Inequalities for the fractional convolution operator on differential forms

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R E S E A R C H

Open Access

Inequalities for the fractional convolution

operator on differential forms

Zhimin Dai

1*

_{, Huacan Li}

2

_{and Qunfang Li}

3

*_{Correspondence:} [email protected]

1_{School of Science, Xi’an}

Technological University, Xi’an, China

Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to derive some Coifman type inequalities for the fractional convolution operator applied to diﬀerential forms. The Lipschitz norm and BMO norm estimates for this integral type operator acting on diﬀerential forms are also obtained.

Keywords: Fractional convolution operator; Coifman type inequality; BMO norm; Lipschitz norm; Diﬀerential form

1 Introduction

As a generalization of functions, the differential form can be regarded as a special kind of vector-valued function. So, if some operators in function spaces are generalized to that in differential forms, similar properties could be obtained as in the function space. In recent years, the research on the generalization of operators from functional spaces to differential forms seems to become a new highlight in the inequalities with differential forms, see [1– 6]. In this paper, we mainly consider the following convolution type fractional integrals operator acting on differential forms and develop some norm inequalities for the fractional convolution operator. Given a nonnegative, locally integrable functionKα andI(y) is a

bounded function with a compactly supported set onRn_{, write}

I(y)∈L∞c . The fractional

convolution operatorFαis deﬁned by a convolution integral

Fα(x) =

I

Rn

Kα(x–y)I(y) dy

dxI, (1.1)

provided this integral exists for almost allRn_{, where}₍_x_{) =}

II(y) dxIis a-form deﬁned

onRn_{, the summation is over all ordered}_-tuples_I_{. The function}_K

αis also assumed to be a wide class of kernels satisfying the following growth condition (see [7]):

(1) Kα∈Sαif there exists a constantC> 0such that

|x|∼s

Kα(x)dx≤Csα; (1.2)

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(2) Kαis said to satisfy theLα,ϕ-Hörmander condition, and writeKα∈Hα,ϕ. If there existc≥1andC> 0(only dependent onϕ) such that, for ally∈Rn_and_R_>_c_|_y_|_,

∞

m=1

2mRn–αKα(·–y) –Kα(·)_ϕ(_|_x_|∼₂m_R₎≤C, (1.3)

whereϕis a Young function deﬁned on[0, +∞),|x| ∼sstands for the set

{s<|x| ≤2s},O(0,s)is a ball with the center at the origin and the radius equal tos, and theϕ-mean Luxemburg norm of a functionf on a cube (or a ball)OinRn_is

given by

fϕ(O)=inf

λ> 0 : 1

|O|

O

ϕ

|f| λ

dx≤1

. (1.4)

Diﬀerential forms can be viewed as an extension of functions. When(x) is a 0-form, the above-mentioned notations are in accord with those of function spaces, and the frac-tional convolution operatorFαwe study in this paper degenerates into the operator which Bernardis discussed in [7]. Namely, for any Lebesgue measurable functionf ∈L∞_c ,Fαis given as follows:

Fαf(x) =

Rn

Kα(x–y)f(y) dy. (1.5)

This degenerated fractional convolution operator was also introduced by Riveros in [8], who presented weighted Coifman type estimates, two weight estimates of strong and weak type for general fractional operators and gave applications to fractional operators pro-duced by a homogeneous function and a Fourier multiplier.

Now we introduce some notations and deﬁnitions. Let be an open subset of Rn

(n≥2) andO be a ball inRn_{. Let} _ρO_{denote the ball with the same center as}_O_and

diam(ρO) =ρdiam(O)(ρ> 0).||is used to denote the Lebesgue measure of a set⊂Rn. Let=(Rn),= 0, 1, . . . ,n, be the linear space of all-forms(x) =_II(x) dxI =

