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تسجيل مستخدم جديدAubin type almost sharp Moser-Trudinger inequality revisited
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Fengbo Hang
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We give a new proof of the almost sharp Moser-Trudinger inequality on compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces. In particular we can lower the smoothness requirement of the metric and apply the same approach to higher order Sobolev spaces and manifolds with boundary under several boundary conditions.
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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{4pi u^{2}}{H_{d}(u)}}dxdyle C(Omega),;;forall uin C^{infty}_{0}(Om
ega), ] where $H_{d}=int_{Omega}| abla u|^{2}dxdy-frac{1}{4}int_{Omega}frac{u^{2}}{d(z,partialOmega)^{2}}dxdy$ and $d(z,partialOmega)=minlimits_{z_{1}inpartialOmega}|z-z_{1}|$.} 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^{4pi u^{2}}-1-4pi u^{2})dV=sup_{|u|_{mathcal{H}}leq 1}int_{mathbb{B}}frac{(e^{4pi u^{2}}-1-4pi 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 $$|u|_{mathcal{H}}=int_{mathbb{B}}| abla u|^{2}dxdy-int_{mathbb{B}}frac{u^{2}}{(1-|z|^{2})^{2}}dxdy.$$ Using this strengthened Trudinger-Moser inequality, we also give a simpler proof of the Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].
A classical result of Aubin states that the constant in Moser-Trudinger-Onofri inequality on $mathbb{S}^{2}$ can be imporved for furnctions with zero first order moments of the area element. We generalize it to higher order moments case. These new in
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