Triebel-Lizorkin space $F_{p,q}^s(\mathbb{R}^d)$, $p=\infty$ One way to define the Triebel-Lizorkin space is using dyadic resolution of unity. Let $\psi$ be a Schwartz function which satisfies $\hat{\psi}(\xi)=1$ when $|\xi|\leq 1$ and $\hat{\psi}(\xi)=0$ when $|\xi|>\frac{3}{2}$. Define $\psi_0:=\psi$ and $\psi_j,j\in\mathbb{N}$ via
$$\widehat{\psi_j}(\xi)=\hat{\psi}(2^{-j}\xi)-\hat{\psi}(2^{-j+1}\xi).$$
Then we can see that $\widehat{\psi_j}$ is supported in the set
$$\{\xi\in\mathbb{R}^d: 2^{j-1}\leq|\xi|\leq 2^{j+1}\}.$$
Moreover:
$$\sum_{j=0}^\infty\widehat{\psi_j}(\xi)=1,\quad\forall \xi\in\mathbb{R}^d.$$
In this case we call $\psi$ a generating function, and $(\psi_j)_{j=0}^\infty$ a dyadic resolution of unity. Note that in this case every tempered distribution $f$ has the following representation
$$f=\sum_{j=0}^\infty\psi_j*f,$$
with the series converging in the space of tempered distributions.
For $s\in\mathbb{R}$, $0<p<\infty$ and $0<q\leq\infty$, the Triebel-Lizorkin space $F_{p,q}^s(\mathbb{R}^d)$ is defined as the spaces of all tempered distribution $f$ such that
$$\|f\|_{F_{p,q}^s}:=\left\|\left(\sum_{j=0}^\infty 2^{jsq}|\psi_j*f|^q\right)^{\frac{1}{q}}\right\|_{L^p}<\infty$$
for $0<q<\infty$, and
$$\|f\|_{F_{p,\infty}^s}:=\left\|\sup_{j\in\mathbb{N}_0}2^{js}|\psi_j*f|\right\|_{L^p}<\infty.$$
By using technical arguments one can prove that dufferent dyadic resolutions of unity will give equivalent norms on the Triebel-Lizorking space. 
What I'm curious about is that, what happens if $p=\infty$? I've read several textbooks , e.g. "Modern Fourier analysis" by Loukas Grafakos, "Theory of Funciton Spaces" by Hans Triebel, in which $p=\infty$ is said to be a case where the above definition does not work. However the authors of those books didn't give more details on this issue. More specifically, my question is, why the above definition fails when $p=\infty$? Can anyone provide me some resources (links) which explains this issue in details? Or give me some key clues if possible?
Thanks in advance.
 A: Let us discuss instead the homogeneous Triebel-Lizorkin spaces $\dot{F}_{p,q}^s$. In the study of these spaces, it is convenient in several scenarios that the representation $f = \sum_{j\in 2^\mathbb{Z}} \psi_j*f$ (defining $\psi_j$ for negative $j$ by the same definitions) to hold not just in the class of tempered distributions, but in $L^p$ as well (in the sense of convergence of the series to $f$ in $L^p$ norm), and this fails exactly for $p=1$ and $p=\infty$. ($p=1$ can be recovered under the additional assumption that $\int f = 0$.) This failure for $p=\infty$ is easy to see: setting $\phi_{M,N} = \sum_{j=M}^N \psi_j$, with $M,N\in 2^{\mathbb{Z}}$, we would require
$$
f*\phi_{M,N}(x) = \int f(x-y)\phi_{M,N}(y)~dy \to f(x)
$$
as $M\to-\infty$, $N\to\infty$ uniformly in a.e. x. But clearly this cannot hold for $f(x) \equiv 1$, as $\phi_{M,N}$ has mean $0$ for all $M$ and $N$.
This might be too philosophical, but I would consider this the fundamental problem behind $p=\infty$ for Triebel-Lizorkin spaces.
While this does not necessarily mean that the given definitions for $F_{p,q}^s$ cannot make sense for $p=\infty$, it should at least indicate that they may not satisfy properties you may desire. For instance, taking $q=2$ and $s=0$ in the definition should give us $L^p$ norms through the Littlewood-Paley square-function theorem. But the square-function theorem fails for $p=1$ and $p=\infty$ due to the reasons given above.
