Are these $L^{p,q}$ Lebesgue spaces?

Question

I have a theorem, deduced mostly from Loukas' Classical Fourier Analysis book 2009:

Let $(X,\mu)$ be a measure space. Then for all $0 <p,q \leq \infty$, the Lebesgue spaces $L^{p,q}(X,\mu)$ over the measure space $(X,\mu)$ are complete with respect to their quasinorm and they are therefore quasi-Banach spaces.

Can you call $L^{p,q}$ Lebesgue spaces?

If you choose $p = 4, q = 4$. Can you call $L^{4,4}$ Lebesgue space?

I do not want to confuse the reader with Lorentz spaces.

Answer 1

It is true that $L^{p,p}=L^{p}$, so when $q=p$, they are Lebesgue spaces.

When $p\neq q$, they are merely Lorentz spaces.