THE WEAK DROP PROPERTY AND THE DE LA VALL´EE POUSSIN THEOREM Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 12 (30), No 3, 2023, pp. 105-118 105

DOI: 10.15393/j3.art.2023.13451

UDC 517.98

H. Kalita

THE WEAK DROP PROPERTY AND THE DE LA VALLEE POUSSIN THEOREM

Abstract. We prove that a closed bounded convex set is uniformly integrable if and only if it has the weak drop property. We extract the weakly compact subsets of the Henstock integrable functions on the H-Orlicz spaces with the weak drop property via de la Vallee Poussin Theorem.

Key words: Young's function, weak drop property, H-Orlicz spaces 2020 Mathematical Subject Classification: 46A55, 46B20

1. Introduction and preliminaries. The weak drop property was introduced by Giles, Sims, and Yorke (see [6]). Rolewicz [18] introduced the sequential approach to the drop property. Qiu established the following fact: in a locally convex space, every weakly sequentially compact convex set possesses the weak drop property, and every weakly compact convex set possesses the quasi-weak drop condition (see [19]). Bhayo et al. [3] introduced convexity and concavity of a function with the identric and Alzer mean. The class of spaces with norm having the drop property resides between the classes of nearly uniformly convex and nearly strictly convex Banach spaces (see [4]). For details of the drop property, one can see [12], [19] and reference therein. It can be traced to as early as 1915, the publications of de-la Vallee Poussin, although it was the Banach space research of 1920's that formally gave birth to what were later called the Orlicz spaces, first proposed by Z. W. Birnbaum and W. Orlicz. Later this space was further studied by Orlicz himself. Orlicz spaces are generalizations of Lp spaces. Further, Luxemburg developed the theory of Orlicz spaces. Details of this theory can be found in [1], [13], [17].

Kurzweil first proposed a solution to the primitives problem in 1957, and then Henstock did the same in 1963. It is a generalized form of the Riemann integral, commonly known as the Henstock-Kurzweil integral

© Petrozavodsk State University, 2023

(HK-integral); Kurzweil and Henstock each proposed this generalization independently. The Henstock-Kurzweil integral is similar to the Riemann integral, but is stronger than the Lebesgue integral. Additionally, it is well known that the HK-integral can resolve the issue of primitives in the real line. The limit of Riemann sums over the appropriate integration domain partitions is referred to as the HK-valued integrals. The HK-integral has a constructive definition. Within the HK-integral, a gauge-like positive function is employed to assess a partition's fineness, rather than a constant as in the Riemann integral: this is the fundamental distinction between the two definitions. One can see [5], [7], [8], [10] and references therein for details of Henstock-Kurzweil integrals.

Hazarika and Kalita [7] introduced the H-Orlicz space with non-absolute integrable functions. They proved that C™ is dense in the H-Orlicz space, but is not dense in the classical Orlicz space. It is known that if a function f is bounded with compact support, then the following statements are equivalent:

(a) f is Henstock-Kurzweil integrable,

(b) f is Lebesgue integrable,

(c) f is Lebesgue measurable.

In general, every Henstock-Kurzweil integrable function is measurable, and f is Lebesgue integrable if and only if both f and | f | are Henstock-Kurzweil integrable. This means that the Henstock-Kurzweil integral can be thought of as a "non-absolutely convergent Lebesgue integral". This fact relies on compact support in H-Orlicz space. Of course, we are working with H-Orlicz spaces without compact support. In this case, H-Orlicz spaces are not equivalent to Classical Orlicz spaces.

Let f be a function defined on [0,1] that has values in a Banach space X. If f has an integrable majorant and X is separable, Caponetti et al. demonstrated in [5] that the limit set Ihk (f) of Henstock-Kurzweil integral sums is non-empty and convex. They provided a detailed explanation of the limit set in the same setting.