Ii1i2···i(x) dxi1 ∧dxi2· · · ∧dxi inR

n_{, where}_I_{= (}_i_1,_i_{2, . . . ,}_i_{), 1}_≤_i

1<i2<· · ·<i≤n, are the ordered-tuples. Moreover, if each of the coeﬃcientsI(x) of(x) is diﬀerential

on, then we call(x) a diﬀerential-form onand useD(,) to denote the space of all diﬀerential-forms on.C∞(,) denotes the space of smooth-forms on. The exterior derivatived:D(,)→D(,+1),= 0, 1, . . . ,n– 1, is given by

d(x) =

I n

j=1

∂i1i2···i(x) ∂xj

dxj∧dxi1∧dxi2∧ · · · ∧dxi (1.6)

for all∈D(,). Lp₍_,

)(1≤p<∞) is a Banach space with the norm p,= (|(x)|p_d_x₎1/p_{= (}

(

I|I(x)|2)p/2dx)1/p<∞. Similarly, the notationsLploc(,

) and

W_loc1,p(,) are self-explanatory.

From [9],is a diﬀerential form in a bounded convex domain, then there is a decom-position

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whereTis called a homotopy operator. For the homotopy operatorT, we know that

Tp,O≤C|O|diam(O)p,O (1.8)

holds for any diﬀerential form∈Lp_loc(,),= 1, 2, . . . ,n, 1 <p<∞. Furthermore, we can deﬁne the-form∈D(,) by

=

⎧ ⎨ ⎩

||–1

(x) dx, = 0,

dT(), = 1, 2, . . . ,n, (1.9)

for all∈Lp(,), 1≤p<∞.

A non-negative functionw∈L1_loc(dx) is called a weight. We recall the deﬁnitions of the Muckenhoupt weights and the reverse Hölder condition (see [10]). For 1 <p<∞, we say thatw∈Apif there exists a constantC> 0 such that, for every ballO⊂Rn,

1

|O|

O

wdx

1

|O|

O

w–p–11 _d_x

p–1

≤C. (1.10)

For the casep= 1,w∈A1if there exists a constantC> 0 such that, for every ballO⊂Rn, 1

|O|

O

wdx≤Cess inf

x∈Ow(x). (1.11)

AlsoA∞=_p_≥₁Ap. It is well known thatAp⊂Aqfor all 1≤p≤q≤ ∞, and also that

for 1 <p≤ ∞, ifw∈Ap, then there existsε> 0 such thatw∈Ap–ε.

A functionϕ: [0,∞)→[0,∞) is a Young function if it is continuous, convex, increasing and satisﬁesϕ(0) = 0 andϕ(t)→ ∞ast→ ∞. Each Young functionϕhas an associated complementary Young functionϕ¯satisfying

t≤ϕ–1(t)ϕ¯–1(t)≤2t (1.12)

for allt> 0, whereϕ–1(t) is the inverse function ofϕ(t) (see [11]).

For each locally integrable functionf and 0≤α<n, the fractional maximal operator associated with the Young functionϕis deﬁned by

Mα,ϕf(x) =sup

x∈O|

O|αn_f

ϕ(O). (1.13)

Forα= 0, we writeMϕinstead ofM0,ϕ. Whenϕ(t) =t, thenMα,ϕ=Mαis the classical fractional maximal operator. Forα= 0 and ϕ(t) =t, we obtainM0,ϕ=Mis the Hardy– Littlewood maximal operator (see [8]).

2 The Coifman type inequalities for the fractional convolution operator

In [7], the inequality for the fractional convolution operator in function with the fractional maximal operator, that is, the Coifman type inequality, is proved.

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Kα∈Sα∩Hα,ϕ,where0 <α<n.Then there exists a constant C such that

Rn

Fαf(x)

p

w(x) dx≤C

Rn

Mα,ϕ¯f(x)

p

w(x) dx (2.1)

for any0 <p<∞and w∈A∞.

Theorem 2.1 Letϕbe a Young function on[0, +∞)and f,g be two functions deﬁned on

Rn_with_|_f₍_x₎_{| ≤ |}_g₍_x₎_|_{for all x}_∈_Rn_._{Then for all cubes O and the Young functions}_ϕ_,

fϕ(O)≤gϕ(O). (2.2)

Proof Sinceϕis a Young function, it follows that

1

|O|

O

ϕ

|f|

gϕ(O)

dx≤ 1 |O|

O

ϕ

|g|

gϕ(O)

dx≤1

⇒ gϕ(O)∈E=

λ> 0 : 1

|O| O ϕ _| f| λ

dx≤1

⇒ fϕ(O)=infE≤gϕ(O). (2.3)

According to Theorem2.1, we can get a similar conclusion to Lemma2.1.

