Generalized Calderón conditions and regular orbit spaces

Hartmut Führ¹

¹ Lehrstuhl A für Mathematik RWTH Aachen 52056 Aachen, Germany

Colloquium Mathematicum, Tome 120 (2010) no. 1, pp. 103-126.

Résumé

The construction of generalized continuous wavelet transforms on locally compact abelian groups $A$ from quasi-regular representations of a semidirect product group $G = A \rtimes H$ acting on ${\rm L}^2(A)$ requires the existence of a square-integrable function whose Plancherel transform satisfies a Calderón-type resolution of the identity. The question then arises under what conditions such square-integrable functions exist.The existing literature on this subject leaves a gap between sufficient and necessary criteria. In this paper, we give a characterization in terms of the natural action of the dilation group $H$ on the character group of $A$. We first prove that a Calderón-type resolution of the identity gives rise to a decomposition of Plancherel measure of $A$ into measures on the dual orbits, and then show that the latter property is equivalent to regularity conditions on the orbit space of the dual action.Thus we obtain, for the first time, sharp necessary and sufficient criteria for the existence of a wavelet inversion formula associated to a quasi-regular representation. As a byproduct and special case of our results we deduce that discrete series subrepresentations of the quasi-regular representation correspond precisely to dual orbits with positive Plancherel measure and associated compact stabilizers. Only sufficiency of the conditions was previously known.

Détail

DOI : 10.4064/cm120-1-8

Keywords: construction generalized continuous wavelet transforms locally compact abelian groups quasi regular representations semidirect product group rtimes acting requires existence square integrable function whose plancherel transform satisfies calder n type resolution identity question arises under what conditions square integrable functions exist existing literature subject leaves gap between sufficient necessary criteria paper characterization terms natural action dilation group character group first prove calder n type resolution identity gives rise decomposition plancherel measure measures dual orbits latter property equivalent regularity conditions orbit space dual action obtain first time sharp necessary sufficient criteria existence wavelet inversion formula associated quasi regular representation byproduct special results deduce discrete series subrepresentations quasi regular representation correspond precisely dual orbits positive plancherel measure associated compact stabilizers only sufficiency conditions previously known