The author continues sharpening of Trudinger’s inequality. In [J. Funct. Anal. 127, No. 2, 259-269 (1995; Zbl 0846.46025)], he proved the following result. Let \(H^{\frac np ,p} (L^p(\mathbb{R}^n))\) be the space of Bessel potentials of order \(\frac{n}{p}\) with densities in \(L^p(\mathbb{R}^n)\), that is \(H^{\frac np ,p} (L^p(\mathbb{R}^n))= (I-\Delta)^{-\frac{n}{2p}}\), \(1 < p < \infty\). There exist positive constants \(\alpha\) and \(C\) such that for all \(f \in H^{\frac np ,p} (L^p(\mathbb{R}^n))\) with \(\| (-\Delta)^\frac{n}{2p}f\| _{L^p}\leq 1\) \[ \int_{\mathbb{R}^n}\Biggl(e^{\alpha | f| ^{p'}}- \sum_{\substack{ 0\leq j\leq p-1\\ j\in \mathbb N }} \frac{1}{j!} (\alpha| f| ^{p'})^j\Biggr)dx \leq C \| f\| ^p_{L^p}. \tag{1} \] Here the restriction \(\| (-\Delta)^\frac{n}{2p}f\| _{L^p}\leq 1\) may be avoided by rescaling \(f\) by \(f/\| (-\Delta)^\frac{n}{2p}f\| _{L^p}\), the inverse statement not being trivial as the author remarks. The author proves that the following three inequalities are equivalent:
the inequality (1), its rescaled form and the Gagliardo-Nirenberg-type inequality \[ \| f\| _{L^q}\leq Mq^{1-\frac{1}{p}} \| (-\Delta)^\frac{n}{2p}f\| _{L^p}^{1-\frac{p}{q}}\| f\| _{L^p}^\frac{p}{q} \] for all \(p\leq q < \infty\) with \(M\) depending only on \(p\) and \(n\),
and he gives the relation between the upper bound for \(\alpha\) and lower bound for \(M\).