Inequalities for the fractional convolution operator on differential forms

Dai et al. Journal of Inequalities and Applications (2018) 2018:176 https://doi.org/10.1186/s13660-018-1768-x

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Inequalities for the fractional convolution operator on differential forms Zhimin Dai1* , Huacan Li2 and Qunfang Li3 *

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 function Kα and I (y) is a bounded function with a compactly supported set on Rn , write I (y) ∈ L∞ c . The fractional convolution operator Fα is defined by a convolution integral

Fα (x) =

I

Rn

Kα (x – y)I (y) dy dxI ,

provided this integral exists for almost all Rn , where (x) =

(1.1) I

I (y) dxI is a -form defined

on R , the summation is over all ordered -tuples I. The function Kα is also assumed to n

be a wide class of kernels satisfying the following growth condition (see [7]): (1) Kα ∈ Sα if there exists a constant C > 0 such that |x|∼s

Kα (x) dx ≤ Csα ;

(1.2)

© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Dai et al. Journal of Inequalities and Applications (2018) 2018:176

Page 2 of 13

(2) Kα is said to satisfy the Lα,ϕ -Hörmander condition, and write Kα ∈ Hα,ϕ . If there exist c ≥ 1 and C > 0 (only dependent on ϕ) such that, for all y ∈ Rn and R > c|y|, ∞

2m R

n–α

Kα (· – y) – Kα (·)

≤ C, ϕ(|x|∼2m R)

(1.3)

m=1

where ϕ is a Young function defined on [0, +∞), |x| ∼ s stands for the set {s < |x| ≤ 2s}, O(0, s) is a ball with the center at the origin and the radius equal to s, and the ϕ-mean Luxemburg norm of a function f on a cube (or a ball)O in Rn is given by 1 |f | dx ≤ 1 . f ϕ(O) = inf λ > 0 : ϕ |O| O λ

(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 fractional convolution operator Fα we study in this paper degenerates into the operator which Bernardis discussed in [7]. Namely, for any Lebesgue measurable function f ∈ L∞ c , Fα is given as follows: Fα f (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 produced by a homogeneous function and a Fourier multiplier. Now we introduce some notations and definitions. Let be an open subset of Rn (n ≥ 2) and O be a ball in Rn . Let ρO denote the ball with the same center as O and diam(ρO) = ρ diam(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) = I I (x) dxI = n I i1 i2 ···i (x) dxi1 ∧ dxi2 · · · ∧ dxi in R , where I = (i1 , i2 , . . . , i ), 1 ≤ i1 < i2 < · · · < i ≤ n, are the ordered -tuples. Moreover, if each of the coefficients I (x) of (x) is differential

on , then we call (x) a differential -form on and use D (, ) to denote the space of

all differential -forms on . C ∞ (, ) denotes the space of smooth -forms on . The

exterior derivative d : D (, ) → D (, +1 ), = 0, 1, . . . , n – 1, is given by d(x) =

n ∂i

1 i2 ···i

I

j=1

∂xj

(x)

dxj ∧ dxi1 ∧ dxi2 ∧ · · · ∧ dxi

(1.6)

for all ∈ D (, ). Lp (, )(1 ≤ p < ∞) is a Banach space with the norm p, =

p ( |(x)|p dx)1/p = ( ( I |I (x)|2 )p/2 dx)1/p < ∞. Similarly, the notations Lloc (, ) and

1,p Wloc (, ) are self-explanatory. From [9], is a differential form in a bounded convex domain , then there is a decomposition = d(T) + T(d),

(1.7)


Page 3 of 13

where T is called a homotopy operator. For the homotopy operator T, we know that Tp,O ≤ C|O| diam(O)p,O

(1.8) p

holds for any differential form ∈ Lloc (,

can define the -form ∈ D (, ) by

), = 1, 2, . . . , n, 1 < p < ∞. Furthermore, we

⎧ ⎨||–1 (x) dx, = 0, = ⎩dT(), = 1, 2, . . . , n,

(1.9)

for all ∈ Lp (, ), 1 ≤ p < ∞. A non-negative function w ∈ L1loc (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 that w ∈ Ap if there exists a constant C > 0 such that, for every ball O ⊂ Rn ,

1 |O|

w dx O

1 |O|

1 – p–1

w

p–1 dx

≤ C.

(1.10)

O

For the case p = 1, w ∈ A1 if there exists a constant C > 0 such that, for every ball O ⊂ Rn , 1 |O|

w dx ≤ C ess inf w(x). x∈O

O

(1.11)

Also A∞ = p≥1 Ap . It is well known that Ap ⊂ Aq for all 1 ≤ p ≤ q ≤ ∞, and also that for 1 < p ≤ ∞, if w ∈ Ap , then there exists ε > 0 such that w ∈ Ap–ε . A function ϕ : [0, ∞) → [0, ∞) is a Young function if it is continuous, convex, increasing and satisfies ϕ(0) = 0 and ϕ(t) → ∞ as t → ∞. Each Young function ϕ has an associated complementary Young function ϕ¯ satisfying t ≤ ϕ –1 (t)ϕ¯ –1 (t) ≤ 2t

(1.12)

for all t > 0, where ϕ –1 (t) is the inverse function of ϕ(t) (see [11]). For each locally integrable function f and 0 ≤ α < n, the fractional maximal operator associated with the Young function ϕ is defined by α

f ϕ(O) . Mα,ϕ f (x) = sup |O| n

(1.13)

x∈O

For α = 0, we write Mϕ instead of M0,ϕ . When ϕ(t) = t, then Mα,ϕ = Mα is the classical fractional maximal operator. For α = 0 and ϕ(t) = t, we obtain M0,ϕ = M is 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. Lemma 2.1 Let ϕ be a Young function on [0, +∞) and f be any n-tuple function on Rn with f ∈ L∞ c . Suppose that the fractional convolution operator Fα = Kα ∗ f and its kernel satisfies


Page 4 of 13

Kα ∈ Sα ∩ Hα,ϕ , where 0 < α < n. Then there exists a constant C such that Rn

Fα f (x)p w(x) dx ≤ C

Rn

p Mα,ϕ¯ f (x) w(x) dx

(2.1)

for any 0 < p < ∞ and w ∈ A∞ . Theorem 2.1 Let ϕ be 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, f ϕ(O) ≤ gϕ(O) .

(2.2)

Proof Since ϕ is a Young function, it follows that 1 |O|

1 |f | |g| dx ≤ dx ≤ 1 ϕ gϕ(O) |O| O gϕ(O) 1 |f | dx ≤ 1 gϕ(O) ∈ E = λ > 0 : ϕ |O| O λ

ϕ O

⇒

f ϕ(O) = inf E ≤ gϕ(O) .

⇒

(2.3)

According to Theorem 2.1, we can get a similar conclusion to Lemma 2.1. Theorem 2.2 Let ϕ be a Young function on [0, +∞), = I I dxI be a differential form on ⊂ Rn , and let all the ordered -tuples I satisfy I ∈ L∞ c . Suppose that Fα is a fractional convolution operator applied to differential forms and its kernel function Kα satisfies Kα ∈ Sα ∩ Hα,ϕ , where 0 < α < n. Then there exists a constant C such that Rn

Fα (x)p w(x) dx ≤ C

Rn

p Mα,ϕ¯ (x) w(x) dx

(2.4)

for any 0 < p < ∞ and w ∈ A∞ . Proof By Lemma 2.1 and the following basic inequality n

|ai | ≤ n s

i=1

n

s |ai |

≤ ns+1

i=1

n

|ai |s ,

i=1

where s > 0 is any constant, it follows that

p

Fα p,w,Rn = =

Rn

Fα (x)p w(x) dx

Rn

I

≤

Rn

= C1

C1

Rn

2 p/2 Kα (x – y)I (y) dy w(x) dx

I

Rn

Rn

Rn

I

p Kα (x – y)I (y) dy w(x) dx p Kα (x – y)I (y) dy w(x) dx

(2.5)


≤ C2

= C2 ≤ C3

Rn

I

Rn

p Mα,ϕ¯ I (x) w(x) dx

p Mα,ϕ¯ I (x) w(x) dx

I

Rn

Page 5 of 13

p Mα,ϕ¯ I (x)

w(x) dx.

