Mixed Wavelet Leaders Multifractal Formalism in a Product of Critical Besov Spaces |
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Authors: | Moez?Ben Abid Email author" target="_blank">Mourad?Ben SlimaneEmail author Ines?Ben Omrane Borhen?Halouani |
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Institution: | 1.High School of Sciences and Technology of Hammam Sousse Sousse University,Sousse,Tunisia;2.Department of Mathematics, Faculty of Science,Al Imam Mohammad Ibn Saud Islamic University (IMSIU),Riyadh,Saudi Arabia;3.Department of Mathematics, College of Science,King Saud University,Riyadh,Saudi Arabia |
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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. |
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