INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL MARIA GIRARDI AND LUZ WEI Abstract. Under fairly mild measurability and integrability conditions on operator-valued kernels, boundedness results for integral operators on Bochner spaces L p X) are given. In particular, these results are applied to convolutions operators. 1. INRODUCION One of the most commonly used boundedness criterion for integral operators states that, for 1 p and σ-finite measure spaces,σ,µ) and, Σ,ν), a measurable kernel k : C defines a bounded linear operator K : L p, C) L p,c) via Kf) ):= k,s) fs) dν s) provided sup s k t, s) dµ t) C and sup k t, s) dνs) C 1.1) t see, e.g. [5, heorem 6.18]). In the theory of evolution equations one frequently uses operatorvalued analogs of this situation, where the kernel k maps into the space B X, Y ) of bounded linear operators from a Banach space X into a Banach space Y and then one desires the boundedness of the corresponding integral operator K : L p, X) L p,y ). uch integral operators appear, for example, in solution formulas for inhomogeneous Cauchy problems see, e.g. [10]) and for Volterra integral equations see, e.g. [11]) as well as in control theory see, e.g. [2]); furthermore, the stability of such solutions is often expressed in terms of the boundedness of these operators. However, difficulties can easily arise since in many situations the kernel k is not measurable with respect to the operator norm because the range of k is not essentially) valued in a separable subspace of B X, Y ). his paper presents boundedness results for integral operators with operatorvalued kernels under relatively mild measurability and integrability conditions on the kernels. Date: 4 August 2003. 2000 Mathematics ubject Classification. Primary 45P05, 47G10, 46E40. econdary 46B20. Key words and phrases. integral operators on Bochner spaces, operator-valued kernels, convolution with operator-

2 GIRARDI AND WEI he first step is to place a mild measurability condition on a kernel k : BX, Y )to guarantee that if f is in the space E, X) of finitely-valued finitely-supported measurable functions then the Bochner integrals Kf) ) := k,s)[fs)] dν s) 1.2) define a measurable function from into Y, thus defining a mapping K : E, X) L 0,Y ). hen, to ensure that K linearly extends to a desired superspace, one adds integability conditions, which replace 1.1) in the scalar case and, roughly speaking, take the form sup k t, s) x Y dµ t) C x X for each x X 1.3) s k t, s) y X dνs) C y Y for each y Y 1.4) sup t along with appropriate measurability conditions see ection 3 for the precise formulations). Assume k has the appropriate measurability conditions. heorem 3.4 shows that if k satisfies 1.3) then K extends to a bounded linear operator from L 1, X) intol 1,Y ); heorem 3.6 shows that if k satisfies 1.4) then K extends to a bounded linear operator from the closure of E, X) in the L -norm into L,Y ). hen heorem 3.8 uses an interpolation argument to show that if k satisfies 1.3) and 1.4) then K extends to a bounded linear operator from L p, X) intol p,y ) for 1 <p<. he case p = is more delicate since E, X) is not necessarily dense in L, X). heorem 3.11 shows that if k satisfies 1.4) then K can be extended to a bounded linear operator from L, X) into the space of w -measurable µ-essentially bounded functions from into Y where the integrals in 1.2) exists a.e) as Dunford integrals for each f L, X); also, sufficient conditions are given to guarantee that K maps L, X) intol,y ). Using ideas from the Geometry of Banach paces, Example 3.13 shows that, without further assumptions, it is necessary to pass to Y in heorem 3.11. As an immediate consequence of these results, Corollary 5.1 gives boundedness results for convolution operators with operator-valued kernels. A similar result, which inspired this paper, was used to obtain operator-valued Fourier multiplier results [8, 7]. 2. NOAION and BAIC hroughout this paper, X, Y, and Z are Banach spaces over the field K of R or C. Also, X is the topological) dual of X and BX) is the closed) unit ball of X. he space B X, Y ) of bounded linear operators from X into Y is endowed with the usual uniform operator topology. A subspace Z of Y τ-norms Y, where τ 1, provided

