A new theorem on the existence of the Riemann Stieltjes integral and an improved version of the Loéve Young inequality

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R E S E A R C H

Open Access

A new theorem on the existence of the

Riemann-Stieltjes integral and an improved

version of the Loéve-Young inequality

Rafał M Łochowski

*

*_{Correspondence:}

[email protected] Department of Mathematics and Mathematical Economics, Warsaw School of Economics,

Madali ´nskiego 6/8, Warsaw, 02-513, Poland

Abstract

Using the notion of the truncated variation we obtain a new theorem on the existence and estimation of the Riemann-Stieltjes integral. As a special case of this theorem we obtain an improved version of the Loéve-Young inequality for the Riemann-Stieltjes integrals driven by irregular signals. Using this result we strengthen some results of Lyons on the existence of solutions of integral equations driven by moderately irregular signals.

MSC: Primary 26A42; secondary 45G10

Keywords: the Loéve-Young inequality; the Riemann-Stieltjes integral; irregular signals

1 Introduction

The purpose of this paper is to investigate the top-down structure of the Riemann-Stieltjes integral and to state some general condition guaranteeing the existence of this integral, expressed in terms of the functional calledtruncated variation.

The simplest (and rather not surprising) case where the Riemann-Stieltjes integral

b

a fdg(RSI in short) exists, is the situation when the integrand and the integrator have no common points of discontinuity, the former is bounded and the latter has ﬁnite total vari-ation. We will prove a general theorem (Theorem ) encompassing this situation as well as the more interesting case when the RSI exists, namely when the integrand and the integra-tor have a possibly unbounded variation, but they have ﬁnitep-variation andq-variation, respectively, withp> ,q>  andp–₊_q–_{> . The latter result is due to Young [], p.,}

theorem on Stieltjes integrability. Forf : [a;b]→Randp> , thep-variation, which we will denote byVp(f; [a;b]), is deﬁned as

Vpf, [a;b]=sup

n

sup

a≤t<t<···<tn≤b

n–

i=

f(ti) –f(ti–)

p .

The aforementioned Theorem  provides also an upper bound for the diﬀerence

b

a

fdg–f(a)g(b) –g(a).

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As a special case of this bound we will obtain the following, improved version of the clas-sical Loéve-Young inequality:

_abfdg–f(a)g(b) –g(a)

≤Cp,q

Vpf, [a;b]–/qf+_osc,[p/q_a–_;_bp_]Vqg, [a;b]/q, ()

whereCp,qis some constant depending onpandqonly andfosc,[a;b]:=supa≤s<t≤b|f(t) –

f(s)|. The original Loéve-Young estimate, published in  in [], reads

b

a

fdg–f(a)g(b) –g(a)≤ζp–+q–Vpf, [a;b]/pVqg, [a;b]/q,

where ζ(r) =+_k_=∞k–r is the famous Riemann zeta function. We say that our bound is improved version of the Loéve-Young inequality since the value of the constantCp,q (al-though possibly greater thanζ(p–₊_q–_{)) is irrelevant in applications while the fact that the}

term (Vp(f, [a;b]))/pin the original Loéve-Young estimate is replaced in our estimate by (Vp₍_f_{, [}_a_;_b_]))–/q_f+p/q–p

osc,[a;b] (notice that /p>  – /qand always (Vp(f, [a;b]))/p–(–/q)≥

f+_osc,[p/q_a–_;_bp_]) makes it possible to obtain stronger results. For example, Proposition  ob-tained in Section  with the help of () is a genuine improvement upon earlier known results of this type.

Let us comment shortly how the results on the existence of the RSI were obtained so far. The original Young’s proof utlilised elementary but clever induction argument for ﬁnite sequences. Another proof of the Young theorem may be found in [], Chapter , where in-tegral estimates based on control function and the Loéve-Young inequality are used. This approach is further applied in the rough-path theory setting. Further generalisations of Young’s theorem are possible, withp-variation replaced by the more generalϕ-variation:

Vϕf, [a;b]=sup

n

sup

a≤t<t<···<tn≤b

n–

i=

ϕf(ti) –f(ti–),

whereϕ: [; +∞)→[; +∞) is a Young function,i.e.a convex, strictly increasing function starting from  (see for example [, ] and for a survey of other results of this type, see [], Chapter , [], Section .).