Theorem 2.2 Letϕbe a Young function on[0, +∞),=_IIdxIbe a diﬀerential form on

⊂Rn,and let all the ordered-tuples I satisfyI∈L∞c .Suppose that Fαis a fractional

convolution operator applied to diﬀerential forms and its kernel function Kαsatisﬁes Kα∈

Sα∩Hα,ϕ,where0 <α<n.Then there exists a constant C such that

Rn

Fα(x)pw(x) dx≤C

Rn

Mα,ϕ(¯ x)pw(x) dx (2.4)

Proof By Lemma2.1and the following basic inequality

n

i=1

|ai|s≤n

_n

i=1

|ai|

s

≤ns+1

n

i=1

|ai|s, (2.5)

wheres> 0 is any constant, it follows that

Fαp_p_,_w_,_Rn=

Rn

Fα(x)pw(x) dx

= Rn I

RnKα(x–y)I(y) dy

2p/2

w(x) dx

≤

RnC1

I

RnK

α(x–y)I(y) dy

p

w(x) dx

=C1

I

Rn

RnKα(x–y)I(y) dy

p

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≤C2

I

Rn

Mα,ϕ¯ I(x)

p

w(x) dx

=C2

Rn

I

Mα,ϕ¯ I(x)

p

w(x) dx

≤C3

Rn

I

Mα,ϕ¯ I(x)

p

w(x) dx. (2.6)

Then, by the deﬁnition of the fractional maximal operator, notice that for anyIsuch that

|I| ≤ ||, we obtain that

I

Mα,ϕ¯ I(x)

=

I

sup

x∈O

|O|α/n_I_¯ ϕ(O)

≤C_nsup

x∈O

|O|α/n

I

_I_¯

ϕ(O)

≤C

nsup x∈O|

O|α/nC

nϕ(¯O)

≤C4Mα,ϕ(¯ x). (2.7)

Combining (2.6) and (2.7), we have

Fαp_p_,_w_,_Rn

≤C3

Rn

I

Mα,ϕ¯ I(x)

p

w(x) dx

≤C3

Rn

C4Mα,ϕ¯ (x)

p

w(x) dx

≤C5

Rn

Mα,ϕ(¯ x) pw(x) dx. (2.8)

⊂Rn_,_{and for all the ordered}_-tuples_,_{let I satisfy}

I∈L∞c .Suppose that Fαis a fractional

convolution operator on diﬀerential forms and its kernel function Kα satisﬁes Kα∈Sα∩

Hα,ϕ,where0 <α<n.Then there exists a constant C such that

O

Fα(x)pdx≤C

O

Mα,ϕ(¯ x)pdx (2.9)

for any0 <p<∞and all the balls O⊂Rn_.

Proof By the deﬁnition of theA∞-weight, there existr0≥1 and a constantC<∞such that, for all the ballsO⊂Rn_{, it follows that}

1

|O|

O

w(x) dx

1

|O|

O

w(x)–r_0–11 _d_x

r0–1

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With the arbitrariness of the conditionw∈A∞of Theorem2.2, now get any ballO0⊂Rn and let

w(x) =χO0(x) =

⎧ ⎨ ⎩

1, x∈O0;

0, x∈/O0.

It is easy to check thatw(x) =χO0(x) satisﬁes (2.10). In fact, we have

1

|O|

O

χO0(x) dx

1

|O|

O

χO0(x)

–_r1

0–1_d_x

r0–1

=

1

|O||O∩O0|

1

|O||O∩O0| r0–1

=

1

|O||O∩O0| r0

≤1. (2.11)

Thus

O0

Fα(x)pdx

=

Rn

Fα(x)pχO0(x) dx

≤C

Rn

Mα,ϕ¯ (x)pχO0(x) dx

≤C

O0

Mα,ϕ(¯ x)

p

dx. (2.12)

If the kernel functionKαand the coeﬃcient functionsIof diﬀerential forms are subject

to some conditions, the following more important conclusion will be obtained.