(2.6)

I

Then, by the definition of the fractional maximal operator, notice that for any I such that |I | ≤ ||, we obtain that

Mα,ϕ¯ I (x)

I

=

I

sup |O|α/n ϕ(O) ¯ x∈O

I

≤ Cn sup |O|α/n x∈O

I

ϕ(O) ¯ I

≤ Cn sup |O|α/n Cn ϕ(O) ¯ x∈O ≤ C4 Mα,ϕ¯ (x).

(2.7)

Combining (2.6) and (2.7), we have p

Fα p,w,Rn p ≤ C3 Mα,ϕ¯ I (x) w(x) dx Rn

≤ C3

p C4 Mα,ϕ¯ (x) w(x) dx

p Mα,ϕ¯ (x) w(x) dx.

Rn

≤ C5

I

Rn

(2.8)

Theorem 2.3 Let ϕ be a Young function on [0, +∞), = I I dxI be a differential form on ⊂ Rn , and for all the ordered -tuples,let I satisfy I ∈ L∞ c . Suppose that Fα is a fractional convolution operator on differential forms and its kernel function Kα satisfies Kα ∈ Sα ∩ Hα,ϕ , where 0 < α < n. Then there exists a constant C such that

Fα (x)p dx ≤ C

O

p Mα,ϕ¯ (x) dx

(2.9)

O

for any 0 < p < ∞ and all the balls O ⊂ Rn . Proof By the definition of the A∞ -weight, there exist r0 ≥ 1 and a constant C < ∞ such that, for all the balls O ⊂ Rn , it follows that

1 |O|

w(x) dx O

1 |O|

w(x) O

– r 1–1 0

r0 –1 dx

≤ C.

(2.10)


Page 6 of 13

With the arbitrariness of the condition w ∈ A∞ of Theorem 2.2, now get any ball O0 ⊂ Rn and let ⎧ ⎨1, x ∈ O ; 0 w(x) = χO0 (x) = ⎩0, x ∈/ O0 . It is easy to check that w(x) = χO0 (x) satisfies (2.10). In fact, we have

r0 –1 1 1 – r 1–1 0 χO (x) dx χO (x) dx |O| O 0 |O| O 0 r0 –1 1 1 |O ∩ O0 | |O ∩ O0 | = |O| |O| r0 1 |O ∩ O0 | = ≤ 1. |O|

(2.11)

Thus

Fα (x)p dx O0

=

Rn

Fα (x)p χO (x) dx 0

≤C

Rn

p Mα,ϕ¯ (x) χO0 (x) dx p Mα,ϕ¯ (x) dx.

≤C

(2.12)

O0

If the kernel function Kα and the coefficient functions I of differential forms are subject to some conditions, the following more important conclusion will be obtained. Theorem 2.4 Let ϕ be a Young function on [0, +∞), = I I dxI be a differential form on ⊂ Rn , and let all the ordered -tuples I satisfy I ∈ L∞ c . Suppose that Fα is a fractional convolution operator on differential forms and its kernel function Kα satisfies Kα ∈ Sα ∩Hα,ϕ and Kα ∈ C0∞ (), where C0∞ () stands for all the C ∞ functions with compactly supported sets in and 0 < α < n. Then there exists a constant C such that

Fα – (Fα )O ≤ C diam(O)|O| Mα,ϕ¯ (d)

p,O p,O

(2.13)

for any 1 < p < ∞ and all the balls with O ⊂ Rn . Proof By the exterior derivative operator d and the fractional convolution operator Fα , we obtain that d =

n ∂I (x) I

Fα (d) =

k=1

∂xk

dxk ∧ dxI ,

n I

k=1

Rn

Kα (x – y)

∂I (y) dy dxk ∧ dxI ∂yk

(2.14)


Page 7 of 13

and dFα () =

n ∂hI (x) I

k=1

∂xk

dxk ∧ dxI ,

(2.15)

where hI (x) =

Kα (x – y)I (y) dy.

