Characterization of Trudinger’s inequality. (English) Zbl 0921.46023
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 \(M\).
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 \(M\).
Reviewer: S.G.Samko (Faro)
MSC:
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
26D10 | Inequalities involving derivatives and differential and integral operators |
46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |