O'Neil Inequality for Convolutions Associated with Gegenbauer Differential Operator and some Applications

Vagif S.Guliyev,E.J.Ibrahimov,S.E.Ekincioglu and S.Ar.Jafarova

1 Department of Mathematics,Dumlupinar University,Kutahya,Turkey;

2 Institute of Mathematics and Mechanics of NASA,AZ 1141 Baku,Azerbaijan;

3 Azerbaijan State Economic University,AZ 1001,Baku,Azerbaijan.

Abstract.In this paper we prove an O'Neil inequality for the convolution operator(G-convolution)associated with the Gegenbauer differential operator Gλ.By using an O'Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the G-convolution.As an application,we obtain necessary and sufficient conditions on the parameters for the boundedness of the G-fractional maximal and G-fractional integral operators from the spaces Lp,λ to Lq,λ and from the spaces L1,λ to the weak spaces WLp,λ.

Key words:Gegenbauer differential operator,G-convolution,O'Neil inequality,G-fractional integral,G-fractional maximal function.

1 Introduction

Denote bythe shift operator(G-shift)(see[9])

generated by Gegenbauer differential operatorGλ

where

The Gegenbauer differential operator was introduced in[5].For the properties of the Gegenbauer differential operator,we refer to[3,4,10-12].

The shift operatorgenerates the corresponding convolution(G-convolution)

The paper is organized as follows.In Section 2,we give some results needed to facilitate the proofs of our theorems.In Section 3,we show that an O'Neil inequality for rearrangements of theG-convolution holds.In Section 4,we prove an O'Neil inequality forG-convolution.In Section 5,we prove the boundedness ofG-fractional maximal andG-fractional integral operators from the spacesLp,λtoLq,λand from the spacesL1,λto the weak spacesWLq,λ.We show that the conditions on the boundedness cannot be weakened.

Furtherdenotes that exists the constantC＞0 such that 0＜A≤CB,moreoverCcan depend on some parameters.SymbolA≈Bdenote thatand

2 Some auxiliary results

In this section we formulate some lemmas that will be needed later.

Lemma 2.1.1)Let1≤p≤∞,f ∈Lp,λ(R+),then for all t∈R+

2)Let1≤p,r≤q≤∞,pp′=p+p′,f ∈Lp,λ(R+),g∈Lr,λ(R+).Then f ⊕g∈Lq,λ(R+)and

(see[9],Lemmas2and4).

Lemma 2.2.For any measurable E?R+the following relation holds

where r=supE.

Proof.First we prove that

Indeed

Then we have

and

then by changing the order of integration we obtain

On the other hand

Then use of the formula(see[2],p.299)

byμ=ν=λ,we have

Further we have

whereχE-characteristic function of the setE?R+,and also

Now we prove that from(2.2)and(2.3)the assertion of lemma follows,i.e.,

Indeed

where(x,t)φ=chx cht?shx shtcosφ,and

Letr=supE.Since|x?t|≤ch(x?t)≤(x,t)φ ≤ch(x+t), then|x?t|＞r ?(x,t)φ ＞r.Therefore,from|x?t|＞rit follows thatAchtχE(chx)=0.

In this way we obtain

Taking in(2.4)cosφ=y,we obtain

Sincech(x?t)≤r≤ch(x+t)then we have?1≤φ(x,t,r)≤1.Therefore we have

Let?1≤φ(x,t,r)≤0.Then

Now let 0≤φ(x,t,r)≤1.Then

Combining(2.7)and(2.8)for?1≤φ(x,t,r)≤1 we get

From(2.6)and(2.9)we get(2.4)and consequently the assertion of Lemma 2.2.

The following two inequalities are analogue of[13]and have an important role in proving our main results.

