- Research Article
- Open Access
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.
- Banach Space
- Integral Operator
- Bounded Linear Operator
- Besov Space
- Fourier Multiplier
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 and .
Here and are Banach spaces over the field and is the dual space of . The space of bounded linear operators from to is endowed with the usual uniform operator topology.
Now, let us note that if is reflexive or separable, then it has the Radon-Nikodym property, which implies that .
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 .
For any and each
(b) defines a measurable function from into .
where denotes the space of measurable functions.
The kernel satisfies the following properties:
(a)a real-valued mapping is product measurable for all ,
for and .
For each there is so that for all ,
(a)a real-valued mapping is measurable for all ,
for and .
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.
It is easy to see that satisfies Conditions 1, 2, and 3 with respect to . Thus, assertion of the corollary follows from Theorem 2.4.
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.
where and denotes the Holder-Zygmund spaces.
where is the smallest . Let us list some important facts:
(i)any Banach space has a Fourier type 1,
(ii) -convex Banach spaces have a nontrivial Fourier type,
(iii)spaces having Fourier type 2 should be isomorphic to a Hilbert spaces.
The following corollary follows from [5, Theorem 3.1].
The uniquely determined operator is the FM operator induced by . Note that if and maps into then satisfies the weak continuity condition (3.11).
For the definition of Besov spaces and their basic properties we refer to .
Corollaries 2.6 and 3.2 can be applied to obtain regularity for (3.10).
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 .
then is a FM from to and for each and .
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).
for each and , then satisfies (3.20).
Using the fact that , the above lemma can be proven in a similar fashion as [5, Lemma 4.10].
Choosing in Lemma 3.6 we get the following corollary.
Corollary 3.7 (Mikhlin's condition).
for each multi-index with , then is a FM from to for each and .
Hence by Lemma 3.6, (3.22) implies assumption (3.20) of Theorem 3.5.
Corollary 3.7 particularly implies the following facts.
(a)if and are arbitrary Banach spaces then ,
(b)if and be Banach spaces having Fourier type and then , suffices for a function to be a FM in .
The author would like to thank Michael McClellan for the careful reading of the paper and his/her many useful comments and suggestions.
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