Mixed Wavelet Leaders Multifractal Formalism for Baire Generic Functions in a Product of Intersections of Hölder Spaces with Non-Continuous Besov Spaces

Abstract

In Ben Slimane (Mediterr J Math, 13(4):1513–1533 (2016)), the second author proved that, generically in the Baire category sense, pairs of functions in ${B_{t_{1}}^{s_{1},\infty}(\mathbb{R}^m) \times B_{t_{2}}^{s_{2},\infty}(\mathbb{R}^m) }$, for ${s_{1} > \frac{m}{t_{1}}}$ and ${s_{2} > \frac{m}{t_{2}}}$, satisfy a mixed multifractal formalism based on wavelet leaders. In this paper, we extend this validity on ${(B_{t_{1}}^{s_{1},\infty}(\mathbb{R}^m) \cap C^{\gamma_{1}}(\mathbb{R}^m)) \times (B_{t_{2}}^{s_{2},\infty}(\mathbb{R}^m) \cap C^{\gamma_{2}}(\mathbb{R}^m)}$, for ${0 < \gamma_{1} < s_{1} < \frac{m}{t_{1}}}$ and ${0 < \gamma_{2} < s_{2} < \frac{m}{t_{2}}}$. The main change is that the wavelet coefficients of the saturating function which generates the residual ${G_\delta}$ set are not everywhere large enough and do not coincide everywhere with the wavelet leaders. Nevertheless, the computation of the wavelet leaders is done everywhere and allows to deduce both mixed spectra and mixed scaling function.