volterra convolution operators with values in rearrangement invariant ...

VOLTERRA CONVOLUTION OPERATORS WITH VALUES IN REARRANGEMENT INVARIANT SPACES GUILLERMO P. CURBERA

A The Volterra convolution operator f(x) l x φ(xky) f( y) dy, where φ(:) is a non-negative non! decreasing integrable kernel on [0, 1], is considered. Under certain conditions on the kernel φ, the maximal Banach function space on [0, 1] on which the Volterra operator is a continuous linear operator with values in a given rearrangement invariant function space on [0, 1] is identified in terms of interpolation spaces. The compactness of the operator on this space is studied.

1. Introduction Consider the classical Volterra integral operator which assigns to a function f on [0, 1], in a suitable class, its antiderivative f(x) l x f(s) ds. Let  be a rearrangement ! invariant function space on [0, 1]. Our aim is to identify the maximal Banach function space on [0, 1] on which the Volterra operator is a continuous linear operator with values in . We will study a more general situation. Let φ : [0, 1] ,-  be a non-negative nondecreasing function. Consider the Volterra convolution operator  defined for a measurable function f : [0, 1] ,-  by f(x) l

&! φ(xky) f( y) dy. x

This type of generalized Volterra operator has been considered by Stepanov [8] in the context of weighted norm inequalities for the Hardy (Volterra) operator. We will restrict our attention to kernels φ such that the associated operator  maps bounded measurable functions into bounded measurable functions (that is, φ is integrable). Given the rearrangement invariant space , we define the space [,  ] to be o f : (Q f Q) ?  q. It is the maximal Banach function space on which the operator  is a continuous linear operator with values in . Okada and Ricker [6] and Ricker [7] have studied this space for the classical Volterra operator (φ l 1) and  l Lp([0, 1]), 1  p  _. Bennett [2] has considered other classical operators on the positive semiaxis (Hilbert, Hardy averaging, Copson) with  l Lp([0, 1]), 1 p _, and has given factorizations of the space [,  ]. The spaces [, L"([0, 1])] and [, L_([0, 1])] are easily identified. Results of interpolation theory show that  is an interpolation space between L"([0, 1]) and L_([0, 1]) obtained via a monotone Riesz–Fischer norm ρ, that is,  l (L", L_)ρ. We Received 25 July 1997 ; final revision 14 October 1998. 1991 Mathematics Subject Classification 47G10, 47B38 (primary), 46E30 (secondary). Research supported in part by DGICYT grant FPB96-1321-C02-01. J. London Math. Soc. (2) 60 (1999) 258–268

  

259

identify the space [,  ] in terms of interpolation spaces. More precisely, we prove that if the kernel φ satisfies a certain property (the ‘ proper quadrature property ’ (see §3)) then the space [,  ] is the interpolation space for the couple ([, L"], [, L_]) obtained via the function norm ρ, that is, [, (L", L_)ρ] l ([, L"], [, L_])ρ. We study the compactness properties of the operator  : [,  ] ,- . We prove that  is not compact regardless of  and φ. We prove that if  is L"([0, 1]) or L_[(0, 1]) then  is not weakly compact. The paper is organized as follows. In §2 we set the preliminaries and notation. In §3 we introduce the proper quadrature property. In §4 we study the spaces [,  ] and prove the main result. In §5 we study the compactness properties of the operator  on the space [,  ]. 2. Preliminaries Throughout this paper  will be a rearrangement invariant space on [0, 1], that is, a vector space of (classes of) measurable functions on [0, 1], which is an order ideal, a Banach space for a norm R:R compatible with the order structure, contains the characteristic functions of measurable sets, satisfies the Fatou property, and if f is in  and g is equimeasurable with f, then g is in  and RgR l R f R (see [1, Definitions I.1.1 and II.4.1]). The associate space of  is the space  h formed by all measurable functions g such that fg is integrable for every f in  (see [1, Definition I.2.3]). The space  h is a norming subspace of the dual space of  (see [1, Theorem I.2.9]). The rearrangement invariant space  is an exact interpolation space between L"([0, 1]) and L_([0, 1]) (see [1, Theorem III.2.2]). The space  can be viewed as the interpolation space between L" and L_ obtained via the K-method of Peetre (see [1, Definition V.1.1]). If (X , X ) is a compatible couple of Banach spaces (see [1, ! " Definition III.1.1]), then the K-functional of f ? X jX is, for t
0, ! " K(t, f ; X , X ) l infoR f RjtR f R : f l f jf ; f ? X , f ? X q. ! " ! " ! " ! ! " " Under certain conditions (see [1, Proposition V.1.15]), which hold in our case, it can be expressed as K(t, f ; X , X ) l ! "