However, as far as we know, Theorem  is a new result on the existence of the RSI. The proof of Theorem  utilises simple properties of the truncated variation and multiple ap-plication of the summation by parts. Similarly, no version of the Loéve-Young inequality as estimate (), as far as we know, has appeared so far (see the detailed historical notes on the Loéve-Young inequality in [], pp.-). We conjecture that using Theorem  one may also obtain a variation of the Loéve-Young inequality forϕ-variation (see [], Theo-rem ., Corollary ., or [], TheoTheo-rem .). We intend to deal with this conjecture in the future.

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construct approximations of integral equations, to consider terms of new type (like Lévy’s area). We believe that the truncated variation approach for such paths is also possible and this will be a topic of our further research.

Let us comment shortly on the organisation of the paper. In the next section we prove a general theorem on the existence of the Riemann-Stieltjes integral, expressed in terms of the truncated variation functionals and derive from it the stronger version of the Loéve-Young inequality. Next, in Section , we deal with the applications of this result to few types of integral equations driven by moderately irregular signals.

2 A theorem on the existence of the Riemann-Stieltjes integral

In this section we will prove a general theorem on the existence of the RSI_abfdg formu-lated in terms of the truncated variation. We will assume that both - integrandf : [a;b]→ Rand integratorg: [a;b]→Rare regulated functions. Let us state the deﬁnition of the truncated variation and recall the deﬁnition of a regulated function.

Forf : [a;b]→Ritstruncated variationwith the truncation parameterδ≥ will be denoted by TVδ(f, [a;b]). It may be simply deﬁned as the greatest lower bound for the total variation of any functiong: [a;b]→R, uniformly approximatingf with the accuracyδ/,

TVδf, [a;b]:=infTVg, [a;b]:f –g∞,[a;b]≤δ/

.

f –g∞,[a;b] denotes heresupa≤t≤b|f(t) –g(t)|and the total variation TV(g, [a;b]) is de-ﬁned as

TVg, [a;b]:=sup

n

sup

a≤t<t<···<tn≤b

n–

i=

g(ti) –g(ti–).

It appears that the truncated variation TVδ₍_f_{, [}_a_;_b_{]) is ﬁnite for any}_δ_{>  iﬀ}_f _is_regulated

(cf.[], Fact .) and then for anyδ>  the following equality holds:

TVδ_f_{, [}_a_;_b_]₌_sup

n

sup

a≤t<t<···<tn≤b

n–

i=

maxf(ti) –f(ti–)–δ, 

()

(cf.[], Theorem ).

Let us recall that a functionh: [a;b]→Risregulatedif there exist one-sided ﬁnite limitslimt→a+h(t) andlimt→b–h(t), and for anyt∈(a;b) there exist one-sided ﬁnite limits limt→x–h(t) andlimt→x+h(t).

We will also need the following result (cf.[], Theorem ): for any regulated function

f : [a;b]→Randδ>  there exists a regulated functionfδ _{: [}_a_;_b_]_→_R_{such that}_f _–

fδ

∞,[a;b]≤δ/ and

TVfδ, [a;b]= TVδf, [a;b].

From equation (), it directly follows that the truncated variation is a superadditive func-tional of the interval,i.e.for anyd∈(a;b)

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Moreover, we also have the following easy estimate of the truncated variation of a function

f perturbed by some other functionh:

TVδf +h, [a;b]≤TVδf, [a;b]+ TVh, [a;b], ()

which stems directly from the inequality, fora≤s<t≤b,

maxf(t) +h(t) –f(s) +h(s)–δ, ≤maxf(t) –f(s)–δ, +h(t) –h(s).

Theorem  Let f,g: [a;b]→Rbe two regulated functions which have no common points of discontinuity.Letη≥η≥ · · · andθ≥θ≥ · · ·be two sequences of non-negative num-bers,such thatηk↓,θk↓as k→+∞.Deﬁneη–:=supa≤t≤b|f(t) –f(a)|and

S:=

+∞

k=

kηk–·TVθk

g, [a;b]+ ∞

k=

kθk·TVηk

f, [a;b].

If S< +∞then the Riemann-Stieltjes integral_abfdg exists and one has the following esti-mate:

_abfdg–f(a)g(b) –g(a)≤S. ()

Remark  The assumption thatf andghave no common points of discontinuity is

neces-sary for the existence of the RSI_abfdg. When more general integrals are considered (e.g.

the Moore-Pollard integral,cf.[], p.), we may weaken this assumption and assume thatf andghave no common one-sided discontinuities.

The proof of Theorem  will be based on the following lemma.

