26A24 Keywords: Rieszp-variation, bounded variation spaces 1

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Vol. 19 (2018), No. 1, pp. 157–170 DOI: 10.18514/MMN.2018.1988

A NOTE ON MERENTES–RIESZ BOUNDED VARIATION SPACES

REN ´E ERL´IN CASTILLO, HUMBERTO RAFEIRO, AND EDUARD TROUSSELOT Received 03 May, 2016

Abstract. In this paper we introduce a function space with some generalization of bounded variation and study some of its properties, like embeddings, decompositions and others.

2010Mathematics Subject Classification: 26A45; 26B30; 26A16; 26A24 Keywords: Rieszp-variation, bounded variation spaces

1. INTRODUCTION

Around two centuries ago C. Jordan introduced the notion of a function of bounded variation and established the relation between these functions and monotonic ones when he was studying convergence of Fourier series. Later on the concept of bounded variation was generalized in various directions by many mathematicians, such has, L.

Ambrosio, R. Caccioppoli, L. Cesari, E. Conway, G. Dal Maso, E. de Giorgi, S. Hud- jaev, J. Musielak, O. Oleinik, W. Orlicz, F. Riesz, J. Smoller, L. Tonelli, A. Vol’pert, N. Wiener, among many others. It is noteworthy to mention that many of these gen- eralizations where motivated by problems in such areas as calculus of variations, convergence of Fourier series, geometric measure theory, mathematical physics, etc.

For many applications of functions of bounded variation in mathematical physics see the monograph [6]. For a thorough exposition regarding bounded variation spaces and related topics, see the recent monograph [1].

In 1992 N. Merentes [4] generalized the concept of boundedp-variation in the sense of Riesz defining the notion of bounded.p; 2/-variation. We say that a function f WŒa; b !Rhas bounded.p; 2/-variation inŒa; bif the number

V_p;2^R .f /D^Vp;2^R .f; Œa; b/

Dsup

˘ n 1

X

jD1

ˇ ˇ ˇ ˇ

f .bj/ f .dj/ bj dj

f .cj/ f .aj/ cj aj

ˇ ˇ ˇ ˇ

p 1

.bj aj/^{p 1}

is finite, wherep1and the supremum is taken on the set of all block partitions of Œa; b.

c 2018 Miskolc University Press

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The set of all bounded .p; 2/-variation functions is denoted by RV.p;2/.Œa; b/

which has an algebra structure. Moreover, it was shown that all functions that have bounded.p; 2/-variation also has bounded second-variation. In this work we gener- alize this concept to obtain the bounded.p; 2; ˛/-variation functions in the sense of Riesz and obtain some characterizations of this new space.

2. PRELIMINARIES

Before introducing the bounded variation space we will need some auxiliary results.

Definition 1. A functionf WŒa; b !Ris said to be an˛-Lipschitz function if there exists a constantM > 0such that

jf .x/ f .y/j Mj˛.x/ ˛.y/j

for allx; y2.a; b/with x¤y. We define the space ˛-LipŒa; bas the space of all

˛-Lipschitz functions. This space is normable, via the norm

kfk˛-LipWD jf .a/j Csup

x¤y

jf .x/ f .y/j j˛.x/ ˛.y/j:

We now introduce the concepts of ˛ absolutely continuous function and ˛- derivative.

Definition 2. A functionf WŒa; b!Ris said to beabsolutely continuous with respect to˛if, for every " > 0, there exists some ı > 0 such that if f.aj; bj/gjⁿD1

are disjoint open subintervals of Œa; b; then

n

X

jD1

j˛.bj/ ˛.aj/j< ı implies

n

X

jD1

jf .bj/ f .aj/j< ":

All functions in˛-ACŒa; bare bounded and form an algebra of functions under point- wise defined standard operations.

Definition 3. Supposef and˛are real-valued functions defined on the same open intervalI and letx02I. We say thatf is˛-differentiable atx0if the following limit exists

xlim!x0

f .x/ f .x0/

˛.x/ ˛.x0/:

If the limit exists we denote its value byf_˛⁰.x0/, which we call the˛-derivative off atx0.

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3. RV.p;2;˛/Œa; bIS A NORMED SPACE

We want to recall the so-called Popoviciu variation (introduced in 1933 by T.