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 3 If Zτ-norms Y, then the natural mapping j : Y Z given by z,jy := y, z for z Z is an isomorphic embedding with 1 τ y Y jy) Z y Y, in which case Y is identified as a subspace of Z.,Σ,µ) and, Σ,ν) are σ-finite positive) measure spaces; Σ finite := {A Σ : νa) < } Σ full := {A Σ : ν \ A) =0}, with similar notation for the corresponding subsets of Σ. E, X) is the space of finitely-valued finitely-supported measurable functions from into X, i.e. { n } E, X) = x i 1 Ai : x i X, A i Σ finite,n N. i=1 Let Γ be a subspace of X. A function f : X is measurable provided there is a sequence f n ) from E, X) so that lim n fs) f n s) X = 0 for ν-a.e. s σ X, Γ)-measurable provided f ),x : K is measurable for each x Γ. he following fact will be used c.f., e.g., [3, Corollary II.1.4]). Fact 2.1 Pettis s Measurability heorem). A function f : X is measurable if and only if i) f is essentially separably valued ii) f is σ X, Γ)-measurable for some subspace Γ of X that 1-norms X. L 0, X) is the space of equivalence classes of) measurable functions from into X. he Bochner- Lebesgue space L p, X), where 1 p, is endowed with its usual norm topology. he space L w,z )ofµ-essentially bounded σ Z,Z)-measurable functions from into Z is endowed with the µ-essential supremum norm, under which it becomes a Banach space. E, X) is norm dense in L p, X) for 1 p<. Let L 0, X) be the closure of E, X) in the L, X)-norm. If X is infinite-dimensional, then L 0, X) L, X) provided Σ contains a countable number of pairwise disjoint sets of strictly positive measure). L 0, X) can be described as follows. Proposition 2.2. Let f L 0, X). hen f L 0, X) if and only if { } 1) inf f1 L,X) \A : A Σfinite = 0 2) there is B Σ full so that the set {f s) :s B}is relatively compact in X.

4 GIRARDI AND WEI Conversely, for ε>0, conditions 1) and 2) give a set G := A B) Σ finite so that f1\g L,X) <ε and {f s) :s G} is relatively compact ; thus allowing one to find, via a finite covering of the set f G) byε-balls, a function f ε E, X), with support in G, so that f f ε <ε. L,X) Lemma 2.3 will help to deal with the fact that E, X) is usually) not norm dense in L, X). Lemma 2.3. Let f L, X) and ε>0. here is a sequence {g n } f s) = g n s) for a.e. s. g n s) X 1 + ε) f L,X) from E, X) so that Proof. Fix a sequence {ε j } j=1 of positive numbers so that ε 1 = 1 and j=1 ε j < 1+ε. Choose a sequence {f j } j=1 from E, X) so that, for a.e. s, f j s) f s) as j f j s) X f s) X for each j N. Find a sequence { k } k=1 each k, of pairwise disjoint sets from Σfinite so that ν \ k=1 k) = 0 and, for f j f uniformly on k f j s) X f s) X for each s k and j N. Hence, on each k, there is a sequence {g k j } j=1 from E k,x) so that f s) = gj k L k,x) gj k s) j=1 for each s k ε j f L,X) for each j N. For n N, let hus g n := k<n g k n1 k ) + n j=1 g n j 1 n. g 1 = g 1 1 1 1 g 2 = g 1 2 1 1 +g 2 1+g 2 2)1 2 1 2 3 3 3

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 5 Note that if s k then and, by the triangle inequality, o clearly the g n s do as they should. g n s) = gj k s) j=1 g n s) X gj k s). X j=1 Let 1 p and 1 p + 1 p = 1. here is a natural isometric embedding of L p,z )into [L p,z)] given by f,g := f t),gt) dµ t) for g L p,z ),f L p,z). here also is a natural isometric embedding of L 1,Y )into [ L w,y ) ] ; indeed, for g = n i=1 y i 1 B i L 1,Y ) and f L w,y ) let n f,g := g t),ft) dµ t) = yi,ft) dµ t) i=1 B i and observe that g [L w,y )] = g L 1,Y ). For a mapping k : BX, Y ) the mapping k : BY,X ) is defined by k t, s) :=[kt, s)]. Non-numerical subscripts on constants indicate dependency. All other notation and terminology, not otherwise explained, are as in [3, 9]. 3. MAIN REUL everal conditions on a kernel k : BX, Y ) will be considered. he first one is a mild measurability condition. Definition 3.1. k : BX, Y ) satisfies condition C 0 ) provided that for each A Σ finite and each x X there is A,x Σ full so that if t A,x then the Bochner integral k t, s) xdνs) exists the mapping A