Lemma  Let f,g: [a;b]→Rbe two regulated functions.Let c=t<t<· · ·<tn=d be any

partition of the interval[c;d]⊂[a;b]and letξ=c andξ, . . . ,ξnbe such that ti–≤ξi≤ti

for i= , , . . . ,n.Then forδ–:=supc≤t≤d|f(t) –f(c)|,δ≥δ≥ · · · ≥δr≥andε≥ε≥

· · · ≥εr≥the following estimate holds:

n

i= f(ξi)

g(ti) –g(ti–) –f(c)

g(d) –g(c)

≤

r

k=

kδk–·TVεk

g, [c;d]+ r

k=

kεk·TVδk

f, [c;d]+nδrεr.

Proof Denoteε=ε, by summation by parts, we have the following equality:

n

i= f(ξi)

g(d) –g(c)

= n

i=

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+ n

i=

f(ξi) –f(c) g(ti) –gε(ti) –

g(ti–) –gε(ti–)

= n

i=

f(ξi) –f(c) gε(ti) –gε(ti–)

+ n

i=

g(d) –gε(d) –g(ti–) –gε(ti–) f(ξi) –f(ξi–) , ()

wheregε_{: [}_c_;_d_]_→_R_{is regulated and such that}

g–gε

∞,[c;d]≤



ε and TV

_gε_{, [}_c_;_d_]_{= TV}ε_g_{, [}_c_;_d_]_.

Similarly, forδ=δwe may write

n

i=

g(d) –gε(d) –g(ti–) –gε(ti–) f(ξi) –f(ξi–)

= n

i=

g(d) –gε(d) –g(ti–) –gε(ti–) fδ(ξi) –fδ(ξi–)

+ n

i=

f(ξi) –fδ(ξi) –

f(c) –fδ(c) g(ti) –gε(ti)

–g(ti–) –gε(ti–) , ()

wherefδ_{: [}_c_;_d_]_→_R_{is regulated and such that}

f –fδ

∞,[c;d]≤



δ and TV

_fδ_{, [}_c_;_d_]_{= TV}δ_f_{, [}_c_;_d_]_.

Since TV(gε_{, [}_c_;_d_{]) = TV}ε

(g, [c;d]), TV(fδ_{, [}_c_;_b_{]) = TV}δ

(f, [c;d]),g–gε

∞,[c;d]≤ε/ and

f–fδ

∞,[c;d]≤δ/, from () and () we have the following estimate: n i= f(ξi)

g(d) –g(c)

≤ sup

c≤t≤d

f(t) –f(c)·TVε_g_{, [}_c_;_d_]₊_ε_·_TVδ_f_{, [}_c_;_d_]₊_nδε_. _()

Denoteg:=g–gε_,_f_:=_f _–_fδ_{on [}_c_;_d_{]. By () and (), instead of the last summand}_nδε_in

() we may write the estimate

n i=

f(ξi) –fδ(ξi) –

f(c) –fδ₍_c₎ _g₍_t

i) –gε(ti)

–g(ti–) –gε(ti–) = n i=

f(ξi) –f(c) g(ti) –g(ti–)

≤ sup

c≤t≤d

f(t) –f(c)·TVε_g_{, [}_c_;_d_]₊_ε·_TVδ_f_{, [}_c_;_d_]₊_nδ_ε

≤δ·TVε_g , [c;d]

+ε·TVδ

f, [c;d]

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where the last but one inequality in () follows by the same reasoning for f andg as inequality () forf andg. Repeating these arguments, by induction we get

n i= f(ξi)

g(d) –g(c)

≤ r k=

δk–·TVεk

gk, [c;d]

+ r

k=

εk·TVδk

fk, [c;d]

+nδrεr, ()

where δ–:=supc≤t≤c|f(t) –f(a)|,g≡g,f≡f and for k= , , . . . ,r, gk:=gk––g

εk–

k–, fk:=fk––f

δk–

k– are deﬁned similarly asgandf.

Sinceεk≤εk–fork= , , . . . ,r, by () and the fact that the functionδ→TVδ(h, [c;d])

is non-increasing, we estimate

TVεk_g

k, [c;d]

= TVεk_g

k––g

εk–

k–, [c;d]

≤TVεk_g

k–, [c;d]

+ TVgεk–

k–, [c;d]

= TVεk_g

k–, [c;d]

+ TVεk–_g

k–, [c;d]

≤TVεk_g

k–, [c;d]

.

