Trace properties in normed spaces established by using of mixed derivatives

R.Mashiyev; Z.Yucedag

In this paper, trace properties of functions in weighted function spaces established by free ${n}+{1}\leq \left\vert {\Sigma }\right\vert \leq {2}^n$ mixed (non-mixed) derivatives defined in an $n$-dimensional domain are studied. We estimate the ${L}_{p}\left( {\Gamma_s}\right) $ norm of the derivatives of the function defined on an ${s}$-dimensional surface via the weighted ${L}_{ {p}}\left( {G}\right) $ norm of these functions. In order to prove th\i s theorem, we use a special form of the integral representation for differentiable functions.

1. Besov O.V., Il'in V.P. and Nikolskii S.M. Integral representation of functions and embedding theorems. -- Moskow, 1996 (in Russian). (English transl. of 1st ed., Vol. 1, 2, Wiley (1979)).

2. Besov O.V. Sobolev's embedding theorem for a domain with irregular boundary// Siberian Math. J. -- 2001. -- V.192, № 3. -- P.323--346.

3. Besov O.V. Integral representations of functions and embedding theorems for a domain with flexible horn condition // Proc. Steklov Inst. Math. -- 1985. -- V.170. -- P.12--30.

4. Dzabrayilov A.D., Mamedov R.S. Integral representations for functions from weighted spaces whose ``indices'' of differential--difference properties are given by $(n+1)$ free vectors // Dokl. Azerb. SSR. -- 1981. -- V.37, № 10. -- P.8--11 (Russian).

5. Dzabrayilov A.D. The properties of functions on the boundary surfaces // Freie Uni. Berlin, Germany. 3rd Internat. ISAAC Congr. (August 20-25, 2001).

6. Sobolev S.L. Applications of functional analysis in mathematical physics. -- NGU, 1963, 3rd ed., Trans. of Math. Monographs, 90, Amer. Math. Soc. provid. RI, 1991.

7. Il'in V.P. On some properties of class of differentiable functions defined in domain // Proc. Steklov Inst. Math. -- 1964. -- V.94.

8. Mashiyev R.A. Integral representations of differentiable functions // AzNIINTI, 4B79, Dep. № 291, 1984.

9. Kudryavtsev L.D. The variation of maddings of regions // in: Metrical Questions of the Theory of Functions and Mappings. № 1. -- Kyiv, 1969. -- P.34--108 (Russian M.R. 44, 6924).

	Trace properties in normed spaces established by using of mixed derivatives
Author	R.Mashiyev, Z.Yucedag mrabil@dicle.edu.tr, zehra@dicle.edu.tr Dicle University, Faculty of Science and Art Mathematics Department, TR 21280 Diyarbakir, Turkey
Abstract	In this paper, trace properties of functions in weighted function spaces established by free ${n}+{1}\leq \left\vert {\Sigma }\right\vert \leq {2}^n$ mixed (non-mixed) derivatives defined in an $n$-dimensional domain are studied. We estimate the ${L}_{p}\left( {\Gamma_s}\right) $ norm of the derivatives of the function defined on an ${s}$-dimensional surface via the weighted ${L}_{ {p}}\left( {G}\right) $ norm of these functions. In order to prove th\i s theorem, we use a special form of the integral representation for differentiable functions.
Keywords	trace property, normed space, mixed derivative
DOI	doi:10.30970/ms.31.1.83-90
Reference	1. Besov O.V., Il'in V.P. and Nikolskii S.M. Integral representation of functions and embedding theorems. -- Moskow, 1996 (in Russian). (English transl. of 1st ed., Vol. 1, 2, Wiley (1979)). 2. Besov O.V. Sobolev's embedding theorem for a domain with irregular boundary// Siberian Math. J. -- 2001. -- V.192, № 3. -- P.323--346. 3. Besov O.V. Integral representations of functions and embedding theorems for a domain with flexible horn condition // Proc. Steklov Inst. Math. -- 1985. -- V.170. -- P.12--30. 4. Dzabrayilov A.D., Mamedov R.S. Integral representations for functions from weighted spaces whose ``indices'' of differential--difference properties are given by $(n+1)$ free vectors // Dokl. Azerb. SSR. -- 1981. -- V.37, № 10. -- P.8--11 (Russian). 5. Dzabrayilov A.D. The properties of functions on the boundary surfaces // Freie Uni. Berlin, Germany. 3rd Internat. ISAAC Congr. (August 20-25, 2001). 6. Sobolev S.L. Applications of functional analysis in mathematical physics. -- NGU, 1963, 3rd ed., Trans. of Math. Monographs, 90, Amer. Math. Soc. provid. RI, 1991. 7. Il'in V.P. On some properties of class of differentiable functions defined in domain // Proc. Steklov Inst. Math. -- 1964. -- V.94. 8. Mashiyev R.A. Integral representations of differentiable functions // AzNIINTI, 4B79, Dep. № 291, 1984. 9. Kudryavtsev L.D. The variation of maddings of regions // in: Metrical Questions of the Theory of Functions and Mappings. № 1. -- Kyiv, 1969. -- P.34--108 (Russian M.R. 44, 6924).
Pages	83-90
Volume	31
Issue	1
Year	2009
Journal	Matematychni Studii
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Table of content of issue	html