6 GIRARDI AND WEI Remark 3.2. Let k : BX, Y ) satisfy condition C 0 ). hen for each f E, X) there is f Σ full so that if t f then the Bochner integral Kf)t) := and 3.1) defines a linear mapping kt, s)[fs)] dν s) exists 3.1) K : E, X) L 0,Y ). 3.2) Next integrability conditions on k are added to ensure that the mapping K in 3.2) extends to the desired superspaces. Condition C 1 ) will be used for the L 1 -case in heorem 3.4. Definition 3.3. k : BX, Y ) satisfies condition C 1 ) provided there is a constant C 1 so that for each x X the mapping t, s) kt, s) x Y R is product measurable there is x Σ full so that for each s x. k t, s) x Y dµ t) C 1 x X Note that the first condition guarantees that the mapping t kt, s) x Y R is measurable for ν-a.e. s. Also, the first condition is often satisfied even though the mapping t, s) k t, s) x Y may not be product measurable. he L 1 -case is a straightforward extension of the scalar-valued situation. heorem 3.4. Let k : BX, Y ) satisfy conditions C 0 ) and C 1 ). operator Kf) ) := extends to a bounded linear operator k,s)fs) dν s) for f E, X) hen the integral K : L 1, X) L 1,Y ) of norm at most the constant C 1 from Definition 3.3. Proof. Fix f = n i=1 x i1 Ai E, X) with the A i s disjoint. By condition C 1 ), for each i, [ ] k t, s) x i 1 Ai s) Y dν s) dµ t) = kt, s) x i Y dµ t) 1 Ai s) dν s) C x 1 s) dν s) 3.3)

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 7 Condition C 0 ) gives that Kf L 0,Y ) and also, combined with 3.3), that n Kf L1,Y ) = k t, s) x i 1 Ai s) dν s) dµ t) i=1 Y n k t, s) x i 1 Ai s) Y dν s) dµ t) his completes the proof. i=1 n C 1 x i X ν A i ) = C 1 f L1,X). i=1 Condition C 0 ) will be used for the L 0 -case in heorem 3.6. Definition 3.5. Let Z be a subspace of Y. hen k : BX, Y ) satisfies condition C 0 ), with respect to Z, provided there is a constant C 0 so that for each y Z there is y Σ full so that for each t y the mapping s k t, s) y X R is measurable k t, s) y X dν s) C 0 y Y. heorem 3.6. Let Z be a subspace of Y that τ-norms Y. Let k: BX, Y ) satisfy conditions C 0 ) and C 0 ) with respect to Z. hen the integral operator Kf) ) := k,s)fs) dν s) for f E, X) extends to a bounded linear operator K : L 0, X) L,Y ) of norm at most τ C 0 where the constant C 0 is from Definition 3.5. Proof. Fix f E, X). Fix y Z. Find the corresponding sets f, y Σ full of conditions C 0 ) and C 0 ). If t f y then Kf)t),y = k t, s) f s) dν s),y f s),k t, s) y dν s) k t, s) y X f s) X dν s) ince f y Σ full and Zτ-norms Y, f L,X) C0 y Y. 0 from the definitions

8 GIRARDI AND WEI Remark 3.7 on heorem 3.6). Note that K maps L 0, X) intol 0,Y ) provided that for each x X and A Σ finite f t k t, s) xdνs) Y 3.4) A defines a function in L 0,Y ). his will be the case, for example, if µ is a Radon measure on a locally compact Hausdorff space e.g., is a Borel subset of R N, endowed with the Lebesgue measure) and 3.4) defines a function in { { g1 } } C 0,Y ) := g: Y g is continuous and inf L \B : B is compact =0 ; indeed, conditions 1) and 2) of Proposition 2.2 are then fulfilled. Interpolating between heorems 3.4 and 3.6 gives the L p -case for 1 <p<. heorem 3.8. Let Z be a subspace of Y that τ-norms Y and 1 <p<.letk: BX, Y ) satisfy conditions C 0 ), C 1 ), and C 0 ) with respect to Z. hen the integral operator Kf) ) := k,s)fs) dν s) for f E, X) extends to a bounded linear operator K : L p, X) L p,y ) of norm at most C 1 ) 1/p τ C 0 ) 1/p where the constants C 1 and C 0 are from Definitions 3.3 and 3.5. Proof. he proof follows directly from heorems 3.4 and 3.6 and Lemma 3.9. he below interpolation lemma is a slight improvement on [1, hm. 5.1.2]. Lemma 3.9. Let the linear mapping satisfy, for each f E, X), K : E, X) L 1,Y )+L,Y ) Kf L1,Y ) c 1 f L1,X) < Kf L,Y ) c f L,X) <. hen, for each 1 <p<, the mapping K extends to a bounded linear operator of norm at most c 1 ) 1/p c ) 1/p. Proof. Fix B Σ finite K : L p, X) L p,y ) and a finite measurable partition π of B. Let Σ 0 := σ B π) betheσ-algebra