&! k(s, f ; X!, X") ds. t

Let ρ be a Riesz–Fischer (rearrangment invariant) norm (see [1, Definition V.1.16]). We write f f if " #

&! f" (s) ds  &! f# (s) ds, t

t

for all t
0,

where f * is the decreasing rearrangement of f (see [1, Definition II.3.5]). The norm is monotone if f f implies that ρ( f )  ρ( f ). If ρ is a monotone Riesz–Fischer norm " # " # the space (X , X )ρ is (see [1, Definition V.1.18]) the space of those f ? X jX such that ! " ! " R f R(X ,X )ρ l ρ(k(:, f ; X , X )) _. ! " ! " Given the rearrangement invariant space , we can define ρ( f ) l R f R, which is a monotone Riesz–Fischer norm (see [1, Corollary II.4.7]). For the couple (L", L_) we have k(:, f ; L", L_) l f * (see [1, Proposition V.1.14]). Thus a function f is in  if and only if R f R l ρ( f *) l ρ(k(:, f ; L", L_)) _. Hence  l (L", L_)ρ.

 . 

260

Given compatible couples (X , X ) and (Y , Y ), the pair ((X , X )ρ, (Y , Y )ρ) has the ! " ! " ! " ! " interpolation property with respect to (X , X ) and (Y , Y ), that is, if T is an operator ! " ! " on X jX with values in Y jY , such that T maps boundedly Xi into Yi, i l 0, 1, then ! " ! " T is a bounded operator from (X , X )ρ into (Y , Y )ρ (see [1, Theorem V.1.19]). ! " ! " In general we follow the notation in the book by Bennett and Sharpley [1]. Occasionally we will use facts from the theory of vector measures, which can be found in the book by Diestel and Uhl [4]. 3. A property for non-decreasing functions Let φ : [0, 1] ,-  be a non-negative non-decreasing integrable function with φ(x)
0 for x
0. Given b in [0, 1], let a be the point in [0, 1] such that

&! φ l (bka) φ(b). b

In other words, the point a has the property that the area of the rectangle with base [a, b] and height φ(b) is the same as the area between the graph of φ and [0, b]. The value a always exists, it is smaller than b, and a as a function of b is non-decreasing, owing to the fact that the function φ is non-negative and non-decreasing. We will say that the function φ has the proper quadrature property if the following condition holds : b φ a
0. inf b ! b"φ(b) φ b φ=(b−a) ! ! We will call this infimum the proper quadrature constant of the function φ. It is easy to see that unbounded functions, defined on [0, 1), do not satisfy this property. On the other hand the following functions do satisfy the property : (a) the functions φ(x) l xα for α  0 ; (b) concave functions, since

& &

& φ  1\2(bka) (φ(a)jφ(b))  1 ; (bka) φ(b) 2 &! φ b

a b

(c) bounded functions with φ(0+) l limx

!+

φ(x)
0, since

& φ  (bka) φ(a)  φ(0 )
0. &! φ (bka) φ(b) φ(1) b

+

a b

Not all bounded functions satisfy this property. Consider the function φ with value 1\2#n in the interval [1\2n, 1\2n−"). For bk l 1\2k we have ak
1\2k+". Thus φ is constant on [ak, bk] and φ(ak) l φ(bk+ ), so we have "

& φ l φ(a ) (b ka ) l φ(a ) φ(b ) " &! φ l φ(b ") l 1 φ(b ) 2# &! φ &! φ &! φ bk

ak bk

k

bk

k

k

k

k

bk

−

bk

k+ k

k

0.

