Orthonormal bases of regular wavelets in spaces of homogeneous type

Details Orthonormal bases of regular wavelets in spaces of homogeneous type Orthonormal bases of regular wavelets in spaces of homogeneous type

2011-10-26

arxiv.org/abs/1110.5766v3

Mathematics

Classical Analysis and ODEs

We have made improvements to section 2 following the referees suggestions. In particular, it now contains full proof of former

Scientific paper

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 H\"older regularity. This is used to build an orthonormal basis of H\"older-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calder\'on 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.

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