One dimensional differential Hardy inequality

R E S E A R C H

Open Access

One-dimensional differential Hardy

inequality

Aigerim Kalybay

*_{Correspondence:}

[email protected] KIMEP University, Abai Ave. 4, Almaty, 050010, Kazakhstan L.N. Gumilyov Eurasian National University, Munaytpasov St. 5, Astana, 010008, Kazakhstan

Abstract

We establish necessary and sufficient conditions for the one-dimensional differential Hardy inequality to hold, including the overdetermined case. The solution is given in terms different from those of the known results. Moreover, the least constant for this inequality is estimated.

MSC: Primary 26D10; secondary 47B38

Keywords: inequality; Hardy-type inequality; weight; differential operator

1 Introduction

LetI= (a,b), –∞ ≤a<b≤ ∞, ≤p,q≤ ∞,_p+_p = , and_q+_q = . Letu≥ andρ> 

be weight functions such thatu∈Lloc_q andρ–≡ _ρ ∈Lloc_p , whereLp≡Lp(I) stands for the

space of measurable functionsf onIwith finite norm

fp=

(_ab|f(t)|p_dt)p_{, }≤_p<∞_,

ess sup_t∈I|f(t)|, p=∞.

LetAC(I) be the set of all functions locally absolutely continuous onI. LetAC(◦ I) be the set of functions fromAC(I) with compact supports onI.

We consider the following Hardy inequality in the differential form

ufq≤Cρf_p, f ∈ ◦

AC(I). ()

In [] it is shown that if

c

a

ρ–p(s)ds=∞ and

b

c

ρ–p(s)ds=∞ ()

for somec∈I, then inequality () does not hold. In the case

c

a

ρ–p(s)ds<∞ and

b

c

ρ–p(s)ds=∞ ()

or

c

a

ρ–p(s)ds=∞ and

b

c

ρ–p(s)ds<∞, ()

then inequality () is satisfied for all functionsf ∈AC(I) such thatf(a) =  orf(b) = , respectively. For example, in case (), it is equivalent to the weighted integral Hardy in-equality (see [])

b

a

u(x)

x

a

f(s)ds q

dx



q

≤C

b

a

v(t)f(t)pdt



p

, ()

which has been studied for all values of the parameters  <p,q≤ ∞(see [, ], and []). In [] it is also shown that in the last case

ρ–_p<∞, ()

inequality () is satisfied for all functionsf ∈AC(I) such thatf(a) =  andf(b) = , that is, it is an overdetermined case. This case is studied in [, ], and [].

In the present work, for ≤p≤q≤ ∞, we establish a criterion for the validity of in-equality () with an estimate of the type

B(u,ρ)≤C≤CB(u,ρ) ()

for the least constantCin (), whereB(u,ρ) is some functional depending onuandρ. Moreover, we present a calculation formula for the least value of C and its two-sided estimate.

We suppose that only condition () does not hold. In case () or (), our criterion co-incides with the well-known Muckenhoupt result. However, our upper estimate in () is worse than the known one (see [–], and Remark . further). In case (), our criterion is given in terms different from those in [, ], and []. The terms in [] are close to ours, but the comparison analysis shows that our results and an estimate of type () are better than in [] (see Remark .).

At the end of the paper, we find a criterion for the compactness of the setM={uf :f ∈ ◦

AC(I),ρfp≤}inLq(I).

2 Auxiliary statements

Lemma . Let <q<∞andϕ(λ) =_λλqq_––_λ– ,λ> .There exists a pointλ≡λ(q)such

that

λ() =  +√

 ,

q

q+ <λ(q) <min{q, } for q= , ()

and

ϕ(λ) = λ

q

 λq_– –

 λ– 

= . ()

In addition, _λλqq_–<_λ– for <λ<λand λ

q

λq_–>_λ– forλ>λ.

Proof Since

ϕ(λ) = 

(λq_{– )(λ– )} λ

the sign of the functionϕ is defined by the value of the functiond(λ) =λq+_{– λ}q_{+ .}

Moreover,ϕ(λ) = ,λ> , if and only ifd(λ) = .

Forq= , we haved(λ) =λ– λ+  = (λ– )(λ–λ– ). This means thatd(λ) =  for λ=+

√   .