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 9 given by K 0 f := E Kf)1 B Σ 0 ) where E Σ 0 ) is the conditional expectation operator relative to Σ 0, satisfies K 0 f L1,Y ) c 1 f L1,X) < K 0 f L,Y ) c f L,X) < for each f E, X). Furthermore, K 0 : L 0, X) L 0,Y ). hus, by [1, hm. 5.1.2], for each p 1, ), the linear mapping K 0 extends to a bounded linear operator from L p, X) to L p,y ) of norm at most c 1 ) 1/p c ) 1/p. Next, fix f E, X) and p 1, ). By assumption, Kf L 1,Y ) L,Y ); thus, Kf L p,y ). Fix B Σ finite. ince is σ-finite, it suffices to show Kf)1 B Lp,Y ) c 1 ) 1/p c ) 1/p f. 3.5) Lp,X) Find a sequence {g n } of functions from E,Y ) that are supported on B and, for µ-a.e. t, g n Kf)t)1 B t) 3.6) g n t) Y Kf)t)1 B t) Y. Let Σ n := σ B g 1,...g n )betheσ-algebra of subsets of B that is generated by {g 1,...,g n }. Note that Kf)1 B is the limit in L p,y )of{g n } by 3.6)) and thus also of {E Kf)1 B Σ n )} since Kf)1 B E Kf)1 B Σ n ) Lp,Y ) Kf)1 B g n Lp,Y ) + E g n Kf)1 B Σ n ) Lp,Y ). But by the previous paragraph, for each n N, E Kf)1 B Σ n ) Lp,Y ) c 1 ) 1/p c ) 1/p f. Lp,X) hus 3.5) holds. Condition C ), a strengthening of condition C 0 ), will be used for the L -case in heorem 3.11. Definition 3.10. Let Z be a subspace of Y. hen k : BX, Y ) satisfies condition C ), with respect to Z, provided there is a constant C and 0 Σ full so that for each t 0 and y Z the mapping s k t, s) y X is measurable k t, s) y X dν s) C y Y. he L -case is more delicate since, in general, E, X) is not norm dense in L, X). heorem 3.11. Let Z be a subspace of Y that τ-norms Y. Let k: BX, Y ) satisfy conditions C 0 ) and C ) with respect to Z. In particular, C ) gives that for each t 0 Σ full and each y Z k t, s) y X dν s) C y Y.

10 GIRARDI AND WEI defined by Kf) ) := k,s)fs) dν s) for f E, X) 3.7) extends identifying Y as a subspace of Z ) to a bounded of norm at most C ) linear operator K : L, X) L w,z ) 3.8) that is given by y, Kf)t) := k t, s) f s),y dν s) for f L, X) and t 0 and y Z. 3.9) Furthermore, the K of 3.8) maps L, X) into L,Y ) provided either or i) Y does not isomorphically) contain c 0 ii) for each t 0 the subset { k t, ) y X : y BZ)} of L 0, R) is equi-integrable. Recall that the subset in ii) is equi-integrable provided if {A n } is a sequence from Σ with A n A n+1 and ν A n) = 0 then lim k t, s) y n X dν s) = 0. 3.10) A n sup y BZ) Remark 3.12 on heorem 3.11). a) here can be advantages in taking a proper norming subspace Z Y over taking Z = Y. First, it eases the assumptions of k. econd, Z may be much smaller than Y and so the conclusion K L, X)) L w,z ) may be more useful than K L, X)) L w,y ). For example, if Y = C [0, 1] then Z := L 1 [0, 1] Y 1-norms Y ; furthermore, Z L [0, 1] is nicer than Y M [0, 1]), which is very large. b) If Y =Y ) is a separable dual space, then Z := Y Y 1-norms Y and, by Pettis s measurability theorem Fact 2.1), one has that L w,z )=L,Y ). c) If ν ) < and for each t 0 there exists q t 1, ] and C t 0, ) such that sup k t, ) y Lqt,X) C t, y BZ) then the equi-integrability condition in ii) holds; indeed, just apply Hölder s inequality. d) Remark 3.7 is valid in this setting also. Proof of heorem 3.11. Fix f L, X). Fix t 0. For each y Z the function k t, ) f ),y : K

