At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg interested in the space BMO? Were their interests motivated by PDEs? Did they initially make the connection with Hardy spaces? Any thoughts/input would be appreciated.
2 Answers
I do not know your location, but the article in question, "On Functions of Bounded Mean Oscillation" (1961), has been scanned and posted by Wiley and is available by subscription from some universities.
The original paper makes no direct mention of Hardy spaces or PDEs. For motivation, they cite a paper by John and a paper by Moser in the same issue of the journal, and they reprove a basic functional analysis result of Weiss and Zygmund. Fritz John's paper was an analysis of the polar decomposition of the derivative of a diffeomorphism of an ndimensional shape. (I mean a polar decomposition in the sense of linear algebra: Every square real matrix is the product of a symmetric matrix and a rotation matrix.) What John had in mind was estimates for deformation strain of solid materials. (Interestingly, he had an Air Force grant. Maybe John or the grant officers were thinking of aircraft parts.) Both John's paper and JohnNirenberg have the flavor of geometric analysis and nutsandbolts L^{p} estimates.
Moser's paper establishes the Harnack inequality for solutions to elliptic PDEs. I suppose that Moser was the one who connected the new definition of BMO to PDEs. At the time, John and Nirenberg were at NYU, and Moser was either there or soon to arrive. So these were people in close correspondence.

$\begingroup$ I should add that what the original paper proved was the JohnNirenberg inequality: If a function on an ndimensional domain has bounded mean oscillation, then the measure on which it is far away from its global mean satisfies a large deviation inequality. At first, this specific result got more attention than the definition of bounded mean oscillation. $\endgroup$ Nov 5, 2009 at 2:04

$\begingroup$ Thank you. This was quite helpful. I'm going to check the paper out this weekend. Thanks again. $\endgroup$– MLeviNov 5, 2009 at 2:41
I think the legend is that an official introduction to $BMO$ was given in the paper "Of Functions of Bounded Mean Oscillation" by F. John and L. Nirenberg , but the initial mention was in the paper by F. John "Rotation and Strain". F. John was looking at mappings and rotations. For example, the class of mappings $f(x)$ satisfying the following: \begin{equation*} 1\epsilon '\leq \frac{f(y)f(x)}{yx} \leq 1+\epsilon ' \end{equation*} for every $\epsilon '>\epsilon,~x\in\mathbb{R},~\exists \delta=\delta (x,\epsilon)$, and fixed $\epsilon <1$.
When looking at approximations for the derivatives of mappings in this class, F. John deduced some inequalities that look suspiciously like the definition for BMO.
Both papers are available online via Communications in Pure and Applied Mathematics.