  

261

4. The Volterra operator with Šalues in rearrangement inŠariant spaces Let φ : [0, 1] ,-  be a non-negative non-decreasing function. Consider the Volterra convolution operator  defined for a measurable function f : [0, 1] ,-  by f(x) l

&! φ(xky) f( y) dy. x

In order to avoid trivial situations we will assume that φ(x)
0 for x
0. We will restrict our attention to kernels φ such that the operator  maps bounded measurable functions into bounded measurable functions. This condition is precisely φ being integrable on [0, 1]. Let  be a rearrangement invariant function space on [0, 1]. We define the space [,  ] to be o f : (Q f Q) ?  q. It is a lattice ideal of measurable functions on [0, 1]. The condition of φ being integrable implies that [,  ] contains the essentially bounded functions. It becomes a Banach function space with the natural norm R f R[,  ] l R(Q f Q)R. It is the maximal Banach function space on which the operator  is continuous and takes values in . We first consider the cases  l L_([0, 1]) and  l L"([0, 1]). A function f is in [, L_] if x " sup φ(xky) Q f( y)Q dy l φ(1ky) Q f( y)Q dy _. ! !x" ! Thus [, L_] is the weighted space L"(ξφ) with the weight ξφ( y) l φ(1ky). Analogously, a function f is in [, L"] if

&

&

&!" &! φ(xky) Q f( y)Q dy dx l &!" Q f( y)Q & " φ(xky) dx dy x

_.

y

Thus [, L"] is the weighted space L"(ωφ) with the weight ωφ( y) l "−y φ(s) ds. ! In this setting we have the following result. T 1. Let  be the Volterra conŠolution operator giŠen by a non-negatiŠe non-decreasing integrable kernel φ. Let  be a rearrangement inŠariant space on [0, 1] giŠen by a monotone Riesz–Fischer norm ρ. If the kernel φ satisfies the proper quadrature property then the space [,  ] is, with equiŠalence of norms, the space (L"(ωφ), L"(ξφ))ρ, for the weight functions ξφ( y) l φ(1ky) and ωφ( y) l "−y φ(s) ds. ! Proof. Let f be in (L"(ωφ), L"(ξφ))ρ. We can assume that f  0. The interpolation property of the pair ((L"(ωφ), L"(ξφ))ρ, (L", L_)ρ) with respect to the couples (L"(ωφ), L"(ξφ)) and (L", L_) implies that f is in (L", L_)ρ l  and Rf R  R f R(L"(ωφ),L"(ξφ))ρ. Hence, f is in [,  ] and R f R[,  ]  R f R(L"(ωφ),L"(ξφ))ρ.

 . 

262

In order to prove the reverse inclusion let f be in L"(ωφ). We first calculate the Kfunctional for f with respect to (L", L_). We can assume that f  0, in which case f is a non-decreasing function. Thus for 0 t  1

&! (f )* (x) dx l & f(1kx) dx ! " l & & φ(1kxky) f( y) dy dx ! ! " l & f( y) & φ(1kxky) dx dy ! ! " " j& f( y) & φ(1kxky) dx dy " ! " " " " l & f( y) 0& φ1 dyj& f( y) 0& φ1 dy. ! " " !

K(t, f ; L", L_) l

t

t

t

−x

t

−t

−y

−t

−t

−y

−y

−y−t

(1)

−t

For t
1, since (f )* l 0 on (1, j_), we have

&!" (f )* (x) dx " l & f(1kx) dx ! " " l & & φ(1kxky) f( y) dy dx ! ! " " l & f( y) 0& φ1 dy. ! !

K(t, f ; L", L_) l

−x

−y

(2)

Equalities (1) and (2) can be written as K(t, f ; L", L_) l

&!" f( y) F(t, y) dy,

(3)

where F(t, y) is the function ωφ( y)kωφ( yjt) 0  y  1kt 1

F(t, y) l 23 4

ωφ( y)

1kt  y  1, 0  y.

The K-functional for f with respect to (L"(ωφ), L"(ξφ)) is given by (see [5, p. 769]) : K(t, f ; L"(ωφ), L"(ξφ)) l R f:minoωφ, t:ξφqRL"

&!" f( y) minoω ( y), t:ξ ( y)q dy " l & f( y) G(t, y) dy, ! l

for the function G(t, y) l minoωφ( y), t:ξφ( y)q.

φ

φ

(4)

  

263

Since φ is non-negative and non-decreasing, a straightforward verification shows that F(t, y)  G(t, y). This corresponds to the previous inclusion (L"(ωφ), L"(ξφ))ρ 9 [,  ], which we obtained via interpolation. Let us assume the following claim, which will be proved later. C 1. The function φ satisfies the proper quadrature property if and only if there exists α
0 such that α:G(t, y)  F(t, y),

for all t
0 and y ? [0, 1].

