Marcinkiewicz spaces, Garsia-Rodemich spaces and the scale of John-Nirenberg self improving inequalities

We extend to n-dimensions a characterization of the Marcinkiewicz $L(p,\mathrm{\infty })$ spaces first obtained by Garsia-Rodemich in the one dimensional case. This leads to a new proof of the John-Nirenberg self-improving inequalities. We also show a related result that provides a still a new characterization of the $L(p,\mathrm{\infty })$ spaces in terms of distribution functions, reflects the self-improving inequalities directly, and also characterizes $L(\mathrm{\infty },\mathrm{\infty }),$ the rearrangement invariant hull of $BMO\mathrm{.}$ We show an application to the study of tensor products with $L(\mathrm{\infty },\mathrm{\infty })$ spaces, which complements the classical work of O’Neil \cite{oneil} and the more recent work of Astashkin \cite{astashkin}.