On Integral Operators with Operator-Valued Kernels
© Rishad Shahmurov. 2010
Received: 17 October 2010
Accepted: 23 November 2010
Published: 6 December 2010
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.
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. For instance, the boundedness of Fourier multiplier operators plays a crucial role in the theory of linear PDE's, especially in the study of maximal regularity for elliptic and parabolic PDE's. For an exposition of the integral operators with scalar-valued kernels see  and for the application of multiplier theorems see .
for . To prove our main result, we shall use some interpolation theorems of spaces. Therefore, we will study and boundedness of integral operator (1.1). The following two conditions are natural measurability assumptions on .
Now, taking into account (2.18) and using the same reasoning as in the proof of [3, Lemma 3.9], one can easily show the assertion of this lemma.
Theorem 2.4 (operator-valued Schur's test).
Combining Theorems 2.1 and 2.2, and Lemma 2.3, we obtain the assertion of the theorem.
Note that choosing we get the original results in .
For estimates (it is more delicate and based on ideas from the geometry Banach spaces) and weak continuity and duality results see . The next corollary plays important role in the Fourier Multiplier theorems.
3. Fourier Multipliers of Besov Spaces
In this section we shall indicate the importance of Corollary 2.6 in the theory of Fourier multipliers (FMs). Thus we give definition and some basic properties of operator valued FM and Besov spaces.
(i)any Banach space has a Fourier type 1,
(iii)spaces having Fourier type 2 should be isomorphic to a Hilbert spaces.
The following corollary follows from [5, Theorem 3.1].
For the definition of Besov spaces and their basic properties we refer to .
for all , so that . Now, taking into account the fact that is continuously embedded in and using the same reasoning as [5, Theorem 4.3] one can easily prove the general case and the weak continuity of .
Taking into consideration Theorem 3.4 one can easily prove the above theorem in a similar manner as [5, Theorem 4.3].
The following corollary provides a practical sufficient condition to check (3.20).
Using the fact that , the above lemma can be proven in a similar fashion as [5, Lemma 4.10].
Corollary 3.7 (Mikhlin's condition).
Hence by Lemma 3.6, (3.22) implies assumption (3.20) of Theorem 3.5.
Corollary 3.7 particularly implies the following facts.
The author would like to thank Michael McClellan for the careful reading of the paper and his/her many useful comments and suggestions.
- Folland GB: Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xiv+350.Google Scholar
- Denk R, Hieber M, Prüss J: -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Memoirs of the American Mathematical Society 2003, 166(788):1–106.View ArticleMathSciNetMATHGoogle Scholar
- Girardi M, Weis L: Integral operators with operator-valued kernels. Journal of Mathematical Analysis and Applications 2004, 290(1):190–212. 10.1016/j.jmaa.2003.09.044MathSciNetView ArticleMATHGoogle Scholar
- Bergh J, Löfström J: Interpolation spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften. Volume 223. Springer, Berlin, Germany; 1976:x+207.Google Scholar
- Girardi M, Weis L: Operator-valued Fourier multiplier theorems on Besov spaces. Mathematische Nachrichten 2003, 251: 34–51. 10.1002/mana.200310029MathSciNetView ArticleMATHGoogle Scholar
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