In several contexts, the de la Vallee-Poussin Theorem has been taken into account. For instance, in the space of measures with values in a countably additive Banach space. The de la Vallee-Poussin Theorem is used to characterise the strongly measurable vector functions that are Pettis integrable through the compactness of a certain set of scalar functions in a specific space of Orlicz, as well as the countably additivity of the Dunford integral of vector functions (see [2]). The characterization

of the countable additivity of the Henstock-Dunford integrable functions with the help of de la Vallee Poussin Theorem was discussed by Kalita and Hazarika in [8]. Recently, Volosivets [20] has studied de la Vallee-Poussin's mean in weighted Orlicz spaces.

The paper is organized as follows: In Section 1, the basic concepts and terminology are introduced together with some definitions and results. In Section 2, we prove that a closed bounded convex set has weak drop property only if it also possesses uniform integrability: Theorem 5. In Section 3, several properties of H-Orlicz spaces with drop property are discussed. In Section 4, with the help of Theorem 5, we find relatively weakly compact subset in H-Orlicz space via weak drop property.

Definition 1. [6] Let C be a closed convex set. A drop D(x, C) determined by a point x E C is the convex hull of the set (xj U C. The set C is said to have drop property if for every non empty closed set A disjoint from C, there exists a point a E A, such that D(a, C) if A = {a}.

Next we recall the concept of weakly sequentially closed set below.

Definition 2. [11] A set A is weakly sequentially closed if every A-valued weakly convergent sequence has its limit in A.

Definition 3. [18] A closed convex set C is said to have the weak drop property if for every weakly sequentially closed set A disjoint from C there is an a E A, such that D(a, C) if A = {a}.

Recalling a closed bounded convex set C, a sequence (xnj in X \ C, such that xn+\ E D(xn, C) Wn E N is called a stream. It is known that the norm || • || has drop property if and only if each stream in X \ B (X) contains a convergent subsequence (see [18]). Giles et al. in [6] showed that the norm || • || has weak drop property if and only if each stream in X \ B(X) contains a weakly convergent subsequence.

Definition 4. [14, Definition 2.5] A subset F of L1(Q) is said to be uniformly integrable if F is a bounded subset of L1(Q), such that

Recall de la Vallée Poussin Theorem from [17], as follows:

Theorem 1. Let F be a family of scalar measurable functions on a finite measure space (Q, T*,^,). Then the following conditions are equivalent:

1) F = {fa ■ & is in an index set} is uniformly integrable.

2) There exists a convex function 9■ R+ — R+, such that 9(0) = 0, 0(-x) = d(x) and 9(x)/x —y to as x — to, in terms of which

sup / 9( fa)sf< to.

« n

Definition 5. [1, Definition 1.1] Let m■ R+ — R+ be a non-decreasing right-continuous and non-negative function satisfying

m(0) = 0, and lim m(t) = to.

A function 9 ■ R+ — R+ is called an N -function if there is a function " m" satisfying the assumptions above and such that

M

9(u) = J m(t)dt. o

Evidently, 9 is an N-function if it is continuous, convex, even, satisfies

lim (u) = ^ and lim —U) = 0.

u u

For example, 9P(x) = xp; p > 1. One can see [9] and references therein for details of N-functions.

Definition 6. ( [17, Definition 7, p 28], [1, Definition 1.5, 1.6])

(a) An N -function 9 is said to satisfy A' condition if there is a k > 0, so that

9(xy) ^ k9(x)9(y) for large values of x and y.

(b) An N-function 9 is said to satisfy A2 condition if there is a k > 0, so that

9(2x) ^ k9(x) for large values of x.

Theorem 2. [18] Every closed bounded convex set with weak drop property is weakly compact.

Remark. All Young's functions are convex. It is easy to claim that every closed bounded set of Young's functions with weak drop property is weakly compact. Also, every bounded closed convex set of N -function with weak drop property is weakly compact.

