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Sharp Hardy-Adams inequalities for bi-Laplacian on hyperbolic space of dimension four. (English) Zbl 1384.46026

Summary: We establish sharp Hardy-Adams inequalities on hyperbolic space \(\mathbb{B}^4\) of dimension four. Namely, we will show that for any \(\alpha > 0\) there exists a constant \(C_\alpha > 0\) such that \[ \mathop{\int}\limits_{\mathbb{B}^4}(e^{32 \pi^2 u^2} - 1 - 32 \pi^2 u^2) d V = 16 \mathop{\int}\limits_{\mathbb{B}^4} \frac{e^{32 \pi^2 u^2} - 1 - 32 \pi^2 u^2}{(1 - | x |^2)^4} d x \leq C_\alpha \] for any \(u \in C_0^\infty(\mathbb{B}^4)\) with \[ \mathop{\int}\limits_{\mathbb{B}^4}(-\Delta_{\mathbb{H}} - \frac{9}{4})(-\Delta_{\mathbb{H}} + \alpha) u \cdot u d V \leq 1 . \] As applications, we obtain a sharpened Adams inequality on hyperbolic space \(\mathbb{B}^4\) and an inequality which improves the classical Adams’ inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two given by G. Wang and D. Ye [Adv. Math. 230, No. 1, 294–320 (2012; Zbl 1241.35007)] and on any convex planar domain by G. Lu and Q. Yang [Calc. Var. Partial Differ. Equ. 55, No. 6, Paper No. 153, 16 p. (2016; Zbl 1369.46029)]. The Fourier analysis techniques on hyperbolic and symmetric spaces play an important role in our work.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
26D10 Inequalities involving derivatives and differential and integral operators
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