Lemma 2.3.Let1＜p ≤q ＜∞and v and w be two functions measurable and positive a.e.on(0,∞).Then there exists a constant C independent of the function φ such that

if and only if

where p+p′=pp′.Moreover,if C is the best constant in(2.1),then

Here the constant k(p,q)in(2.12)can be written in various forms.For example(see[7])

Proof.Necessity.Ifφ≥0 and suppφ∈[0,r],then

For this we have

i.e.,

Suppose

Then by(2.13)

From this it follows that

Sufficiency.Suppose

By Holder inequality we have

Now we prove that ifφ,ψ≥0,r≥1 then

Indeed,since expression on the left hand in(2.17)is equal to

whereχ[u,∞)is the characteristic function of the[u,∞),by Minkowsky inequality we have

According to(2.17)right-hand(2.16)is estimate expression

Take into account(2.15)in(2.18)we obtain

Suppose

we have

then the integral(2.19)is equal to

From(2.11)it follows that

Therefore

From this and(2.20)it follows that

Suppose

then

From this,(2.21),(2.11)and(2.15)we obtain

From this and(2.16)it follows that the inequality(2.10)holds with constant

This completes the proof of the lemma.

Lemma 2.4.Let1＜p≤q＜∞and let v and w be two functions measurable and positive a.e.on(0,∞).Then there exists a constant C independent of the function φ such that

if and only if

Moreover,the best constant C in(2.22)satisfies the inequalities B1≤C≤k(p,q)B1.

Proof.Necessity.Ifφ≥0 and suppφ∈[r,∞),then

From this according to(2.22)we have

i.e.

Suppose

Then by(2.24)

From this it follows that

Sufficiency.Suppose

By H?lder inequality we have

Now we prove that ifφ,ψ≥0,r≥1 then

Indeed,since expression on the left hand in(2.28)is equal to

whereχ(0,u)-is the characteristic function on the(0,u)and by Minkowsky inequality is less than

According to(2.28)right-hand(2.27)is estimate by expression

Take into account(2.26)in(2.29)we obtain

Suppose

we have

but then the integral(2.30)is equal to

From(2.23)it follows that

but then

From this and(2.31)it follows that

Suppose

then

From this,(2.32),(2.23)and(2.26)we obtain

Therefore the expression(2.29)is less than

3 O'Neil inequality for rearrangements of G-convolution

In this section, we will establish a relation between shift operatorandλ-rearrangement off.We show that for theG-convolution an O'Neil inequality for rearrangements holds.Letf:R+→R be a measurable function and for any measurable setWe defineλ-rearrangement offin decreasing order by

wheref?denotes theλ-distribution function offgiven by

Further we need some properties ofλ-rearrangement of functions which are analogous from[1,7].

Observe thatf?depends only on the absolute value|f|of the functionf,andf?may assume the value+∞.

Proposition 3.1.Let f,g,fn,(n=1,2,...)measurable and nonnegative functions onR+.Then

(i)f?is decreasing and right-continuous on[0,∞).

(ii)If|f(chx)|≤|g(chx)|μ?a.e.,then f?(u)≤g?(u)for u≥0.

The proof of this properties is precisely the same how the Proposition 1.7 from[7].

Proposition 3.2.The following equality is valid

where m is the Lebesgue measure.

Proof.Sincef?is a decreasing function by Proposition 3.1(i)it follows that

Hence we get

The next proposition establish some properties of the decreasing rearrangement.

Proposition 3.3.The following properties holds.

(ii)f and f ?are equimeasurable,that is,for all

(iii)If E∈R+,then

(iv)If u≥0and f?(u)＜∞,then

If t≥0and f ?(cht)＜∞,then

Thusf ?(cht)＞u.

Now assume that

(ii)Letmbe the Lebesgue measure on R+.Then by(i)we get

(iii)Since(f χE)(chx)≤f(chx)for allx∈Ewe have by Proposition 3.1(ii)and Proposition 3.2 that

On the other hand,since

we have

Combining these two estimates we can conclude that

(iv)Assume thatf?(u)＜∞.Sincef?is a decreasing function then suppose by assuming thatcht=f?(u)we get

Also,for allε＞0

Now assume thatf ?(cht)＜∞,then

by the right-continuity off?.Furthermore,for allε＞0 by(ii)we have

This completes the proof of the proposition.

Proposition 3.4.For any E?Rthe following equalities are valid

Proof.We first prove(3.1)for simple positive functions.Letfbe a positive simple function onEof the form

where=1,...,nandαn+1=0.

Thus we have

Further since

then

As,then

From this,(3.1)and(3.4)it follows that(3.1)is satisfied for simple functions.The general case follows from Proposition 3.1(iii),Proposition 3.2 and the monotone convergence theorem.