Hence, by recursion, fork= , , . . . ,r,

TVεk_g

k, [c;d]

≤kTVεk_g_{, [}_c_;_d_]_. Similarly, fork= , , . . . ,r, we have

TVδk_f

k, [c;d]

≤kTVδk_f_{, [}_c_;_d_]_.

By () and last two estimates we get the desired estimate.

Remark  Notice that starting in () from the summation by parts, then splitting the diﬀerencef(ξi) –f(ξi–):

n

i= f(ξi)

g(d) –g(c)

= n

i=

g(d) –g(ti–) f(ξi) –f(ξi–) =

n

i=

g(d) –g(ti–) fδ(ξi) –fδ(ξi–)

+ n

i=

g(d) –g(ti–) f(ξi) –fδ(ξi) –

f(ξi–) –fδ(ξi–)

and proceeding similarly to the proof of Lemma  we get the symmetric estimate

n i= f(ξi)

g(d) –g(c)

≤ r k=

kεk–·TVδk

f, [c;d]+ r

k=

kδk·TVεk

g, [c;d]+nδrεr, ()

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Remark  Setting in Lemma ,n=  for anyξ∈[c;d] we get the estimate

f(ξ) –f(c)g(d) –g(c)

≤

r

k=

k_δ

k–·TVεk

f, [c;d]+ r

k=

k_ε k·TVδk

g, [c;d]+nδrεr, ()

and similarly, setting in Remark n=  we get a similar estimate, where the right side of () is replaced by the right side of ().

Now we proceed to the proof of Theorem .

Proof It is enough to prove that for any two partitions

π={a=a<a<· · ·<al=b},

ρ={a=b<b<· · ·<bm=b},

whereνi∈[ai–;ai],ξj∈[bj–;bj],i= , , . . . ,l,j= , , . . . ,m, the diﬀerence

l

i= f(νi)

g(ai) –g(ai–) –

m

j= f(ξj)

g(bj) –g(bj–)

is as small as we please, provided that the meshes of the partitionsπandρ, deﬁned as

mesh(π) := max

i=,,...,l(ai–ai–), mesh(ρ) :=j=,,...,maxm(bj–bj–),

respectively, are suﬃciently small. Deﬁne

σ=π∪ρ={a=s<s<· · ·<sn=b} and fori= , , . . . ,lconsider

f(νi)

g(ai) –g(ai–) –

k:sk–,sk∈[ai–;ai]

f(sk–)

g(sk) –g(sk–).

We estimate

f(νi)

g(ai) –g(ai–) –

f(sk–)

g(sk) –g(sk–)

≤f(νi)

g(ai) –g(ai–) –f(ai–)

g(ai) –g(ai–)

+

f(sk–)

g(sk) –g(sk–) –f(ai–)

g(ai) –g(ai–).

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they are regulated), hence the integral_abfdg exists. Thus we may and will assume that

ηN>  andθN>  for allN= , , , . . . .

ChooseN= , , . . . . By the assumption thatf andghave no common points of discon-tinuity, for suﬃciently smallmesh(π), fori= , , . . . ,lwe have

sup

ai–≤s≤ai

f(s) –f(ai–)≤ηN– ()

or

sup

ai–≤s≤ai

g(ai) –g(s)≤θN–. ()

To see this, assume that for everyh> , there exist [ah;bh]⊂[a;b] such thatbh–ah≤hand

sup_x_,_y∈_[_a

h;bh]|f(y) –f(y)|>ηN–andsupx,y∈[ah;bh]|g(x) –g(y)|>θN–. We choose a convergent subsequence of the sequence (a/n+b/n)/,n= , , . . . , and we see that the limit of this sequence is a point of discontinuity for bothf andg, which is a contradiction with the assumption thatf andghave no common points of discontinuity.

LetIbe the set of all indicesi= , , . . . ,lfor which () holds. Now, fori∈I, setδj–:= ηN+j–,εj:=θN+j,j= , , , . . . , and deﬁne

Si:=

+∞

j=

jηj–·TVθj

g, [ai–;ai]

+

+∞

j=

jθj·TVηj

f, [ai–;ai]

.

By Lemma  we estimate

f(sk–)

g(sk) –g(sk–) –f(ai–)

≤

+∞

j=

jδj–·TVεj

g, [ai–;ai]

+

+∞

j=

jεj·TVδj

f, [ai–;ai]

≤

+∞

j=

j_η

N+j–·TVθN+j

g, [ai–;ai]

+

+∞

j=

j_θ

N+j·TVηN+j

f, [ai–;ai]

≤–NSi.

