Multifractal analysis studies the relationships between the pointwise and the global regularity of functions. The sharpest results require to introduce new function spaces which are either fitted to pointwise regularity (they generalize the $T^p_u (x)$ conditions of Calderon and Zygmund) or to global regularity (oscillation spaces, which generalize Besov spaces). We determine the wavelet characterizations of these spaces and we show how they allow to study some geometric properties of functions. Applications to the analysis of domains with fractal boundaries will be developed.
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