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An alternative proof of the Brezis-Wainger inequality. (English) Zbl 0688.46016
The author gives a localized version of the inequality \[ \| f\|_{\infty}\leq c(1+\log (1+\| f\|_{s,q}))^{1-1/p},\quad k,s\in (0,\infty),\quad p\in (1,\infty),\quad q\in [1,\infty),\quad kp=n<sq, \] due to H. Brezis and S. Wainger, where \(f\in W^{s,q}({\mathbb{R}}^ n)\cap W^{k,p}({\mathbb{R}}^ n)\) with \(\| f\|_{k,p}\leq 1\). This version relates to the case of an open subset \(\Omega\) of \({\mathbb{R}}^ n\) with the cone property and it is given in terms of \(\| \omega \|_{x,p}=\| \omega \|_{L^ p(B_ 1(x)\cap \Omega)}.\) It has the form \[ | f(x)| \leq c(\| \nabla^ kf\|_{x,p}(\log (1+\frac{\| \nabla^ sf\|_{x,q}}{\| \nabla^ kf\|_{x,p}}))^{1-1/p}+\| f\|_{x,1}),\quad 1\leq p,q<\infty,\quad kp=n<sq, \] but k and s being integers.
Reviewer: S.G.Samko

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