On Integral Operators with Operator-Valued Kernels

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.

1. Introduction

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 [1] and for the application of multiplier theorems see [2].

Girardi and Weis [3] recently proved that the integral operator

(1.1)

defines a bounded linear operator

(1.2)

provided some measurability conditions and the following assumptions

(1.3)

are satisfied. Inspired from [3] we will show that (1.1) defines a bounded linear operator

(1.4)

if the kernel satisfies the conditions

(1.5)

where

(1.6)

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 state some important notations from [3]. A subspace of -norms , where , provided

(1.7)

It is clear that if -norms then the canonical mapping

(1.8)

is an isomorphic embedding with

(1.9)

Let and be -finite (positive) measure spaces and

(1.10)

will denote the space of finitely valued and finitely supported measurable functions from into , that is,

(1.11)

Note that is norm dense in for . Let be the closure of in the norm. In general (see [3, Proposition 2.2] and [3, Lemma 2.3]).

A vector-valued function is measurable if there is a sequence converging (in the sense of topology) to and it is -measurable provided is measurable for each . Suppose and . There is a natural isometric embedding of into given by

(1.12)

Now, let us note that if is reflexive or separable, then it has the Radon-Nikodym property, which implies that .

2. Estimates for Integral Operators

In this section, we identify conditions on operator-valued kernel , extending theorems in [3] so that

(2.1)

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 .

Condition 1.

For any and each

(a)there is so that if then the Bochner integral

(2.2)

(b) defines a measurable function from into .

Note that if satisfies the above condition then for each , there is so that the Bochner integral

(2.3)

and (1.1) defines a linear mapping

(2.4)

where denotes the space of measurable functions.

Condition 2.

The kernel satisfies the following properties:

(a)a real-valued mapping is product measurable for all ,

(b)there is so that

(2.5)

for and .

Theorem 2.1.

Suppose and the kernel satisfies Conditions 1 and 2. Then the integral operator (1.1) acting on extends to a bounded linear operator

(2.6)

Proof.

Let be fixed. Taking into account the fact that and using the general Minkowski-Jessen inequality with the assumptions of the theorem we obtain

(2.7)

Hence, .

Condition 3.

For each there is so that for all ,

(a)a real-valued mapping is measurable for all ,

(b)there is so that

(2.8)

for and .

Theorem 2.2.

Let be a separable subspace of that -norms . Suppose and satisfies Conditions 1 and 3. Then integral operator (1.1) acting on extends to a bounded linear operator

(2.9)

Proof.

Suppose and are fixed. Let , be corresponding sets due to Conditions 1 and 3. By separability of , we can choose a countable set of satisfying the above condition (note that since is a sigma algebra, the union of these countable sets still belongs to and the intersection of these sets should be nonempty). If then, by using Hölder's inequality and assumptions of the theorem, we get

(2.10)

Since, and -norms

(2.11)

Hence, .

In [3, Lemma 3.9], the authors slightly improved interpolation theorem [4, Theorem 5.1.2]. The next lemma is a more general form of [3, Lemma 3.9].

Lemma 2.3.

Suppose a linear operator

(2.12)

satisfies

(2.13)

Then, for and the mapping extends to a bounded linear operator

(2.14)

with

(2.15)

Proof.

Let us first consider the conditional expectation operator

(2.16)

where is a -algebra of subsets of . From (2.13) it follows that

(2.17)

Hence, by Riesz-Thorin theorem [4, Theorem 5.1.2], we have

(2.18)

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).

Let be a subspace of that -norms and for . Suppose satisfies Conditions 1, 2, and 3 with respect to . Then integral operator (1.1) extends to a bounded linear operator

(2.19)

with

(2.20)

Proof.

Combining Theorems 2.1 and 2.2, and Lemma 2.3, we obtain the assertion of the theorem.

Remark 2.5.

Note that choosing we get the original results in [3].

For estimates (it is more delicate and based on ideas from the geometry Banach spaces) and weak continuity and duality results see [3]. The next corollary plays important role in the Fourier Multiplier theorems.

Corollary 2.6.

Let be a subspace of that -norms and for . Suppose is strongly measurable on , is strongly measurable on and

(2.21)

Then the convolution operator defined by

(2.22)

satisfies .

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.

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.

Consider some subsets and of given by

(3.1)

Let us define the partition of unity of functions from . Suppose is a nonnegative function with support in , which satisfies

(3.2)

Note that

(3.3)

Let and . The Besov space is the set of all functions for which

(3.4)

is finite; here and are main and smoothness indexes respectively. The Besov space has significant interpolation and embedding properties:

(3.5)

where and denotes the Holder-Zygmund spaces.

Definition 3.1.

Let be a Banach space and . We say has Fourier type if

(3.6)

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].

Corollary 3.2.

Let be a Banach space having Fourier type and . Then the inverse Fourier transform defines a bounded operator

(3.7)

Definition 3.3.

Let be one of the following systems, where :

(3.8)

A bounded measurable function is called a Fourier multiplier from to if there is a bounded linear operator

(3.9)

such that

(3.10)

(3.11)

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 [5].

Since (3.10) can be written in the convolution form

(3.12)

Corollaries 2.6 and 3.2 can be applied to obtain regularity for (3.10).

Theorem 3.4.

Let and be Banach spaces having Fourier type and , so that . Then there is a constant depending only on and so that if

(3.13)

then is a FM from to with

(3.14)

where

(3.15)

Proof.

Let and . Assume that . Then . Since , choosing an appropriate and using (3.7) we obtain

(3.16)

where depends only on . Since we have and . Thus, in a similar manner as above, we get

(3.17)

for some constant depending on . Hence by (3.16)-(3.17) and Corollary 2.6

(3.18)

satisfies

(3.19)

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 .

Theorem 3.5.

Let and be Banach spaces having Fourier type and be so that . Then, there exist a constant depending only on and so that if satisfy

(3.20)

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).

Lemma 3.6.

Let and . If and

(3.21)

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).

Let and be Banach spaces having Fourier type and . If satisfies

(3.22)

for each multi-index with , then is a FM from to for each and .

Proof.

It is clear that for

(3.23)

Moreover, for we have

(3.24)

which implies

(3.25)

Hence by Lemma 3.6, (3.22) implies assumption (3.20) of Theorem 3.5.

Remark 3.8.

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 .