BMO-Interpolations and Jump Detection for Functions in $BV\cap BMO$

Abstract

A generalization of the John--Nirenberg inequality is established. As a consequence, local and global $BMO$--interpolation inequalities in Lorentz spaces $L^{q,{\hat{I}}^3}$ are obtained for the full range $0<q<\infty$ and $0<{\hat{I}}^3\le \infty$. These inequalities yield interpolation results in Besov spaces, fractional Sobolev spaces, and the space $BV$, including corresponding weak spaces. As a geometric application, consequences for the jump set of functions in $BV\cap BMO$ are derived.