Optimal AB-program for Poincaré--Sobolev and logarithmic Sobolev inequalities on hyperbolic spaces

Abstract

In this paper, we prove that we can subtract the full spectral gap $\left(\frac{N-1}{p}\right)^{p}$ of the $p$-Laplacian and keep the sharp Sobolev constant $S(N,p)$ of $\mathbb{R}^N$ at the same time in a large family of $p$. For $p=2$, this was proved by Benguria, Frank and Loss in dimension three, and it was shown by Hebey to be false in dimensions $N\geq4$. For $p\neq2$, it was proved by Nguyen for $N\geq4$ and $\frac{2N}{N-1}\leq p<N$. By employing, among other tools, a suitable conjugation that maps the radial $p$-Dirichlet integral on $\mathbb{H}^N$ onto its Euclidean counterpart on $\mathbb{R}^N$ while preserving the $L^p$ norm, together with a one-dimensional tangent line inequality, we establish such sharp Poincaré--Sobolev inequalities for $N=3$ and $2\leq p<3$, for $N=4$ and $\frac{1+\sqrt{13}}{2}\leq p<4$, and for $N\geq5$ and $\min\{3,\frac{1+\sqrt{4N-3}}{2}\}\leq p<N$. In particular, in all cases we also obtain an explicit remainder term. Moreover, since the change of function keeps the $L^p$ norm, we prove the sharp logarithmic Sobolev inequalities with the full spectral gap and the sharp Euclidean constant for $N=3$ and $2\leq p\leq3$, and for $N\geq4$ on an explicit interval of $p$. For $p\geq3$, we also use the change of variable $t=\sinhρ$ and the Hardy inequality, which give a second remainder term. At the endpoint $p=N$, this remainder gives a Hardy--Trudinger--Moser inequality on $\mathbb{H}^N$ under a weaker constraint than the one of Liang, Lu, Wang and Yang, in every dimension $N\geq3$. Finally, we show that for $1<p<\frac{1+\sqrt{4N-3}}{2}$, which includes all $1<p<2$, the full spectral gap can still be subtracted with some positive constant in front of the Sobolev term, and that this constant is smaller than $S(N,p)^{p}$ when $p<\min\{2,\sqrt N\}$. Together with our other results, this shows that the full spectral gap can be subtracted for every $1<p<N$.