  In this work we generalize the spaces T^{p}_{u} introduced by Calder\'on and
Zygmund using a pointwise version of conditions defining the generalized Besov
spaces and give conditions binding the functions belonging to these spaces and
the wavelet coefficients of such functions. Next, we propose a multifractal
formalism based on the new spaces wich generalize the so-called wavelet leaders
method and show that it is satisfied on a prevalent set.