Proposition 3.5.Let0＜p＜∞.Then

Proof.Sincefisμ-measurable function,‖f‖pis aμ-measurable function for 0＜p＜∞.By Proposition 3.3(ii)it follows that|f|pand(f ?)pis equimeasurable,then by Proposition 3.4 we have

This completes the proof of the proposition.

Proposition 3.6.For any measurable E ?R+such that|E|λ ≤t the following inequalities are valid

Proof.Ift=∞,then the inequality is true by Proposition 3.4.Assume thatt＜∞.Then by Proposition 3.4 and Proposition 3.3(iii)we obtain

From Proposition 3.6 we immediately obtain the inequality

Proposition 3.7.Let f and g be measurable functions onR+.Then the following inequality is valid

Proof.We prove this inequality for positive simple functions and the general results will follow by Proposition 3.1(iii),Proposition 3.2 and the monotone convergence theorem for measurable functions on R+.Letfbe a simple function of the form

whereα1＞α2＞...＞αnandLet

andβj=αj?αj+1,αn+1=0.By Proposition 3.6 we get

Proposition 3.8.For any t＞0the following equality is valid

and so

i.e.,

Therefore,let

Then

and by the equimeasurability offandf ?(see Proposition 3.3(ii))we have

that is

From(3.6)and(3.7)sinceεwas arbitrary,we have

foru≥0.Hencef χEandare equimeasurable and because by(3.1)we obtain

Takewe obtain(3.5).The case wheretis not the range off?prove the same when of Lemma 2.5 from[7].

The functionf ??on R+is defined by

Sincef ?is decreasing then

We denote byWLp,λ(R+)the weakLp,λspace of all measurable functionsfwith finite norm

Lemma 3.1.For any measurable set E?R+and for any y∈R+

These inequalities immediate follows from Lemma 2.2 and(2.28).

The following theorem is one of our main results which shows that an O'Neil inequality for rearrangements of theG-convolution holds.The methods of the proof used here are close to those[6].

Theorem 3.1.Let f,g be positive measurable functions onR+.Then for all0＜t＜∞

Proof.Fort＞0 we choose a measurable setEtsuch that

Let

For any measurable set A?R+with measure|A|λ=t,we have

Hence by Lemma 3.1 we obtain

Thus by(3.5)we have

hence by Proposition 3.7 we get

Consequently by(3.5)we have

Therefore we obtain(3.8).

Theorem 3.2.If g∈WLr,λ(R+),1＜r＜∞,then

Proof.Sincef ∈WLr,λ(R+),we have

Taking into account inequality(3.8)we get the inequality(3.10).

4 O'Neil inequality for the G-convolution

In this section we prove O'Neil inequality for theG-convolution.

Theorem 4.1.1)Let1＜p ＜q ＜∞,,f ∈Lp,λ(R+),g ∈WLr,λ(R+).Then f ⊕g ∈Lq,λ(R+)and

2)Let p=1,1＜q＜∞,f ∈L1,λ(R+),g∈WLq,λ(R+).Then f ⊕g∈WLq,λ(R+)and

Proof.1)LetandFrom Proposition 3.5 and inequality(3.10)applied Minkowski inequality we get

By Lemma 2.3,for the validity of the inequality

it is necessary and sufficient that

it is necessary and sufficient condition that

2)Letp=1,1＜q ＜∞,f ∈L1,λ(R+)andg ∈WLq,λ(R+). By inequality(3.10)and Proposition 3.5 we have

This complete the proof.

5 Boundedness of G-fractional integral operator in Lp,λ

We define theG-fractional maximal function by

theG-fractional integral by

whereH(0,r)=(0,r),and

The following relation holds(see[5],Lemma 1.1)

and sincesht≤cht≤2shtfort≥1,then

Taking into account(5.4)we have

Now let 2≤x＜∞,then from(5.3)we get

with it follows from(5.4).

From(5.5)and(5.6)it follows that

From(5.7)we have

Sincethen from(5.8)we obtain

By definition off ??,we get

On the other hand

From(5.10)and(5.11)it follows that

Corollary 5.1.Let0＜α＜2λ+1.Then the following inequalities hold

Indeed by the definition of convolution we have

From this,Theorem 3.2 and(5.7)we have(5.13).