Letq= . Letλ=  +ε,ε> . Using the Lagrange finite-increment formula, we have

ϕ(λ) = ( +ε)

q

( +ε)q_{– }–

 ε≥

( +ε)q qε( +ε)q––

 ε=



qε ε– (q– )

.

This gives thatϕ(λ) =ϕ( +ε)≥ forε≥q– , that is,ϕ(λ) >  forλ>q.

Let us find an extremum of the functiondforλ> . We have thatd(λ) = (q+ )λq– qλq–=λq–((q+ )λ– q). This gives thatd(λ) =  forλ=_q_+q,d(λ) >  forλ> _q_+q, andd(λ) <  for  <λ< _q_+q. Therefore, the functiond(λ) decreases for  <λ≤ _q_+q and increases forλ>_q_+q. Moreover, it has a minimum at_q_+q. Sinced() =  > ,d(λ) >  for λ≥. Hence,ϕ(λ) >  forλ> .

Thus,

ϕ(λ) >  forλ>min{q, }. ()

Sinced() =  anddhas a minimum at_q_+q, it follows thatd(_q_+q) <  forλ>  andd(λ) <  for  <λ≤_q_+q. Therefore,ϕ(_q_+q) < , and

ϕ(λ) <  for  <λ≤ q

q+ . ()

In view of the continuity ofϕforλ> , from () and () there follows the existence of a point that satisfies () and ().

The last statement of Lemma . follows from the intersection of graphs of two decreas-ing and concave upward functions λq

λq_–and_λ– at the pointλ=λ. The proof of Lemma .

is complete.

Lemma . Let <q<∞and f(λ) =λ(λq–)



q

λ– ,λ> .Then

f λ()=

√ + 

√

 – 

 

, inf

λ>f(λ) =f(λ) = λ_

(λ– )



q

, ()

and,for q= ,we have the estimate

γ<f(λ) <min{γ,γ, }, ()

whereγ=_q_+q(_q_+q)



q₍q

q–)



q_,γ

=_q_+qq



q₍q

q–)



q_,andγ

=qq



q_(q)



q_.

Proof By Lemma . forq=  we haveλ() =+

√ 

 , so thatf(λ()) = ( _√√+

–)

 .

Let q = . The function f is continuous when λ > , and limλ→+f(λ) = ∞ and

limλ→∞f(λ) =∞. Therefore, it has a minimum. Since f(λ) = (λq–)



q

λ– ( λq

λq_– – _λ– ), by

is, the function f decreases for  <λ<λ, increases for λ>λ, and has a minimum at λ=λ. Thus,infλ>f(λ) =f(λ). Again by Lemma . we have that

λq_ λq_–=



λ–. Substituting

this equality into the expression off(λ), we get (). The functiong(x) = λ

(λ–)



q

has a minimum at the pointλ=_q_+q. Therefore,

f(λ) =g(λ) >g

q q+ 

= q

q+ 

q q+ 



q _q

q– 



q

. ()

By Lemma . we have that _q_+q <λ<min{q, }. Hence,

f(λ) <min

f

q q+ 

,f(q),f()

. ()

It is easy to see thatf() < . Sinceλ<q, we have q

q

qq_–> _q_– orq(q– )



q _{> (q}q_{– )}q_.

Therefore,

f(q) =q(q

q_{– )}q

q–  <

q

(q– )



q

=qqq _q



q_{. ()}

Let us estimatef(_q_+q):

f

q q+ 

= q

q– 

 +q– 

q+ 

q

– 



q

≤ q q– 

qq–  q+ 

q q+ 

q–_q

= q

q– q



q

q– 

q+ 



q _q

q+ 



q

= q

q+ q



q

q

q– 



q

. ()

From (), (), (), and (), taking into account thatf() < , we have (). The proof of

Lemma . is complete.

3 Main results

Leta<c<b,d=d(a,c,b) =min{c–a,b–c}. Assume that

Ap,q(c) = sup

<h<d

uq,(c–h,c+h) (ρ––p

p,(a,c–h)+ρ– –p p,(c+h,b))



p

, ≤p<∞,

Ap,q(c) = sup

<h<d

uq,(c–h,c+h)

ρ––

,(a,c–h)+ρ––,(c+h,b)

, p=∞,

and

Ap,q=sup c∈I

Ap,q(c).

Theorem . Let≤p≤q<∞.Inequality()holds if and only if Ap,q<∞.Moreover,for

the least constant C in(),we have the estimate

In turn,for f(λ),by Lemma.we have f(λ()) = ( √

+ √

–)



 _{and the estimateγ<f(λ) <}

min{γ,γ, }for q= .

