Orthonormal bases of regular wavelets in spaces of homogeneous type

Pascal Auscher (1) , Tuomas Hytönen (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Department of Mathematics and Statistics

Adapting the recently developed randomized dyadic structures, we introduce the notion of spline function in geometrically doubling quasi-metric spaces. Such functions have interpolation and reproducing properties as the linear splines in Euclidean spaces. They also have Hölder regularity. This is used to build an orthonormal basis of Hölder-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calderón reproducing formula to study and develop function space theory and singular integrals. We discuss the examples of $L^p$ spaces, BMO and apply this to a proof of the $T(1)$ theorem. As no extra condition {(like 'reverse doubling', 'small boundary' of balls, etc.)} on the space of homogeneous type is required, our results extend a long line of works on the subject.

Mots clés: Geometrically doubling quasi-metric space; space of homogenous type; spline function; wavelet; orthonormal basis; dyadic cube; random geometric construction; T(1) theorem

Domaines: Analyse classique [math.CA]; Géométrie métrique [math.MG]; Analyse fonctionnelle [math.FA]

Pascal Auscher, Tuomas Hytönen. Orthonormal bases of regular wavelets in spaces of homogeneous type. Applied and Computational Harmonic Analysis, 2013, 34 (2), pp.266-296. ⟨hal-00635900v3⟩