Let us look at the definition of Besov spaces from [Bergh and L"ofstr"om, 1976]. Suppose $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ is a Schwartz class function satisfying

1. the support of $\varphi$ is contained in $\{ \omega : 2^{-1} \leq |\omega| \leq 2 \}$

2. $\varphi(\omega)>0$ for $2^{-1} <|\omega| <2$

3. $\sum_{k\in\mathbb{Z} } \varphi(2^{-k}\omega) =1$ for $\omega \neq 0$

Then the Besov norm of a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is defined as

\begin{align*} \lVert f \rVert_{B_{p,q}^s} &= \left\lVert \mathcal{F}^{-1} \left\{ \widehat{f} \left( 1-\sum_{k=1}^{\infty} \varphi(2^{-k} \cdot) \right) \right\} \right\rVert_{L_p} \\ & \quad + \left( \sum_{k=1}^{\infty} \left( 2^{sk} \left\lVert \mathcal{F}^{-1} \left\{ \widehat{f} \varphi(2^{-k} \cdot) \right\} \right\rVert_{L_p} \right)^q \right)^{1/q}, \end{align*} Where $\widehat{f}$ denotes the Fourier transform of $f$, and $\mathcal{F}^{-1}$ is the inverse Fourier transform operator.

The functions $\varphi(2^{-k} \cdot)$ form a partition of unity in the Fourier domain. The Besov norm uses this partition of unity to group the frequency information of $f$ into overlapping dyadic regions. The function $1-\sum_{k=1}^{\infty} \varphi(2^{-k} \cdot)$ is compactly supported in a neighborhood of the origin, so the first term of the Besov norm is the $L_p$ norm of the low frequency content of $f$.

The second term, depending on $q$, measures the high frequency content of $f$. The $L_p$ content on each region $2^{k-1}\leq |\omega| \leq 2^{k}$ (cut out by $\varphi(2^{-k}\cdot)$) is measured individually, and the total is combined using an $\ell_q$ norm. The factor $2^{sk}$ gives us the primary rate of decay of the sequence; e.g. if $q=\infty$, then there is a constant $C>0$ such that \begin{equation*} \left\lVert \mathcal{F}^{-1} \left\{ \widehat{f} \varphi(2^{-k} \cdot) \right\} \right\rVert_{L_p} \leq C 2^{-sk}. \end{equation*} Equivalently, we would have \begin{equation*} \left\lVert \mathcal{F}^{-1} \left\{ \widehat{f} 2^{sk} \varphi(2^{-k} \cdot) \right\} \right\rVert_{L_p} \leq C , \end{equation*} which indicates a rate of decay for $\widehat{f}$ and provides a link with Sobolev regularity.

In the general case, we consider the $\ell_q$ norm of this sequence, and we know that $\ell_{q_1}(\mathbb{N}) \subset \ell_{q_2}(\mathbb{N})$ for $0<q_1<q_2\leq \infty$, which means that $\lVert \cdot \rVert_{\ell_{q_2}} \leq \lVert \cdot \rVert_{\ell_{q_1}}$. Therefore, the Besov parameter $q$ can be viewed as a fine tuning parameter for regularity with \begin{equation*} B_{p,q_1}^s \subset B_{p,q_2}^s \subset B_{p,\infty}^s \end{equation*} \begin{equation*} \lVert \cdot\rVert_{B_{p,\infty}^s} \leq \lVert \cdot\rVert_{B_{p,q_2}^s} \leq \lVert \cdot\rVert_{B_{p,q_1}^s} \end{equation*} Furthermore, for arbitrary $0<q_3,q_4\leq \infty$ and $\epsilon>0$, we have the inclusion \begin{equation*} B_{p,q_3}^{s+\epsilon} \subset B_{p,q_4}^s. \end{equation*} This last fact explains why the Besov spaces $B_{p,q}^s$ are typically represented by the single point $(1/p,s)$ in function space diagrams (cf. http://terrytao.wordpress.com/2010/03/11/a-type-diagram-for-function-spaces/).

