Abstract
This paper is devoted to the construction of a logarithmic Littlewood-Paley decomposition. The approach we adopted to carry out this construction is based on the notion introduced in (Bahouri, Trends Math pp 1–15 (2013), [3]) of being \(\log \)-oscillating with respect to a scale. The relevance of this theory is illustrated on several examples related to Orlicz spaces.
In memory of M. Salah Baouendi
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Notes
- 1.
Where \(\widehat{u} \) denotes the Fourier transform of u defined by: \( \displaystyle \widehat{u }(\xi )= \int _{{\mathbb {R}}^{2N}}\, \mathrm{e}^{- i \, x \cdot \xi } \, u(x)\, dx\,\).
- 2.
We recall that \( {\mathcal F}(\Theta (D)u) (\xi )= \Theta (\xi ) {\mathcal F}( u) (\xi )\), with \({\mathcal F}\) the Fourier transform.
- 3.
Where obviously \( \xi =|\xi |\cdot \omega \), with \(\omega \in {\mathbb S}^{2N-1}\).
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Bahouri, H. (2015). Logarithmic Littlewood-Paley Decomposition and Applications to Orlicz Spaces. In: Baklouti, A., El Kacimi, A., Kallel, S., Mir, N. (eds) Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 127. Springer, Cham. https://doi.org/10.1007/978-3-319-17443-3_4
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