Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID pot

Volume 2010, Article ID 850125, 12 pagesdoi:10.1155/2010/850125 Research Article On Integral Operators with Operator-Valued Kernels Rishad Shahmurov1, 2 1 Department of Mathematics, Univ

Volume 2010, Article ID 850125, 12 pages

doi:10.1155/2010/850125

Research Article

On Integral Operators with

Operator-Valued Kernels

Rishad Shahmurov1, 2

1 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA

2 Vocational 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

Copyrightq 2010 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

get L q S; X → L p T; Y estimates under some natural conditions on the kernel k : T × S →

B X, Y, where X and Y are Banach spaces, and T,T , μ  and S,S , ν 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



Kf

· 



S

defines a bounded linear operator

K : L p S, X −→ L p T, Y 1.2

provided some measurability conditions and the following assumptions

sup

s ∈S



T

sup

t ∈T



S

k∗t, sy∗

X∗dν s ≤ C2y∗

Y∗, ∀y∗∈ Y∗

1.3

K : L q S, X −→ L p T, Y 1.4

sup

s ∈S



T

kt, sx θ

Y dt

1/θ

sup

t ∈T



S

k∗t, sy∗θ

X∗ds

1/θ

≤ C2y∗

Y∗, ∀y∗∈ Y∗,

1.5

where

1

q−1

space BX, Y of bounded linear operators from X to Y is endowed with the usual uniform

operator topology

x∗∈BY |x∗x| ∀x ∈ X. 1.7

It is clear that if Y τ-norms X then the canonical mapping

u : X −→ Y∗ with

y, ux  x, y 1.8

is an isomorphic embedding with

1

LetT,T , μ  and S,S , ν  be σ-finite positive measure spaces and

finite

S



A∈

S

: νA < ∞ ,

full

S



A∈

S

from S into X, that is,

ε S, X 

i1

x i1A i : x i ∈ X, A i∈finite

S

Lemma 2.3

f, g 



T

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

2. Lq → Lp Estimates for Integral Operators

K L q S,X → L p T,Y ≤ C 2.1

Condition 1 For any A∈finite

a there is T A,x∈full



A

T

so that the Bochner integral



S

K : ε S, X −→ L0T, Y, 2.4

Condition 2 The kernel k : T × S → BX, Y satisfies the following properties:

b there is S x∈full

Theorem 2.1 Suppose 1 ≤ θ < ∞ and the kernel k : T × S → BX, Y satisfies Conditions 1 and 2 Then the integral operator1.1 acting on εS, X extends to a bounded linear operator

K : L1S, X −→ L θ T, Y. 2.6

Proof Let f  n

and using the general Minkowski-Jessen inequality with the assumptions of the theorem we

Kft L θ T,Y≤

⎡

T



S





k t, s

n

i1

x i1A i s





Y

dν s

dμ t

⎤

⎦

1/θ

≤



S

⎛

T





k t, s

n

i1

x i1A i s





θ

Y

dμ t

⎞

⎠

1/θ

dν s

≤



S

⎡

T

i1

1A i skt, sx iY

dμ t

⎤

⎦

1/θ

dν s

≤



S

n

i1

1A i s



T

kt, sx iθ

Y dμ t

1/θ

dν s

≤



S

n

i1

1A i skt, sx iL θ T,Y dν s ≤ C1

n

i1

x iX



S

1A i sdνs

 C1

n

i1

x iX ν A i   C1f

L1S,X

2.7

Hence,K L1→ L θ ≤ C1

Condition 3 For each y∗∈ Z there is T y∗∈full

T so that for all t ∈ T y∗,

X∗is measurable for all x∗∈ X∗,

b there is S x∈full

k∗t, sy∗

L θ S,X∗ ≤ C2y∗

Theorem 2.2 Let Z be a separable subspace of Y∗ that τ-norms Y Suppose 1 ≤ θ < ∞ and k :

