A sharp Trudinger-Moser inequality on any bounded and convex planar domain

Abstract

Wang and Ye conjectured in [22]: Let $\Omega$ be a regular, bounded and convex domain in ${\mathbb{R}}^2$. There exists a finite constant $C(\Omega )>0$ such that \[ \int_Ωe^{\frac{4πu^{2}}{H_{d}(u)}}dxdy\le C(Ω),\;\;\forall u\in C^{\infty}_{0}(Ω), \] where $H_d={\int }_{\Omega }|\nabla u{|}^{2}dxdy-\frac{1}{4}{\int }_{\Omega }\frac{u^2}{d(z,\partial \Omega {)}^2}dxdy$ and $d(z,\partial \Omega )=\underset{z_1\in \partial \Omega }{\min }|z-z_1|$.} The main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in ${\mathbb{R}}^2$ via the Riemann mapping theorem (the smoothness of the boundary of the domain is thus irrelevant). We also give a rearrangement-free argument for the following Trudinger-Moser inequality on the hyperbolic space $\mathbb{B}=\{z=x+iy:|z|=\sqrt{x^2+y^2}<1\}$: \[ \sup_{\|u\|_{\mathcal{H}}\leq 1} \int_{\mathbb{B}}(e^{4πu^{2}}-1-4πu^{2})dV=\sup_{\|u\|_{\mathcal{H}}\leq 1}\int_{\mathbb{B}}\frac{(e^{4πu^{2}}-1-4πu^{2})}{(1-|z|^{2})^{2}}dxdy< \infty, \] by using the method employed earlier by Lam and the first author [9, 10], where $\mathcal{H}$ denotes the closure of $C_0^{\infty }(\mathbb{B})$ with respect to the norm $$\Vert u{\Vert }_{\mathcal{H}}={\int }_{\mathbb{B}}|\nabla u{|}^{2}dxdy-{\int }_{\mathbb{B}}\frac{u^2}{(1-|z{|}^2{)}^2}dxdy.$$ Using this strengthened Trudinger-Moser inequality, we also give a simpler proof of the Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].