On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\frac1q\left(\int_0^\infty t^{q-1}\mu(|f|>t)^\frac qpdt\right)^{1/q}.$$
In many literature (e.g. Grafakos' Classical Fourier analysis) the space $L^{\infty,q}(\mathbb R^n)$ is defined to be $\{0\}$ due to the fact $\lim_{p\to\infty}\|f\|_{L^{p,q}(\mathbb R^n)}=+\infty$ when $q<\infty$ and $f\neq0$.
My question is, whether it is possible to seek for an alternative definition so that $L^{\infty,q}(\mathbb R^n)$ is nonempty. And say we still have real interpolation $(L^{p,q_0}(\mathbb R^n),L^{\infty,q_1}(\mathbb R^n))_{\theta,q}=L^{p/(1-\theta),q}(\mathbb R^n)$ for $0<p<\infty$ and $0<q_0,q_1,q\le\infty$?
It seems the definition of endpoint Triebel-Lizorkin spaces faces a similar situration. For $0<p,q<\infty$ and $s\in\mathbb R$, the space $\scr F_{pq}^s(\mathbb R^n)$ is given by $$\|f\|_{{\scr F}_{pq}^s(\phi)}:=\|(2^{js}\phi_j\ast f)_{j=0}^\infty\|_{L^p(\mathbb R^n;\ell^q)}=\left(\int_{\mathbb R^n}\left(\sum_{j=0}^\infty2^{jsq}|\phi_j\ast f(x)|^q\right)^{p/q}dx\right)^{1/p}.$$ Here $(\phi_j)_{j=0}^\infty$ is a suitable family of Schwartz functions (usually given by some Fourier support conditions). One can check that the norms are equivalent among different (reasonable) choice of $\phi$.
If we keep the same notations for $p=\infty$ the space is not well-defined since the norms are no longer equivalent among different $\phi$. Nevertheless, the Morrey-type definition $$\|f\|_{{\scr F}_{\infty q}^s(\phi)}=\|f\|_{{\scr F}_{q,q}^{s,1/q}(\phi)}:=\sup_{y\in\mathbb R^n;J\ge0}2^{J\frac nq}\left(\int_{B(y,2^{-J})}\sum_{j=\max(0,J)}^\infty2^{jsq}|\phi_j\ast f(x)|^qdx\right)^{1/q}$$ is useful and becomes the standard definition for the endpoint Triebel-Lizorkin spaces.
At least if we focus only on $\mathbb R^n$, is there reasonable way to adapt such Morrey-type characterization on the Lorentz spaces?
Updated: On sequential space we have equivalent norm $$\|u\|_{\ell^{p,q}}\approx\Big(\sum_{k=1}^\infty k^{q/p-1}u^*(k)^q\Big)^{1/q},$$ where $u^*(k)=\inf\big\{c>0:\#\{i:|u(i)|>c\}< k\big\}$ is the decreasing rearrangement.
If we take $p=\infty$ the definition is still legitmate. Moreover $\ell^{1/\epsilon}\subset\ell^{\infty,q-\epsilon}\subset\ell^{\infty,q}$ still holds for all $q$ and $\epsilon>0$.