⊂Rn_,_{and let all the ordered}_{-tuples I satisfy}

convolution operator on diﬀerential forms and its kernel function Kαsatisﬁes Kα∈Sα∩Hα,ϕ

and Kα∈C0∞(),where C0∞()stands for all the C∞functions with compactly supported

sets inand0 <α<n.Then there exists a constant C such that

Fα– (Fα)O_p_,_O≤Cdiam(O)|O|Mα,ϕ(d¯ )_p_,_O (2.13)

for any1 <p<∞and all the balls with O⊂Rn_.

Proof By the exterior derivative operatordand the fractional convolution operatorFα, we obtain that

d=

I n

k=1

∂I(x)

∂xk

dxk∧dxI,

Fα(d) =

I n

k=1

RnKα(x–y)

∂I(y)

∂yk

dy

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and

dFα() =

I n

k=1

∂hI(x)

∂xk

dxk∧dxI, (2.15)

where

hI(x) =

RnKα(x–y)I(y) dy. (2.16)

According to (1.7)–(1.9), it follows that

_F_α_{– (}_F_α)_O

p,O=T

d(Fα) _p_,_O≤C1diam(O)|O|d(Fα)_p_,_O. (2.17) Now we will give theLp_{-norm estimation of}_d₍_F_{α). With}_K

α∈C0∞() and considering the deﬁnition of the general partial derivative (see [12]), we obtain

d(Fα)p_p_,_O

= O I n k=1 ∂hI(x)

∂xk 2 p/2 dx = O I n k=1

∂RnKα(x–y)I(y) dy

∂xk 2 p/2 dx = O I n k=1 _Rn

∂Kα(x–y)

∂xk

I(y) dy

2 p/2 dx = O I n k=1 – Rn

∂Kα(x–y)

∂yk I

(y) dy

2p/2 dx = O I n k=1 _Rn

∂I(y)

∂yk

Kα(x–y) dy

2p/2 dx

=Fα(d)p_p_,_O, (2.18)

that is

d(Fα)_p_,_O=Fα(d)_p_,_O. (2.19)

Combining (2.17) and (2.19), we obtain that

Fα– (Fα)O_p_,_O

≤C1diam(O)|O|d(Fα)_p_,_O =C1diam(O)|O|Fα(d)_p_,_O

≤C1diam(O)|O|Mα,ϕ(¯ d)_p_,_O. (2.20)

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Since a new function is obtained when the diﬀerential form is taken as a model, we can get a global inequality in theLp₍_m_{) domain with Theorem}_{2.4. Now recall the deﬁnition}

of theLp₍_m_{) domain introduced by Staples (see [13]).}

Deﬁnition 2.1 Letbe a real subdomain inRn_{. If, for all the functions}_f_∈_Lp

loc(), there exists a constantCsuch that

||–1/pf–fO0p,≤Csup

O⊂|

O|–1/pf–fOp,O, (2.21)

thenis called anLp₍_m_{)-average domain, where}_O

0is a ﬁxed ball ofandp≥1.

and Kα∈C0∞(),where0 <α<n.Then there exists a constant C such that

Fα– (Fα)O0_p_,≤C||diam()Mα,ϕ¯d(x)_p_, (2.22)

for any1 <p<∞and O0is a ﬁxed ball in.

Proof By the deﬁnition of theLp₍_m_{)-average domain and noticing that 1 – 1/}_p_≥_{0, we}

have

Fα– (Fα)O0p,

≤C1||1/psup

O⊂

|O|–1/pFα– (Fα)O_p_,_O

≤C1||1/psup

O⊂

|O|–1/pC2diam(O)|O|Mα,ϕ(¯ d)_p,O

≤C3||1/psup

O⊂

|O|1–1/pdiam(O)Mα,ϕ(¯ d)_p_,_O

≤C3||1/psup

O⊂

||1–1/pdiam()Mα,ϕ(¯ d)_p_,

=C3|diam()Mα,ϕ(¯ d)_p_,. (2.23)

3 The Lipschitz and BMO norm inequalities for the fractional convolution operator

It is well known that Lipschitz and BMO norms are two kinds of important norms in diﬀerential forms, which can be found in [14]. Now we recall these deﬁnitions as follows. Let∈L1_loc(,),= 0, 1, . . . ,n. We write∈locLip_k(,), 0≤k≤1, if

locLipk,= sup ρO⊂

|O|–(n+k)/n–O1,O<∞ (3.1)

for someρ≥1.

