Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 850125, 12 pages
doi:10.1155/2010/850125
Research Article

On Integral Operators with Operator-Valued Kernels

Rishad Shahmurov1,2

1Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA
2Vocational High School, Okan University, Istanbul 34959, Turkey

Received 17 October 2010; Revised 18 November 2010; Accepted 23 November 2010

Academic Editor: Martin Bohner

Copyright © 2010 Rishad Shahmurov. 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

Here, we study the continuity of integral operators with operator-valued kernels. Particularly we get 𝐿 𝑞 ( 𝑆 ; 𝑋 ) → 𝐿 𝑝 ( 𝑇 ; 𝑌 ) estimates under some natural conditions on the kernel 𝑘 ∶ 𝑇 × 𝑆 → 𝐵 ( 𝑋 , 𝑌 ) , where 𝑋 and 𝑌 are Banach spaces, and ∑ ( 𝑇 , 𝑇 , 𝜇 ) and ∑ ( 𝑆 , 𝑆 , 𝜈 ) are positive measure spaces: Then, we apply these results to extend the well-known Fourier Multiplier theorems on Besov spaces.