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_{\Omega}e^{\frac{4\pi u^{2}}{H_{d}(u)}}dxdy\le C(\Omega),\;\;\forall u\in C^{\infty}_{0}(\Omega), ] where $H_{d}=\int_{\Omega}|\nabla u|^{2}dxdy-\frac{1}{4}\int_{\Omega}\frac{u^{2}}{d(z,\partial\Omega)^{2}}dxdy$ and $d(z,\partial\Omega)=\min\limits_{z_{1}\in\partial\Omega}|z-z_{1}|$.} 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^{4\pi u^{2}}-1-4\pi u^{2})dV=\sup_{|u|_{\mathcal{H}}\leq 1}\int_{\mathbb{B}}\frac{(e^{4\pi u^{2}}-1-4\pi u^{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 $C^{\infty}_{0}(\mathbb{B})$ with respect to the norm $$|u|{\mathcal{H}}=\int{\mathbb{B}}|\nabla u|^{2}dxdy-\int_{\mathbb{B}}\frac{u^{2}}{(1-|z|^{2})^{2}}dxdy.$$ Using this strengthened Trudinger-Moser inequality, we also give a simpler proof of the Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].