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| We investigate concentration and vanishing phenomena concerning Moser type inequalities in the whole plane which involve complete and reduced Sobolev norms. In particular we show that the critical Ruf inequality, which involves the complete Sobolev norm, is equivalent to an improved version of the subcritical Adachi-Tanaka inequality which we prove to be attained. Then, we consider the limiting space $\mathcal{D}^{1,2}(\mathbb{R}^2)$, completion of smooth compactly supported functions with respect to the Dirichlet norm $\|\nabla\cdot\|_2$, and we prove an optimal Lorentz-Zygmund type inequality from which can be derived classical inequalities in $H^1(\mathbb{R}^2)$. |
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