Rn

(2.16)

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

Fα – (Fα )O = T d(Fα ) ≤ C1 diam(O)|O| d(Fα ) . p,O p,O p,O

(2.17)

Now we will give the Lp -norm estimation of d(Fα ). With Kα ∈ C0∞ () and considering the definition of the general partial derivative (see [12]), we obtain

d(Fα ) p p,O

p/2 n ∂hI (x) 2 = dx ∂x O

I

k=1

I

k=1

I

k=1

I

k=1

I

k=1

k

p/2 n ∂ Rn Kα (x – y)I (y) dy 2 dx = ∂x O

k

n = O

Rn

n – = O

n = O

2 p/2 ∂Kα (x – y) I (y) dy dx ∂xk

Rn

Rn

2 p/2 ∂Kα (x – y) I (y) dy dx ∂yk

2 p/2 ∂I (y) Kα (x – y) dy dx ∂yk

p = Fα (d) p,O ,

(2.18)

that is

d(Fα ) = Fα (d) . p,O p,O

(2.19)

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

Fα – (Fα )O

p,O

≤ C1 diam(O)|O| d(Fα ) p,O

= C1 diam(O)|O| Fα (d) p,O

≤ C1 diam(O)|O| Mα,ϕ¯ (d)

p,O

.

(2.20)


Page 8 of 13

Since a new function is obtained when the differential form is taken as a model, we can get a global inequality in the Lp (m) domain with Theorem 2.4. Now recall the definition of the Lp (m) domain introduced by Staples (see [13]). p

Definition 2.1 Let be a real subdomain in Rn . If, for all the functions f ∈ Lloc (), there exists a constant C such that ||–1/p f – fO0 p, ≤ C sup |O|–1/p f – fO p,O ,

(2.21)

O⊂

then is called an Lp (m)-average domain, where O0 is a fixed ball of and p ≥ 1. Theorem 2.5 Let ϕ be a Young function on [0, +∞), = I I dxI be a differential form on ⊂ Rn , and let all the ordered -tuples I satisfy I ∈ L∞ c . Suppose that Fα is a fractional convolution operator on differential forms and its kernel function Kα satisfies Kα ∈ Sα ∩Hα,ϕ and Kα ∈ C0∞ (), where 0 < α < n. Then there exists a constant C such that

Fα – (Fα )O ≤ C|| diam() Mα,ϕ¯ d(x)

0 p, p,

(2.22)

for any 1 < p < ∞ and O0 is a fixed ball in . Proof By the definition of the Lp (m)-average domain and noticing that 1 – 1/p ≥ 0, we have

Fα – (Fα )O

0 p,

≤ C1 ||1/p sup |O|–1/p Fα – (Fα )O p,O O⊂

≤ C1 ||1/p sup |O|–1/p C2 diam(O)|O| Mα,ϕ¯ (d) p,O O⊂

≤ C3 ||1/p sup |O|1–1/p diam(O) Mα,ϕ¯ (d) p,O O⊂

≤ C3 ||

1/p

sup ||1–1/p diam() Mα,ϕ¯ (d) p,

O⊂

= 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 ∈ L1loc (, ), = 0, 1, . . . , n. We write ∈ locLipk (, ), 0 ≤ k ≤ 1, if locLipk , = sup |O|–(n+k)/n – O 1,O < ∞

(3.1)

ρO⊂

for some ρ ≥ 1.