Popoviciu in [5]) for a partition ˘ D faDx1< x2< < xmDbgand a function f WŒa; b !Ris given by

Var_k;1.f; ˘; Œa; b/D

m kC1

X

jD1

jf Œxj; : : : ; x_j_C_{k 1} f Œxj 1; : : : ; x_j_C_{k 2}j wheref Œ; : : : ;is defined recursively in the following way:

f Œx0WDf .x0/;

f Œx0; x1WDf Œx1 f Œx0 x1 x0

f Œx0; x1; x2WDf Œx1; x2 f Œx0; x1 x2 x0

f Œx0; x1; : : : ; x_kWDf Œx1; x2; : : : ; x_k f Œx0; x1; : : : ; x_{k 1} x_k x0

:

In the following we will consider a block partition˘ of the intervalŒa; b. It will be taken in the following way

˘ D faDx1;1< x1;26x1;3< x1;4Dx2;1< x2;26x2;3< x2;4

Dx3;1< < xn 1;4Dxn;1< xn;26xn;3< xn;4Dbg; (3.1) in place of the regular partition.

Definition 4. Let f be a real-valued function defined on Œa; b, ˘ be a block partition ofŒa; band˛be an increasing function. Let

_.p;2;˛/^R .f; ˘ /D

n

X

jD1

ˇˇf˛Œxj;4; xj;3 f˛Œxj;2; xj;1ˇ ˇ

p

j˛.xj;4/ ˛.xj;1/j^{p 1} where

f˛Œa; bD f .b/ f .a/

˛.b/ ˛.a/

and

V^R_.p;2;˛/.f; Œa; b/D^V^R_.p;2;˛/.f /Dsup

˘

_.p;2;˛/^R .f; ˘ /;

where the supremum is taken on all block partitions of Œa; b. To the number V^R_.p;2;˛/.f; Œa; b/ we call theRiesz.p; 2; ˛/-variation of the functionf inŒa; b. If V^R_.p;2;˛/.f; Œa; b/ <1, then we say that f hasbounded Riesz .p; 2; ˛/-variation.

The set of all functions is denoted byRV.p;2;˛/.Œa; b/.

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Remark1. WhenpD1we observe thatBV.2;˛/Œ.a; b/D^RV.1;2;˛/.Œa; b/.

In the following result we will show that iff has Riesz bounded.p; 2; ˛/-variation, thenf has bounded˛-second variation.

Theorem 1. We haveRV.p;2;˛/.Œa; b/ ,!^BV.2;˛/Œ.a; b/, with

V_.2;˛/.f; Œa; b/.˛.b/ ˛.a//^p^p¹V^R_.p;2;˛/.f; Œa; b/: (3.2) Proof. Let˘ be a block partition of type (3.1). Using the H¨older inequality we obtain

n

X

jD1

ˇˇf˛Œxj;4; xj;3 f˛Œxj;2; xj;1ˇ ˇ j˛.xj;4/ ˛.xj;1/j^p^p¹

j˛.xj;4/ ˛.xj;1/j^p^p¹

0

@

n

X

jD1

ˇˇf˛Œxj;4; xj;3 f˛Œxj;2; xj;1ˇ ˇ

p

j˛.xj;4/ ˛.xj;1/j^{p 1} 1 A

1 p0

@

n

X

jD1

j˛.xj;4/ ˛.xj;1/j 1 A

p 1 p

V^R_.p;2;˛/.f; Œa; b/_p¹

.˛.b/ ˛.a//^p^p¹:

Since the obtained inequality holds for all block partitions we obtain the desired

inequality (3.2).

Remark 2. We know (see [3]) that if f 2^BV.2;˛/Œ.a; b/, then there exists the right and left˛-derivativef_˛⁰_C.x0/andf_˛⁰ .x0/on eachx02.a; b/andf_˛⁰_C.a/and f_˛⁰ .b/. The last result allow us to conclude that this is also true if f 2 RV.p;2;˛/.Œa; b/. In particular, there existsf_˛⁰_C.a/which we write asf_˛⁰.a/.

