Journal of Inequalities and Applications
Volume 2005 (2005), Issue 4, Pages 329-345

Embedding operators and maximal regular differential-operator equations in Banach-valued function spaces

Veli B. Shakhmurov

Department of Electric-Electronics Engineering, Faculty of Engineering, Istanbul University, Avcilar, Istanbul 34850, Turkey

Received 11 November 2003; Revised 27 December 2004

Copyright © 2005 Veli B. 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.


This study focuses on anisotropic Sobolev type spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E, depending of α and order of spaces are found that mixed derivatives Dα belong with values; the boundedness and compactness of differential operators Dα from this space to Eα-valued Lp spaces are proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp regularity uniformly with respect to these parameters.