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 $B_s^{p,2}(\mathbb{R}^n)$ and $F_s^{p,2}(\mathbb{R}^n)$ are both equivalent to the Sobolev space $W_s^{p}(\mathbb{R}^n)$, and one can justify the integrability/regularity interpretations when $q \neq 2$.

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

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'?