Lemma 1. Letf WŒa; b !Rbe a function such thatf 2^RV.p;2;˛/.Œa; b/and V^R_.p;2;˛/.f; Œa; b/D0. Then there exists; 2Rsuch thatf .x/D˛.x/C for allx2Œa; b.

Proof. SinceV^R_.p;2;˛/.f; Œa; b/D0we have that_.p;2;˛/^R .f; ˘ /D0for all block partitions ˘ of Œa; b. Let us consider the particular partition given by ˘0 D fa < xx < bg,

_.p;2;˛/^R .f; ˘₀/D ˇ ˇ ˇ ˇ

f .b/ f .x/

˛.b/ ˛.x/

f .x/ f .a/

˛.x/ ˛.a/

ˇ ˇ ˇ ˇ

p 1

j˛.b/ ˛.a/j^{p 1} D0;

where

ˇ ˇ ˇ ˇ

f .b/ f .x/

˛.b/ ˛.x/

f .x/ f .a/

˛.x/ ˛.a/

ˇ ˇ ˇ ˇ

p

D0;

Direct calculations show that

f .x/Df .b/ f .a/

˛.b/ ˛.a/˛.x/Cf .a/˛.b/ f .b/˛.a/

˛.b/ ˛.a/

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and now takingandin the following manner Df .b/ f .a/

˛.b/ ˛.a/; and Df .a/˛.b/ f .b/˛.a/

˛.b/ ˛.a/

we have the desired result.

Remark3. The setRV.p;2;˛/.Œa; b/can be equipped with a linear space structure considering the operator f 7!

f j^RV.p;2;˛/.Œa; b/

defined in the space RV_.p;2;˛/.Œa; b/given in the following way:

WD jf .a/j C jf_˛⁰.a/j C

V^R_.p;2;˛/.f; Œa; b/_p¹

; (3.3) forf 2^RV.p;2;˛/.Œa; b/.

Theorem 2. The operator f 7!

is a norm in the space RV_.p;2;˛/.Œa; b/.

Proof. Let us takef such that

D0. From (3.3) this means that jf .a/j D 0, jf_˛⁰.a/j D 0 and V^R_.p;2;˛/.f / D0. Since V^R_.p;2;˛/.f /D0 from Lemma1we deduce that

f .x/D˛.x/C; ; 2R:

Sincef_˛⁰.x/Dforx2Œa; bwe conclude thatf_˛⁰ is a constant function which is null atxDaand thusD0, from which we get thatf 0sincef .a/D0.

It is straighforward to see that for2Rwe have that V^R_.p;2;˛/.f /D jj^p^V^R.p;2;˛/.f /

and now by the definition of the operator (3.3) implies the homogeneity of the operator under consideration.

Let us now prove the triangle inequality. We now introduce the following notation ^j_˛.f /Df˛Œxj;4; xj;3 f˛Œxj;2; xj;1:

Letf; g2^RV.p;2;˛/.Œa; b/and˘ be a block partition ofŒa; bas in (3.1), then _.p;2;˛/^R .f Cg; ˘ /D

n

X

jD1

j^j˛.f Cg/j^p j˛.xj;4/ ˛.xj;1/j^{p 1} D

n

X

jD1

j^j˛.f /C^j˛.g/j^p j˛.xj;4/ ˛.xj;1/j^{p 1}

n

X

jD1

j^j˛.f /C^j˛.g/j^{p 1}

j˛.xj;4/ ˛.xj;1/j^{p 1}j^j_˛.f /j

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C

n

X

jD1

j^j˛.f /C^j˛.g/j^{p 1}

j˛.xj;4/ ˛.xj;1/j^{p 1}j^j_˛.g/j

n

X

jD1

j^j˛.f /C^j˛.g/j^{p 1} j˛.xj;4/ ˛.xj;1/j^.p^p^1/2

j^j˛.f /j j˛.xj;4/ ˛.xj;1/j^p^p¹ C

n

X

jD1

j^j˛.f /C^j˛.g/j^{p 1} j˛.xj;4/ ˛.xj;1/j^.p^p^1/2

j^j˛.g/j j˛.xj;4/ ˛.xj;1/j^p^p¹

2 4

n

X

jD1

3 5

p 1

p 2

4

n

X

jD1

j^j˛.f /j^p j˛.xj;4/ ˛.xj;1/j^{p 1}

3 5

1 p

C 2 4

n

X

jD1

3 5

p 1

p 2

4

n

X

jD1

j^j˛.g/j^p j˛.xj;4/ ˛.xj;1/j^{p 1}

3 5

1 p

:

