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Volume 2010, Article ID 850125,12pages doi:10.1155/2010/850125

Research Article

On Integral Operators with

Operator-Valued Kernels

Rishad Shahmurov

1, 2

1Department 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;XLpT;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

(2)

provided some measurability conditions and the following assumptions

sup

sS

T

kt, sxYdμtC1xX,xX,

sup

tT

S

k∗t, sy∗XdνsC2yY,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×SBX, Ysatisfies the conditions

sup

sS

T

kt, sxθ

Ydt

1

C1xX,xX,

sup

tT

S

kt, syθ Xds

1

C2yY,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 ofXτ-normsX, whereτ≥1, provided

xXτ sup

xBY|x

x| ∀xX.

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

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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 :xiX, Ai∈ finite

S

, nN

. 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 : SX is measurable if there is a sequence fnn1 ⊂

εS, Xconvergingin the sense ofX topologytof and it isσX,Γ-measurable provided

f·, x∗:S Kis measurable for eachxΓX. Suppose 1p≤ ∞and 1/p 1/p1. There is a natural isometric embedding ofLpT, Y∗intoLpT, Y∗given by

f, g

T

ft, gtdμt forgLpT, 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

p

Estimates for Integral Operators

In this section, we identify conditions on operator-valued kernel k : T ×SBX, Y, extending theorems in3so that

KLqS,XLpT,YC 2.1

for 1≤qp. To prove our main result, we shall use some interpolation theorems ofLpspaces.

Therefore, we will study L1S, X → T, Y andS, X → L∞T, Yboundedness of

integral operator1.1. The following two conditions are natural measurability assumptions

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Condition 1. For anyA∈finiteS and eachxX

athere isTA,x

full

T so that iftTA,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 eachfεS, 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×SBX, Ysatisfies the following properties:

aa real-valued mappingkt, sxθXis product measurable for allxX,

bthere isSx∈fullS so that

kt, sxLθT,YC1xX 2.5

for 1≤θ <∞andxX.

Theorem 2.1. Suppose1≤θ <and the kernelk:T×SBX, Ysatisfies Conditions1and2. Then the integral operator1.1acting onεS, Xextends to a bounded linear operator

K:L1S, X−→T, Y. 2.6

Proof. Let f ni1xi1Ais ∈ εS, X be fixed. Taking into account the fact that 1 ≤ θ

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obtain

Kft

LθT,Y≤ ⎡ ⎣ T S kt, s n

i1

xi1Ais Y dνs θ dμt ⎤ ⎦ 1S ⎛ ⎝ T kt, s n

i1

xi1Ais θ Y dμt ⎞ ⎠ 1 dνsS ⎡ ⎣ T n

i1

1Aiskt, sxiY

θ dμt ⎤ ⎦ 1 dνsS n

i1

1Ais

T

kt, sxiθYdμt

1 dνsS n

i1

1Aiskt, sxiLθT,Y dνsC1

n

i1

xiX

S

1Aisdνs

C1 n

i1

xiXνAi C1fL1S,X.

2.7

Hence,KL1LθC1.

Condition 3. For eachy∗∈Zthere isTy∗∈

full

T so that for alltTy∗,

aa real-valued mappingk∗t, sx∗θX∗is measurable for allx∗∈X∗,

bthere isSx∈fullS so that

kt, sy

LθS,X∗≤C2y

Y∗ 2.8

for 1≤θ <∞andxX.

Theorem 2.2. LetZbe a separable subspace ofYthatτ-normsY. Suppose1 ≤ θ <and k :

T×SBX, Ysatisfies Conditions1and3. Then integral operator1.1acting onεS, Xextends

to a bounded linear operator

K:S, X−→L∞T, Y. 2.9

Proof. SupposefεS, 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

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countable sets still belongs tofullT and the intersection of these sets should be nonempty. If

tTfTy∗then, by using H ¨older’s inequality and assumptions of the theorem, we get

y,KftY

y,

S

kt, sfsdνs

S

kt, syfsdνs

k∗t, sy∗LθS,XfsLθS,X

C2yfLθS,X.

2.10

Since,TfTy∗∈fullT andZ τ-normsY

Kf

LT,YC2τfLθS,X. 2.11

Hence,KL

θL∞≤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−→T, Y L∞T, Y 2.12

satisfies

KfL

θT,YC1fL1S,X, KfLT,YC2fLθS,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

KLqLp ≤C1θ/pC21−θ/p. 2.15

Proof. Let us first consider the conditional expectation operator

K0f

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whereis aσ-algebra of subsets ofB∈finiteT . From2.13it follows that K0f

LθT,YC1fL1S,X<, K0f

LT,YC2fL

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 ofYthatτ-normsYand1/q

1/p1−1/θfor1≤q < θ/θ−1≤ ∞. Supposek:T ×SBX, Ysatisfies Conditions1,2, and3with respect toZ. Then integral operator1.1extends to a bounded linear operator

K:LqS, X−→LpT, Y 2.19

with

KLqLp ≤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.

