## 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:

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

 [1] C. Bennett and R. Sharpley, Interpolation of Operators, Acad. Press Inc. (1988). · Zbl 0647.46057 [2] S. L. Sobolev, ?On a theorem of functional analysis,? Mat. Sb.4(46), No. 3, 471-497 (1938). [3] O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1975). [4] V. A. Solonnikov, ?On some inequalities for functions from the classesW p (? n ,? Notes of Scientific Seminars, Leningrad Section of the Mathematics Institute, Academy of Sciences of the USSR,27, 194-210 (1972). [5] J. Bergh and J. Löfström, ?Interpolation Spaces: an Introduction,? Springer-Verlag, Berlin-New York (1976). [6] V. I. Kolyada, ?On relations between moduli of continuity in various metrics,? Trudy MIAN SSSR,181, 117-136 (1988). [7] V. I. Kolyada, ?On the embedding of some classes of functions of several variables,? Sib. Mat. Zh.,14, No. 4, 766-790 (1973). [8] S. G. Krein, Yu. I. Petunin, and E. M. Semenov, ?Interpolation of Linear Operators,? American Mathematical Society, Providence, R. I. (1982). [9] H. Hadwiger, Lectures on Volume, Surface Area, and Isoperimetry [in Russian], Nauka, Moscow (1966). In German: Vorlesungen uber Inhalt, Oberflache und Isoperimetrie, Springer-Verlag, Berlin (1957). [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ (1970). · Zbl 0207.13501 [11] O’Neil, ?Convolution operators andL(p, q) spaces,? Duke Math. J.,30, 129-142 (1963). · Zbl 0178.47701 [12] L. D. Kudryavtsev and S. M. Nikol’skii, ?Spaces of differentiable functions of several variables and embedding theorems,? Itogi Nauki i Tekhniki. Sovremennye Problemy Matematiki, Moscow, VINITI, 26 (1988). [13] E. M. Stein, ?The differentiability of functions in ? n ,? Ann. Math.,113, 383-385 (1981). · Zbl 0531.46021
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