Volume 2010, Article ID 850125,12pages doi:10.1155/2010/850125
Research Article
On Integral Operators with
Operator-Valued Kernels
Rishad Shahmurov
1, 21Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA 2Vocational High School, Okan University, Istanbul 34959, Turkey
Correspondence should be addressed to Rishad Shahmurov,shahmurov@hotmail.com
Received 17 October 2010; Revised 18 November 2010; Accepted 23 November 2010
Academic Editor: Martin Bohner
Copyrightq2010 Rishad Shahmurov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Here, we study the continuity of integral operators with operator-valued kernels. Particularly we getLqS;X → LpT;Yestimates under some natural conditions on the kernelk : T ×S →
BX, Y, whereXand Y are Banach spaces, andT,T, μ andS,
S, νare 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 see1and for the application of multiplier theorems see2.
Girardi and Weis3recently proved that the integral operator
Kf·
S
k·, sfsdνs 1.1
defines a bounded linear operator
provided some measurability conditions and the following assumptions
sup
s∈S
T
kt, sxYdμt≤C1xX, ∀x∈X,
sup
t∈T
S
k∗t, sy∗X∗dνs≤C2y∗Y∗, ∀y∗∈Y∗
1.3
are satisfied. Inspired from3we will show that1.1defines a bounded linear operator
K:LqS, X−→LpT, Y 1.4
if the kernelk:T×S → BX, Ysatisfies the conditions
sup
s∈S
T
kt, sxθ
Ydt
1/θ
≤C1xX, ∀x∈X,
sup
t∈T
S
k∗t, sy∗θ X∗ds
1/θ
≤C2y∗Y∗, ∀y∗∈Y∗,
1.5
where
1
q−
1
p 1−
1
θ 1.6
for 1≤q < θ/θ−1≤ ∞andθ∈1,∞.
HereXandY are Banach spaces over the fieldCandX∗is the dual space ofX. The spaceBX, Yof bounded linear operators fromXtoY is endowed with the usual uniform operator topology.
Now let us state some important notations from3. A subspaceY ofX∗ τ-normsX, whereτ≥1, provided
xX ≤τ sup
x∗∈BY|x
∗x| ∀x∈X.
1.7
It is clear that ifY τ-normsXthen the canonical mapping
u:X−→Y∗ with y, ux x, y 1.8
is an isomorphic embedding with
1
LetT,T, μandS,S, νbeσ-finitepositivemeasure spaces and
finite
S
A∈
S
:νA<∞
,
full
S
A∈
S
:νS\A 0
. 1.10
εS, Xwill denote the space of finitely valued and finitely supported measurable functions
fromSintoX, that is,
εS, X
n
i1
xi1Ai :xi∈X, Ai∈ finite
S
, n∈N
. 1.11
Note thatεS, Xis norm dense inLpS, Xfor 1 ≤ p < ∞. Let L0∞S, X be the closure of
εS, Xin theL∞S, Xnorm. In generalL0∞S, X/L∞S, X see3, Proposition 2.2and3,
Lemma 2.3.
A vector-valued functionf : S → X is measurable if there is a sequence fn∞n1 ⊂
εS, Xconvergingin the sense ofX topologytof and it isσX,Γ-measurable provided
f·, x∗:S → Kis measurable for eachx∗∈Γ⊂X∗. Suppose 1≤p≤ ∞and 1/p 1/p1. There is a natural isometric embedding ofLpT, Y∗intoLpT, Y∗given by
f, g
T
ft, gtdμt forg∈LpT, Y∗, f ∈LpT, Y. 1.12
Now, let us note that ifXis reflexive or separable, then it has the Radon-Nikodym property, which implies thatEX∗E∗X∗.
2.
L
q→
L
pEstimates for Integral Operators
In this section, we identify conditions on operator-valued kernel k : T ×S → BX, Y, extending theorems in3so that
KLqS,X→LpT,Y≤C 2.1
for 1≤q≤p. To prove our main result, we shall use some interpolation theorems ofLpspaces.
Therefore, we will study L1S, X → LθT, Y andLθS, X → L∞T, Yboundedness of
integral operator1.1. The following two conditions are natural measurability assumptions
Condition 1. For anyA∈finiteS and eachx∈X
athere isTA,x∈
full
T so that ift∈TA,xthen the Bochner integral
A
kt, sxdνs exists, 2.2
bTA,x :t →
Akt, sxdνsdefines a measurable function fromT intoY.
Note that ifksatisfies the above condition then for eachf ∈εS, X, there isTf ∈fullT
so that the Bochner integral
S
kt, sfsdνs exists 2.3
and1.1defines a linear mapping
K:εS, X−→L0T, Y, 2.4
whereL0denotes the space of measurable functions.
