The sharp Hardy--Moser--Trudinger inequality in dimension $n$

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Abstract

In this paper, we prove a Hardy--Moser--Trudinger inequality in the unit ball $\mathbb B^n$ in $\mathbb R^n$ which improves both the classical singular Moser--Trudinger inequality and the classical Hardy inequality at the same time. More precisely, we show that for any $\beta \in [0,n)$ there exists a constant $C>0$ depending only on $n$ and $\beta$ such that \[ \sup_{u\in W^{1,n}_0(\mathbb B^n), \mathcal H(u) \leq 1}\int_{\mathbb B^n} e^{(1-\frac\beta n)\alpha_n |u|^{\frac n{n-1}}} |x|^{-\beta} dx \leq C \] where $\alpha_n = n \omega_{n-1}^{\frac1{n-1}}$ with $\omega_{n-1}$ being the surface area of the unit sphere $S^{n-1} = \partial \mathbb B^n$, and \[ \mathcal H(u) = \int_{\mathbb B^n} |\nabla u|^n dx -\left(\frac{2(n-1)}n\right)^n \int_{\mathbb B^n} \frac{|u|^n}{(1-|x|^2)^n} dx. \] This extends an inequality of Wang and Ye in dimension two to higher dimensions and to the singular case as well. The proof is based on the method of transplantation of Green's functions and without using the blow-up analysis method. As a consequence, we obtain a singular Moser--Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev \cite[Conjecture $5.2$]{MST}. We also propose an inequality which extends the singular Hardy--Moser--Trudinger inequality to any bounded convex domain in $\mathbb R^n$ which is analogue of the conjecture of Wang and Ye in higher dimensions.