(5)

From Claim 1 and (3) and (4) we have α:K(t, f ; L"(ωφ), L"(ξφ))  K(t, f ; L", L_). Hence, for every t
0 α: so

&! k(s, f ; L"(ω ), L"(ξ )) ds  &! k(s, f ; L", L ) ds, t

t

φ

φ

_

α:k(:, f ; L"(ωφ), L"(ξφ))

k(:, f ; L", L_).

Owing to the monotonicity of ρ we then have α:ρ(k(:, f ; L"(ωφ), L"(ξφ)))  ρ(k(:, f ; L", L_)). If the function f is in [,  ], then f is in , so we have ρ(k(:, f ; L", L_)) _. Hence ρ(k(:, f ; L"(ωφ), L"(ξφ))) _, and thus f is in (L"(ωφ), L"(ξφ))ρ and α:R f R(L"(ωφ),L"(ξφ))ρ  Rf R l R f R[,  ].

The proof is complete.

Proof of Claim 1. Consider first the case 1kt  y  1. Then F(t, y) l ωφ( y). Since the function φ is non-decreasing and 1ky  t, we have ωφ( y) l

&!"

−y

φ(s) ds  (1ky) φ(1ky)  t:ξφ( y),

so G(t, y) l ωφ( y). Thus in this case the functions F and G coincide. Now consider the case 0  y  1kt. Then F(t, y) l ωφ( y)kωφ( yjt). There are two alternatives. We can have ωφ( y)  t:ξφ( y). In this case G(t, y) l ωφ( y), so condition (5) is ωφ( y)kωφ( yjt)  α:ωφ( y), which is

&"" &!"

−y

inf y :φ"−t "−y! φ t ("−y) !  

φ

−y−t −y

φ

 α
0.

(6)

The other possibility is that ωφ( y)
t:ξφ( y). Then G(t, y) l t:ξφ( y), so condition (5) is ωφ( y)kωφ( yjt)  α:t:ξφ( y), which is "−y φ −y−t " inf  α
0. (7) t:φ(1ky) y −t " "−y! φ
:φ t ("−y) !

&

 . 

264

Owing to the fact that the function φ is non-negative and non-decreasing, the infimum in (6) and (7) corresponds to the case in which "−y φ l t:φ(1ky). Hence (6) ! and (7) can be jointly rewritten as

& φ
0. &! φ b

a inf b  b ! " b φ=(b−a) φ(b)

!

The previous infimum being strictly positive is precisely the condition of φ satisfying the proper quadrature property. The claim is proved. R 1. From the proof of Theorem 1 it follows that the isomorphism constant between the spaces [,  ] and (L"(ωφ), L"(ξφ))ρ is given by the proper quadrature constant of the kernel φ, which is independent of the space . R 2. The classical Volterra operator f(x) l x f( y) dy corresponds to the ! kernel φ l 1, which satisfies the proper quadrature property with α l 1. In this case [,  ] and (L"(ωφ), L"(ξφ))ρ are isomorphically isometric spaces. If we impose the condition F(t, y) l G(t, y) then φ is constant and the operator  is a multiple of the classical Volterra operator. The converse to Theorem 1 holds in the following sense. T 2. Let  be the Volterra conŠolution operator giŠen by a non-negatiŠe non-decreasing integrable kernel φ. If, for eŠery rearrangement inŠariant space  giŠen by a monotone Riesz–Fischer norm ρ the spaces (L"(ωφ), L"(ξφ))ρ and [,  ] are isomorphic with the same isomorphism constant, then the kernel φ satisfies the proper quadrature property. Proof. There exists α
0 such that if  is any rearrangement invariant space and ρ is its associated monotone Riesz–Fischer norm, then for every f in [,  ] we have α:R f R(L"(ωφ),L"(ξφ))ρ  R f R[,  ], that is α:ρ(k(:, f ; L"(ωφ), L"(ξφ)))  ρ(k(:, f ; L", L_)). (8) We will consider f to be the characteristic function of any measurable set. Fix t
0. Let us consider the sequence of monotone Riesz–Fischer norms

&

t 1 ρn,t( f ) l :sup f *(s)j f *, for n ? . n s
! ! By hypothesis we have (8) for every ρn,t. Since the functions in (8) are non-increasing, by passing to the limit we have

α: which is

&! k(s, f ; L"(ω ), L"(ξ )) ds  &! k(s, f ; L", L ) ds, t

t

φ

φ

_

α:K(t, f ; L"(ωφ), L"(ξφ))  K(t, f ; L", L_).