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 11 b) in L 1, K) with k t, s) f s),y dν s) hus by the Closed Graph heorem, applied to the mapping the mapping k t, s) y X f s) X dν s) 3.11) C y Z f. L,X) Z y kt, ) f ),y L 1, K), Z y k t, s) f s),y dν s) K defines an element Kf)t)ofZ that satisfies 3.9). Let f E, X). By condition C 0 ), there is f Σ full such that if t f then k t, ) f ) L 1, Y ). For each t f 0 Σ full and each y Z k t, s) f s) dν s),y = kt, s) f s),y dν s) = y,kf)t). Hence 3.7) holds. hus, by heorem 3.6, K maps E, X) intol,y ). Fix f L, X). o see that Kf is σ Z,Z)-measurable, fix a sequence {f n } n N from E, X) that converges a.e. to f and f n L,X) f L,X) for each n N. hen, by the Lebesgue Dominated Convergence heorem, for each y Z and for a.e. t, y, Kf)t) = fs),k t, s) y dν s) = lim f n s),k t, s) y dν s) = lim n n y, Kf n )t) and the latter functions y, Kf n ) ) are µ-measurable functions by condition C 0 ). Furthermore, by 3.11) sup Kf)t) Z = sup t 0 t 0 sup y BZ) k t, s) f s),y dν s) C f ; L,X) thus, Kf L w,z ) and the K of 3.8) is of norm at most C Proof of i) Assume that c 0 does not isomorphically embed into Y. Fix f L, X). By Lemma 2.3, there is a sequence {f n } from E, X) so that f s) = f n s) f n s) X 2 f L,X) for a.e. s. ince each f n is in E, X), by 3.7) and condition C 0 ), there is 1 Σ full, with 1 0, so that for each t 1 and each n N Kf )t) = kt,s)f s) dν s) Y

12 GIRARDI AND WEI Fix t 1. By the Lebesgue Dominated Convergence heorem, since, for each y Z, y, Kf)t) = Kf)t) = = lim m lim m m Kf n )t) in the σ Z,Z)-topology 3.13) fs),k t, s) y dν s) m f n s),k t, s) y dν s) = lim m m Kf n )t),y. It suffices to show that the series in 3.13) converges also in the Z -norm topology; thus, since Y does not contain c 0, it suffices to show that Kf n )t),y < for each y Y 3.14) by a theorem of Bessaga and Pe lczyński cf., e.g., [4, hm. V.8]). For each y Z, Kf n )t),y = k t, s) f n s),y dν s) f n s),k t, s) y dν s) k t, s) y X f n s) X dν s) ) k t, s) y X f n s) X dν s) 2 f L,X) C y Y. hus the mapping Z y U { Kfn )t),y } l 1 is a bounded linear operator. Fix {α n } B l ) and m N. Ify BZ) then m α n Kf n )t),y m = α n Kf n )t),y U BZ,l 1 ) and so m α n Kf n )t) τ U BZ,l1 ). Y

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 13 Fix f L, X). Choose a sequence {f n } from E, X) such that lim f n s) = f s) for a.e. s n f n L,X) f L,X) for each n N. As in the proof of i), since the f n s are in E, X), there is 1 Σ full, with 1 0, so that 3.12) holds for each t 1 and each n N. It suffices to show that {Kf n )t)} converges in Z -norm to Kf)t) for each t 1. Fix t 1. Fix δ>0 and let B n := {s : f s) f n s) X >δ}and A n := k=n B k. hen Kf)t) Kf n )t) Z = sup k t, s)fs) f n s)),y dν s) y BZ) sup y BZ) k t, s) y X f s) f n s) X dν s) [ ] sup δ k t, s) y X dν s) + 2 f L,X) k t, s) y X dν s) y BZ) Bn C B n [ ] δc + 2 f L,X) sup y BZ) k t, s) y X A n dν s). Note that A n {s :f n s) does not converge to f s)}. hus, by the equi-integrability assumption, {Kf n )t)} converges in Z -norm to Kf)t). he following example illustrates the limitations on the conclusions in heorems 3.6 and 3.11. Example 3.13. Let X = C and Y = c 0.hus Let = = R. Define BX, Y ) c 0 and B Y,X ) l. k 0 : R c 0 k 0 ) := k : BX, Y ) e n 1 In ) k t, s) := k 0 t s) where {e n } is the standard unit vector basis of c 0 and I n =[n 1,n). ince k 0 L R,c 0 ), for each f L 1, X) the Bochner integral Kf)t) := kt, s) f s) ds = k 0 t s) f s) ds = k 0 f)t) exists for each t ; furthermore, Kf L,Y ) and Kf is uniformly continuous. From this it follows that k satisfies condition C 0 ). he kernel k also satisfies condition C ) with Z = Y and 0 = since for each y Y l 1 and t k t, s) y = y e n )1 In t s) for each s R

