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

Abstract

We find the best possible constant $C$ in the inequality $\|\varphi\|_{L^r}\leq C\|\varphi\|_{L^p}^{\frac{p}{r}}\|\varphi\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$, where $2 \leq r$ 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.