We therefore have

_.p;2;˛/^R .f Cg; ˘ /_p¹

_.p;2;˛/^R .f; ˘ /_p¹ C

_.p;2;˛/^R .g; ˘ /_p¹ and since this is true for all partitions, we have

V^R_.p;2;˛/.f Cg; Œa; b/^V^R.p;2;˛/.f; Œa; b/C^V^R.p;2;˛/.g; Œa; b/;

and now it easily follows the triangle inequality.

4. EMBEDDING WITHRV.p;2;˛/.Œa; b/

We will show that if p < q, then there exists an embedding between the spaces RV_.p;2;˛/.Œa; b/andRV_.q;2;˛/.Œa; b/. We will need this fact to show completeness ofRV.p;2;˛/.Œa; b/.

Theorem 3. If1 < q < p <1, thenRV.p;2;˛/.Œa; b/ ,!^RV.q;2;˛/.Œa; b/, with f j^RV.q;2;˛/.Œa; b/

max n

1; .˛.b/ ˛.a//¹^q ^p¹o

; forf 2^RV.p;2;˛/.Œa; b/.

Proof. Letf 2^RV.p;2;˛/.Œa; b/and˘ be a block partition of Œa; bas in (3.1).

Let us consider

_.q;2;˛/^R .f; ˘ /D

n

X

jD1

j^j˛.f /j^q j˛.xj;4/ ˛.xj;1/j^{q 1} D

n

X

jD1

j^j˛.f /j^q

j˛.xj;4/ ˛.xj;1/j^.p^p^1/qj˛.xj;4/ ˛.xj;1/j^p^p^q:

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Applying the H¨older inequality we obtain

_.q;2;˛/^R .f; ˘ / 2 4

n

X

jD1

j^j_˛.f /j^p j˛.xj;4/ ˛.xj;1/j^{p 1}

3 5

q p2

4

n

X

jD1

j˛.xj;4/ ˛.xj;1/j 3 5

p q p

Dh

_.p;2;˛/^R .f; ˘ /i_p^q

.˛.b/ ˛.a//^p^p^q; from which

h

V^R_.q;2;˛/.f; Œa; b/i¹_q

.˛.b/ ˛.a//^q¹ ^p¹h

V^R_.p;2;˛/.f; Œa; b/i_p¹ whence

f j^RV.q;2;˛/.Œa; b/

maxn

1; .˛.b/ ˛.a//¹^q ^p¹o

;

forf 2^RV.p;2;˛/.Œa; b/.

Remark4. The proof of the above theorem remains valid ifqD1.

Corollary 1. Forp1we haveRV.p;2;˛/.Œa; b/ ,!^BV.2;˛/Œ.a; b/;with f j^BV.2;˛/Œ.a; b/

max n

1; .˛.b/ ˛.a//^p^p¹o

:

The following corollary follows from the previous results and from results from [2].

Corollary 2. If1 < q < p <1, then

RV_.p;2;˛/Œa; b ,!^RV.q;2;˛/Œa; b ,!^V.2;˛/Œa; b ,!˛-LipŒa; b ,!

,!^RV.p;˛/Œa; b ,!^RV.q;˛/Œa; b ,!˛-ACŒa; b ,!^VŒa; b ,!^BŒa; b:

5. RV_.p;2;˛/.Œa; b/IS ABANACH SPACE

Theorem 4. The space RV.p;2;˛/.Œa; b/;

j^RV.p;2;˛/.Œa; b/

is a Banach space.