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L1 loc(,

),= 0, 1, . . . ,n, we write∈BMO(,) if

,= sup ρO⊂

|O|–1–O1,O<∞ (3.2)

for someρ≥1.

Whenis a 0-form, Eq. (3.2) reduces to the classical deﬁnition ofBMO().

Lemma 3.1(see [10]) Let0 <p,q<∞and1/s= 1/p+ 1/q.If f and g are two measurable functions onRn_,_then

fgs,≤ fp,gq, (3.3)

for any⊂Rn_.

and Kα∈C0∞(),where0 <α<n.Then,for any1 <p<∞,there exists a constant C such

that

FαlocLip_k,≤CMα,ϕ(¯ d)_p_,, (3.4)

where k is a constant with0≤k≤1.

Proof By Theorem2.4, we obtain

Fα– (Fα)O_p_,_O≤C|O|diam(O)Mα,ϕ¯d(x)_p_,_O. (3.5)

By Lemma3.1with 1 = 1/p+ (p– 1)/p, for any ball withO(O⊂), we have

Fα– (Fα)O_1,_O

=

O

Fα– (Fα)Odx

≤

O

Fα– (Fα)O p

dx 1/p

O

1pp–1_d_x

(p–1)/p

=|O|(p–1)/p_F_α_{– (}_F_α₎ O_p_,_O

=|O|1–1/pFα– (Fα)O_p_,_O

≤ |O|1–1/pC1|O|diam(O)Mα,ϕ¯d(x)_p_,_O

≤C2|O|2–1/p+1/nMα,ϕ¯d(x)_p_,_O. (3.6)

By the deﬁnition of the Lipschitz norm and 2 – 1/p+ 1/n– 1 –k/n= 1 – 1/p+ 1/n–k/n> 0, we obtain

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= sup ρO⊂

|O|–(n+k)/nFα– (Fα)O_1,_O

= sup ρO⊂|

O|–1–k/nFα– (Fα)O_1,_O

≤ sup ρO⊂

|O|–1–k/nC2|O|2–1/p+1/nMα,ϕ¯d(x)_p_,_O

= sup ρO⊂

C2|O|1–1/p+1/n–k/nMα,ϕ¯d(x)_p_,_O

≤ sup ρO⊂

C2||1–1/p+1/n–k/nMα,ϕ¯d(x)_p_,_O

≤C3 sup ρO⊂

Mα,ϕ¯d(x)_p_,_O

≤C3Mα,ϕ¯d(x)_p_,. (3.7)

Lemma 3.2(see [14]) If the diﬀerential form∈locLip_k(,_),_{= 0, 1, . . . ,}_n_{, 0}_≤_k_≤_1,

is deﬁned in a bounded convex domain,then∈BMO(,)and there exists a constant C such that

,≤ClocLipk,. (3.8)

By Theorem3.1and Lemma3.2, we get the following conclusion.

and Kα∈C0∞(),where0 <α<n.Then,for any1 <p<∞,there exists a constant C such

that

Fα,≤CMα,ϕ(¯ d)_p_,. (3.9)

4 Applications

With regard to the applications of the fractional convolution operator, we will point out that Theorem2.2has diﬀerent expression forms.

Deﬁnition 4.1(see [7]) LetKα(x) be a function deﬁned onRn_{, if there exist two constants}

c≥1 andC> 0 such that

Kα(x–y) –Kα(x)≤C |y|

|x|n+1–α, |x|>c|y|, (4.1) then the kernel functionKαis said to satisfy theHα,∗∞-condition.

Lemma 4.1(see [7]) Letϕbe any Young function deﬁned on[0, +∞),then H_α,∗_∞⊂Hα,ϕ.

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Proof Firstly prove thatKα∈Sα. By the deﬁnition ofKα, we have

|x|∼s

Kα(x)dx≤

O(0,2s) 1

|x|n–αdx≤σn(2s)

n_·₍_s₎α–n_{= 2}n_σ

nsα, (4.2)

whereσnis the volume of a unit sphereninRn. ThusKα∈Sα.

