Sharp mutliplicative inequalities with $\mathrm{BMO}$ $\mathrm{I}$

Abstract

We find the best possible constant $C$ in the inequality $\Vert \phi {\Vert }_{L^r}\le C\Vert \phi {\Vert }_{L^p}^{\frac{p}{r}}\Vert \phi {\Vert }_{\mathrm{BMO}}^{1-\frac{p}{r}}$, where $2\le r$ and $p<r$. We employ the Bellman function technique to solve this problem in the case of an interval and then transfer our results to the circle and the line.

17 pages