Henstock-Kurzweil integral is defined as follows:

Definition 7. [8] A function f: [a,b] ^ R is said to be Henstock-Kurzweil integrable on a set [a, b] if there is an element E R, such that for every t > 0 there is a positive function 5 (called gauge) on [a, b] with

\S(f,P) - I[a,b]\<t whenever P = {a = x0 ^ x1 ^ x2... ^ xn = b} is a 5-fine partition of

n

[a, b] and S(f, P) = f (ti)(xi — xi-1) is a Riemann sum.

i=i

Through out the article, the space of Henstock-Kurzweil integrable functions is denoted by HK(^). This space is not complete with the Alexiewicz semi-norm

II/IU = sup | f f |,

[a,6]CR J a

where the integral is in the sense of Henstock-Kurzweil.

Theorem 3. [7, Theorem 1.1] If f: I0 C [0,1] ^ R is a measurable function with the gauge 5: I0 ^ R, then g = 9(f): [0,1] ^ R+ is Henstock-Kurzweil integrable.

Let (X, £, y) be a measure space and X be a Banach space. The Orlicz space L*(X, or L*(^) is defined as follows: if 9(f) = (L) J 9(f )dy, then

Le(u) = {/ measurable: 9(f) < ro}. (1)

The space Le(y) is absolutely convex, i.e., if f,g E Le(y) and a, ft are scalars, such that |a| + | ^ 1, then af + ¡3g E Le(y). Also, if 9 E A2, then Ld(y) is a linear space (see [17, Theorem 2]). If 9: R+ ^ R+ is a Young function, then it can be represented as

+

9(x) = <t>{x)dx, x E R

where 0(0) = 0, 0: R+ ^ R+ is non-decreasing left-continuous, and if 0(x) = +ro for x ^ a, x ^ a > 0, then 9(x) = +ro for x ^ a. The function 0 is called the complementary function of 9 (see [17, Corollary 2]). If 0 denotes the complementary in Young's sense function to 9, and

Ld*(^) = |f measurable: \ (L) fgd^\ < ro for all g E

X

the collection (¡) is then a linear space. The collection (l^(¡1), || ■ is a Banach space called an Orlicz space, where

\Le = sup{| fgdiI ■ 4>(g) ^ ^.

Moreover, let || ■ ||(#) be the Minkowski functional associated with the convex set {f e L\(ß): 0(f) ^ l}. In the sequel, || ■ ||(g) is an equivalent norm on L®(ß), called the Luxemberg norm. Indeed,

^ HfL* < 2||/||w for all/ eLl (ß).

For details of Orlicz spaces, see [1], [9], [13], [17] and references therein. It is clear that A2 conditions of the Young function play crucial role for the reflexivity of the Orlicz space (see [9], [13], [17]). Now, we recall reflexivity of Orlicz space.

Theorem 4. [1], [9] The Orlicz space L® (¡1) is reflexive if and only if 0 e A2.

2. Uniform integrability and weak drop property. In this section, we prove that a closed and bounded convex set of HK(¡) is uniform integrable if and only if the set has weak drop property. Consider a finite

collection P = |(Aj, ti) ■ i = 1, 2,... ,p|, where A» are non-overlapping

p

intervals in [0,1], U e Aj and (J Aj = [0,1].

i=l

A strictly positive function i on I0 C [0,1] is called a gauge on I0. Given a gauge 8 on [0,1], a partition P = |(Aj, ti) ■ i = 1, 2,...,p | of [0,1] is called ¿-fine if Aj C (ti — 8(ti), U + 8(ti)) for i = 1, 2,...,p. Given a partition P = |(Aj, t^^ i = 1, 2,...,p | of [0,1], Henstock-Kurzweil integral sum S(P, g) = Y?i=i fl,(*i)|Ai|.

Definition 8. [5] Let <?■ [0,1] — X,x e X. We say x is a Henstock-Kurzweil point of g, if for every e > 0 and for every gauge 8 on [0,1] there is a 8-flne partition P of [0,1], such that ||S(P, g) — x|| < t. We denote Ihk (g) the set of all Henstock-Kurzweil points of g.

