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Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. (English) Zbl 0955.46019
Authors’ abstract: Let $$m$$ and $$n$$ be positive integers with $$n\geq 2$$ and $$1\leq m\leq n-1$$. We study rearrangement-invariant quasinorms $$\varrho_R$$ and $$\varrho_D$$ on functions $$f:(0, 1)\to \mathbb{R}$$ such that to each bounded domain $$\Omega$$ in $$\mathbb{R}^n$$, with Lebesgue measure $$|\Omega|$$, there corresponds $$C= C(|\Omega|)> 0$$ for which one has the Sobolev imbedding inequality $$\varrho_R(u^*(|\Omega|t))\leq C\varrho_D(|\nabla^m u|^*(|\Omega|t))$$, $$u\in C^m_0(\Omega)$$, involving the nonincreasing rearrangements of $$u$$ and a certain $$m$$th order gradient of $$u$$. When $$m= 1$$ we deal, in fact, with a closely related imbedding inequality of Talenti, in which $$\varrho_D$$ need not be rearrangement-invariant, $\varrho_R(u^*(|\Omega|t))\leq C\varrho_D ((d/dt) \int_{\{x\in\mathbb{R}^n:|u(x)|> u^*(|\Omega|t)\}}|(\nabla u)(x)|dx), \qquad u\in C^1_0(\Omega).$ In both cases we are especially interested in when the quasinorms are optimal in the sense that $$\varrho_R$$ cannot be replaced by an essentially larger quasinorm and $$\varrho_D$$ cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Brézis, and Wainger.
Reviewer: Josef Wloka (Kiel)

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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