  

265

From (3) and (4) we get, for every t
0

&!" f( y) G(t, y) dy  &!" f( y) F(t, y) dy.

α:

Since f is the characteristic function of any measurable set, and the functions F(t, y) and G(t, y) are non-increasing in y, we conclude that α:G(t, y)  F(t, y) for every t
0 and y ? [0, 1]. From Claim 1 we deduce that φ satisfies the proper quadrature property. The result is proved. 5. Compactness properties of the Volterra operator We study the compactness properties of the Volterra operator  : [,  ] ,- . The case of the classical Volterra operator (φ l 1) into  l Lp([0, 1]) was studied by Okada and Ricker [6] for p l 1 and by Ricker [7] for 1  p  _. Our proofs are suitable adaptations of their proofs. We prove that  is not compact regardless of  and for a general non-negative non-decreasing integrable kernel φ. T 3. Let φ be a non-negatiŠe non-decreasing integrable kernel and let  be a rearrangement inŠariant space on [0, 1]. The Volterra conŠolution operator  : [, ] ,-  giŠen by the kernel φ is not compact. Proof. We first reduce the problem to the study of the Volterra operator on an L"-space. Consider the vector measure ν on the Lebesgue σ-algebra in [0, 1] with values in  defined by ν(A) l (χA) ? . If the measure is considered to take values in L_([0, 1]), then

&

&

x

Rν(A)R_ l sup φ(xky) QχA( y)Q dy l φ(1ky) dy. A !x" ! Thus, as an L_-valued measure ν is countably additive, has finite total variation and is absolutely continuous with respect to Lebesgue measure. Since L_([0, 1]) is continuously included in , the same properties hold for ν which is considered as an -valued measure. We will denote its variation by QνQ. The space L"(QνQ) is continuously included in [,  ]. To see this let f be a non-negative function in L"(QνQ) and (sn) be a sequence of simple functions increasing to f. Then (sn) is a sequence in  that increases to f. Since RsnR  RsnRL"(QνQ)  R f RL"(QνQ) _, the Fatou property of  ensures that f is in , and Rf R  R f RL"(QνQ). Hence f is in [,  ] and R f R[,  ]  R f RL"(QνQ). Thus we can restrict the operator  to L"(QνQ). We will denote the restricted operator by , so that  : L"(QνQ) ,- . Let us assume by way of contradiction that " "  : [,  ] ,-  is compact. By the ideal property of compact operators, the operator  is compact ; thus  is representable (see [4, Theorem III.2.2]). Thus there " " exists a Bochner integrable function
: [0, 1] ,-  such that fl "

& ! " f( y)
( y) dQν Q, 

for f ? L"(QνQ),

[ , ]

with the range of
being essentially relatively compact in .

(9)

 . 

266

Let us identify the function
. Let f be in L"(QνQ) and g be in  h. Then

&!" g(x) " f(x) dx " l & g(x) & φ(xky) f( y) dy dx ! ! " " l & f( y) & φ(xky) g(x) dx dy ! " φ , gg dy, l & f( y) fχ " !

f f, gg l "

x

y

(10)

[y, ] y

where φy is the function φy(x) l φ(xky) for y  x  1 and zero elsewhere. On the other hand from (9) and the fact that QνQ is absolutely continuous with respect to Lebesgue measure

& ! " f( y) f
( y), gg d Qν Q l & ! " f( y) -
( y) ddyQν Q , g. dy.

(11)

-
( y) dQνdy Q , g. l fχ

(12)

f f, gg l "





[ , ]

[ , ]

Fix g in  h. In (10) and (11) we can consider f to be the characteristic function of an arbitrary measurable set in [0, 1], which is in L"(QνQ). We deduce that 

φ , gg, for y @ Z(g),

[y, "] y

where Z(g) has Lebesgue measure zero and depends on the function g. Let g be in the countable set of characteristic functions of intervals in [0, 1] with rational endpoints. Since the functions in (12) are in L"([0, 1]), we deduce that
( y)

dQνQ l χ[y, ] φy, for y @ Z, " dy

(13)

where Z has Lebesgue measure zero. From (9) and (13), if A is a measurable set we have ν(A) l

&χ

so QνQ(A) l

& Rχ A

Hence we have

φ dy,

[y, "] y

A

φ R dy.

