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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:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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