Lemma 5.1.Let0＜α＜2λ+1.Then

Proof.From(5.2)we have

where

Further we have

In this way

Note that

But

then

and therefore from(5.15)we have

If we take the supremum with respect toj∈Z in the both sides of the above inequality,then we get

On the other hand,from(5.1)we have

Thus,the assertion of Lemma 5.1 follows from(5.14),(5.16)and(5.17).

Corollary 5.2.Let0＜α＜2λ+1,then for0＜t＜∞

Corollary 5.3.Let0＜α＜2λ+1.Then

1)If1＜p＜,f ∈Lp,λ(R+)and

Indeed,from(5.5)it follows that

Supposer=and use(5.9)in(4.1)we have the assertion 1)of Corollary 5.3.From the condition 2)it follows thatq=Therefore the assertion 2)follows from(5.9)and(4.2).

Theorem 5.1.Let0＜α＜2λ+1.Then

1)if1＜p＜,then the conditionis necessary and sufficient for the bound-edness of Jα,λ

G from Lp,λ(R+)to Lq,λ(R+).

2)if p=1,then the condition1?is necessary and sufficient for the boundedness offrom L1,λ(R+)to WLq,λ(R+).

Proof.Sufficiencyof Theorem 5.1 follows from Theorem 4.1.

Necessity 1).Let 1＜p ＜f ∈Lp,λ(R+)andbe bounded fromLp,λ(R+)toLq,λ(R+)i.e.,

Moreover assume thatf(x)＞0 is increasing.We define the dilation functionft(chx)as follows:

From(5.18)for 0＜t＜1 we have

where in the second step we used the transformation(cth t)x=uanddx=(tht)du.On the other hand,

Let 1≤t＜∞,then from(5.19)we have

On the other hand,

Combining(5.20)-(5.23)we obtain

Further from(5.2)for 0＜t＜1 from(5.19),we get

Analogously

Now let 1≤t＜∞.Then from(5.19)we have

On the other hand

Combining(5.25)-(5.28)we obtain

Taking into account(5.18)and also(5.29)and(5.24),we get

0＜t＜1 we have

On the other hand from(5.19)we get

We consider the case 1≤t＜∞.From(5.19)we obtain

On the other hand,

From(5.30)-(5.33)for allt＞0,we have

Let the operatorbe bounded fromL1,λ(R+)toWLq,λ(R+),i.e.,

then from(5.24)and(5.34),we have

Recently,in the work[5]the Gegenbauer-Riesz(G-Riesz)potential

is introduced,where

is Gegenbauer function. The following inequality(see[5]),Corollary 3.1)is valid

From this it follows that Corollary 5.3 and Theorem 5.1 are valid forG-Riesz potential

From Corollary 5.1 and(5.35)we get

Corollary 5.4.Let0＜α＜2λ+1.Then the following inequalities hold

Corollary 5.5.Let0＜α＜2λ+1.Then

1)If1＜p＜,then the conditionis necessary and sufficient for the bound-edness of from Lp,λ(R+)to Lq,λ(R+).

2)If p=1,then the conditionis necessary and sufficient for the boundedness offrom L1,λ(R+)to WLq,λ(R+).

Proof.Sufficiency of Corollary 5.5 follows from Theorem 5.1 and Corollary 5.3.

Necessity 1).Letbe bounded fromLp,λ(R+)toLq,λ(R+)fori.e.,

Analogously of(5.25)it can be easily shown that

Taking into account(5.25),(5.36)and(5.37),we get

2)Suppose that the operatoris bounded fromL1,λ(R+)toWLq,λ(R+),i.e.,

From(5.38)we obtain

Now from(5.38),(5.39)and(5.24),we have

Corollary 5.6.Let0＜α＜2λ+1.Then

Proof.Sufficiency follows from Theorem 5.1 and Lemma 5.1.

Remark 5.1.We note that the results of this paper are analogues in[3].

Acknowledgements

The authors would like to express their gratitude to the referees for their very valuable comments and suggestions.The research of V.S.Guliyev and E.Ibragimov was partially supported by the grant of 1st Azerbaijan-Russia Joint Grant Competition(Agreement number no.EIF-BGM-4-RFTF-1/2017-21/01/1).