It can be seen from the Theorem 3 that we get a class of subsets of HK(¡1) as follows:

[lHK(g):g= f d(f) E HK(ß)}.

[0,1]

Let F = [Ihk(g)'- 9 = ) G HK(^)} be a bounded closed subset of HK(y). Clearly, F is a convex set.

Definition 9. Let g - [0,1] ^ X be an arbitrary function. We will say that a point x G X belongs to the weak limit set WIhk (g) of g, if for any weak neighborhood U of x and any partition P of [0,1] there exists a finer partition Q > P, such that S(Q,g) G U.

For t > 0, we set

v(F,e) = sup{(##) J №: Í G F, A G E, y(A) ^ e}

A

and define the modulus of uniform integrability v(F) of F by

v(F) = lim v(F, e) = inf v(F, e).

e>0

Lemma 1. Let F be a subset of HK(y). Then the following are equivalent:

1) F has weak drop property.

2) F is weakly sequentially compact in HK(y).

3) F is a bounded subset of HK(y) satisfying the two properties: v(F) = 0 and for every t > 0 there exists [0,1]e G E, such that ^([0,1]e) < to and

sup(HK) \£\dp ^ e.

?eF J

[0,l]\[0,l]e

Proof. For (1) (2)- Suppose F has weak drop property. The [15,

Proposition 4.4.7] states that F is weakly compact. Clearly, Eberlein-Smulian theorem says that F is weakly sequentially compact in HK(^). (2) (3) follows from [14, Theorem 2.3].

For (3) (1) - Given F is a bounded subset of HK(y) satisfying the two properties: v(F) = 0 and for every t > 0 there exist [0,1]e G E, such that ^([0,1]e) < to and sup(HK) f[0 ^^ 1] \£^ e. Since v(F) = 0 and

F is a bounded subset of HKwe can confirm from [14, Proposition 2.6] that F is uniformly integrable. Hence, from Prokhorov's theorem (see [16, Theorem 1.12]) F = (HK) Jj0 (HK) Jj0 is weakly sequentially

pre-compact. The definition of weakly sequentially pre-compact is: every sequence {in F has a subsequence {} that converges weakly to some

£ e HK(¡).

Suppose F does not have weak drop property. Then there exists a weakly sequentially closed set C of HK (¡) disjoint from F, such that

c e C, inf jd(x, F): x e C H D(c, F) j = 0. So, there exists a sequence

{xn} in C, such that xn+1 e D(xn, F) and there exists a sequence {yn} in F, such that \\xn — yn\\ ^ 0. As a stream {xn} has a subsequence {xnk} weakly convergent to some x0 and C is weakly sequentially closed, x0 e C, but \\xnk — ynk\\ ^ 0. So, {ynk} is weakly convergent to x0. Since F is closed and convex, in both cases x0 e F. This is a contradiction to the fact that C & F are disjoint. Hence, F has weak drop property. □

Theorem 5. A closed bounded convex subset F of HK(¡) is uniformly integrable if and only if F has weak drop property.

Proof. The proof follows directly from Lemma 1. □

Proposition 1. A subset F of HK(¡) is bounded and uniformly integrable if and only if there is an N-function 9, such that

sup {Ihk(g): g = d( f) e HK(¡)} < to.

3. H-Orlicz spaces and weak drop property. In this section, we discuss the reflexivity of the H-Orlicz spaces with the drop property. For an N-function 9 and a measurable f define g = (HK) Jj0 1j 9(f)d¡. Let '

He = | f me asura ble: g < to| .

If 0 is the complement of 9, we assume

h:

*

|/ me a s urabl e: |( HK) ff | < to f or all f E H |.

The collection HI is a linear space. For f E HI, define

e = supj |( HK) J ffdßl : g ^ lj, g = (HK)J W)dß.