Condition 2. The kernelk:T×S → BX, Ysatisfies the following properties:
aa real-valued mappingkt, sxθXis product measurable for allx∈X,
bthere isSx∈fullS so that
kt, sxLθT,Y ≤C1xX 2.5
for 1≤θ <∞andx∈X.
Theorem 2.1. Suppose1≤θ <∞and the kernelk:T×S → BX, Ysatisfies Conditions1and2. Then the integral operator1.1acting onεS, Xextends to a bounded linear operator
K:L1S, X−→LθT, Y. 2.6
Proof. Let f ni1xi1Ais ∈ εS, X be fixed. Taking into account the fact that 1 ≤ θ
obtain
Kft
LθT,Y≤ ⎡ ⎣ T S kt, s n
i1
xi1Ais Y dνs θ dμt ⎤ ⎦ 1/θ ≤ S ⎛ ⎝ T kt, s n
i1
xi1Ais θ Y dμt ⎞ ⎠ 1/θ dνs ≤ S ⎡ ⎣ T n
i1
1Aiskt, sxiY
θ dμt ⎤ ⎦ 1/θ dνs ≤ S n
i1
1Ais
T
kt, sxiθYdμt
1/θ dνs ≤ S n
i1
1Aiskt, sxiLθT,Y dνs≤C1
n
i1
xiX
S
1Aisdνs
C1 n
i1
xiXνAi C1fL1S,X.
2.7
Hence,KL1→Lθ≤C1.
Condition 3. For eachy∗∈Zthere isTy∗∈
full
T so that for allt∈Ty∗,
aa real-valued mappingk∗t, sx∗θX∗is measurable for allx∗∈X∗,
bthere isSx∈fullS so that
k∗t, sy∗
LθS,X∗≤C2y
∗
Y∗ 2.8
for 1≤θ <∞andx∈X.
Theorem 2.2. LetZbe a separable subspace ofY∗ thatτ-normsY. Suppose1 ≤ θ < ∞and k :
T×S → BX, Ysatisfies Conditions1and3. Then integral operator1.1acting onεS, Xextends
to a bounded linear operator
K:LθS, X−→L∞T, Y. 2.9
Proof. Supposef ∈ εS, Xand y∗ ∈ Zare fixed. LetTf,Ty∗ ∈
full
T be corresponding sets
due to Conditions1and3. By separability ofZ, we can choose a countable set ofTy∗ ∈fullT
countable sets still belongs tofullT and the intersection of these sets should be nonempty. If
t∈Tf ∩Ty∗then, by using H ¨older’s inequality and assumptions of the theorem, we get
y∗,KftY
y∗,
S
kt, sfsdνs
≤
S
k∗t, sy∗fsdνs
≤ k∗t, sy∗LθS,X∗fsLθS,X
≤C2y∗fLθS,X.
2.10
Since,Tf ∩Ty∗∈fullT andZ τ-normsY
Kf
L∞T,Y≤C2τfLθS,X. 2.11
Hence,KL
θ→L∞≤τC2.
In3, Lemma 3.9, the authors slightly improved interpolation theorem4, Theorem 5.1.2. The next lemma is a more general form of3, Lemma 3.9.
Lemma 2.3. Suppose a linear operator
K:εS, X−→LθT, Y L∞T, Y 2.12
satisfies
KfL
θT,Y≤C1fL1S,X, KfL∞T,Y≤C2fLθS,X. 2.13
Then, for1/q−1/p 1−1/θand1 ≤ q < θ/θ−1≤ ∞the mappingKextends to a bounded linear operator
K:LqS, X−→LpT, Y 2.14
with
KLq→Lp ≤C1θ/pC21−θ/p. 2.15
Proof. Let us first consider the conditional expectation operator
K0f
whereis aσ-algebra of subsets ofB∈finiteT . From2.13it follows that K0f
LθT,Y≤C1fL1S,X<∞, K0f
L∞T,Y ≤C2fL
θS,X<∞.
2.17
Hence, by Riesz-Thorin theorem4, Theorem 5.1.2, we have
K0fLpT,Y≤C1 θ/pC
21−θ/pfLqS,X. 2.18
Now, taking into account2.18and using the same reasoning as in the proof of3, Lemma 3.9, one can easily show the assertion of this lemma.
Theorem 2.4operator-valued Schur’s test. LetZbe a subspace ofY∗thatτ-normsYand1/q−
1/p1−1/θfor1≤q < θ/θ−1≤ ∞. Supposek:T ×S → BX, Ysatisfies Conditions1,2, and3with respect toZ. Then integral operator1.1extends to a bounded linear operator
K:LqS, X−→LpT, Y 2.19
with
KLq→Lp ≤C1θ/pτC21−θ/p. 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 choosingθ1 we get the original results in3.