Secondly prove thatKα∈Hα,ϕ. According to Lemma4.1, we only need to prove that

Kα∈H_α,∗_∞. If we choose|x|> 2|y|(c= 2≥1) andy=O= (0, . . . , 0) (fory=O, it is clearly established), it follows that

|x–y|/|x| ∈ ⎧ ⎨ ⎩

(1₂, 1), x,yeach component has the same sign;

(1,3₂), x,yeach component has the diﬀerent sign,

wherex= (x1, . . . ,xn),y= (y1, . . . ,yn). Considering that each component ofx,yis greater

than zero, other cases may be considered similarly. By Lagrange’s mean value theorem

Kα(x–y) –Kα(x)

=|x–y|α–n–|x|α–n

=|x|α–n _|

x–y| |x|

α–n

– 1

≤(n–α)|x|α–n _|

y| |x|

|ξ| α–n–1

≤2n+1–α(n–α) |y|

|x|n+1–α, (4.3)

where

|x|=

n

i=1

x2_i,ξ= (ξ1, . . . ,ξn),

1 2<min

|x–y|

|x| , 1

<|ξ|<max

|x–y|

|x| , 1

<3 2.

ThusKα∈Hα,∗∞.

Theorems2.2and4.1yield the following.

Theorem 4.2 Letϕbe any Young function deﬁned on[0, +∞)and Kα(x) =_|_x_|1n–α,then the fractional convolution operator Fαin(1.1)becomes the classical Riesz potential operator

Iα(x) =

I

Rn 1

|x–y|n–αI(y) dy

dxI, (4.4)

where=_IIdxI is a diﬀerential form inRnand such thatI∈L∞c for all the ordered

-tuples I.Then there exists a constant C such that

Rn

Iα(x)pw(x) dx≤C

Rn

Mα,ϕ(¯ x)pw(x) dx (4.5)

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Lemma 4.2(see [7]) Denote by Sn–1 _{the unit sphere of}_Rn_,_{is a homogeneous function}

deﬁned on Sn–1_with₍_x_{) =}₍_x₎_{and the kernel function K}_α(_x_{) =}₍_x_)/_|_x_|n–α₍_x_{= 0),}_where

x=x/|x|(x= 0).Given a Young functionϕ,we deﬁne the Łϕ_{-modulus of continuity of}_as

ϕ(t) =sup

|y|≤t

(·+y) –(·)_ϕ(_Sn–1₎ (4.6)

and write∈Łϕ(Sn–1).If 1

0

ϕ(t)dt

t <∞, (4.7)

then Kα∈Sα∩Hα,ϕ.

By Theorem2.2and Lemma4.2, we have the following.

Theorem 4.3 Letϕbe a Young function,be a homogeneous function in Sn–1_with₍_x_{) =}

(x)and∈Łϕ₍_Sn–1_). _{Suppose that F}

α is the fractional convolution operator with its

kernel function Kα(x) =(x)/|x|n–α. Let=_IIdxI be a diﬀerential form inRn with

I ∈L∞c for all the ordered-tuples I.If

1 0ϕ(t)

dt

t <∞,then there exists a constant C

such that

Rn

Fα(x)pw(x) dx≤C

Rn

Mα,ϕ(¯ x)pw(x) dx (4.8)

Funding

This work is supported by Xi’an Technological University president Fund Project (XAGDXJJ16019). The authors wish to thank the anonymous referees for their time and thoughtful suggestions.

Competing interests

The authors declare that there are no competing interests regarding the publication of this article.

Authors’ contributions

All results and investigations of this manuscript were due to the joint efforts of all authors. ZD finished the proof and the writing work. HL gave ZD some excellent advices in the proof and writing. QL took part in the original conceiving and discussion, and carefully checked the second amendment of the English writing and the proof in this article. All authors read and approved the final manuscript.

Author details

1_{School of Science, Xi’an Technological University, Xi’an, China.}2_{School of Science, Jiangxi University of Science and} Technology, Ganzhou, China.3_{Department of Mathematics, Ganzhou Teachers’ College, Ganzhou, China.}

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Received: 9 December 2017 Accepted: 11 June 2018

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