Local Sharp Maximal Functions

Abstract

In considering the resistance of materials to certain types of deformations, F. John was led to the study of quasi-isometric mappings. The setting is essentially as follows. Let f be a continuous function defined on a cube $$Q_{0} \subset \mathbb{R}^{n}$$ . Assume that to each subcube Q of Q 0 with sides parallel to those of Q 0 there is assigned a constant c Q and let μ Q be the function of the real variable M given by $$\displaystyle{\mu _{Q}(M) = \frac{\vert \{y \in Q: \vert f(y) - c_{Q}\vert > M\}\vert } {\vert Q\vert } \,.}$$ Let $$\phi (M) =\sup _{Q\subset Q_{0}}\mu _{Q}(M)$$ , 0 < s < 1∕2, and λ a number such that ϕ(λ) ≤ s. Then under these assumptions $$\displaystyle{\phi (M) \leq Ae^{-BM/\lambda }}$$ holds for all nonnegative M where A, B are universal functions of s and the dimension n. Thus the space of functions of bounded mean oscillation (BMO) was introduced and the John–Nirenberg inequality established [18].