Proof Necessity. Let inequality () hold with the least constantC> .

Suppose that  <p<∞. Letc∈I,  <h<d, anda<α<c–h<c+h<β<b. We introduce the following function:

fc,h(t) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

t

αρ–p

(s)ds(_αc–hρ–p_(s)ds)–_{, α≤t≤c–h,}

, c–h≤t≤c+h,

β

t ρ

–p_(s)ds(β

c+hρ

–p_(s)ds)–_{, c+h≤t≤β,}

, t∈I\(α,β).

It is obvious thatfc,h∈ ◦

AC(I). Substitutingfc,hinto (), we get

uq,(c–h,c+h)≤C ρ– –p

p,(α,c–h)+ρ ––p

p,(c+h,β)



p_.

From the last inequality, taking into account that its left-hand side does not depend onα, βsuch thata<α<β<b, we have

uq,(c–h,c+h)≤C ρ– –p

p,(a,c–h)+ρ ––p

p,(c+h,b)



p _()

for allc∈Iand  <h<d.

In the casep= , we constructf in the following way. Let numberscandhbe defined as before,δ> , anda<x–δ<x+δ≤c–h<c+h≤y–δ<y+δ<b. Assume that

fc,h(t) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

t x–δρ

–_(s)ds(x+δ

x–δ ρ

–_(s)ds)–_{, x–δ≤t≤x+δ,}

, x+δ≤t≤y–δ,

y+δ

t ρ–(s)ds(

y+δ

y–δ ρ–(s)ds)–, y–δ≤t≤y+δ, , t∈I\(x–δ,y+δ).

Substitutingfc,hinto (), we get

uq,(x+δ,y–δ)≤C

 δ

x+δ

x–δ

ρ–(s)ds

– +

 δ

y+δ

y–δ

ρ–(s)ds

– .

Taking the limit in this inequality asδ→, we get

uq,(x,y)≤C ρ(x) +ρ(y)

for almost allx: (a<x≤c–h) and almost ally: (c+h≤y<b).

Forα> , there exist pointsx: (a<x≤c–h) andy: (c+h≤y<b) such that

ρ–∞,(a,c–h)

α ≤ρ

–_{(x) and} ρ–∞,(c+h,b)

α ≤ρ

–_(y).

Then

uq,(x,y)≤αC ρ–_{∞,(a,c–h)}

–

+ ρ–_∞ ,(c+h,b)

This, together withuq,(x,y)≥ uq,(c–h,c+h), yields that

uq,(c–h,c+h)≤αCρ– –

∞,(a,c–h)+ρ ––

∞,(c+h,b)

.

Since the left-hand side of this inequality does not depend onα> , lettingα→, we get () forp= . Thus, for all ≤p≤q<∞, we have that

Ap,q≤C. ()

Sufficiency. LetAp,q<∞be correct. Letf be a nontrivial function from ◦

AC(I). Without loss of generality, we assume thatf ≥. Letλ> . For any integerk, we assume thatTk=

{t∈I:f(t) >λk},Tk=Tk\Tk+. Due to the boundedness of the functionf, there exists an integern=n(f) such thatTn= ∅andTn+=∅. It is obvious thatI=

k≤nTk=

k≤nTk.

The setTkis open. Therefore, there exists a family of mutually disjoint intervals{Jjk},Jjk=

(ck_j,d_jk), such thatTk=_jJ_jk. Forn– ≥k> –∞, we assume thatM_jk =Tk+∩Jjk. For Mk_j =∅, we defineαk_j =infMk_j andβ_jk=supMk_j. Then

Tk+⊂

j

αk_j,β_jk and Tk⊃ j

ck_j,α_jk∪ β_jk,d_jk. ()

In view of the continuity of the functionf, we get thatf(α_jk) =f(β_jk) =λk+ andf(ck_j) =

f(dk_j) =λk. Hence,

λk(λ– ) =λk+–λk=

α_jk

ck_j

f(s)ds= –

dk_j

β_jk

f(s)ds. ()

From () by Hölder’s inequality we have that

λkpρ––p

p,(ck_j,αk_j)≤

ρfp p,(ck_j,αk_j)

(λ– )p , ()

λkpρ––p

p,(β_jk,dk_j)≤

ρfp p,(β_jk,dk_j)

