Is the Trudinger-Moser nonlinearity a true critical nonlinearity?

While the critical nonlinearity $\int |u|^{2^*}$ for the Sobolev space $H^1$ in dimension $N>2$ lacks weak continuity at any point, Trudinger-Moser nonlinearity $\int e^{4πu^2}$ in dimension $N=2$ is weakly continuous at any point except zero. In the former case the lack of weak continuity can be attributed to invariance with respect to actions of translations and dilations. The Sobolev space $H_0^1$ of the unit disk $\mathbb D\subset\R^2$ possesses transformations analogous to translations (Möbius transformations) and nonlinear dilations $r\mapsto r^s$. We present improvements of the Trudinger-Moser inequality with sharper nonlinearities sharper than $\int e^{4πu^2}$, that lack weak continuity at any point and possess (separately), translation and dilation invariance. We show, however, that no nonlinearity of the form $\int F(|x|,u(x))\mathrm{d}x$ is both dilation- and Möbius shift-invariant. The paper also gives a new, very short proof of the conformal-invariant Trudinger-Moser inequality obtained recently by Mancini and Sandeep and of a sharper version of Onofri-type inequality of Beckner.

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