In all harmonic analysis literature I've seen, the dilation factor people always use in the Littlewood-Paley decomposition is $2$, i.e. decompose the function dyadically. To be specific here's what happens:
Let $\mathcal{S}(\mathbb{R}^d)$ be the Schwartz space and $\mathcal{S}'(\mathbb{R}^d)$ denotes the space of tempered distributions. Choose $\varphi_0,\varphi\in\mathcal{S}(\mathbb{R}^d)$ be such that
- $\text{supp}(\widehat{\varphi_0})\subseteq\{\xi:|\xi|<1\}$ and $\text{supp}(\hat{\varphi})\subseteq\{\xi:\frac{1}{2}<|\xi|<2\}$, where $\text{supp}(\cdot)$ denotes the support of a function and $\hat{\varphi}$ denotes the Fourier transform of $\varphi$.
- $\sum_{j=0}^\infty\widehat{\varphi_j}(\xi)=1$ for all $\xi\in\mathbb{R}^d$, where $$\widehat{\varphi_j}(\xi)=\hat{\varphi}(2^{-j}\xi),\quad\forall\xi\in\mathbb{R}^d, j\in\mathbb{N}.$$ Then it can be shown that every $f\in\mathcal{S}'(\mathbb{R}^d)$ can be decomposed in the following way: $$f=\sum_{j=0}^\infty f*\varphi_j,\quad(*)$$ with the series converging in $\mathcal{S}'(\mathbb{R}^d)$. This is a classical way to dyadically decompose a function.
My question is, what happens if we use other dilation factors? Say let $M$ be an invertible $d\times d$ such that all its eigenvalues are greater than $1$ in magnitudes. Choose $\varphi_0,\varphi\in\mathcal{S}(\mathbb{R}^d)$ be such that $$\sum_{j=0}^\infty\widehat{\varphi_{j;M}}(\xi)=1$$ for all $\xi\in\mathbb{R}^d$ with $$\widehat{\varphi_{0;M}}(\xi)=\widehat{\varphi_0}(\xi),\quad \widehat{\varphi_{j;M}}(\xi)=\hat{\varphi}((M^T)^{-j}\xi),\forall j\in\mathbb{N}.$$ I believe in this case we still have the decomposition $(*)$ with $\varphi_j$ being replaced by $\varphi_{j;M}$. However if we use the dilation factor $M$ instead of $2$ to define function spaces, say for example we define $$\|f\|_{B_{p,q}^s;M}:=\left(\sum_{j=0}^\infty (|\text{det}(M)|^{js}\|f*\varphi_{j;M}\|_{L^p(\mathbb{R}^d)})^q\right)^{\frac{1}{q}},$$ and for $s\in\mathbb{R}, 0<p,q<\infty$ we define $$B_{p,q}^s(\mathbb{R}^d;M):=\{f\in\mathcal{S}'(\mathbb{R}^d):\|f\|_{B_{p,q}^s;M}<\infty\}.$$ Does the space $B_{p,q}^s(\mathbb{R}^d;M)$ coincide with the Besov space $B_{p,q}^s(\mathbb{R}^d)$? Moreover, are the semi-norms on these spaces equivalent?
Side remark: the Besov space $B_{p,q}^s(\mathbb{R}^d)$ is defined via $$B_{p,q}^s(\mathbb{R}^d):=\{f\in\mathcal{S}'(\mathbb{R}^d):\|f\|_{B_{p,q}^s}<\infty\},$$ where $$\|f\|_{B_{p,q}^s}:=\left(\sum_{j=0}^\infty (2^{jsd}\|f*\varphi_{j}\|_{L^p(\mathbb{R}^d)})^q\right)^{\frac{1}{q}}.$$