ForL∞estimatesit is more delicate and based on ideas from the geometry Banach spacesand weak continuity and duality results see3. The next corollary plays important role in the Fourier Multiplier theorems.
Corollary 2.6. LetZbe a subspace ofY∗thatτ-normsY and1/q−1/p 1−1/θfor1 ≤q < θ/θ−1 ≤ ∞. Supposek :Rn → BX, Yis strongly measurable onX,k∗ : Rn → BY∗, X∗is
strongly measurable onZand
kxLθRn,Y≤C1xX, ∀x∈X,
k∗y∗
LθRn,X∗ ≤C2y
∗
Y∗, ∀y∗∈Y∗.
2.21
Then the convolution operator defined by
Kft
Rn
kt−sfsds fort∈Rn 2.22
satisfiesKLq→Lp ≤C1 θ/pC
It is easy to see thatk :Rn → BX, Ysatisfies Conditions1,2, and3with respect to
Z. Thus, assertion of the corollary follows fromTheorem 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{Jk}∞k0and{Ik}∞k0ofRngiven by
J0{t∈Rn:|t| ≤1}, Jk t∈Rn: 2k−1<|t| ≤2k
!
fork∈N,
I0{t∈Rn:|t| ≤2}, Ik t∈Rn: 2k−1<|t| ≤2k 1
!
fork∈N.
3.1
Let us define the partition of unity{ϕk}k∈N0of functions fromSR
n, R. Supposeψ∈SR, R
is a nonnegative function with support in2−1,2, which satisfies
∞
k−∞
ψ2−ks1 for s∈R\ {0},
ϕkt ψ
2−k|t|, ϕ0t 1−
∞
k1
ϕkt fort∈Rn.
3.2
Note that
suppϕk⊂Ik, suppϕk⊂Ik. 3.3
Let 1 ≤ q ≤ r ≤ ∞ands ∈R. The Besov space is the set of all functionsf ∈ SRn, Xfor
which
fBs
q,rRn,X:
2ks"ϕˇk∗f
#∞
k0
lrLqRn,X
≡
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(∞
k0
2ksrϕˇ
k∗frLqRn,X )1/r
ifr /∞
sup
k∈N0 *
2ksϕˇ
k∗fLqRn,X +
ifr∞
is finite; hereqand sare main and smoothness indexes respectively. The Besov space has significant interpolation and embedding properties:
Bsq,rRn;X LqRn;X, WqmRd;X
s/m,r,
Wql 1X →Bsq,rX →WqlX →LqX, wherel < s < l 1,
B∞s,1X →CsX →Bs∞,∞X fors∈Z,
Bp,d/p1 Rd, X →L∞Rd, X fors∈Z,
3.5
wherem∈NandCsXdenotes the Holder-Zygmund spaces.
Definition 3.1. LetXbe a Banach space and 1≤u≤2. We sayXhas Fourier typeuif FfL
uRn,X≤CfLuRn,X for eachf∈S
RN, X, 3.6
where 1/u 1/u1,Fu,nXis the smallestC∈0,∞. Let us list some important facts:
iany Banach space has a Fourier type 1,
iiB-convex Banach spaces have a nontrivial Fourier type,
iiispaces having Fourier type 2 should be isomorphic to a Hilbert spaces.
The following corollary follows from5, Theorem 3.1.
Corollary 3.2. LetX be a Banach space having Fourier typeu ∈ 1,2and1 ≤ θ ≤ u. Then the inverse Fourier transform defines a bounded operator
F−1:Bn1/θ−1/u
u,1 Rn, X−→LθRn, X. 3.7
Definition 3.3. LetE1Rn, X, E2Rn, Ybe one of the following systems, where 1≤q≤p ≤
∞:
LqX, LpY
or Bsq,rX, Bsp,rY. 3.8
A bounded measurable functionm:Rn → BX, Yis called a Fourier multiplier fromE
1X
toE2Yif there is a bounded linear operator
Tm:E1X−→E2Y 3.9
such that
Tm
fF−1m·Ff· for eachf∈SX, 3.10
Tmisσ
E1X, E∗1X∗
toσE2Y, E∗2Y∗
The uniquely determined operator Tm is the FM operator induced by m. Note that if
Tm ∈BE1X, E2YandTm∗ mapsE∗2Y∗intoE1∗X∗thenTmsatisfies the weak continuity
condition3.11.
For the definition of Besov spaces and their basic properties we refer to5. Since3.10can be written in the convolution form
Tm
ft
Rn
ˇ
mt−sfsds, 3.12
Corollaries2.6and3.2can be applied to obtainLqRn, X → LpRn, Yregularity for3.10.