to a bounded linear operator

K : L θ S, X −→ L∞T, Y. 2.9

Proof Suppose f ∈ εS, X and y∗ ∈ Z are fixed Let T f ,T y∗ ∈ full

T

countable sets still belongs tofull

 y∗,

Kf

t Y y∗,



S

k t, sfsdνs



≤



S

k∗t, sy∗

f sdν s

≤ k∗t, sy∗L θ S,X∗ fs L S,X

≤ C2y∗f L S,X

2.10

Since, T f ∩ T y∗∈full

Kf

L∞T,Y ≤ C2τf L S,X 2.11

5.1.2 The next lemma is a more general form of 3, Lemma 3.9

Lemma 2.3 Suppose a linear operator

K : ε S, X −→ L θ T, Y L∞T, Y 2.12

satisfies

Kf

L θ T,Y ≤ C1f

L1S,X , Kf

L∞T,Y ≤ C2f

L S,X 2.13

Then, for 1/q − 1/p  1 − 1/θ and 1 ≤ q < θ/θ − 1 ≤ ∞ the mapping K extends to a bounded

linear operator

K : L q S, X −→ L p T, Y 2.14

with

K L q → L p ≤ C1θ/p C21−θ/p. 2.15

Proof Let us first consider the conditional expectation operator



K0f

Kf

1 |

K0f

L θ T,Y ≤ C1f

L1S,X < ∞,

K0f

L∞T,Y ≤ C2f

L S,X < ∞. 2.17

K0fL p T,Y ≤ C1θ/p C21−θ/pf

L q S,X 2.18

3.9, one can easily show the assertion of this lemma

Theorem 2.4 operator-valued Schur’s test Let Z be a subspace of Y∗that τ-norms Y and 1/q−

and 3 with respect to Z Then integral operator1.1 extends to a bounded linear operator

K : L q S, X −→ L p T, Y 2.19

with

K L q → L p ≤ C1θ/p τC21−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 choosing θ 1 we get the original results in 3

spaces and weak continuity and duality results see 3 The next corollary plays important role in the Fourier Multiplier theorems

Corollary 2.6 Let Z be a subspace of Y∗that τ-norms Y and 1/q − 1/p  1 − 1/θ for 1 ≤ q <

strongly measurable on Z and

kx L θ R n ,Y≤ C1x X , ∀x ∈ X,

k∗y∗

LθRn,X∗ ≤ C2y∗

Then the convolution operator defined by



Kf

t 



R n

satisfies K L → L ≤ C1θ/p C21−θ/p.

It is easy to see that k : R n → BX, Y satisfies Conditions1,2, and3with respect to

Z Thus, assertion of the corollary follows fromTheorem 2.4

3 Fourier Multipliers of Besov Spaces

FM and Besov spaces

Consider some subsets{J k}∞k0and{I k}∞k0of R ngiven by

J0 {t ∈ R n:|t| ≤ 1}, J k t ∈ R n: 2k−1< |t| ≤ 2 k!

I0 {t ∈ R n:|t| ≤ 2}, I k t ∈ R n: 2k−1< |t| ≤ 2 k 1!

3.1

∞

k−∞

ψ

2−k s

 1 for s ∈ R \ {0},

ϕ k t  ψ2−k |t|, ϕ0t  1 −∞

k1

ϕ k t for t ∈ R n

3.2

Note that

which

f

B s q,r R n ,X:2ks"

 ˇϕ k ∗ f#∞k0

l r L q R n ,X

≡

⎧

⎪

⎪

⎪

⎪

(∞

k0

2ksrϕˇk ∗ fr

L q R n ,X

)1/r

if r / ∞

sup

k ∈N

*

2ksϕˇk ∗ f

L q R n ,X

+

3.4

is finite; here q and s are main and smoothness indexes respectively The Besov space has

significant interpolation and embedding properties:

B s q,r R n ; X L q R n ; X, W m

q R d ; X

s/m,r ,

W q l 1X → B s

q,r X → W l

q X → L q X, where l < s < l 1,

B ∞,1 s X → C s X → B s

B p,1 d/p

R d , X

→ L∞R d , X

3.5

Definition 3.1 Let X be a Banach space and 1 ≤ u ≤ 2 We say X has Fourier type u if

Ff L R n ,X≤ Cf

L u R n ,X for each f ∈ SR N , X

i any Banach space has a Fourier type 1,

ii B-convex Banach spaces have a nontrivial Fourier type,

iii spaces having Fourier type 2 should be isomorphic to a Hilbert spaces

Corollary 3.2 Let X be a Banach space having Fourier type u ∈ 1, 2 and 1 ≤ θ ≤ u Then the inverse Fourier transform defines a bounded operator

F−1: B n u,1 1/θ−1/uR n , X  −→ L θ R n , X . 3.7

Definition 3.3 Let E1R n , X , E2R n , Y  be one of the following systems, where 1 ≤ q ≤ p ≤

