Abstract
In this paper, we define the multilinear Calderón–Zygmund operators on differential forms and prove the end-point weak type boundedness of the operators. Based on nonhomogeneous A-harmonic tensor, the Poincaré-type inequalities for multilinear Calderón–Zygmund operators on differential forms are obtained.
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1 Introduction
The multilinear Calderón–Zygmund theory was originally introduced by Coifman and Meyer [1–3] in their study of certain singular integral operators, such as Calderón commutators, paraproducts, and pseudodifferential operators. Afterwards, the multilinear Calderón–Zygmund theory has been further developed by many scholars in the last few decades. For example, Grafakos and Torres studied systematically on the multilinear Calderón–Zygmund operators in [4]. They proved an end-point weak type estimate and obtained the strong type \(L^{p_{1}}\times \cdots \times L^{p_{m}}\rightarrow L^{p} \) boundedness results for multilinear Calderón–Zygmund operators by the classical interpolation method. In [5] and [6], the maximal operator associated with multilinear Calderón–Zygmund singular integrals was introduced and used to obtain the weighted norm estimates for multilinear singular integrals. More recently, Stockdale and Wick [7] provided an alternative proof of the weak-type \((1,\ldots ,1;1/m)\) estimates for m-multilinear Calderón–Zygmund operators on \(\mathbb{R}^{n}\) first proved by Grafakos and Torres. Subsequent results on end-point estimates of the bilinear Calderón–Zygmund operators can be found in [8]. Motivated by the research and considering that differential forms as the generalizations of functions are widely used in physics systems, differential geometry, and PDEs, we aim to establish the boundedness of multilinear Calderón–Zygmund operators on differential forms. In this paper, the definition of multilinear Calderón–Zygmund operators on differential forms is set forth. Moreover, by combining the Calderón–Zygmund decomposition with some skillful techniques, we establish the end-point weak type boundedness of multilinear Calderón–Zygmund operators on differential forms which includes the result of multilinear Calderón–Zygmund operators on functions in [4] as a special case. Unfortunately, it is difficult to apply the complex variable theory to differential forms. So the T1 theorem and strong type boundedness of multilinear Calderón–Zygmund operators on differential forms are still some open questions. But with the help of decomposition theorem of differential forms, we derive the Poincaré-type inequalities for multilinear Calderón–Zygmund operators on A-harmonic tensors which are the generalized solutions to A-harmonic equations on differential forms. Based on the Poincaré-type inequalities, we can make a further study of the multilinear Calderón–Zygmund operators in Orlicz spaces and establish the \(L^{\varphi }\) norm inequalities. More results on the Poincaré-type inequalities and singular integral operators on differential forms can be found in [9–13].
This work is organized as follows. To state our results, we first recall some necessary notations and lemmas in Sect. 2. Then, in Sect. 3, we define the multilinear Calderón–Zygmund operators on differential forms and prove the end-point weak type boundedness of multilinear Calderón–Zygmund operators on differential forms in Theorem 1. Using the weak type inequality, we derive the Poincaré-type inequality for multilinear Calderón–Zygmund operator on a local domain in Theorem 2 in Sect. 4. Finally, the result is extended to obtain the Poincaré-type inequality for the multilinear Calderón–Zygmund operator on a bounded convex domain in Theorem 3.
2 Preliminaries
Throughout this paper, let \(\Omega \subset \mathbb{R}^{n}\) be a bounded domain, \(n\geq 2\), B and σB be the balls with the same center and \(\operatorname{diam}(\sigma B)=\sigma \operatorname{diam}(B)\). We use \(|E|\) to denote the Lebesgue measure of a set \(E\subset \mathbb{R}^{n}\). Let \(\Lambda ^{l}(\mathbb{R}^{n})=\Lambda ^{l}\), \(l=1,2,\ldots ,n\), be the set of all l-forms \(u(x)=\sum_{I}u_{I}(x)\,dx_{I}=\sum u_{i_{1} \cdots i_{l}}(x)\,dx_{i_{1}} \wedge \cdots \wedge {dx_{i_{l}}}\) with summation over all ordered l-tuples \(I=(i_{1},i_{2},\ldots ,i_{l})\), \(1\leq {i_{1}}<\cdots <i_{l}\leq {n}\). \(D^{\prime }(\Omega ,\Lambda ^{l})\) is the space of all differential l-forms on Ω, namely, the coefficient of the l-forms is differential on Ω. The operator \(\star :\Lambda ^{l}(\mathbb{R}^{n})\rightarrow \Lambda ^{n-l}( \mathbb{R}^{n})\) is the Hodge–Star operator as usual, and the linear operator \(d:D^{\prime }(\Omega ,\Lambda ^{l})\rightarrow {D^{\prime }(\Omega ,\Lambda ^{l+1})}\), \(0\leq l\leq n-1\), is called the exterior differential operator. The Hodge codifferential operator \(d^{\star }:D^{\prime }(\Omega ,\Lambda ^{l+1})\rightarrow {D^{\prime }(\Omega , \Lambda ^{l})}\), the formal adjoint of d, is defined by \(d^{\star }=(-1)^{nl+1}\star \,d\star \), see [14] for more introduction. We shall denote by \(L^{p}(\Omega ,\Lambda ^{l})\) the space of differential l-forms with the coefficients in \(L^{p}(\Omega ,\mathbb{R}^{n})\) and with the norm \(\|u\|_{p,\Omega }= (\int _{\Omega } (\sum_{I}|u_{I}(x)|^{2} )^{ \frac{p}{2}}\,dx )^{\frac{1}{p}}\). The homotopy operator \(T:C^{\infty }(\Omega ,\Lambda ^{l})\rightarrow C^{\infty }(\Omega , \Lambda ^{l-1})\) is a very important operator in the theory of differential forms, given by
where \(\psi \in C^{\infty }_{0}(\Omega )\) is normalized by \(\int _{\Omega }\psi (y)\,dy=1\), and \(K_{y}\) is a linear operator defined by
See [15] for more of the function ψ and the operator \(K_{y}\). About the homotopy operator T, we have the following decomposition:
for any differential form \(u\in L^{p}(\Omega ,\Lambda ^{l})\), \(1\leq p<\infty \). We also call it the decomposition theorem for differential form which will be used repeatedly in the proof of this paper. A closed form \(u_{\Omega }\) is defined by \(u_{\Omega }=d(Tu)\), \(l=1, \ldots , n\), and when u is a differential 0-form, \(u_{\Omega }=|\Omega |^{-1}\int _{\Omega }u(y)\,dy\).
