Generalized spaces of pointwise regularity: toward a general framework for the WLM

In this work we generalize the spaces ${T}_{u}^{p}$ introduced by Calderón and Zygmund using pointwise conditions emanating from generalized Besov spaces. We give conditions binding the functions belonging to these spaces and their wavelet coefficients. Next, we propose a multifractal formalism based on such spaces which generalizes the so-called wavelet leaders method and show that it is satisfied on a prevalent set.