Proof. Let.f_k/_k₂_Nbe a Cauchy sequence inRV_.p;2;˛/.Œa; b/. Given" > 0there existsN"2Nsuch that

fq frj^RV.p;2;˛/.Œa; b/

< "

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ifq; r > N", from which we get the following system of inequalities 8

ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ :

jfq.a/ fr.a/j< ";

j.fq/⁰_˛.a/ .fr/⁰_˛.a/j< ";

V^R_.p;2;˛/.fq fp/ < "^p:

From Corollary2we conclude that.fk/k2Nis a Cauchy sequence in˛-LipŒa; b;thus we have

fq fr j˛-LipŒa; b

< "K withKDmaxn

1; .˛.b/ ˛.a//^p^p¹o , hence jfq.a/ fr.a/j Csup

x¤y

ˇ ˇ ˇ ˇ

.fq fr/.x/ .fq fr/.y/

˛.x/ ˛.y/

ˇ ˇ ˇ ˇ

< "K and so

jfq.x/ fr.x/j< K".1C˛.b/ ˛.a//; x2Œa; b:

This tell us that for eachx2Œa; b .f_k.x//_k₂_Nis a Cauchy sequence inR. SinceR is a complete space, we can definef WŒa; b !Rasx7!f .x/WDlimk!1f_k.x/.

We are about to prove that:

(i) f 2^RV.p;2;˛/.Œa; b/, and

(ii) .f_k/_k₂_Nconverges tof in theRV_.p;2;˛/.Œa; b/-norm.

(i) Let˘ be a block partition ofŒa; bas in (3.1) Since.f_k/_k₂_Nis a Cauchy sequence inRV.p;2;˛/.Œa; b/the norm sequence

f_kj^RV.p;2;˛/.Œa; b/

k2Nis bounded, that is, there existsM > 0such that

M fork2N. From this fact we have

2 4

n

X

jD1

ˇ ˇ ˇ ˇ

f .xj;4/ f .xj;3/

˛.xj;4/ ˛.xj;3/

f .xj;2/ f .xj;1/

˛.xj;2/ ˛.xj;1/ ˇ ˇ ˇ ˇ

p 1

j˛.xj;4/ ˛.xj;1/j^{p 1} 3 5

1 p

D lim

k!1

2 4

n

X

jD1

ˇ ˇ ˇ ˇ

fk.xj;4/ fk.xj;3/

˛.xj;4/ ˛.xj;3/

fk.xj;2/ fk.xj;1/

˛.xj;2/ ˛.xj;1/ ˇ ˇ ˇ ˇ

p 1

j˛.xj;4/ ˛.xj;1/j^{p 1} 3 5

1 p

lim

k!1

h

V^R_.p;2;˛/.fk; Œa; b/i¹_p lim

k!1

fkj^RV.p;2;˛/.Œa; b/

M

for all partitions˘, then

V^R_.p;2;˛/.f /_p¹

M and thusf 2^RV.p;2;˛/.Œa; b/. Using embedding we may observe thatf_˛⁰.a/exists.

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(ii) One more time let us consider˘ to be a block partition of Œa; bas in (3.1).

Letq; r > N", then

n

X

jD1

ˇ ˇ ˇ ˇ

.fq fr/.xj;4/ .fq fr/.xj;3/

˛.xj;4/ ˛.xj;3/

.fq fr/.xj;2/ .fq fr/.xj;1/

˛.xj;2/ ˛.xj;1/

ˇ ˇ ˇ ˇ

p

1

j˛.xj;4/ ˛.xj;1/j^{p 1} ^V^R.p;2;˛/..fq fr/; Œa; b/ < "^p:

Lettingr! 1the above expression becomes

n

X

jD1

ˇ ˇ ˇ ˇ

.fq f /.xj;4/ .fq f /.xj;3/

˛.xj;4/ ˛.xj;3/

.fq f /.xj;2/ .fq f /.xj;1/

˛.xj;2/ ˛.xj;1/

ˇ ˇ ˇ ˇ

p

1

j˛.xj;4/ ˛.xj;1/j^{p 1} < "^p:

This holds for any partition ofŒa; b, thereforeV^R_.p;2;˛/.fq f / < "^pwhenq > N". Leth2R^Cbe such thata < aCh < s < t b, then

