On Integral Operators with Operator-Valued Kernels

Rishad Shahmurov

doi:10.1155/2010/850125

Abstract

Here, we study the continuity of integral operators with operator-valued kernels. Particularly we get estimates under some natural conditions on the kernel , where and are Banach spaces, and and are positive measure spaces: Then, we apply these results to extend the well-known Fourier Multiplier theorems on Besov spaces.

Highlights

It is well known that solutions of inhomogeneous differential and integral equations are represented by integral operators
To investigate the stability of solutions, we often use the continuity of corresponding integral operators in the studied function spaces
Girardi and Weis 3 recently proved that the integral operator

Summary

Research Article

We study the continuity of integral operators with operator-valued kernels. We get Lq S; X → Lp 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: we apply these results to extend the well-known Fourier Multiplier theorems on Besov spaces

Introduction

1Ai s

Then the convolution operator defined by