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Register a new userImproved Moser-Trudinger-Onofri inequality under constraints
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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 inequalities bear similarity to a sequence of Lebedev-Milin type inequalities on $mathbb{S}^{1}$ coming from the work of Grenander-Szego on Toeplitz determinants (as pointed out by Widom). We also discuss the related sharp inequality by a perturbation method.
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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.
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}(Omega), ] 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].
Suppose $F: mathbb{R}^{N} rightarrow [0, +infty)$ be a convex function of class $C^{2}(mathbb{R}^{N} backslash {0})$ which is even and positively homogeneous of degree 1. We denote $gamma_1=inflimits_{uin W^{1, N}_{0}(Omega)backslash {0}}frac{int_{Omega}F^{N}( abla u)dx}{| u|_p^N},$ and define the norm $|u|_{N,F,gamma, p}=bigg(int_{Omega}F^{N}( abla u)dx-gamma| u|_p^Nbigg)^{frac{1}{N}}.$ Let $Omegasubset mathbb{R}^{N}(Ngeq 2)$ be a smooth bounded domain. Then for $p> 1$ and $0leq gamma <gamma_1$, we have $$ sup_{uin W^{1, N}_{0}(Omega), |u|_{N,F,gamma, p}leq 1}int_{Omega}e^{lambda |u|^{frac{N}{N-1}}}dx<+infty, $$ where $0<lambda leq lambda_{N}=N^{frac{N}{N-1}} kappa_{N}^{frac{1}{N-1}}$ and $kappa_{N}$ is the volume of a unit Wulff ball. Moreover, by using blow-up analysis and capacity technique, we prove that the supremum can be attained for any $0 leqgamma <gamma_1$.
We give a new proof of Aubins improvement of the Sobolev inequality on $mathbb{S}^{n}$ under the vanishing of first order moments of the area element and generalize it to higher order moments case. By careful study of an extremal problem on $mathbb{S}^{n}$, we determine the constant explicitly in the second order moments case.
We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $displaystyle Lu:=-r^{-theta}(r^{alpha}vert u(r)vert^{beta}u(r))$, where $theta, betageq 0$ and $alpha>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Polya-Szeg{o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
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