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Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth. (English) Zbl 1101.22006
The Littlewood-Paley decompositions are a powerful tool in investigating deep properties of function spaces of distributions. To recall briefly the classical construction in \(\mathbb{R}^n,\) let \(\varphi\in C^{\infty}(R)\) be an even function such that \(0\leq\varphi\leq1,\;\varphi=1\) in \(\left[ 0,\frac {1}{4}\right] \) and \(\varphi=0\) in \([1,\infty).\) Let \(\psi(\lambda )=\varphi\left( \frac{\lambda}{4}\right) -\varphi(\lambda)\) so that \(\text{*supp}\,\psi\subset\left\{ \frac{1}{4}\leq\left| \lambda\right| \leq4\right\} .\) We have the following partition of unity on the frequency space of the Fourier transform:
\[ 1=\varphi\left( \left| \zeta\right| ^{2}\right) + {\displaystyle\sum\limits_{j=0}^{\infty}} \psi\left( 2^{-2j}\left| \zeta\right| ^{2}\right) ,\;\;\zeta \in\mathbb{R}^{n}. \] This gives the identity
\[ \hat{u}=\varphi\left( \left| \cdot\right| ^{2}\right) \hat{u}+ {\displaystyle\sum\limits_{j=0}^{\infty}} \psi\left( 2^{-2j} \left| \cdot\right| ^{2}\right)\hat{u},\;\;u\in S^{\prime}(\mathbb{R}^{n}), \] in \(S^{\prime}(\mathbb{R}^n).\) If we set \[ \widehat{S_{0}u}=\varphi\left( \left| \cdot\right| ^{2}\right) \hat{u}\;\;\text{\;and\;\;\;}\widehat{\Delta_{j}u}=\psi\left( 2^{-2j} \left| \cdot\right| ^{2}\right) \hat{u}, \] we obtain the Littlewood-Paley decomposition in \(S^{\prime}(\mathbb{R}^n):\) \[ u=S_{0}u+ {\displaystyle\sum\limits_{j=0}^{\infty}} \Delta_{j}u,\;\;\;u\in S^{\prime}(\mathbb{R}^{n}).\tag{1.1} \] The Littlewood-Paley theorem may be stated as follows: Let \(1<p<\infty\) and \(u\in S^{\prime }(\mathbb{R}^n).\) Then \(u\in L^{p}(\mathbb{R}^n)\) if and only if \(S_{0}u\in L^{p}(\mathbb{R}^n)\) and \(\left( \sum_{j=0}^{\infty}\left| \Delta _{j}u\right| ^{2}\right) ^{\frac{1}{2}}\in L^{p}(\mathbb{R}^n).\) Moreover there exists a constant \(C_{p}>1,\) which depends only on \(p\), such that \[ C_{p}^{-1}\left\| u\right\| _{L^{p}(\mathbb{R}^{n})}\leq\left\| S_{0}u\right\| _{L^{p}(\mathbb{R}^{n})}+\left\| \left( {\displaystyle\sum\limits_{j=0}^{\infty}} \left| \Delta_{j}u\right| ^{2}\right) ^{\frac{1}{2}}\right\| _{L^{p}(\mathbb{R}^{n})}\leq C_{p}\left\| u\right\| _{L^{p} (\mathbb{R}^{n})}, \] where \( u\in L^{p}(\mathbb{R}^{n}).\)
Now let \(G\) be a connected Lie group, and fix a left-invariant Haar measure \(dx\) on \(G.\) Denote by \(\left| A\right| \) the measure of a measurable subset \(A\) of \(G.\) Let \(G\) have polynomial volume growth, i.e., if \(U\) is a compact neighborhood of the identity element \(e\) of \(G,\) there is a constant \(C>0\) such that \(\left| U^{n}\right| \leq Cn^{C},n\in\mathbb{Z}_{+}.\) Then \(G\) is unimodular and there exists \(D\in\mathbb{N},\) which does not depend on \(U,\) such that \[ \left| U^{n}\right| \sim n^{D}\;\;\;for\;\;\;n\longrightarrow\infty. \] In this article the authors introduce a Littlewood-Paley decomposition on a Lie group \(G\) of polynomial volume growth and then prove a Littlewood-Paley theorem in this general setting and conclude their paper by deriving a dyadic characterization of Besov spaces \(B_p^{s,q}(G)\) similar to the classical Besov spaces on \(\mathbb{R}^{n}.\)

22E30 Analysis on real and complex Lie groups
43A80 Analysis on other specific Lie groups
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