14 GIRARDI AND WEI heorem 3.6 gives that K L 0, X) ) L,Y ). However, K E, X)) L 0,Y ). Indeed, K1 I1 )n)=e n for each n N and so, since K1 I1 is uniformly continuous, K1 I1 does not satisfy 2) of Proposition 2.2 and so K1 I1 / L 0,Y ). heorem 3.11 gives that K L, X)) L w,y ). However, K L, X)) L,Y ). Indeed, consider f =1,0) L, X). If n N and 0 <δ 1 then Kf)n δ)=δe n + j N e n+j Y \ Y. hus Kf / L,Y ). 4. REMARK ON DUALIY AND WEAK CONINUIY he remarks in this section explore the duality and weak continuity of K. For this, dual versions of the four conditions in ection 3 are needed. Definition 4.1. Let C) possibly with respect to a subspace Z of Y ) be one of the four conditions in ection 3 on a kernel k : BX, Y ). hen k satisfies condition C ) possibly with respect to a subspace Z of X ) provided the mapping k : BY,X ) ks, t) := [kt, s)] satisfies condition C) possibly with respect to a subspace Z of X ). For example, k : BX, Y ) satisfies condition C 0 ) provided that for each B Σfinite and each y Y there is B,y Σ full so that if s B,y then the Bochner integral k t, s) y dµ t) exists the mapping B B,y s defines a measurable function from into X. Remark 4.2. Let 1 <p< and 1 p + 1 p B k t, s) y dµ t) X = 1. Let k : BX, Y ) be so that i) k satisfies conditions C 0 ), C 1 ), C 0 ) with respect to Z = Y, and C 0 ) ii) for each y Y and each x X, the mapping t, s) kt, s) x, y Kis

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 15 By heorem 3.8 with Z = Y ), there is a bounded linear operator K : L p, X) L p,y ) 4.1) defined by Kf) ) = Note that k satisfies k,s)fs) dν s) L p,y ) for f E, X). iv) C 1 ) by iii) and the fact that k satisfies C0 ) with respect to Z = Y v) C 0 ) with respect to Z = X since k satisfies C 1 ). o by heorem 3.8 with Z = X), there is a bounded linear operator K : L p,y ) L p, X ) defined by Note that Kg ) ) = k t, ) g t) dµ t) L p, X ) for g E,Y ). K g = Kg for each g L p,y ) since if g = y 1 B E,Y ) L p,y ) and f = x1 A E, X) L p, X) f,k g = Kf, g = k t, s) x1 A s) dν s),y 1 B t) dµ t) = k t, s) x1 A s),y 1 B t) dν s) dµ t) = x1 A s),k t, s) y 1 B t) dµ t) dν s) = x1 A s), k t, s) y 1 B t) dµ t) dν s) = f, Kg 4.2) where assumption ii) helps justify the use of Fubini s theorem. hus K g) ) = k t, ) g t) dµ t) L p, X ) for g E,Y ) and K maps L p,y )intol p, X ). hus the K in 4.1) is σ L p, X),L p, X ) ) to σ L p,y ),L p,y ) ) continuous. Remark 4.3. Let k : BX, Y ) be so that