ForLestimatesit 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 ofYthatτ-normsY and1/q−1/p 1−1/θfor1 ≤q < θ/θ−1 ≤ ∞. Supposek :Rn BX, Yis strongly measurable onX,k: Rn BY, Xis

strongly measurable onZand

kxLθRn,YC1xX,xX,

ky

LθRn,X∗ ≤C2y

Y,y∗∈Y.

2.21

Then the convolution operator defined by

Kft

Rn

ktsfsds fortRn 2.22

satisfiesKLqLp ≤C1 θ/pC

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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{tRn:|t| ≤1}, Jk tRn: 2k−1<|t| ≤2k

!

forkN,

I0{tRn:|t| ≤2}, Ik tRn: 2k−1<|t| ≤2k 1

!

forkN.

3.1

Let us define the partition of unity{ϕk}kN0of functions fromSR

n, R. SupposeψSR, R

is a nonnegative function with support in2−1,2, which satisfies

k−∞

ψ2−ks1 for sR\ {0},

ϕkt ψ

2−k|t|, ϕ0t 1−

k1

ϕkt fortRn.

3.2

Note that

suppϕkIk, suppϕkIk. 3.3

Let 1 ≤ qr ≤ ∞andsR. The Besov space is the set of all functionsfSRn, Xfor

which

fBs

q,rRn,X:

2ks"ϕˇkf

#

k0

lrLqRn,X

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(

k0

2ksrϕˇ

kfrLqRn,X )1/r

ifr /

sup

kN0 *

2ksϕˇ

kfLqRn,X +

ifr

(9)

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,

Bs,1X →CsX →Bs,X forsZ,

Bp,d/p1 Rd, XLRd, X forsZ,

3.5

wheremNandCsXdenotes the Holder-Zygmund spaces.

Definition 3.1. LetXbe a Banach space and 1≤u≤2. We sayXhas Fourier typeuif FfL

uRn,XCfLuRn,X for eachfS

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 ≤ θu. Then the inverse Fourier transform defines a bounded operator

F−1:Bn1−1/u

u,1 Rn, X−→Rn, X. 3.7

Definition 3.3. LetE1Rn, X, E2Rn, Ybe one of the following systems, where 1≤qp

∞:

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 eachfSX, 3.10

Tmisσ

E1X, E∗1X∗

toσE2Y, E∗2Y∗

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The uniquely determined operator Tm is the FM operator induced by m. Note that if

TmBE1X, 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

mtsfsds, 3.12

Corollaries2.6and3.2can be applied to obtainLqRn, XLpRn, 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

mBu,n11/u 1/p−1/qRn, BX, Y 3.13

thenmis a FM fromLqRn, XtoLpRn, Ywith

TmLqRn,XLpRn,YCMum, 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/p1−1and 1≤q < θ/θ−1≤ ∞. Assume thatmSBX, Y. Then ˇ

mSBX, Y. SinceF−1ma·xs anms/axˇ , choosing an appropriateaand using

3.7we obtain

mxˇ LθYann/θma·x∨L

θY

C1an/θ

ma·Bn1−1/u

u,1

xX

≤2C1Mp,qu mxX,

3.16

where C1 depends only on Fu,nY. Since mSBX, Y we have m∗∨ mˇ ∗ ∈

SBY, XandMp,q

u m M p,q

u m∗. Thus, in a similar manner as above, we get

mˇ ·y

LθY≤2C2M p,q

u myY∗ 3.17

for some constantC2depending onFu,nX∗. Hence by3.16-3.17andCorollary 2.6

Tmf

t

Rn

ˇ

(11)

satisfies

Tmf

LpRn,YCM 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 casemBu,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·mBu,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,rBp,rsCAfor eachsR 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< lNandθ∈u,∞. IfmClRn, BX, Yand

DαmLθI0A,

2k−1n1/q−1/pDαmkLθI1≤A, mk· m

2k−1·,

3.21

for eachαNn,|α| ≤landkN, thenmsatisfies3.20.

Using the fact thatWl

uRn, BX, YB

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. IfmClRn, BX, Ysatisfies

1 |t||α| n1/q−1/pDαm

LRn,BX,YA 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 eachsRandr∈1,.

Proof. It is clear that fortI0

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Moreover, fortI1we 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/pDαmkLI1≤1 |t||α| n1/q−1/pDαmtLRn. 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.

References

1 G. B. Folland,Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1984.

2 R. Denk, M. Hieber, and J. Pr ¨uss, “R-boundedness, Fourier multipliers and problems of elliptic and parabolic type,”Memoirs of the American Mathematical Society, vol. 166, no. 788, pp. 1–106, 2003.

3 M. Girardi and L. Weis, “Integral operators with operator-valued kernels,”Journal of Mathematical Analysis and Applications, vol. 290, no. 1, pp. 190–212, 2004.

4 J. Bergh and J. L ¨ofstr ¨om,Interpolation spaces. An Introduction, vol. 223 ofGrundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1976.

References

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