ˇ ˇ ˇ ˇ

.fq f /.t / .fq f /.s/

˛.t / ˛.s/

.fq f /.aCh/ .fq f /.a/

˛.aCh/ ˛.a/

ˇ ˇ ˇ ˇ

p 1

j˛.t / ˛.s/j^{p 1} ^V^R.p;2;˛/.fq f / < "^p ifq > N". Lettingh!0we have

ˇ ˇ ˇ ˇ

.fq f /.t / .fq f /.s/

˛.t / ˛.s/ .fq f /⁰_˛.a/

ˇ ˇ ˇ ˇ

p 1

j˛.t / ˛.s/j^{p 1} "^p from which

ˇ ˇ ˇ ˇ

.fq f /.t / .fq f /.s/

˛.t / ˛.s/ .fq/⁰_˛.a/C.f /⁰_˛.a/

ˇ ˇ ˇ ˇ

p

"^pj˛.t / ˛.s/j^{p 1}: Sincefq; f 2^RV.p;2;˛/.Œa; b/ ,!˛-LipŒa; b, we have

j.fq/⁰_˛.a/ .f /⁰_˛.a/j "j˛.b/ ˛.a/j^p^p¹C ˇ ˇ ˇ ˇ

.fq f /.t / .fq f /.s/

˛.t / ˛.s/

ˇ ˇ ˇ ˇ "j˛.b/ ˛.a/j^p^p¹C

fq f j˛-LipŒa; b

"j˛.b/ ˛.a/j^p^p¹C"KD QK"

ifq > N", where

KQ Dmax n

K; .˛.b/ ˛.a//^p^p¹o

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since

fq f j˛-LipŒa; b

Dlimr!1

fq frj˛-LipŒa; b

< K". Finally, for q > N"we obtain

jfq.a/ f .a/j C j.fq/⁰_˛.a/ .f /⁰_˛.a/j Ch

V^R_.p;2;˛/.fq f; Œa; b/i_q¹

.KQC2/";

in other words

fq f j^RV.p;2;˛/.Œa; b/

.KQC2/"

ifq > N"which means that.f_k/_k₂_Nconverges tof 2^RV.p;2;˛/.Œa; b/in the norm

j^RV.p;2;˛/.Œa; b/

.

6. RV.p;2;˛/.Œa; b/IS ABANACHALGEBRA

We are going to show thatRV.p;2;˛/.Œa; b/is closed under the multiplication of functions.

Theorem 5. Letf; g2^RV.p;2;˛/.Œa; b/. Thenfg2^RV.p;2;˛/.Œa; b/.

Proof. Let˘ be a block partition ofŒa; bas in (3.1). Let us consider _.p;2;˛/^R .f g; ˘ /D

n

X

jD1

ˇˇ.fg/_˛Œx_j;4; x_j;3 .fg/_˛Œx_j;2; x_j;1ˇ ˇ

p

j˛.xj;4/ ˛.xj;1/j^{p 1} D

n

X

jD1

ˇˇ.fg/˛Œxj;4; xj;3 .fg/˛Œxj;2; xj;1ˇ ˇ

p 1

j˛.xj;4/ ˛.xj;1/j^{p 1} ˇ

ˇ.fg/˛Œxj;4; xj;3 .fg/˛Œxj;2; xj;1ˇ

ˇ: (6.1) Observe that the termˇ

ˇ.fg/˛Œxj;4; xj;3 .fg/˛Œxj;2; xj;1ˇ

ˇDWT can be written as T D jf .xj;4/g˛Œxj;4; xj;3Cg.xj;3/f˛Œxj;4; xj;3

f .xj;2/g˛Œxj;2; xj;1 g.xj;1/f˛Œxj;2; xj;1j and now adding and subtracting appropriate terms and grouping the terms we obtain

T D jf .xj;4/.g˛Œxj;4; xj;3 g˛Œxj;2; xj;1/Cg.xj;3/.f˛Œxj;4; xj;3 f˛Œxj;2; xj;1/

C.f .xj;4/ f .xj;2//g˛Œxj;2; xj;1C.g.xj;3/ g.xj;1//f˛Œxj;2; xj;1j: (6.2) From Corollary2we have

RV.p;2;˛/.Œa; b/ ,!˛-LipŒa; b and RV.p;2;˛/.Œa; b/ ,!^BŒa; b:

This let us write

jf .xj;4/j kfk₁; jg.xj;3/j kgk₁ and

f ./ f ./

˛./ ˛./ ^Lip˛.f /; g./ g./

˛./ ˛./ ^Lip˛.g/;