16 GIRARDI AND WEI By heorem 3.4, there is a bounded linear operator defined by Kf) ) = K : L 1, X) L 1,Y ) 4.3) k,s)fs) dν s) L 1,Y ) for f E, X). ince k satisfies condition C 1 ), it satisfies condition C 0 ) with respect to Z = X; thus, by heorem 3.6, there is a bounded linear operator K : L 0,Y ) L, X ) defined by Note that Kg ) ) = k t, ) g t) dµ t) L, X ) for g E,Y ). K g = Kg for each g L 0,Y ) since if g = y 1 B E,Y ) L 0,Y ) and f = x1 A E, X) L 1, X) then the calculation in 4.2) shows that f,k g = f, Kg.hus K g) ) = k t, ) g t) dµ t) L, X ) for g E,Y ) and K maps L 0,Y )intol, X ). hus the K in 4.3) is σ L 1, X),L, X )) to σ L 1,Y ),L 0,Y ) ) continuous. Remark 4.4. Let k : BX, Y ) be so that k satisfies conditions C 0 ), C 1 ), C 0 ), and C ) with respect to Z = X condition ii) of Remark 4.2 holds X does not contain c 0. By heorem 3.4, there is a bounded linear operator K : L 1, X) L 1,Y ) 4.4) defined by Kf) ) = By heorem 3.11, there is a bounded linear operator k,s)fs) dν s) L 1,Y ) for f E, X). K : L,Y ) L, X ) defined by, for some 0 Σ full, ) x, Kg s) = x, k t, s) g t) dµ t)

f,k g = Kf, g INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 17 Note that K g = Kg for each g L,Y ) since if g L,Y ) and f = x1 A E, X) L 1, X) then f,k g = Kf, g = k t, s) x1 A s) dν s),gt) dµ t) = k t, s) x, gt) 1 A s)dν s) dµ t) = x, k t, s) g t) 1 A s) dµ t) dν s) ) = x, Kg s) 1 A s) dν s) = f, Kg where assumption ii) helps justify the use of Fubini s theorem. hus, by 3.7), K g) ) = k t, ) g t) dµ t) L, X ) for g E,Y ) and K maps L,Y )intol, X ). hus the K in 4.4) is σ L 1, X),L, X )) to σ L 1,Y ),L,Y )) continuous. Remark 4.5. Let k : BX, Y ) be so that k satisfies conditions C 0 ), C ) with respect to Z = Y, and C 0 ) conditions ii) and iii) of Remark 4.2 hold. hen 3.9) of heorem 3.11, with Z = Y, defines a bounded linear operator K : L, X) L w,y ). 4.5) Note that k satisfies condition C 1 ) by iii) and the fact that k satisfies condition C ) with respect to Z = Y. o by heorem 3.4 ) Kg ) := k t, ) g t) dµ t) L 1, X ) for g E,Y ) defines a bounded linear operator K : L 1,Y ) L 1, X ). Note that Kg = K g for each g L 1,Y ) since for g = y 1 B E,Y ) L 1,Y ) and f L, X)

18 GIRARDI AND WEI = = = = = [ ] k t, s) f s),y dν s) 1 B t) dµ t) f s),k t, s) y 1 B t) dµ t) dν s) f s), k t, s) y 1 B t) dµ t) dν s) ) f s), Kg s) dν s) f, Kg where assumption ii) helps justify the use of Fubini s theorem. hus K g) ) = k t, ) g t) dµ t) L 1, X ) for each g E,Y ) and K maps L 1,Y )intol 1, X ). o the K of 4.5) is ) σ L, X),L 1, X )) to σ L w,y ),L 1,Y ) continuous. hus, since E, X) resp. the chwartz class R N,X ) in the case = R N )is σl, X),L 1, X ))-dense in L, X), many of the properties of K are determined by its restriction to E, X) resp. R N,X ) ). If furthermore Y does not contain c 0, then 3.9) of heorem 3.11, with Z = Y, defines a bounded linear operator K : L, X) L,Y ) that is σ L, X),L 1, X )) to σ L,Y ),L 1,Y )) continuous. 5. CONVOLUION OPERAOR he results thus far are now applied to convolution operators on = = R N endowed with the Lebesgue measure). Corollary 5.1. Let Z be a subspace of Y that τ-norms Y. Let k: R N BX, Y ) be strongly measurable on X and k : R N BY,X ) be strongly measurable on Z and k s) x Y ds C 1 x X < for each x X 5.1) R N k s) y X ds C y Y R N < for each y Z. 5.2) hen the convolution operator K : E R N,X ) L 0 R N,Y ) defined by Kf)t) := R N kt s)f s) ds for f E R N,X ) 5.3)

