Wavelets and Local Regularity

Abstract: We treat a number of topics related to wavelets and the description of local regularity properties of functions. First we prove several different exact characterizations of pointwise Hölder spaces, and use these characterizations to improve a theorem by Jaffard on Hölder exponents. Next we generalize the two-microlocal spaces of Bony to a Triebel-Lizorkin setting, and show that these new spaces can be characterized by wavelet coefficients. We also prove two alternative characterizations, formulated in terms of local norms and weighted spaces. Then we construct a new type of basis that combines the active segmentation properties of local trigonometric bases with the ability of wavelets to analyze a multitude of function spaces. Finally, we study the set of functions such that all the wavelet coefficients with respect to a compactly supported wavelet basis vanish, and show how this relates to characterization of homogeneous smoothness spaces and large-scale renormalization of homogeneous wavelet expansions.