Further, we write Lipk (, ) for those forms whose coefficients are in the usual Lipschitz space with exponent k and write Lipk , for this norm. Similarly, for ∈


L1loc (,

Page 9 of 13

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

) if

, = sup |O|–1 – O 1,O < ∞

(3.2)

ρO⊂

for some ρ ≥ 1. When is a 0-form, Eq. (3.2) reduces to the classical definition of BMO(). Lemma 3.1 (see [10]) Let 0 < p, q < ∞ and 1/s = 1/p + 1/q. If f and g are two measurable functions on Rn , then fgs, ≤ f p, gq,

(3.3)

for any ⊂ Rn . Theorem 3.1 Let ϕ be a Young function on [0, +∞), = I I dxI be a differential form on ⊂ Rn , and let all the ordered -tuples I satisfy I ∈ L∞ c . Suppose that Fα is a fractional convolution operator on differential forms and its kernel function Kα satisfies Kα ∈ Sα ∩Hα,ϕ and Kα ∈ C0∞ (), where 0 < α < n. Then, for any 1 < p < ∞, there exists a constant C such that

Fα locLipk , ≤ C Mα,ϕ¯ (d) p, ,

(3.4)

where k is a constant with 0 ≤ k ≤ 1. Proof By Theorem 2.4, we obtain

Fα – (Fα )O ≤ C|O| diam(O) Mα,ϕ¯ d(x) . p,O p,O

(3.5)

By Lemma 3.1 with 1 = 1/p + (p – 1)/p, for any ball with O(O ⊂ ), we have

Fα – (Fα )O

1,O = Fα – (Fα )O dx O

≤

Fα – (Fα )O p dx

1/p

p

(p–1)/p

1 p–1 dx

O

= |O|(p–1)/p Fα – (Fα )O p,O

= |O|1–1/p Fα – (Fα )O

O

p,O

≤ |O|1–1/p C1 |O| diam(O) Mα,ϕ¯ d(x) p,O

≤ C2 |O|2–1/p+1/n Mα,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 Fα locLipk ,


Page 10 of 13

= sup |O|–(n+k)/n Fα – (Fα )O 1,O ρO⊂

= sup |O|–1–k/n Fα – (Fα )O 1,O ρO⊂

≤ sup |O|–1–k/n C2 |O|2–1/p+1/n Mα,ϕ¯ d(x) p,O ρO⊂

= sup C2 |O|1–1/p+1/n–k/n Mα,ϕ¯ d(x) p,O ρO⊂

≤ sup C2 ||1–1/p+1/n–k/n Mα,ϕ¯ d(x) p,O ρO⊂

≤ C3 sup Mα,ϕ¯ d(x) p,O ρO⊂

≤ C3 Mα,ϕ¯ d(x)

p,

.

(3.7)

Lemma 3.2 (see [14]) If the differential form ∈ locLipk (, ), = 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 Theorem 3.1 and Lemma 3.2, we get the following conclusion. Theorem 3.2 Let ϕ be a Young function on [0, +∞), = I I dxI be a differential form on ⊂ Rn , and let all the ordered -tuples I satisfy I ∈ L∞ c . Suppose that Fα is a fractional convolution operator on differential forms and its kernel function Kα satisfies Kα ∈ Sα ∩Hα,ϕ and Kα ∈ C0∞ (), where 0 < α < n. Then, for any 1 < p < ∞, there exists a constant C such that

Fα , ≤ C Mα,ϕ¯ (d) p, .

(3.9)

4 Applications With regard to the applications of the fractional convolution operator, we will point out that Theorem 2.2 has different expression forms. Definition 4.1 (see [7]) Let Kα (x) be a function defined on Rn , if there exist two constants c ≥ 1 and C > 0 such that Kα (x – y) – Kα (x) ≤ C

|y| , |x|n+1–α

|x| > c|y|,

(4.1)

∗ then the kernel function Kα is said to satisfy the Hα,∞ -condition. ∗ Lemma 4.1 (see [7]) Let ϕ be any Young function defined on [0, +∞), then Hα,∞ ⊂ Hα,ϕ .

Theorem 4.1 Let Kα (x) =

1 |x|n–α

and 0 ≤ α < n, then Kα ∈ Sα ∩ Hα,ϕ .


