Sharp multiplicative inequalities with $\mathrm{BMO}$ $\mathrm{II}$

TL;DR

This paper determines the optimal constant in a multiplicative inequality involving $L^p$, $L^r$, and BMO norms using Bellman functions, providing explicit formulas and extending results to various domains.

Contribution

It introduces explicit Bellman function formulas for the inequality's best constant across different domains and parameter ranges, advancing the understanding of BMO-related inequalities.

Findings

Explicit Bellman function formulas derived for the inequality

Optimal constants identified for various domains and parameters

Extended results to multi-dimensional cases

Abstract

We find the best possible constant $C$ in the inequality $$\|\varphi\|_{L^r}^{\phantom{\frac{p}{r}}}\leq C\|\varphi\|_{L^p}^{\frac{p}{r}}\|\varphi\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$$ for all possible values of parameters $p$ and $r$ such that $1 \le p < r < +\infty$. We employ the Bellman function technique to solve this problem. The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. First, we solve the problem on an interval and then transfer our results to the circle and the line. We also obtain explicit estimates in multi-dimensional cases.