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 19 K 0 : L 0 R N,X ) L 0 R N,Y ), and, if Y does not contain c 0, then to K : L R N,X ) L R N,Y ) satisfying y, K f)t) = kt s)fs),y dν s) R N for f L R N,X ) and t R N and y Z. Furthermore, for 1 p and K 0 L 0 L 0 τc. K p Lp L p C 1 ) 1 p τc ) 1 p Remark 5.2. In Corollary 5.1, if Z = Y and either 1 <p< p= 1 and X does not contain c 0 p = and Y does not contain c 0, then the dual operator K p : [ L p R N,Y )] [ Lp R N,X )] has the form K p g ) s) = k t s)gt) dt L p R N,X ) for g E R N,Y ) R N and Kp maps L p R N,Y ) into L p R N,X ) and thus K p is σ L p R N,X ),L p R N,X )) to σ L p R N,Y ),L p R N,Y )) continuous. Proof of Corollary 5.1 and Remark 5.2. If f = x1 A E R N,X ), then Kf =[k )x] 1 A with k ) x L 1 R N,Y ) and 1 A L R N, R ). hus for each f E R N,X ) : the Bochner integral in 5.3) exists for each t in R N, Kf is a uniformly continuous function from R N to Y, and Kf vanishes at infinity. hus K E R N,X )) L 0 R N,Y ). It is straightforward to verify that the kernel k 0 : R N R N BX, Y ) k 0 t, s) := kt s) satisfies conditions: C 0 ), C 1 ), C ) with respect to Z with 0 = R N,C 0 ), C ) with respect to Z = X, and ii) and iii) of Remark 4.2. he corollary now follows from: heorems 3.4, 3.6, 3.8, 3.11 and Remarks 4.2, 4.4, 4.5. Remark 5.3 on Corollary 5.1). a) he proof shows that if k is strongly measurable on X and 5.1) holds, then one has the

20 GIRARDI AND WEI b) Under the stronger assumption that k L 1 R N, B X, Y ) ), for 1 p the Bochner integrals K p f)t) := kt s)f s) ds R N f L p R N,X ) L R N,X ) and t R N exist and define a bounded linear operator K p : L p R N,X ) L p R N,Y ). his fact is well-known and easy to show; indeed, for f L p R N,X ) L R N,X ) and f s t) := ft s), k t s) f s) Y R N ds = k s) f s t) Y R N ds k L1 R N,BX,Y )) f L R N,X) for each t R N and Kf) ) LpR N,Y ) k s) f s ) LpR N,Y ) ds R N k s) BX,Y ) f s ) LpR N,X) ds R N = k L1 R N,BX,Y )) f L pr N,X) ; thus, K p Lp Lp k L1 R N,BX,Y )). If, in addition, k satisfies 5.1) and k satisfies 5.2) with Z = Y, then it was shown in [8, Lemma 4.5] that K p Lp Lp C 1 ) 1 p C ) 1 p. References [1] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, pringer-verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223. [2] Ruth F. Curtain and Hans Zwart, An introduction to infinite-dimensional linear systems theory, pringer-verlag, New York, 1995. [3] J. Diestel and J. J. Uhl, Jr., Vector measures, American Mathematical ociety, Providence, R.I., 1977, With a foreword by B. J. Pettis, Mathematical urveys, No. 15. [4] Joseph Diestel, equences and series in Banach spaces, Graduate exts in Mathematics, vol. 92, pringer-verlag, New York, 1984. [5] Gerald B. Folland, Real analysis, Pure and Applied Mathematics, John Wiley & ons Inc., New York, 1984, Modern techniques and their applications, A Wiley-Interscience Publication. [6] José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Publishing Co., Amsterdam, 1985, Notas de Matemática [Mathematical Notes], 104. [7] Maria Girardi and Lutz Weis, Operator-valued Fourier multiplier theorems on L px) and geometry of Banach spaces, Journal of Functional Analysis 204 no. 2 2003), 320 354. [8] Maria Girardi and Lutz Weis, Operator-valued Fourier multiplier theorems on Besov spaces, Math. Nachr. 251 2003), 34 51. [9] Joram Lindenstrauss and Lior zafriri, Classical Banach spaces. I, pringer-verlag, Berlin, 1977, equence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. [10] A. Pazy, emigroups of linear operators and applications to partial differential equations, pringer-verlag, New York, 1983. [11] Jan Prüss, Evolutionary integral equations and applications, Birkhäuser Verlag, Basel, 1993.

INEGRAL OPERAOR WIH OPERAOR-VALUED KERNEL 21 Mathematisches Institut I, Universität Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany E-mail address: Lutz.Weis@math.uni-karlsruhe.de