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'?



Heuristic interpretation of the 'third index' for Besov and Triebel-Lizorkin spaces
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 $B^{p,p}\_s$).


