We prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of Devyver, Fraas, and Pinchover (2014), namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator P_{\lambda }:= -\Delta _{{\open H}^{N}} - \lambda $P_{\lambda }:= -\Delta _{{\open H}^{N}} - \lambda$ where 0 ⩽ λ ⩽ λ1(ℍN) and λ1(ℍN) is the bottom of the L2 spectrum of -\Delta _{{\open H}^{N}} $-\Delta _{{\open H}^{N}}$, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for the operator P_{\lambda _{1}({\open H}^{N})}$P_{\lambda _{1}({\open H}^{N})}$. A different, critical and new inequality on ℍN, locally of Hardy type is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities, we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator P_\lambda.$P_\lambda.$

An a priori Campanato type regularity condition is established for a class of W1X local minimisers \overline{u}$\overline{u}$ of the general variational integral \int_{\Omega} F(\nabla u(x))\,{\rm d}x$\int_{\Omega} F(\nabla u(x))\,{\rm d}x$ where \Omega \subset \mathbb{R}^n$\Omega \subset \mathbb{R}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition F(\xi)\leq c(1+|\xi|^p)$F(\xi)\leq c(1+|\xi|^p)$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space \mathcal{L}^{p,\mu} (\Omega,\mathbb{R}^{N\times n})$\mathcal{L}^{p,\mu} (\Omega,\mathbb{R}^{N\times n})$ , µ < n, Campanato space \mathcal{L}^{p,n}(\Omega,\mathbb{R}^{N\times n})$\mathcal{L}^{p,n}(\Omega,\mathbb{R}^{N\times n})$ and the space of bounded mean oscillation {\rm BMO}\Omega,\mathbb{R}^{N\times n})${\rm BMO}\Omega,\mathbb{R}^{N\times n})$ . The admissible maps u\colon \Omega \to \mathbb{R}^N$u\colon \Omega \to \mathbb{R}^N$ are of Sobolev class W1,p , satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W1BMO local minimisers is extended from Lipschitz maps to this admissible class.

In this paper, we establish the existence of positive solution of a nonlinear subelliptic equation involving the critical Sobolev exponent on the Heisenberg group, which generalizes a result of Brezis and Nirenberg in the Euclidean case.