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

Mixed Wavelet Leaders Multifractal Formalism in a Product of Critical Besov Spaces 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 $$B_{t_{1}}^{\frac{m}{t_{1}},q_{1}}(\mathbb {R}^m) \times B_{t_{2}}^{\frac{m}{t_{2}},q_{2}}(\mathbb {R}^m)$$ B t 1 m t 1 , q 1 ( R m ) × B t 2 m t 2 , q 2 ( R m ) , for $$t_1,t_2>0$$ t 1 , t 2 > 0 , $$q_1 \le 1$$ q 1 ≤ 1 and $$q_2 \le 1$$ q 2 ≤ 1 . Contrary to product spaces $$B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \times B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m)$$ B t 1 s 1 , ∞ ( R m ) × B t 2 s 2 , ∞ ( R m ) with $$s_{1} > \frac{m}{t_{1}}$$ s 1 > m t 1 and $$s_{2} >\frac{m}{t_{2}}$$ s 2 > m t 2 (Ben Slimane in Mediterr J Math, 13(4):1513–1533, 2016) and $$(B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \cap C^{\gamma _{1}}(\mathbb {R}^m)) \times (B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m) \cap C^{\gamma _{2}}(\mathbb {R}^m)$$ ( B t 1 s 1 , ∞ ( R m ) ∩ C γ 1 ( R m ) ) × ( B t 2 s 2 , ∞ ( R m ) ∩ C γ 2 ( R m ) with $$0<\gamma _{1}<s_{1}<\frac{m}{t_{1}}$$ 0 < γ 1 < s 1 < m t 1 and $$0<\gamma _{2}<s_{2}<\frac{m}{t_{2}}$$ 0 < γ 2 < s 2 < 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

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

, Volume 14 (4) – Jul 31, 2017
20 pages

Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-017-0964-0
Publisher site
See Article on Publisher Site

### 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 $$B_{t_{1}}^{\frac{m}{t_{1}},q_{1}}(\mathbb {R}^m) \times B_{t_{2}}^{\frac{m}{t_{2}},q_{2}}(\mathbb {R}^m)$$ B t 1 m t 1 , q 1 ( R m ) × B t 2 m t 2 , q 2 ( R m ) , for $$t_1,t_2>0$$ t 1 , t 2 > 0 , $$q_1 \le 1$$ q 1 ≤ 1 and $$q_2 \le 1$$ q 2 ≤ 1 . Contrary to product spaces $$B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \times B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m)$$ B t 1 s 1 , ∞ ( R m ) × B t 2 s 2 , ∞ ( R m ) with $$s_{1} > \frac{m}{t_{1}}$$ s 1 > m t 1 and $$s_{2} >\frac{m}{t_{2}}$$ s 2 > m t 2 (Ben Slimane in Mediterr J Math, 13(4):1513–1533, 2016) and $$(B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \cap C^{\gamma _{1}}(\mathbb {R}^m)) \times (B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m) \cap C^{\gamma _{2}}(\mathbb {R}^m)$$ ( B t 1 s 1 , ∞ ( R m ) ∩ C γ 1 ( R m ) ) × ( B t 2 s 2 , ∞ ( R m ) ∩ C γ 2 ( R m ) with $$0<\gamma _{1}<s_{1}<\frac{m}{t_{1}}$$ 0 < γ 1 < s 1 < m t 1 and $$0<\gamma _{2}<s_{2}<\frac{m}{t_{2}}$$ 0 < γ 2 < s 2 < 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.

### Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Jul 31, 2017

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