Similarly,

f(νi)

g(ai) –g(ai–) –f(ai–)

g(ai) –g(ai–) ≤–NSi. Hence

f(νi)

g(ai) –g(ai–) –

f(sk–)

g(sk) –g(sk–)≤–NSi. ()

The truncated variation is a superadditive functional of the interval, from which we have

i∈I

TVθj_g_{, [}_a

i–;ai]

≤TVθj_g_{, [}_a_;_b_]_,

i∈I

TVηj_f_{, [}_a

i–;ai]

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By () and the last two inequalities, summing overi∈Iwe get the estimate

i∈I

f(νi)

g(ai) –g(ai–) –

f(sk–)

≤–N i∈I

Si≤–NS. ()

Now, letJbe the set of all indices, for which () holds. Fori= , , . . . ,ldeﬁne

Ti:=

+∞

j=

jθj·TVηj

f, [ai–;a]

+

+∞

j=

jηj·TVθj+

g, [ai–;ai]

.

Fori∈J, by the summation by parts and then by Lemma  we get

f(ai)

g(ai) –g(ai–) –

f(sk–)

=

g(sk)

f(sk) –f(sk–) –g(ai–)

f(ai) –f(ai–)

≤

+∞

j=

jθN+j–·TVηN+j

f, [ai–;ai]

+

+∞

j=

jηN+j·TVθN+j

g, [ai–;ai]

≤–NTi≤–NSi.

Similarly, by Lemma ,

f(ai)

g(ai) –g(ai–) –f(νi)

=g(ai–)

f(νi) –f(ai–) +g(ai)

f(ai) –f(νi) –g(ai–)

f(ai) –f(ai–)

≤–NSi.

From the last two inequalities we get

f(νi)

g(ai) –g(ai–) –

f(sk–)

≤f(ai)

g(ai) –g(ai–) –f(νi)

+f(ai)

g(ai) –g(ai–) –

f(sk–)

≤–NSi.

Summing overi∈Jand using the superadditivity of the truncated variation as a function of the interval, we get the estimate

i∈J

f(νi)

g(ai) –g(ai–) –

f(sk–)

≤–N i∈J

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Finally, from () and () we get

f(νi)

g(b) –g(a) – n

k= f(sk–)

≤·–NS.

A similar estimate holds for

m

i=j

f(ξj)

g(bj) –g(bj–) –

n

k= f(sk–)

g(sk) –g(sk–) ,

provided that the mesh(ρ) is suﬃciently small. Hence

l

i= f(νi)

g(ai) –g(ai–) –

m

i=j

f(ξj)

g(bj) –g(bj–)

≤·–NS,

provided thatthe meshes(π) and (ρ) are suﬃciently small. SinceNmay be arbitrarily large, we get the convergence of the approximating sums to an universal limit, which is the Riemann-Stieltjes integral. The estimate () follows directly from the proved convergence of approximating sums to the Riemann-Stieltjes integral and Lemma .

Using Remark  and reasoning similarly to the proof of Theorem , we get the symmetric result.

Theorem  Let f,g: [a;b]→Rbe two regulated functions which have no common points of discontinuity.Letη≥η≥ · · · andθ≥θ≥ · · ·be two sequences of non-negative num-bers,such thatηk↓,θk↓as k→+∞.Deﬁneθ–:=supa≤t≤b|g(b) –g(t)|and

˜

S:=

+∞

k=

kθk–·TVηk

f, [a;b]+ ∞

k=

kηk·TVθk

g, [a;b].

IfS˜< +∞then the Riemann-Stieltjes integral_abfdg exists and one has the following esti-mate:

_abfdg–f(a)g(b) –g(a)≤ ˜S. ()

From Theorem , Theorem  and Remark  we also have the following.

Corollary  Let f,g: [a;b]→Rbe two regulated functions which have no common points of discontinuity,ξ ∈[a;b]and S andS be as in Theorem˜ and Theorem,respectively.

