A new look at the John-Nirenberg and John-Stromberg theorems for BMO. Lecture Notes

We develop some techniques for studying various versions of the function space BMO. Special cases of one of our results give alternative proofs of the celebrated John- Nirenberg inequality and of related inequalities due to John and to Wik. Our approach enables us to pose a simply formulated "geometric" question, for which an affirmative answer would lead to a version of the John-Nirenberg inequality with dimension free constants. A more detailed summary of the main ideas and results of this paper can be found at http://www.math.technion.ac.il/~mcwikel/bmo/CwikSaghShvaSummary.pdf