Theorem 3.4. LetXandY be Banach spaces having Fourier typeu∈1,2andp,q∈1,∞so that
0≤1/q−1/p≤1/u. Then there is a constantCdepending only onFu,nXandFu,nYso that if
m∈Bu,n11/u 1/p−1/qRn, BX, Y 3.13
thenmis a FM fromLqRn, XtoLpRn, Ywith
TmLqRn,X→LpRn,Y≤CMum, 3.14
where
Mp,qu m inf
,
an1/q−1/pma·Bn1/u1/p−1/q
u,1 Rn,BX,Y :a >0
-. 3.15
Proof. Let 1/q−1/p1−1/θand 1≤q < θ/θ−1≤ ∞. Assume thatm∈SBX, Y. Then ˇ
m ∈SBX, Y. SinceF−1ma·xs a−nms/axˇ , choosing an appropriateaand using
3.7we obtain
mxˇ LθYan−n/θma·x∨L
θY
≤C1an/θ
ma·Bn1/θ−1/u
u,1
xX
≤2C1Mp,qu mxX,
3.16
where C1 depends only on Fu,nY. Since m ∈ SBX, Y we have m∗∨ mˇ ∗ ∈
SBY∗, X∗andMp,q
u m M p,q
u m∗. Thus, in a similar manner as above, we get
mˇ ·∗y∗
LθY≤2C2M p,q
u my∗Y∗ 3.17
for some constantC2depending onFu,nX∗. Hence by3.16-3.17andCorollary 2.6
Tmf
t
Rn
ˇ
satisfies
Tmf
LpRn,Y≤CM p,q
u mfLqRn,X 3.19
for all p, q ∈ 1,∞ so that 0 ≤ 1/q− 1/p ≤ 1/u. Now, taking into account the fact
that SBX, Y is continuously embedded in Bu,n11/u 1/p−1/qBX, Y and using the same
reasoning as5, Theorem 4.3one can easily prove the general casem ∈ Bu,n11/u 1/p−1/qand the weak continuity ofTm.
Theorem 3.5. LetXandY be Banach spaces having Fourier typeu∈1,2andp, q∈1,∞be so that0≤1/q−1/p≤1/u. Then, there exist a constantCdepending only onFu,nXandFu,nYso
that ifm:Rn → BX, Ysatisfy
ϕk·m∈Bu,n11/u 1/p−1/qRn, BX, Y, Mp,qu
ϕk·m
≤A 3.20
thenmis a FM from Bs
q,rRn, XtoBsp,rRn, YandTmBs
q,r→Bp,rs ≤ CAfor eachs ∈ R andr ∈
1,∞.
Taking into consideration Theorem 3.4one can easily prove the above theorem in a similar manner as5, Theorem 4.3.
The following corollary provides a practical sufficient condition to check3.20.
Lemma 3.6. Letn1/u 1/p−1/q< l∈Nandθ∈u,∞. Ifm∈ClRn, BX, Yand
DαmLθI0≤A,
2k−1n1/q−1/pDαmkLθI1≤A, mk· m
2k−1·,
3.21
for eachα∈Nn,|α| ≤landk∈N, thenmsatisfies3.20.
Using the fact thatWl
uRn, BX, Y ⊂ B
n1/u 1/p−1/q
u,1 Rn, BX, Y, the above lemma
can be proven in a similar fashion as5, Lemma 4.10.
Choosingθ∞inLemma 3.6we get the following corollary.
Corollary 3.7Mikhlin’s condition. LetXandYbe Banach spaces having Fourier typeu∈1,2
and0≤1/q−1/p≤1/u. Ifm∈ClRn, BX, Ysatisfies
1 |t||α| n1/q−1/pDαm
L∞Rn,BX,Y≤A 3.22
for each multi-indexαwith|α| ≤ln1/u 1/p−1/q 1, thenmis a FM fromBs
q,rRn, Xto
Bs
p,rRn, Yfor eachs∈Randr∈1,∞.
Proof. It is clear that fort∈I0
Moreover, fort∈I1we have
2k−1n1/q−1/pDαm
ktBX,Y
2k−1|α| n1/q−1/pm2k−1t
BX,Y
≤2k−1t|α| n1/q−1/pm2k−1t
BX,Y,
3.24
which implies
2k−1n1/q−1/pDαmkL∞I1≤1 |t||α| n1/q−1/pDαmtL∞Rn. 3.25
Hence byLemma 3.6,3.22implies assumption3.20ofTheorem 3.5.
Remark 3.8. Corollary 3.7particularly implies the following facts.
aifXandY are arbitrary Banach spaces thenln1/p 1/q 1,
bifX andY be Banach spaces having Fourier typeu∈ 1,2and 1/q−1/p 1/u
thenl1, suffices for a function to be a FM inBs
q,rRn, X, Bp,rs Rn, Y.
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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