If S< +∞orS˜< +∞then the Riemann-Stieltjes integral_abfdg exists and one has the following estimate:

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2.1 Young’s theorem and the Loéve-Young inequality

Let, forp> ,Vp_([_a_;_b_{]) denote the family of functions}_f _{: [}_a_;_b_]_→_R_{with ﬁnite}_p_-variation. Note that iff ∈Vp_([_a_;_b_{]) then}_f _{is regulated. The additional relation we will use is the} following one: iff ∈Vp([a;b]) for somep≥, then for everyδ> ,

TVδf, [a;b]≤Vpf, [a;b]δ–p. ()

As far as we know, the ﬁrst result of this kind, namely, TVδ(f, [a;b])≤Cfδ–pfor a contin-uous functionf ∈Vp_([_a_;_b_{]) and some constant}_C

f < +∞depending onf, was proven in [], Section . In [], TVε(f, [a;b]) is calledε-variation and is denoted byVf(ε). However, being equipped with equation () we see that equation () follows immediately from the inequality: for anya≤s<t≤b,

maxf(t) –f(s)–δ, ≤|f(t) –f(s)| p

δp– ,

which is an obvious consequence of the estimate:

δp–max|x|–δ, ≤

⎧ ⎨ ⎩

 ifδ≥ |x|,

|x|p–_max_{|_x_|_–_δ_{, }_} _{if  <}_δ_<_|_x_|≤ |x|

p

for anyδ>  and any realx. Let us denote

fp–var,[a;b]:=

Vpf, [a;b]/p ()

and recall thatfosc,[a;b]=supa≤s<t≤b|f(t) –f(s)|. Now we are ready to state a corollary

stemming from Theorem , which was one of the main results of []. The second part of this corollary is an improved version of the Loéve-Young inequality.

Corollary  Let f,g: [a;b]→Rbe two functions with no common points of discontinuity.

If f ∈Vp_([_a_;_b_])_{and g} _∈_Vq_([_a_;_b_]),_{where p}_{> ,}_q_{> ,}_p–₊_q–_{> ,}_{then the} Riemann-Stieltjes_abfdg exists.Moreover,there exists a constant Cp,q,depending on p and q only,

such that

b

a

fdg–f(a)g(b) –g(a)≤Cp,qfpp–var,[–p/qa;b]f +p/q–p

osc,[a;b]gq–var,[a;b].

Proof By Theorem  it is enough to prove that for some positive sequencesη≥η≥ · · ·

andθ≥θ≥ · · ·, such thatηk↓,θk↓ ask→+∞andη–=supa≤t≤b|f(t) –f(a)|one has

S:=

+∞

k=

g, [a;b]+

+∞

k=

kθk·TVηk

f, [a;b]

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The proof will follow from the proper choice of (ηk) and (θk). Sincep–+q–> , we have (q– )(p– ) < . We choose

α∈(q– )(p– ); , β= sup

a≤t≤b

f(t) –f(a), γ> ,

and, fork= , , . . . , deﬁne

ηk–=β·–(α

_/[(_q_–)(_p_–)])k_+ ,

θk=γ ·–(α

_/[(_q_–)(_p_–)])k_α_/(_q_–) .

By () we estimate

ηk–·TVθk

g, [a;b]≤β·–(α/[(q–)(p–)])k+

×Vqg, [a;b]γ·–(α/[(q–)(p–)])kα/(q–)–q = –(–α)(α/[(q–)(p–)])k+Vqg, [a;b]βγ–q,

and similarly

θk·TVηk

f, [a;b]≤γ·–(α/[(q–)(p–)])kα/(q–)

×Vpf, [a;b]β·–(α/[(q–)(p–)])k++–p = –(–α)(α/[(q–)(p–)])k_α_/(_q_{–)+–}_p

Vpf, [a;b]β–pγ.

Hence

S =

+∞

k=

g, [a;b]+

+∞

k=

kθk·TVηk

f, [a;b]

≤

₊_∞

k=

k–(–α)(α/[(q–)(p–)])k+

Vqg, [a;b]βγ–q

+

₊_∞

k=

k–(–α)(α/[(q–)(p–)])k_α_/(_q_{–)+–}_p

Vpf, [a;b]β–pγ.

Sinceα<  andα_/[(_q_{– )(}_p_{– )] > , we easily infer that}_S_{< +}_∞_{, from which we see that}

the integral_abfdgexists. Moreover, denoting

Cp,q=max

₊_∞

k=

k+–(–α)(α/[(q–)(p–)])k ,

+∞

k=

k+–(–α)(α/[(q–)(p–)])k_α_/(_q_–)–_p

we get

S≤

Cp,q

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Setting in this expressionγ = (Vq₍_g_{, [}_a_;_b_])/_Vp₍_f_{, [}_a_;_b_]))/q_βp/q_{we obtain}

S≤Cp,q

Vqg, [a;b]/qVpf, [a;b]–/qβ+p/q–p

≤Cp,qgq–var,[a;b]f_pp–_–var,[p/q_a_;_b_]f_osc,[+p/q_a–_;_bp_].

