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Journal of Function Spaces and Applications
Volume 2012, Article ID 819321, 15 pages
http://dx.doi.org/10.1155/2012/819321
Research Article

Embedding Operators in Vector-Valued Weighted Besov Spaces and Applications

Department of Electronics Engineering and Communication, Okan University, Akfirat Beldesi, Tuzla, 34959 Istanbul, Turkey

Received 12 July 2011; Accepted 9 January 2012

Academic Editor: Lars-Erik Persson

Copyright © 2012 Veli Shakhmurov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The embedding theorems in weighted Besov-Lions type spaces 𝐡 𝑙 , 𝑠 𝑝 , π‘ž , 𝛾 ( Ξ© ; 𝐸 0 , 𝐸 ) in which 𝐸 0 , 𝐸 are two Banach spaces and 𝐸 0 βŠ‚ 𝐸 are studied. The most regular class of interpolation space 𝐸 𝛼 between 𝐸 0 and E is found such that the mixed differential operator 𝐷 𝛼 is bounded from 𝐡 𝑙 , 𝑠 𝑝 , π‘ž , 𝛾 ( Ξ© ; 𝐸 0 , 𝐸 ) to 𝐡 𝑠 𝑝 , π‘ž , 𝛾 ( Ξ© ; 𝐸 𝛼 ) and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results, the uniform separability of degenerate abstract differential equations with parameters and the maximal B-regularity of Cauchy problem for abstract parabolic equations are obtained. The infinite systems of the degenerate partial differential equations and Cauchy problem for system of parabolic equations are further studied in applications.