[y, "] y

d QνQ l Rχ[y, ] φyR, " dy

and we conclude from (13) that χ[y, ] φy " . (14) Rχ[y, ] φyR " Assume that
([0, 1]) has compact closure in . By successive extraction of subsequences, if necessary, we can obtain a sequence ( yn) strictly increasing to 1 and a function } ?  such that the sequence (
( yn)) converges to } in the norm of  and pointwise almost everywhere. Since the functions B( yn) are non-negative, } is nonnegative.
( y) l

   Let 0

a

267

1 and set g l χ[ ,a], which is an element in  h. Then ! lim f
( yn), gg l f}, gg l

n

_

where 1

f
( yn), gg l

&! }, a

χ R & ! a

(x) φ(xkyn) dx. [yn, "] Rχ[y , ] φy  n" n Since ( yn) increases to 1 we have yn
a for n  n , so the previous expression is zero ! for n  n . Thus a } l 0 for every 0 a 1, and so } l 0. ! ! On the other hand, R
( yn)R converges to R}R. From (14) we have R
( y)R l 1 for every y ? [0, 1), so R}R l 1. We have arrived at a contradiction, which establishes the result. R 3. The relation between the spaces L"(QνQ) and [,  ] is precise. The space [,  ] is the space of real functions which are integrable with respect to the valued vector measure ν (see [3]). We now consider weak compactness of the operator  : [,  ] ,- . T 4. Let φ be a non-negatiŠe non-decreasing integrable kernel and let  be a rearrangement inŠariant space on [0, 1]. Let  : [,  ] ,-  be the Volterra conŠolution operator giŠen by the kernel φ. If  is L"([0, 1]) or L_([0, 1]), then the operator  is not weakly compact. Proof. In the case  l L"([0, 1]), just consider the functions fn l (n\ωφ):χ[ − /n, ] " " " which are in [, L"] l L"(ωφ). Since fn and fn are supported in [1k1\n, 1], and R fnRL"(ωφ) l RfnR l "

&" "

[ − /n, "]

Vfn l 1,

it follows that the image by  of the unit ball of the space L"(ωφ) is not uniformly integrable in L"([0, 1]). Hence  is not weakly compact. In the case  l L_([0, 1]) we have [, L_] l L"(ξφ). Let us assume by way of contradiction that the operator  : L"(ξφ) ,- L_ is weakly compact. Then it is representable as in (9) by a Bochner integrable function
: [0, 1] ,- L_ which has essentially relatively weakly compact range (see [4, Theorem III.2.12]). From (13) [y, ] φy χ φ " l [y, "] y . Rχ[y, ] φyR_ φ(1ky) " Consider a sequence ( yn) strictly increasing to 1. By the Eberlein–S3 mulian theorem there exists a subsequence of (
( yn)) which is weakly convergent. We will still denote this subsequence by (
( yn)). Direct computation shows that (
( yn)) converges to zero in the weak* topology of L_. Hence (
( yn)) converges weakly to zero in L_. By a theorem of Mazur, there exists a sequence of convex combinations of the functions
( yn) that converges to zero in the norm topology of L_. However, this is not so, since φ(tkyn ) i l  α l 1.  αi
( yn ) l sup  αi χ[y , ](t) i " i _ n φ(1kyn ) i i i !t" i i Hence  is not weakly compact.
( y) l

))

))

)

)

268

   References

1. C. B and R. S, Interpolation of operators (Academic Press, 1988). 2. G. B, Factorizing the classical inequalities, American Mathematical Society Memoirs 576 (1996). 3. G. P. C, ‘ When L" of a vector measure in an AL-space ’, Pacific J. Math. 162 (1994) 287–303. 4. J. D and J. J. U J, Vector measures, American Mathematical Society Surveys 15 (1977). 5. L. M and L. E. P, ‘ Real interpolation between weighted Lp and Lorentz spaces ’, Bull. Polish Acad. Sci. Math. 35 (1987) 766–778. 6. S. O and W. J. R, ‘ Non weak compactness of the integration map for vector measures ’, J. Austral. Math. Soc. 54 (1993) 287–303. 7. W. J. R, ‘ Compactness properties of extended Volterra operators in Lp([0, 1]) for 1  p  _ ’, Arch. Math. 66 (1996) 132–140. 8. V. D. S, ‘ Weighted inequalities for a class of Volterra convolution operators ’, J. London Math. Soc. 45 (1992) 232–242.

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