For the usual Lebesgue spaces $L^p (\mu)$ ($p \in [1,\infty]$) on a ($\sigma$-finite) measure space $(X,\mu)$, it is well-known that one has the characterization $$ L^p (\mu) = \left\{f : X \to \Bbb{C} \,:\, \int |f(x) g(x)| d\mu(x) < \infty \quad \forall g \in L^{p'} (\mu)\right\}, $$ where $p'$ is conjugate to $p$.

I am interested in a similar characterization for a more general setting.

Precisely, let $(X,\mu),(Y,\nu)$ be two ($\sigma$-finite) measure spaces, and consider the measure space $Z = X \times Y$ with the product measure. For $p,q \in [1,\infty]$, consider the iterated (or mixed) Lebesgue spaces $$ L^{p,q} := \{ f : X \times Y \to \Bbb{C} \,:\, \left\| y \mapsto \| f(\cdot, y) \|_{L^p (\mu)} \right\|_{L^q (\nu)} < \infty \}. $$ In a sense, these are vector-valued Lebesgue spaces.

Note: These are not the Lorentz spaces $L^{p,q}$.

For an article I am writing, I am interested in a "dual characterization" of the space $L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1}$. I can show that (at least if the atoms of $X,Y$ have a bounded measure), then \begin{align*} &L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1} \\ &= \left\{ f : Z \to \Bbb{C} \,:\, \int |f(z)g(z)| d(\mu \otimes \nu)(z) < \infty \quad \forall g \in L^1 \cap L^\infty \cap L^{1,\infty} \cap L^{\infty, 1}\right\}. \end{align*} However, to carefully prove this, I invoke quite heavy machinery from the theory of solid Banach function spaces (in particular the Lorentz-Luxemburg theorem, and a special function space norm for the space $L^1 + L^\infty$ which was originally introduced in [Schäffer] and [Gould] ), and the whole proof takes about 4 pages.

Since I only need this "dual characterization" as an auxiliary result, it would be very nice if such a result was already known, and I could just cite it.

Therefore, my question is:

Does anybody know whether the "dual characterization" of $L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1}$ from above is well-known, or a special case of some more general theorem? If so, I would appreciate any reference.


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