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for all; 2Œa; b. Using these estimates we obtain

jf .xj;4/ f .xj;2/jjg˛Œxj;2; xj;1j C jg.xj;3 g.xj;1/jjf˛Œxj;2; xj;1j Djf .xj;4/ f .xj;2/j

j˛.xj;4/ ˛.xj;2/jjg.xj;2/ g.xj;1/j

j˛.xj;2/ ˛.xj;1/jjj˛.xj;4/ ˛.xj;1/jj Cjf .xj;2/ f .xj;1/j

j˛.xj;2/ ˛.xj;1/j jg.xj;3/ g.xj;1/j

j˛.xj;3/ ˛.xj;1/jj˛.xj;4/ ˛.xj;1/j .Lip_˛f /.Lip_˛g/ ˛.xj;4 ˛.xj;2/C˛.xj;3/ ˛.xj;1//

replacing all this into (6.2) we have ˇˇ.fg/˛Œxj;4; xj;3 .fg/˛Œxj;2; xj;1ˇ

ˇ kfk₁ˇ

ˇg˛Œxj;4; xj;3 g˛Œxj;2; xj;1ˇ

ˇC kgk₁ˇ

ˇf˛Œxj;4; xj;3 f˛Œxj;2; xj;1ˇ ˇ C.Lip_˛f /.Lip_˛g/ ˛.xj;4/ ˛.xj;2/C˛.xj;3/ ˛.xj;1//

:

Now replacing this last estimate into (6.1), separating summations and fixing expo- nents to apply the H¨older inequality we have

_.p;2;˛/^R .fg; ˘ / kfk₁

n

X

jD1

p 1

j˛.xj;4/ ˛.xj;1/j^.p^p^1/2

ˇˇg˛Œxj;4; xj;3 g˛Œxj;2; xj;1ˇ ˇ j˛.xj;4/ ˛.xj;1/j^p^p¹ C kgk₁

n

X

jD1

p 1

j˛.xj;4/ ˛.xj;1/j^.p^p^1/2

ˇˇf˛Œxj;4; xj;3 f˛Œxj;2; xj;1ˇ ˇ j˛.xj;4/ ˛.xj;1/j^p^p¹ C.Lip_˛f /.Lip_˛g/

n

X

jD1

p 1

j˛.xj;4/ ˛.xj;1/j^.p^p^1/2 j˛.xj;4/ ˛.xj;2/C˛.xj;3/ ˛.xj;1/j

j˛.xj;4/ ˛.xj;1/j^p^p¹

:

Applying in each summation the H¨older inequality we obtain _.p;2;˛/^R .fg; ˘ / kfk1

h

_.p;2;˛/^R .fg; ˘ /i^p_p¹h

_.p;2;˛/^R .g; ˘ /i_p¹ C kgk₁h

_.p;2;˛/^R .fg; ˘ /i^p_p¹h

_.p;2;˛/^R .f; ˘ /i_p¹

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C.Lip_˛f /.Lip_˛g/h

_.p;2;˛/^R .fg; ˘ /i^p_p¹

2 4

n

X

jD1

j˛.xj;4/ ˛.xj;2/C˛.xj;3/ ˛.xj;1/j^p j˛.xj;4/ ˛.xj;1/j^{p 1}

3 5

1 p

:

Simplifying we have h

_.p;2;˛/^R .fg; ˘ / i_p¹

kfk₁h

_.p;2;˛/^R .g; ˘ / i¹_p

C kgk₁h

_.p;2;˛/^R .f; ˘ / i_p¹

C.Lip_˛f /.Lip_˛g/

2 4

n

X

jD1

3 5

1 p

(6.3) Simplifying the last bracket in (6.3), forj D1; : : : ; n, we can observe that

D ˇ ˇ ˇ ˇ

˛.xj;4/ ˛.xj;2/C˛.xj;3/ ˛.xj;1/

˛.xj;4/ ˛.xj;1/

ˇ ˇ ˇ ˇ

p 1

j˛.xj;4/ ˛.xj;2/C˛.xj;3/ ˛.xj;1/j D

ˇ ˇ ˇ ˇ

1C˛.xj;3/ ˛.xj;2/

˛.xj;4/ ˛.xj;1/ ˇ ˇ ˇ ˇ

p 1

j˛.xj;4/ ˛.xj;2/C˛.xj;3/ ˛.xj;1/j: (6.4) Sincexj;1< xj;2xj;3< xj;4we have

˛.xj;3/ ˛.xj;2/ < ˛.xj;4/ ˛.xj;1/I

˛.xj;4/ ˛.xj;2/ > 0I

˛.xj;3/ ˛.xj;1/ > 0:

Substituting this into (6.4)

2^{p 1} ˛.xj;4/ ˛.xj;2/C˛.xj;3/ ˛.xj;1/ and now summing we obtain

n

X

jD1

j˛.xj;4/ ˛.xj;2/C˛.xj;3/ ˛.xj;1/j^p

j˛.xj;4/ ˛.xj;1/j^{p 1} 2^p.˛.b/ ˛.a//:

Substituting this inequality into (6.3) we have h

_.p;2;˛/^R .fg; ˘ /i_p¹

kfk₁h

V^R_.p;2;˛/.g; Œa; b/i_p¹

C kgk₁h

V^R_.p;2;˛/.f; Œa; b/i_p¹ C2.˛.b/ ˛.a//^p¹.Lip_˛f /.Lip_˛g/;

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this last inequality holds for all partition˘ ofŒa; b, then h

V^R_.p;2;˛/.fg; Œa; b/i_p¹

kfk₁h

V^R_.p;2;˛/.g; Œa; b/i_p¹ C kgk₁h

V^R_.p;2;˛/.f; Œa; b/i_p¹

C2.˛.b/ ˛.a//^p¹.Lip_˛f /.Lip_˛g/ <1; (6.5)

from which it follows thatfg2^RV.p;2;˛/.Œa; b/.

We now obtain the following corollary.

Corollary 3. Letf; g2^RV.p;2;˛/.Œa; b/, then fgj^RV.p;2;˛/.Œa; b/

P

gj^RV.p;2;˛/.Œa; b/

C kfk₁

C kgk₁

: withP D2maxn

.˛.b/ ˛.a//^p¹; .˛.b/ ˛.a//^2p^p²o . Proof. Note that

Lip_˛.f / kf j˛-LipŒa; bk

f j^V.2;˛/Œa; b

max

n

1; .˛.b/ ˛.a//^p^p¹o

and similarly tog. Then (6.5) can be written as

h

V^R_.p;2;˛/.fg; Œa; b/i_p¹

kfk₁h

V^R_.p;2;˛/.g; Œa; b/i_p¹

C kgk₁h

V^R_.p;2;˛/.f; Œa; b/i_p¹ C2 .˛.b/ ˛.a//^p¹ h

max n

1; .˛.b/ ˛.a//^p^p¹oi2

:

REFERENCES

[1] J. Appell, J. Bana´s, and N. J. Merentes,Bounded variation and around. Berlin: de Gruyter, 2014.

doi:10.1515/9783110265118.

[2] R. E. Castillo and E. Trousselot, “On functions of.p; ˛/-bounded variation,”Real Anal. Exchange, vol. 34, no. 1, pp. 49–60, 2009.

[3] R. E. Castillo, H. Rafeiro, and E. Trousselot, “Space of functions with some generalization of bounded variation in the sense of de La Vall´ee Poussin.”J. Funct. Spaces, vol. 2015, p. 9, 2015, doi:10.1155/2015/605380.

[4] N. Merentes, “On functions of bounded.p; 2/-variation.”Collect. Math., vol. 43, no. 2, pp. 117–

123, 1992.

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[5] T. Popoviciu, “Sur quelques propriétés des fonctions d’une variable réelle convexes d’ordre supérieur,”Bul. Soc. S¸tiint¸. Cluj, vol. 7, pp. 254–282, 1933.

[6] A. Vol’pert and S. Khudyaev,Analysis in classes of discontinuous functions and equations of mathematical physics. Kluwer Academic Publishers, 1985.

Authors’ addresses

Ren´e Erl´ın Castillo

Universidad Nacional de Colombia, Colombia E-mail address:recastillo@unal.edu.co

Humberto Rafeiro

Pontificia Universidad Javeriana, Colombia E-mail address:silva-h@javeriana.edu.co

Eduard Trousselot

Universidad de Oriente, Venezuela

E-mail address:eddycharles2007@gmail.com