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

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