∞:



B s q,r X, B s

such that

T m



f

 F−1

T m is σ

E1 X, E∗

1X∗ to σ

E2 Y, E∗

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

m maps E∗2Y∗ into E∗

T m

f

t 



R n

ˇ

Theorem 3.4 Let X and Y be Banach spaces having Fourier type u ∈ 1, 2 and p, q ∈ 1, ∞ so that

m ∈ B n 1/u 1/p−1/q

then m is a FM from L q R n , X  to L p R n , Y  with

T mL q R n ,X  → L p R n ,Y≤ CM u m, 3.14

where

M p,q u m  inf

,

a n 1/q−1/p ma· B n 1/u 1/p−1/q

u,1 R n ,B X,Y : a > 0

Proof Let 1/q − 1/p  1 − 1/θ and 1 ≤ q < θ/θ − 1 ≤ ∞ Assume that m ∈ SBX, Y Then

ˇ

3.7 we obtain

 ˇmx L θ Y  a n −n/θ ma·x∨

L θ Y

≤ C1a n/θ ma· B n

≤ 2C1M p,q u mx X ,

3.16

 ˇm·∗y∗

L θ Y ≤ 2C2Mp,q u my∗



T m f

t 



R n

ˇ

T m f

L p R n ,Y≤ CM p,q

u mf

Theorem 3.5 Let X and Y be Banach spaces having Fourier type u ∈ 1, 2 and p, q ∈ 1, ∞ be so

that 0 ≤ 1/q − 1/p ≤ 1/u Then, there exist a constant C depending only on F u,n X and F u,n Y so

that if m : R n → BX, Y satisfy

ϕ k · m ∈ B n 1/u 1/p−1/q

then m is a FM from B s

q,r R n , X  to B s

p,r R n , Y  and T mB s

q,r → B s

1, ∞.

The following corollary provides a practical sufficient condition to check 3.20

Lemma 3.6 Let n1/u 1/p − 1/q < l ∈ N and θ ∈ u, ∞ If m ∈ C l R n , B X, Y and

D α mL θ I0≤ A,

2k−1n 1/q−1/p

D α m kL θ I1≤ A, m k ·  m2k−1·,

3.21

for each α ∈ N n , |α| ≤ l and k ∈ N, then m satisfies 3.20.

u R n , B X, Y ⊂ B u,1 n 1/u 1/p−1/q R n , B X, Y, the above lemma

Corollary 3.7 Mikhlin’s condition Let X and Y be Banach spaces having Fourier type u ∈ 1, 2

and 0 ≤ 1/q − 1/p ≤ 1/u If m ∈ C l R n , B X, Y satisfies

for each multi-index α with |α| ≤ l  n1/u 1/p − 1/q 1, then m is a FM from B s

q,r R n , X  to

B s

p,r R n , Y  for each s ∈ R and r ∈ 1, ∞.

Proof It is clear that for t ∈ I0

Moreover, for t ∈ I1we have



2k−1n 1/q−1/p D α m k t B X,Y 2k−1|α| n1/q−1/pm2 k−1t

B X,Y

≤2k−1t|α| n1/q−1/pm2 k−1t

B X,Y ,

3.24

which implies

2k−1n 1/q−1/p

D α m kL∞I1≤ 1 |t| |α| n1/q−1/p D α m t L∞R n. 3.25

Remark 3.8. Corollary 3.7particularly implies the following facts

b if X and Y be Banach spaces having Fourier type u ∈ 1, 2 and 1/q − 1/p  1/u

q,r R n , X , B s

p,r R n , 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 of Grundlehren der Mathematischen

Wissenschaften, Springer, Berlin, Germany, 1976.

5 M Girardi and L Weis, “Operator-valued Fourier multiplier theorems on Besov spaces,” Mathematische

Nachrichten, vol 251, pp 34–51, 2003.

References

1 G B Folland, Real Analysis: Modern Techniques and Their Applications, Pure and. .. Girardi and L Weis, “Integral operators with operator-valued kernels,” Journal of Mathematical

Analysis and Applications, vol 290, no 1, pp 190212, 2004.

4 J Bergh and. .. NY, USA, 1984

2 R Denk, M Hieber, and J Pr ăuss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type,” Memoirs of the American Mathematical Society, vol 166,