Let us look at the definition of Besov spaces from [Bergh and Löfström, 1976]. Suppose $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ is a Schwartz class function satisfying

1. the support of $\varphi$ is contained in $\{ \omega : 2^{-1} \leq |\omega| \leq 2 \}$

2. $\varphi(\omega)>0$ for $2^{-1} <|\omega| <2$

3. $\sum_{k\in\mathbb{Z} } \varphi(2^{-k}\omega) =1$ for $\omega \neq 0$

Then the Besov norm of a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is defined as

\begin{align*} \lVert f \rVert_{B_{p,q}^s} &= \left\lVert \mathcal{F}^{-1} \left\{ \widehat{f} \left( 1-\sum_{k=1}^{\infty} \varphi(2^{-k} \cdot) \right) \right\} \right\rVert_{L_p} \\ & \quad + \left( \sum_{k=1}^{\infty} \left( 2^{sk} \left\lVert \mathcal{F}^{-1} \left\{ \widehat{f} \varphi(2^{-k} \cdot) \right\} \right\rVert_{L_p} \right)^q \right)^{1/q}, \end{align*} Where $\widehat{f}$ denotes the Fourier transform of $f$, and $\mathcal{F}^{-1}$ is the inverse Fourier transform operator.

The functions $\varphi(2^{-k} \cdot)$ form a partition of unity in the Fourier domain. The Besov norm uses this partition of unity to group the frequency information of $f$ into overlapping dyadic regions. The function $1-\sum_{k=1}^{\infty} \varphi(2^{-k} \cdot)$ is compactly supported in a neighborhood of the origin, so the first term of the Besov norm is the $L_p$ norm of the low frequency content of $f$.

The second term, depending on $q$, measures the high frequency content of $f$. The $L_p$ content on each region $2^{k-1}\leq |\omega| \leq 2^{k}$ (cut out by $\varphi(2^{-k}\cdot)$) is measured individually, and the total is combined using an $\ell_q$ norm. The factor $2^{sk}$ gives us the primary rate of decay of the sequence; e.g. if $q=\infty$, then there is a constant $C>0$ such that \begin{equation*} \left\lVert \mathcal{F}^{-1} \left\{ \widehat{f} \varphi(2^{-k} \cdot) \right\} \right\rVert_{L_p} \leq C 2^{-sk}. \end{equation*} Equivalently, we would have \begin{equation*} \left\lVert \mathcal{F}^{-1} \left\{ \widehat{f} 2^{sk} \varphi(2^{-k} \cdot) \right\} \right\rVert_{L_p} \leq C , \end{equation*} which indicates a rate of decay for $\widehat{f}$ and provides a link with Sobolev regularity.

In the general case, we consider the $\ell_q$ norm of this sequence, and we know that $\ell_{q_1}(\mathbb{N}) \subset \ell_{q_2}(\mathbb{N})$ for $0<q_1<q_2\leq \infty$, which means that $\lVert \cdot \rVert_{\ell_{q_2}} \leq \lVert \cdot \rVert_{\ell_{q_1}}$. Therefore, the Besov parameter $q$ can be viewed as a fine tuning parameter for regularity with \begin{equation*} B_{p,q_1}^s \subset B_{p,q_2}^s \subset B_{p,\infty}^s \end{equation*} \begin{equation*} \lVert \cdot\rVert_{B_{p,\infty}^s} \leq \lVert \cdot\rVert_{B_{p,q_2}^s} \leq \lVert \cdot\rVert_{B_{p,q_1}^s} \end{equation*} Furthermore, for arbitrary $0<q_3,q_4\leq \infty$ and $\epsilon>0$, we have the inclusion \begin{equation*} B_{p,q_3}^{s+\epsilon} \subset B_{p,q_4}^s. \end{equation*} This last fact explains why the Besov spaces $B_{p,q}^s$ are typically represented by the single point $(1/p,s)$ in function space diagrams (cf. http://terrytao.wordpress.com/2010/03/11/a-type-diagram-for-function-spaces/).