Remark  Letf,g,p,qandCp,qbe the same as in Corollary . Using Theorem  instead of Theorem , we get the following, similar estimate:

b

a

fdg–f(a)g(b) –g(a)≤Cp,qfp–var,[a;b]gqq––var,[q/pa;b]g +q/p–q

osc,[a;b].

From Corollary  and the obtained estimates, we also have for anyξ∈[a;b]

b

a

fdg–f(ξ)g(b) –g(a)

≤Cp,qfp–var,[a;b]gq–var,[a;b]min

_f+p/q–p

osc,[a;b]

f+_p_–var,[p/q–p_a_;_b_],

g+_osc,[q/p_a–_;_bq_]

g+_q_–var,[q/p–_aq_;_b_]

.

Remark  From Corollary , reasoning in a similar way to [], p., we get the following important estimate of theq-variation of the functiont→_atfdg:

·

a

fdg

q–var,[a;b]

≤Cp,qfpp––var,[p/qa;b]f +p/q–p

osc,[a;b]+f∞,[a;b]

gq–var,[a;b]

≤Cp,qfp–var,[a;b]+f∞,[a;b]

gq–var,[a;b],

wheref,g,p,qandCp,qare the same as in Corollary .

3 Integral equations driven by moderately irregular signals

Letp∈(; ). The preceding section provides us with tools to solve integral equations of the following form:

y(t) =y+

t

a

Fy(s)dx(s), ()

wherexis a continuous function from the spaceVp_([_a_;_b_{]) and}_F_:_R_→_R_is_α_-Lipschitz. The functional·var,p,[a;b]:Vp([a;b])→[; +∞) deﬁned as

fvar,p,[a;b]:=f(a)+fp–var,[a;b]

is a norm and the spaceVp_([_a_;_b_{]) equipped with this norm is a Banach space. For our} purposes it will be enough to work with the following deﬁnition of a locally or globallyα -Lipschitz function whenα∈(; ]. Forx= (x, . . . ,xn)∈Rnwe denotex=maxi=,...,n|xi|.

Deﬁnition  LetF:Rn_→_R_and_α_∈_{(; ]. For any}_R_{>  we deﬁne its}_local_α-Lipschitz

parameter Kα

F(R)as

K_F(α)(R) :=sup _|

F(y) –F(x)|

y–xα :x,y∈R

n_,_x₌_y_,_x_≤_R_,_y_≤_R

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and itsglobalα-Lipschitz parameter K_F(α) as K_F(α):=lim_R→+∞KF(R) < +∞. The function

F will be calledlocallyα-Lipschitzif for everyR> ,K_F(α)(R) < +∞and it will be called

globallyα-LipschitzifK_F(α)< +∞.

In the case when there is no ambiguity on what is the value of the parameterαand what is the functionF, we will writeKF(R),KF,K(R) or evenK.

First we will consider the casep–  <α< . In this case we have the existence but no uniqueness result. We will obtain a stronger result than similar results [], Lemma, p., or [], Theorem .. Namely, we will prove that there exists a solution to ()which is an element of the spaceVp([a;b]), not only an element of the spaceVq_([_a_;_b_{]) for arbitrary} chosenq>p. This will be possible with the use of Remark .

Proposition  Let p∈(; ),y∈R,x be a continuous function from the spaceVp_([_a_;_b_])

and F:R→Rbe globallyα-Lipschitz where p–  <α< .Equation()admits a solu-tion y,which is an element ofVp_([_a_;_b_])._Moreover_,_y

var,p,[a;b]≤R,where R> satisﬁes the equality

R= (Cp/α,p+ )KF(α)xp–var,[a;b]Rα+|y|+F()xp–var,[a;b], with Cp/α,pbeing the same as in Corollaryand Remark.

Now we proceed to the proof of Proposition . We will proceed in a standard way, but with the more accurate estimate of Remark  we will be able to obtain the ﬁniteness of

·var,p,[a;b]norm of the solution.

Proof Letf ∈Vp_([_a_;_b_{]). By [], Lemma .,}_F₍_f₍_·₎₎_∈_Vp/α_([_a_;_b_{]) and since}_α_/_p_{+ /}_p_{> ,}

we may apply Remark  and deﬁne the operatorT:Vp_([_a_;_b_])_→_Vp_([_a_;_b_]),

Tf :=y+

·

a

Ff(t)dx(t).

