Pointwise regularity associated with function spaces and multifractal analysis
Volume 72 / 2006
Abstract
The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) $f$ has a given pointwise regularity exponent $H$. This notion has many variants depending on the global hypotheses made on $f$; if $f$ locally belongs to a Banach space $E$, then a family of pointwise regularity spaces $C^\alpha_E (x_0)$ are constructed, leading to a notion of pointwise regularity with respect to $E$; the case $E = L^\infty$ corresponds to the usual Hölder regularity, and $E = L^p$ corresponds to the $T^p_\alpha (x_0)$ regularity of Calderón and Zygmund. We focus on the study of the spaces $T^p_\alpha (x_0)$; in particular, we give their characterization in terms of a wavelet basis and show their invariance under standard pseudodifferential operators of order 0.
