subset of C+ of fixed hyperbolic measure nu(Delta), are exactly the discs corresponding to the pseudo-hyperbolic metric of the upper half space. This resolves a question raised by Abreu and Dorfler in Abreu and Dorfler (Inverse Problems 28 (2012) 16). Our approach utilizes 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 necessitates the use of a hyperbolic rearrangement function and the hyperbolic isoperimetric inequality in our analysis.