Then ( Hi, || ■ ) is a Banach space called an H-Orlicz space. Moreover, letting || ■ ||(0) be the Minkowski functional associated with the convex set

e

{/ E H*: g ^ 1}, we have the fact that || ■ ||(#) is an equivalent norm on HI, called the Luxemberg norm.

In fact, ||/ ||w ^ If Ho ^ 2Hf ||w, for all f E Hi.

Theorem 6. The H-Orlicz space Hf (y) is reflexive if and only if 9 E A2.

Proof. The proof is similar to the proof [13, Theorem 5]. □

Theorem 7. Let 9 be a N-function and Eg be the closure of the bounded functions in H*. Then the conjugate space (Eg, || ■ |(e}) is isometrically isomorphic to (Hf, || ■ where is the complement of 9.

Proof. Let vg E H?; define f: Ee ^ R by

f (ug) = (HK) < V0(t),U0(t) > dp. J [0,1]

Then f is well defined. Also ug, vg are limits of simple functions, so that < vg ,ug > is measurable. So,

(HK) | <Vg, Ug > \d» ^ (HK) Hvg(t) || hk.Hug(t) Hhkd^

J [0,1] J[0,1]

<

e.H'ue H<t>.

Consequently, f (vg) = (HK) Jj01] < vg(t),ug(t) > d^ is linear and

I hk ^ 11 ^ | ^. Consider ve = ^ XiXSi, xi E X, i = 1, 2,..., and }°==1

1 1

is a countable partition of [0,1] with yJ(£i) > 0, i = 1, 2.... Now, for t > 0, \vg\ E H*so there exists a non-negative h E E^(^) C HK(y), 0 < HhH </, ^ 1, such that

Hvg ||e- 2< (HK) J Hvg (¿)|hk h(t)dp.

[0,1]

If ||xj || = 1, so that

K*||- 2HhHHK ^ x*(Xi)

IX

and ug = Xih\£i, then for a > 0 we have

i=1

9{Haug (t)H)dv = HK) 6(Haxih(t)H)dv = [0,1] i=1 S

= HK) J 9(a\h(t)\)dß = (HK) J 9(a\h(t))dß < to. i=1 [0,1]

Thus, ue E Ee and

\\ue||e = inf [a> 0: (HK)J 9^M) C i} =

[0,1]

= inf [a> 0: £( HK) J Ml)dß C i} =

C i.

Now,

f(ue) =< we,ue >= (HK) < ve,ue > dß =

•Vi]

„ ^

= ( HK) J h(t) ^x*(xi)xeidß 2

[0,1] 1=1

2 ( HK) J h(t) jr (\\x*\\- 2HhH],)dß = [0,1] t=1

( HK) J h(t)dß

= ( HK) / h(i) \ |ue(i)Wdß - 2.-¡ihr-2

J 2 \\h\Hi

[0,1]

2 \\Ve\\f - 2 - 2.

So, HfUHK 2 \M\f. Hence, HIUhk = \\vq\\f and 0 C \\/ - fn\\HK C C \\ve - ven\\f ^ 0. Therefore, \\ IHhk = lim \\ fn\\ = lim \\ven\\f =

n—y^o n—y^o

So, Hf is isometrically isomorphic to a subspace of Ee. □

Theorem 8. The H-Oriicz space He(ß) has weak drop property if and only if 9 E A2.

Proof. The H-Orlicz space Hq(ß) is reflexive if and only if 9 E A2. We can conclude the complete proof with a similar way of the proof of the [6, Theorem 5]. □

e

4. Some consequences for relatively weakly compact subset

in H-Orlicz spaces. In this section, we discuss the subsets of HK(y) with weak drop property that are relatively weakly compact in H*

Theorem 9. A closed and bounded convex subset F of HK(y) has weak drop property if and only if there is an N-function 9 with A' condition, such that F is relatively weakly compact in H* (y).

Proof. Suppose a closed and bounded convex subset F of HKhas weak drop property. From the Theorem 5, F is uniform integrable.

