Heuristic interpretation of the 'third index' for Besov and Triebel-Lizorkin spaces

Question

For $p,q \in (0,\infty)$ and $s \in \mathbb{R}$, one can define certain function spaces, $B_s^{p,q}(\mathbb{R}^n)$ and $F_s^{p,q}(\mathbb{R}^n)$, the Besov and Triebel-Lizorkin spaces respectively. The definition of these spaces is a bit complicated, and it probably won't offer any new insight to anybody who hasn't already considered these spaces, but it can be found for example in Triebel's 'Theory of Function Spaces II' (section 2.3.1) or Grafakos' 'Modern Fourier Analysis' (section 6.5.1).

For both of these spaces, one can rightfully refer to $p$ as the 'integrability index' and to $s$ as the 'regularity index'; for example, the spaces $F_s^{p,2}(\mathbb{R}^n)$ are equivalent to the Sobolev spaces $W_s^{p}(\mathbb{R}^n)$, and one can justify the integrability/regularity interpretations when $q \neq 2$ or for the spaces $B^{p,q}_s(\mathbb{R}^n)$.

Is there a simple way of describing the role of the $q$ index? Or are stuck with referring to it as 'the third index'?

Answer 1

This is really a comment with possibly relevant references, but I would prefer to avoid creating an account. The additional parameter is sometimes referred to as the "microscopic parameter" in the literature (this may help in searching google and MathSciNet). Two references of interest are:

W. Sickel, L. Skrzypczak and J. Vybiral. On the Interplay of Regularity and Decay in Case of Radial Functions I. Inhomogeneous spaces. arXiv:1201.5007 Commun. Contemp. Math 14 (2012) No. 1

(along with reference list of this article -- there is also a followup paper II. Homogeneous spaces in J. Fourier Anal. Appl. 2012).

as well as the work of Brezis and Mironescu (J. Evol. Eq. 1, 2001; see Section III).

As a tangential point related to the statement of the question, while $F^{p,2}_s$ is certainly the Bessel potential space $L^{s,p}$, unless I miss something this space is not the same as $B^{p,2}_s$ (note that there are many conventions for notation in the literature, and many authors use $W^{s,p}$ for the Besov space $B^{p,p}_s$).

Answer 2

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/).