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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 $$\mathbb{R}^{n}.$$

##### MSC:
 22E30 Analysis on real and complex Lie groups 43A80 Analysis on other specific Lie groups
##### Keywords:
Lie groups; Littlewood-Paley decompositions; Besov spaces
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