# Motivation/Intuition behind Lorentz spaces

My current understanding is that the Lorentz spaces $L^{p,q}$ arise naturally as interpolation spaces between $L^1$ and $L^\infty$, but then people often describe them heuristically by saying something along the lines of "Lorentz spaces provide a finer control than $L^p$ spaces", and this is where I'm lost - what does that really mean?

It certainly seems like a reasonable claim, if only because you now have an extra parameter to tweak, and since $L^{p,p}=L^p$, well the Lorentz spaces are simply a larger class of spaces amongst which your classical $L^p$ spaces live, so sure, they are "better" because there's more of them, so I can give more nuanced descriptions, but I don't really understand where the nuance lies, I don't understand what extra control the Lorentz spaces provide you that the usual $L^p$ spaces do not.

I feel like my question is very vague overall, so feel free to ask for clarifications. As an example of the type of answer that I think there might be to what I am asking is the following cryptic (to me anyway) comment on the wikipedia page for "Lorentz spaces": "The Lorentz norms provide tighter control over both qualities than the $L^{p}$ norms, by exponentially rescaling the measure in both the range (p) and the domain (q)". I have no idea what that means, if anyone does, please let me know, but it seems like, after clarification, it would provide a nice intuitive explanation for precisely how Lorentz spaces provide finer control than $L^p$ spaces do.

Note that if $$F$$ is the distribution function of $$f$$, and $$f = H \cdot \mathbf{I}_E$$, where $$E$$ is a measurable set with $$|E| = W$$, then $$F = W \cdot \mathbf{I}_{[0,H]}$$. Thus, in some sense, the distribution function switches the domain and range of a function, so that the 'range' of $$f$$, is the 'domain' of $$F$$, and vice versa. In particular, the $$L^p$$ norms

$$\| f \|_p = \left( \int |f(x)|^p\; dx \right)^{1/p} \sim \left( \int_0^\infty F(t) t^p \frac{dt}{t} \right)^{1/p}$$

try to understand the distribution of $$f$$ by scaling it's range, or by scaling the `domain' of $$F$$ (changing the power of $$t$$ in the equation). Conversely, the Lorentz norm

$$\| f \|_{p,q} \sim \left( \int_0^\infty (tF(t)^{1/p})^q \frac{dt}{t} \right)^{1/q}$$

have two separate powers $$p$$ and $$q$$. Here $$p$$ scales the domain of $$f$$, and $$q$$ scales the domain and range of $$f$$ simultaneously. We changed $$F(t)$$ from being linear to being a power of $$1/p$$, but this is only slightly confusing because

$$\left( \int_0^\infty (t^pF(t))^{1/q} \frac{dt}{t} \right)^{1/q} \sim \| f \|_{p,q/p}$$

The reason that $$q$$ needs to scale the domain and range simultaneously is so that it acts as a second order parameter for the family of quasinorms, when compared to the primary exponent, which is $$p$$.

Maybe, it is related the embedding $L^{p,p} \subseteq L^{p,q}$ for every $q\ge p$ with $$\|f\|_{L^{p,q}} \lesssim \|f\|_{L^{p,p}}.$$