Page 11 of 13

Proof Firstly prove that Kα ∈ Sα . By the definition of Kα , we have |x|∼s

Kα (x)dx ≤

O(0,2s)

1 dx ≤ σn (2s)n · (s)α–n = 2n σn sα , |x|n–α

(4.2)

where σn is the volume of a unit sphere n in Rn . Thus Kα ∈ Sα . Secondly prove that Kα ∈ Hα,ϕ . According to Lemma 4.1, we only need to prove that ∗ . If we choose |x| > 2|y|(c = 2 ≥ 1) and y = O = (0, . . . , 0) (for y = O, it is clearly Kα ∈ Hα,∞ established), it follows that ⎧ ⎨( 1 , 1), x, y each component has the same sign; |x – y|/|x| ∈ 2 ⎩(1, 3 ), x, y each component has the different sign, 2

where x = (x1 , . . . , xn ), y = (y1 , . . . , yn ). Considering that each component of x, y is 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 α–n α–n |x – y| = |x| – 1 |x| |y| α–n–1 |ξ | ≤ (n – α)|x|α–n |x| ≤ 2n+1–α (n – α)

|y| , |x|n+1–α

(4.3)

where n |x| = x2i , ξ = (ξ1 , . . . , ξn ), i=1

|x – y| |x – y| 3 1 < min , 1 < |ξ | < max ,1 < . 2 |x| |x| 2 ∗ Thus Kα ∈ Hα,∞ . Theorems 2.2 and 4.1 yield the following.

1 , then the Theorem 4.2 Let ϕ be any Young function defined on [0, +∞) and Kα (x) = |x|n–α fractional convolution operator Fα in (1.1) becomes the classical Riesz potential operator

Iα (x) =

I

Rn

1 (y) dy dxI , I |x – y|n–α

(4.4)

where = I I dxI is a differential form in Rn and such that I ∈ L∞ c for all the ordered -tuples I. Then there exists a constant C such that Rn

Iα (x)p w(x) dx ≤ C

Rn

for any 0 < p < ∞ and w ∈ A∞ .


(4.5)


Page 12 of 13

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 Kα (x) = (x)/|x|n–α (x = 0), where x = x/|x|(x = 0). Given a Young function ϕ, we define the Łϕ -modulus of continuity of as

(· + y) – (·) n–1 ϕ (t) = sup ϕ(S ) |y|≤t

(4.6)

and write ∈ Łϕ (Sn–1 ). If

1

ϕ (t) 0

dt < ∞, t

(4.7)

then Kα ∈ Sα ∩ Hα,ϕ . By Theorem 2.2 and Lemma 4.2, we have the following. Theorem 4.3 Let ϕ be a Young function, be a homogeneous function in Sn–1 with (x) = (x ) and ∈ Łϕ (Sn–1 ). Suppose that Fα is the fractional convolution operator with its kernel function Kα (x) = (x)/|x|n–α . Let = I I dxI be a differential form in Rn with 1 dt I ∈ L ∞ c for all the ordered -tuples I. If 0 ϕ (t) t < ∞, then there exists a constant C such that Rn

Fα (x)p w(x) dx ≤ C

Rn


(4.8)

for any 0 < p < ∞ and w ∈ A∞ . 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.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 9 December 2017 Accepted: 11 June 2018 References 1. Bi, H.: Weighted inequalities for potential operators on differential forms. J. Inequal. Appl. 2010, Article ID 713625 (2010). https://doi.org/10.1155/2010/713625 2. Tang, Z., Zhu, J.: Carathéodory operator of differential forms. J. Inequal. Appl. 2011, Article ID 88 (2011). https://doi.org/10.1186/1029-242X-2011-88 3. Tang, Z., Zhu, J.: Estimates for the composition of the Carathéodory and homotopy operators. J. Inequal. Appl. 2012, Article ID 188 (2012). https://doi.org/10.1186/1029-242X-2012-188 4. Dai, Z., Xing, Y., Ding, S., Wang, Y.: Inequalities for the composition of Green’s operator and the potential operator. J. Inequal. Appl. 2012, Article ID 271 (2012). https://doi.org/10.1186/1029-242X-2012-271


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