We investigate several topics related to the local behavior of functions:
pointwise Holder regularity, local scaling invariance and very oscillatory
"chirp-like" behaviors. Our main tool is to relate these notions to two-
microlocal conditions which are defined either on the Littlewood-Paley
decomposition or on the wavelet transform. We give characterizations
and the main properties of these two-microlocal spaces and we give sev-
eral applications, such as bounds on the dimension of the set of Holder
singularities of a function, Sobolev regularity of trace functions, and
chirp expansions of specific functions.
AMS Classification: 26A16, 26A30, 26A69, 26B35, 42A16
Key words: Wavelets, Littlewood-Paley decomposition, Two-microlo-
calization, Modulus of continuity, Hausdorff dimension, Chirps, Selfsim-
ilarity, Riemann function, Trace functions.
Received by the editor July 21, 1993, and in revised form June 5, 1995.