Inequalities for the fractional convolution operator on differential forms

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.


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][2][3][4][5][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 K α and I (y) is a bounded function with a compactly supported set on R n , write I (y) ∈ L ∞ c . The fractional convolution operator F α is defined by a convolution integral provided this integral exists for almost all R n , where (x) = I I (y) dx I is a -form defined on R n , the summation is over all ordered -tuples I. The function K α is also assumed to be a wide class of kernels satisfying the following growth condition (see [7]): (1) K α ∈ S α if there exists a constant C > 0 such that |x|∼s K α (x) dx ≤ Cs α ; (1.2) (2) K α is said to satisfy the L α,ϕ -Hörmander condition, and write K α ∈ H α,ϕ . If there exist c ≥ 1 and C > 0 (only dependent on ϕ) such that, for all y ∈ R n and R > c|y|, 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 R n is given by 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 F α we study in this paper degenerates into the operator which Bernardis discussed in [7]. Namely, for any Lebesgue measurable function f ∈ L ∞ c , F α is given as follows: 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 R n (n ≥ 2) and O be a ball in R n . 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 ⊂ R n . Let = (R n ), = 0, 1, . . . , n, be the linear space of all -forms (x) = I I (x) dx I = . . , i ), 1 ≤ i 1 < i 2 < · · · < 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 Similarly, the notations L p loc ( , ) and W 1,p loc ( , ) are self-explanatory. From [9], is a differential form in a bounded convex domain , then there is a decomposition where T is called a homotopy operator. For the homotopy operator T, we know that holds for any differential form ∈ L p loc ( , ), = 1, 2, . . . , n, 1 < p < ∞. Furthermore, we can define the -form ∈ D ( , ) by for all ∈ L p ( , ), 1 ≤ p < ∞.
A non-negative function w ∈ L 1 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 that w ∈ A p if there exists a constant C > 0 such that, for every ball O ⊂ R n , 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 functionφ satisfying 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 (1.13) For α = 0, we write M ϕ instead of M 0,ϕ . When ϕ(t) = t, then M α,ϕ = M α is the classical fractional maximal operator. For α = 0 and ϕ(t) = t, we obtain M 0,ϕ = M is the Hardy-Littlewood maximal operator (see [8]).

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
Suppose that the fractional convolution operator F α = K α * f and its kernel satisfies 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 Then for all cubes O and the Young functions ϕ, Proof Since ϕ is a Young function, it follows that 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 dx I be a differential form on ⊂ R n , and let all the ordered -tuples I satisfy I ∈ L ∞ c . Suppose that F α is a fractional convolution operator applied to differential forms and its kernel function K α satisfies K α ∈ S α ∩ H α,ϕ , where 0 < α < n. Then there exists a constant C such that for any 0 < p < ∞ and w ∈ A ∞ .
Proof By Lemma 2.1 and the following basic inequality where s > 0 is any constant, it follows that Then, by the definition of the fractional maximal operator, notice that for any I such that | I | ≤ | |, we obtain that Combining (2.6) and (2.7), we have

Theorem 2.3
Let ϕ be a Young function on [0, +∞), = I I dx I be a differential form on ⊂ R n , and for all the ordered -tuples,let I satisfy I ∈ L ∞ c . Suppose that F α is a fractional convolution operator on differential forms and its kernel function for any 0 < p < ∞ and all the balls O ⊂ R n .
Proof By the definition of the A ∞ -weight, there exist r 0 ≥ 1 and a constant C < ∞ such that, for all the balls O ⊂ R n , it follows that With the arbitrariness of the condition w ∈ A ∞ of Theorem 2.2, now get any ball O 0 ⊂ R n and let It is easy to check that w(x) = χ O 0 (x) satisfies (2.10). In fact, we have If the kernel function K α 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 dx I be a differential form on ⊂ R n , and let all the ordered -tuples I satisfy I ∈ L ∞ c . Suppose that F α is a fractional convolution operator on differential forms and its kernel function K α satisfies K α ∈ S α ∩H α,ϕ and K α ∈ C ∞ 0 ( ), where C ∞ 0 ( ) stands for all the C ∞ functions with compactly supported sets in and 0 < α < n. Then there exists a constant C such that for any 1 < p < ∞ and all the balls with O ⊂ R n .
Proof By the exterior derivative operator d and the fractional convolution operator F α , we obtain that and According to (1.7)-(1.9), it follows that Now we will give the L p -norm estimation of d(F α ). With K α ∈ C ∞ 0 ( ) and considering the definition of the general partial derivative (see [12]), we obtain Since a new function is obtained when the differential form is taken as a model, we can get a global inequality in the L p (m) domain with Theorem 2.4. Now recall the definition of the L p (m) domain introduced by Staples (see [13]).

Definition 2.1
Let be a real subdomain in R n . If, for all the functions f ∈ L p loc ( ), there exists a constant C such that then is called an L p (m)-average domain, where O 0 is a fixed ball of and p ≥ 1.

Theorem 2.5
Let ϕ be a Young function on [0, +∞), = I I dx I be a differential form on ⊂ R n , and let all the ordered -tuples I satisfy I ∈ L ∞ c . Suppose that F α is a fractional convolution operator on differential forms and its kernel function K α satisfies K α ∈ S α ∩H α,ϕ and K α ∈ C ∞ 0 ( ), where 0 < α < n. Then there exists a constant C such that for any 1 < p < ∞ and O 0 is a fixed ball in .
Proof By the definition of the L p (m)-average domain and noticing that 1 -1/p ≥ 0, we have (2.23)

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 ∈ L 1 loc ( , ), = 0, 1, . . . , n. We write ∈ locLip k ( , ), 0 ≤ k ≤ 1, if for some ρ ≥ 1. Further, we write Lip k ( , ) for those forms whose coefficients are in the usual Lipschitz space with exponent k and write Lip k , for this norm. Similarly, for ∈ L 1 loc ( , ), = 0, 1, . . . , n, we write ∈ BMO( , ) if for some ρ ≥ 1.
When is a 0-form, Eq. (3.2) reduces to the classical definition of BMO( ).

3)
for any ⊂ R n .
Theorem 3.1 Let ϕ be a Young function on [0, +∞), = I I dx I be a differential form on ⊂ R n , and let all the ordered -tuples I satisfy I ∈ L ∞ c . Suppose that F α is a fractional convolution operator on differential forms and its kernel function K α satisfies K α ∈ S α ∩H α,ϕ and K α ∈ C ∞ 0 ( ), where 0 < α < n. Then, for any 1 < p < ∞, there exists a constant C such that

4)
where k is a constant with 0 ≤ k ≤ 1.
Theorem 3.2 Let ϕ be a Young function on [0, +∞), = I I dx I be a differential form on ⊂ R n , and let all the ordered -tuples I satisfy I ∈ L ∞ c . Suppose that F α is a fractional convolution operator on differential forms and its kernel function K α satisfies K α ∈ S α ∩H α,ϕ and K α ∈ C ∞ 0 ( ), where 0 < α < n. Then, for any 1 < p < ∞, there exists a constant C such that (3.9)

Applications
With regard to the applications of the fractional convolution operator, we will point out that Theorem 2.2 has different expression forms.
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 S n-1 with (x) = (x ) and ∈ Ł ϕ (S n-1 ). Suppose that F α is the fractional convolution operator with its kernel function K α (x) = (x)/|x| n-α . Let = I I dx I be a differential form in R n with I ∈ L ∞ c for all the ordered -tuples I. If 1 0 ϕ (t) dt t < ∞, then there exists a constant C such that 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.