Mixed Wavelet Leaders Multifractal Formalism in a Product of Critical Besov Spaces

Abstract

In this paper, we will prove (resp. study) the Baire generic validity of the upper-Hölder (resp. iso-Hölder) mixed wavelet leaders multifractal formalism on a product of two critical Besov spaces $B_{t_{1}}^{\frac{m}{t_{1}},q_{1}}(\mathbb {R}^m) \times B_{t_{2}}^{\frac{m}{t_{2}},q_{2}}(\mathbb {R}^m)$, for $t_1,t_2>0$, $q_1 \le 1$ and $q_2 \le 1$. Contrary to product spaces $B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \times B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m)$ with $s_{1} > \frac{m}{t_{1}}$ and $s_{2} >\frac{m}{t_{2}}$ (Ben Slimane in Mediterr J Math, 13(4):1513–1533, 2016) and $(B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \cap C^{\gamma _{1}}(\mathbb {R}^m)) \times (B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m) \cap C^{\gamma _{2}}(\mathbb {R}^m)$ with $0<\gamma _{1}<s_{1}<\frac{m}{t_{1}}$ and $0<\gamma _{2}<s_{2}<\frac{m}{t_{2}}$ (Ben Abid et al. in Mediterr J Math, 13(6):5093–5118, 2016), all pairs of functions in the obtained generic set are not uniform Hölder. Nevertheless, the characterization of the upper bound of the Hölder exponent by decay conditions of local wavelet leaders suffices for our study.