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On an embedding of Sobolev spaces. (English. Russian original) Zbl 0821.46043

Math. Notes 54, No. 3, 908-922 (1993); translation from Mat. Zametki 54, No. 3, 48-71 (1993).
Using estimates of the nonincreasing rearrangement \(f^*\) in terms of the derivatives \(D^{r_ i}_ i f= \partial^{r_ i} f/\partial x^{r_ i}_ i\), the author derives imbeddings of the anisotropic space \[ W^{r_ 1,\dots, r_ n}_ p (\mathbb{R}^ n)= \bigl\{f\in L_ p(\mathbb{R}^ n);\;D^{r_ i}_ i f\in L_ p(\mathbb{R}^ n),\;i= 1,2,\dots, n\bigr\} \] into Lorentz and Besov spaces. His results fill up some gaps, in particular for the case \(p= 1\). It is shown that for \(1\leq p< {n\over r}\) with \(r= n(1/ r_ 1+\cdots+ 1/r_ n)^{-1}\) and \(q^*= {np\over n- rp}\), \(W^{r_ 1,\dots, r_ n}_ p (\mathbb{R}^ n)\) is imbedded into the Lorentz space \(L_{q^* p}(\mathbb{R}^ n)\) and, for \(1\leq p< q< \infty\) with \({1\over p}- {1\over q}< {r\over n}\), into \(L_{q\theta} (\mathbb{R}^ n)\) for any \(\theta> 0\) and into the Besov space \(B^{\alpha_ 1,\dots, \alpha_ n}_{qp} (\mathbb{R}^ n)\) with \(\alpha_ i= r_ i\left[1- {n\over r}\left({1\over p}- {1\over q}\right)\right]\), \(i= 1,2,\dots, n\).
Reviewer: A.Kufner

MSC:

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