TITLE: Motivation for BMO
QUESTION [13 upvotes]: 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.

REPLY [4 votes]: 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)}{|y-x|} \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.