DenoteK=K_F(α). Using Remark  and [], Lemma ., we estimate

Tfvar,p,[a;b]= y+

·

a

Ff(t)dx(t)

var,p,[a;b]

≤

·

a

Ff(t)–Ff(a) dx(t) p–var,[a;b]

+Ff(a)xp–var,[a;b]+|y|

≤Cp/α,pF

f(·)_p_/_α_–var,[_a_;_b_]+Ff(·)–Ff(a)_∞_,[_a_;_b_]+Ff(a)

× xp–var,[a;b]+|y|

≤(Cp/α,p+ )F

f(·)_p_/_α_–var,[_a_;_b_]+Ff(a)xp–var,[a;b]+|y|

≤(Cp/α,p+ )Kfαp–var,[a;b]+F

f(a)xp–var,[a;b]+|y|. ()

By the Lipschitz property,

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Denoting

A= (Cp/α,p+ )Kxp–var,[a;b] and B=|y|+F()xp–var,[a;b], ()

from () we get

Tfvar,p,[a;b]≤Afαvar,p,[a;b]+B. ()

Forα<  letRbe the least positive solution of the inequalityR≥A·Rα₊_B₍_{i.e. R}₌

A·Rα₊_B_{). From () we see that the operator}_T_{maps the closed ball}_B₍_R_{) =}_{_f _∈_Vp_([_a_;_b_{]) :}

fvar,p,[a;b]≤R}to itself.

Now, forf,g∈Vp_([_a_;_b_{]) we are going to investigate the diﬀerence}_Tf _–_Tg_{. Again, using} Remark  and the Lipschitz property we estimate

Tf–Tgvar,p,[a;b]

=

·

a

Ff(t)–Fg(t) dx(t) p–var,[a;b]

≤

_·

a

Ff(t)–Fg(t)–Ff(a)–Fg(a) dx(t) p–var,[a;b]

+Ff(a)–Fg(a)xp–var,[a;b]

≤(Cp/α,p+ )F

f(·)–Fg(·)_p(p_/–)/_α_–var,[α _a_;_b_]Ff(·)–Fg(·)(_osc,[α+–_a_;p_b)/_]α

× xp–var,[a;b]+F

f(a)–Fg(a)xp–var,[a;b]

≤(Cp/α,p+ )K

fα

p–var,[a;b]+gαp–var,[a;b] (p–)/α

f –gα_osc,[+–_ap_;_b_]

× xp–var,[a;b]+Kf(a) –g(a)

α

xp–var,[a;b]. ()

From () we see thatT is continuous. Moreover, from the ﬁrst inequality in Remark  and the continuity ofxwe see that functions belonging to the imageT(B(R)) are equicon-tinuous. LetU be the closure of the convex hull ofT(B(R)) (in the topology induced by the norm·var,p,[a;b]). It is easy to see that functions belonging toUare also

equicontinu-ous. Moreover,U⊂B(R) (sinceT(B(R))⊂B(R) andB(R) is convex) andT(U)⊂U(since

T(U)⊂T(B(R))). Now, letV =T(U). From the equicontinuity ofU, the Arzela-Ascoli theorem and () we see that the setV is compact in the topology induced by the norm

·var,p,[a;b]. Thus, by the ﬁxed-point theorem of Schauder, we see that there exists a point

y∈U such thatTy=y.

Now we will consider the caseα= .

Fact  Letp∈(; ),y∈R, xbe a continuous function from the space Vp_([_a_;_b_{]) and}

F:R→Rbe globally -Lipschitz. Equation () admits a solutiony, which is an element ofVp([a;b]).

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deﬁned in display (), holds for suﬃciently large R. This may be achieved by splitting the interval [a;b] into small intervals, such thatA<  on each of these intervals, and then solving equation () on each of these intervals with the initial condition being equal the terminal value of the solution on the preceding interval. This is possible since for any

ε>  there existsδ>  such that for any [c;c+δ]⊂[a;b],xp–var,[c;c+δ]≤ε, which is a

consequence of the fact that the function [a;b]t→ xp–var,[a;t] is continuous and we

havexp–var,[a;c+δ]≥ xp–var,[a;c]+xp–var,[c;c+δ].

Competing interests

The author declares to have no competing interests.

Acknowledgements

I would like to thank Raouf Ghomrasni for drawing my attention to paper [11] and Professor Terry Lyons for drawing my attention to his paper [7].

Received: 5 August 2015 Accepted: 21 November 2015 References

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