(λ– )p . ()

In view off(t) <λk+fort∈Tkandλqk= ( –λ–q)_i≤kλqi, we have that

ufq_q=

k≤n–

ufq_q,T

k+≤

k

λq(k+)uq_q,T

k+

≤λq λq– 

k

uq_q,Tk+ i≤k

λqi=λq λq– 

i

λqi

k≥i

uq_q,Tk+

=λq λq– 

i

λqiuq_q,T

i+

(by ())

≤λq λq– 

i

λqi

j

βi_j

αi_j

uq(s)ds=λq λq– 

i

λqi

j uq

SinceAp,q<∞, from () and (), taking into account thatq_p≥, this gives

ufq_q≤λq λq– Aq_p,q i

j

λpiρ––p

p,(ci_j,α_ji)+λ

pi_ρ––p p,(β_ji,di_j)

q p

≤λq(λq– )

(λ– )q A q p,q

i

j

ρfp_p,(ci j,αji)+

ρfp_p,(βi j,dij)

qp

(by the second relation from ())

≤λq(λq– )

(λ– )q A q p,q

i

ρfp_p,T

i

q p

≤λq(λq– )

(λ– )q A q p,qρf

q p.

Therefore,

ufq≤

λ(λq– )q

(λ– ) Ap,qρf

p. ()

Since the left-hand side of this inequality does not depend onλ> , by Lemma . we have

ufq≤f(λ)Ap,qρf_p,

that is, inequality () holds with the estimateC≤f(λ)Ap,qfor the least constantCin ().

This fact, together with (), gives (). The proof of Theorem . is complete.

Remark . Let us notice that in [], for inequality (), an estimate of the type () has been obtained in the case  <q=p<∞.

Remark . If () or () is correct, then by Theorem . there follows the correctness of the corresponding integral Hardy inequality (see []). For example, if () holds, then

Ap,q=sup z∈I

b

z

uq(x)dx



q z

a

v–p(s)ds



p

,

where the conditionAp,q<∞coincides with the Muckenhoupt condition (see []), and

inequality () is equivalent to the integral Hardy inequality (). However, our upper esti-mate in () is worse than that in the known result (see e.g. [], Thm. ). For example, in case () withp=q= , from () we haveAp,q≤C≤(

√ + √

–)



Ap_,q≈.Ap_,q, but from

Theorem  of [] it follows thatAp,q≤C≤Ap,q.

Theorem . Let =p=q.Inequality()holds if and only if Ap,q<∞.Moreover,Ap,q=C,

where C is the least constant in().

Proof From () we have that ufq ≤ λAp,qρfp. Taking λ→, we get ufq ≤

Ap,qρfp, that is, inequality () holds with the estimateAp,q≤C, which, together with

Theorem . Let≤p≤q=∞.Inequality()holds if and only if Ap,q<∞.Moreover, Ap,q≤C≤Ap,q,where C is the least constant in().

Proof Let ≤p<q=∞. The necessity follows from Theorem .. Let us prove the

suffi-ciency. LetAp,q<∞. For ≤f ∈ ◦

AC(I), we have

ufq=sup k

ufq,Tk+≤λ

_sup

k

λkuq,Tk+

≤λsup

k

λkuq,Tk+≤λ

_sup

k,i

λku_q,(αk i,βik)

≤λAp,qsup k,i

λpkρ––p

p,(ck_i,αk_i)+λ

pk_ρ––p p,(β_ik,dk_i)



p

≤ λ

λ– Ap,qsupk

ρf_p,T

k≤

λ

λ– Ap,qρf

p,

that is,

ufq≤inf

λ> λ

λ– Ap,qρf

p= Ap,qρf

p, ()

which, as before, means that

C≤Ap,q.

Letp=q=∞.

Sufficiency. From () we have

λkρ––_,(ck i,αki)≤

 λ– ρf

p,(ck_i,α_ik),

λkρ––_,(βk i,dki)≤

 λ– ρf

p,(β_ik,dk_i).

Using these relations instead of () and (), we have

ufq≤λsup k,i

λku_q,(αk i,βki)≤λ

_Ap ,qsup

k,i

λkρ––_,(ck i,αik)+λ

k_ρ–– ,(β_ik,dk_i)

≤ λ

λ– Ap,qsupk

ρf_p,T

k≤

λ

λ– Ap,qρf

p.