Take an N-function 9 e A' with 9(9(x)) ^ 91(x), where 91 is an Y-function for large values of x.

Then sup\(HK) J 0(0(£))d£: £ e f) < to. So, by de la Vallee 1 [0,1] J

Poussin's theorem, F = \ Ihk(g): 9 = f 9(f ) > is uniformly integrable in

1 [0,i] J HK(^). The modulus of uniform integrability vanishes: v(F) = 0 in H*. Since F has weak drop property, so, F is relatively compact in HK(^). It is also relatively compact in the topology of convergence in measure. Hence, F is relatively compact in H*(^). □

Theorem 10. Let 9 e A2 and suppose that its complement satisfies lim = oo for some c> 0; then there is a closed bounded convex set

F C Hi (p) that

is relatively weakly compact if and only if v (F) = 0.

Proof. Let us consider the closed bounded convex set F C HI(y) that is relatively compact. If possible, assume v(F) = 0; then there is an e0 > 0, a sequence (fn) C F, and a measurable sequence (£n) with ^(£n) ^ 0, so \\xen fn\\ > £0 for each n e N.

Using the Eberlein-Smulian theorem, we can have f e Hi and a subsequence (fnk) of (fn), so that fnk ^ f weakly in Hi(^). We can state that \\ • \\# has weak drop from [6, Theorem 5]. The Lemma 1 gives v(F) = 0.

Conversely, suppose v(F) = 0 and F C Eg. If (fn) is a sequence in F, it is bounded in HK(^)-norm as (fn) is bounded in H*-norm. Now, by Komlos's theorem, there is a subsequence (fnk) that has ^-a.e. convergent arithmetic means to f. When 0 is the complement of 9, we have, for any measurable subset £ C [0,1] and g e Hi with \\g\\,p ^ 1:

^(F) = l(HK) j gX£ fd^l ^ (HK) j lgX£ f ^

I l n 1 n l

C Ii minf(HK) \gX£- ^ fnk\dß C sup - £(HK) \gXs fnk\dß C

n

- n -

k=i n k=i

1 n

C sup- ll5rN(^)-HX£fnk Ü0 C sup{ llXf h\\e: h E Ff = 0

n - k=i 1 J

Thus,

\\Xsf ||0 C sup{\( HK) igxsfdß\

C n C

C supj ||X£h\\e : h E f} = 0

So, f E HI(j) with absolutely continuous norm. Assume an = n hk. Since the inclusion map: Hf ^ HK(j) is continuous, there is a K > 0, such that Hgll^x ^ K||g\\f V g E Hf. Now for any e > 0 choose 8 > 0, so

that sup j Ixsh\\o: h E f| = 0 whenever j(£) < 8. By Egorov's theorem,

there is a measurable set £ with j([0,1] \£) < 8, so that an ^ f uniformly on £. Choose N E N, so that \\xe(an — /)||^ —^ 0 whenever n ^ N. Then, for g E Hf with \lg\lf ^ 1 and n ^ N, |(HK) J g(an — f)dj\ ^ 0. So, F is a Banach-Saks set in H®. Since fnk ^ f weakly in Hf, so, by Eberlein Smulian theorem, F is relatively weakly compact in Hf. □

Conclusions. In this article, we have studied uniform integrability of a closed bounded convex sets with weak drop property. Relatively weakly compactness of some subsets of HK(¡) with weak drop property that are relatively weakly compact in Hf (¡) have been also discussed.

Acknowledgments. I am very grateful to the Academic Editor and the anonymous Referees for their valuable suggestions in the improvement of this paper.

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Received February 19, 2023. In revised form, May 16, 2023. Accepted May 28, 2023. Published online July 6, 2023.

Hemanta Kalita

Mathematics Division, VIT Bhopal University, Indore-Bhopal Highway, Sehore, Madhya Pradesh, India E-mail: [email protected]