An estimate related to the Gagliardo-Nirenberg inequality

Abstract

Let Ω be a domain of unit volume in ℝⁿ satisfying the cone condition, If u ∈ L _∞ (Ω) and ∥u∥ _{L ∞} ≠ 0, denote

$$ O(u): = {\left\| u \right\|_{{{L_{\infty }}}}}/{\left\| u \right\|_{{{L_1}}}}. $$

(1.1)

If u ≥ 0, the quantity O(u) is a measure of the ‘amplitude’ of u, i.e. of its deviation from a constant.