Integral Operator With Continuous Kernel Being Nuclear
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On Integral Operators with Operator-Valued Kernels
Journal of Inequalities and Applications volume 2010, Article number:850125 (2010) Cite this article
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
.
References
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Folland GB: Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xiv+350.
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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.
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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.2003.09.044
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Bergh J, Löfström J: Interpolation spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften. Volume 223. Springer, Berlin, Germany; 1976:x+207.
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Girardi M, Weis L: Operator-valued Fourier multiplier theorems on Besov spaces. Mathematische Nachrichten 2003, 251: 34–51. 10.1002/mana.200310029
Acknowledgment
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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Shahmurov, R. On Integral Operators with Operator-Valued Kernels. J Inequal Appl 2010, 850125 (2010). https://doi.org/10.1155/2010/850125
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DOI : https://doi.org/10.1155/2010/850125
Keywords
- Banach Space
- Integral Operator
- Bounded Linear Operator
- Besov Space
- Fourier Multiplier
Source: https://journalofinequalitiesandapplications.springeropen.com/articles/10.1155/2010/850125
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