This gives that

ufq≤Ap,qρf_p

and

C≤Ap,q.

Necessity. Substituting the functionfc,hinto (), we have

uq,(c–h,c+h)≤C ρ– –

,(α,c–h)+ρ ––

,(c+h,β)

which means that

Ap,q≤C.

Therefore,

Ap,q≤C≤Ap,q.

The proof of Theorem . is complete.

Remark . The obtained results can be compared with the results of Theorem . of [],

where it is proved that, for ≤p≤q≤ ∞, the validity of () is equivalent to the condition

Bp,q= sup

(c,d)⊂I

uq,(c,d),minρ–_p,(d,b),ρ–_p,(a,c)

<∞.

Moreover, for the least constantCin (), we have the estimates

–p_B≤_C≤ _inf

<λ<B, ≤q<∞, ()

and

–p_B≤_C≤_{B, q=}∞_{. ()}

It is easy to see thatAp,q<Bp,q. Moreover, the estimates for the least constantCin ()

obtained in Theorems . and . are obviously better than in () and (), respectively.

4 Compactness

LetI= (–∞, +∞) andM={uf :f ∈AC(◦ I),ρfp≤}.

Let

A+_p,q(z) =sup

h>

(_zz+huq_(t)dt)q

(ρ––p

p,(–∞,z)+ρ– –p p,(z+h,∞))



p

,

A–_p,q(z) =sup

h>

(_zz_–huq_(t)dt)q

(ρ––p

p,(–∞,z–h)+ρ– –p p,(z,∞))



p

.

Theorem . Let≤p≤q<∞.The set M is relatively compact in Lq(I)if and only if Ap,q<∞and

lim

|x|→∞Ap,q(x) = . ()

Proof Necessity. LetMbe relatively compact inLq(I). Then by Theorem . we have that

Ap,q<∞. Letfc,h,α,β≡fc,hbe the function introduced in the necessary part of Theorem ..

We assume that

g_z+_,h,α,β(t) =f_z+_,h,α,β(t) ρ––p

p,(α,z)+ρ ––p

p,(z+h,β)

–_p ,

g–

z,h,α,β(t) =fz–,h,α,β(t) ρ– –p

p,(α,z–h)+ρ ––p

p,(z,β)

–_p .

Sinceg_z±_,h,α,β∈AC(◦ I) andρ(g_z±_,h,α,β)p ≤, we havegz±,h,α,β ∈M, and by the Frechet-Kolmogorov theorem [], p. we get

 = lim

N→∞sup_f∈M

|t|>N |uf|qdt



q

≥ lim

N→∞_zsup_,h,α,β

t>N

ug_z+_,h,α,βqdt



q

≥ lim

N→∞supz>N

sup

h>

(_zz+huq_(t)dt)q

(ρ––p

p,(–∞,z)+ρ– –p p,(z+h,∞))



p

= lim

z→∞supA

+

p,q(z).

Therefore,

lim

z→∞A

+

p,q(z) = . ()

Similarly, working with the functiong–

z,h,α,β, we getlimz→–∞A–p,q(z) = , which, together

with (), gives ().

Sufficiency. LetAp,q<∞and () hold. Then, on the basis of Theorem ., the setMis

bounded inLq(I). Therefore, by the Frechet-Kolmogorov theorem it suffices to show that

lim

N→∞_fsup_∈M

|t|>N |uf|qdt



q

= . ()

Let

u(t) =

u(t), t<N,

, t≥N, and u(t) =

, t≤N,

u(t), t>N.

Hence, by Theorem . we have

∞

–∞|uf|

q_dt



q

≤f(λ)sup

x<N

Ap,q(x) forf ∈M,

∞

–∞

|uf|qdt



q

≤f(λ)sup

x>N

Ap,q(x) forf ∈M.

Then

sup

f∈M

|t|>N |uf|qdt



q

≤f(λ)sup |x|>N

Ap,q(x).

This, together with (), gives (). The proof of Theorem . is complete.

Competing interests

Acknowledgements

The author would like to thank Professor Ryskul Oinarov and reviewers for their generous suggestions, which have improved this paper. The paper was written under financial support by the Scientific Committee of the Ministry of Education and Science of Kazakhstan, Grant No.5495/GF4, on priority area “Intellectual potential of the country.”

Received: 22 July 2016 Accepted: 4 January 2017

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