The following lemma is the \(L^{p}\) estimate for the homotopy operator T from [15].
Lemma 1
Let \(u\in L^{p}_{loc}(\Omega ,\Lambda ^{l})\), \(l=1,2,\ldots ,n,1< p<\infty \), be a differential form in Ω, and T be the homotopy operator defined on differential forms. Then there exists a constant C, independent of u, such that
The following nonlinear partial differential equation for differential forms
is called nonhomogeneous A-harmonic equation, where \(A: \Omega \times \Lambda ^{l}(\mathbb{R}^{n}) \to \Lambda ^{l}( \mathbb{R}^{n})\) and \(B: \Omega \times \Lambda ^{l}(\mathbb{R}^{n}) \to \Lambda ^{l-1}( \mathbb{R}^{n})\) satisfy the conditions:
for \(x \in \Omega \) a.e. and all \(\xi \in \Lambda ^{l} (\mathbb{R}^{n})\). Here, \(p>1\) is a constant related to equation (1), and \(a, b >0\).
In general, we call the differential form satisfying nonhomogeneous A-harmonic equation the nonhomogeneous A-harmonic tensor. In the proof of the strong type inequality for the multilinear Calderón–Zygmund operator on differential forms, we let the differential form be the nonhomogeneous A-harmonic tensor to get the desired result. See [16–20] for a list of recent results on the A-harmonic equations and related topics. We also need the following weak inverse Hölder inequality for nonhomogeneous A-harmonic tensor, see [21] for more introduction.
Lemma 2
Let \(u\in \Omega \) satisfy equation (1), \(\sigma >1\), \(0 < s, t < \infty \). Then there exists a constant C, independent of u, such that
for \(\sigma B \subset \Omega \).
3 End-point estimate
In the classical theory of singular integrals, it is important to prove the end-point weak type inequality which is the core link to get the strong boundedness of singular integral operators. So the focus of this section is to establish the end-point weak type inequality of multilinear Calderón–Zygmund operators on differential forms. Before giving the definition of multilinear Calderón–Zygmund operators on differential forms, we first define the kernel function in multilinear Calderón–Zygmund operators, see [4] for more introduction about the kernel function.
Definition 1
Let \(K(x,y_{1},\ldots ,y_{m})\) be a locally integrable function which is defined on
and satisfy the following conditions:
\((1)\) For some \(A>0\) and all points in the domain of definition, the function \(K(x,y_{1},\ldots ,y_{m})\) satisfies
\((2)\) For \(\varepsilon >0\), we have
where \(|x-x'|\le \frac{1}{2}\max_{1\le j\le m}|x-y_{j}|\). For other \(y_{j}\), we also have
where \(|y_{j}-y'_{j}|\le \frac{1}{2}\max_{1\le j\le m}|x-y_{j}|\). For convenience, we call conditions \((1)\), \((2)\) for kernel function the \(m-\operatorname{CZK}(A,\varepsilon )\) conditions.
Next, we give the definition of multilinear Calderón–Zygmund operators on differential forms.
Definition 2
The operator \(\mathcal{L}:\Lambda ^{l}(\mathbb{R}^{n})\times \Lambda ^{l}( \mathbb{R}^{n})\times \cdots \times \Lambda ^{l}(\mathbb{R}^{n}) \rightarrow \Lambda ^{l}(\mathbb{R}^{n})\) is called multilinear operator on differential forms if
where \(x\notin \bigcap^{m}_{i=1}\operatorname{supp} u^{(i)}_{I}\), \({\xi }=(\xi _{1},\xi _{2},\ldots ,\xi _{l})\).
Now, we establish the end-point weak type inequality for multilinear Calderón–Zygmund operators on differential forms.
Theorem 1
Let \(\mathcal{L}\) be a multilinear Calderón–Zygmund operator, the kernel function K satisfies the \(m-\operatorname{CZK}(A,\varepsilon )\) conditions, and \(u^{(1)},\ldots ,u^{(m)}\subset D'(\Omega ,\Lambda ^{l})\). Assume that, for \(1\leq q_{1},q_{2},\ldots ,q_{m}\leq \infty \) and \(0< q<\infty \) with
if the operator \(\mathcal{L}\) is weak-\((q_{1},\ldots ,q_{m},q)\), that is,
for all \(\lambda >0\), then \(\mathcal{L}\) is also a weak-\((1,1,\ldots ,1, \frac{1}{m})\) operator. Especially, we have
where \([\cdot ]\) means the weak norm of an operator and \(C>0\) is a constant that depends only on the parameters n, m.
Proof
Without loss of generality, suppose
In order to prove the conclusion, we need to show that there exists a constant C independent of \(u^{(1)},\ldots ,u^{(m)}\) such that
According to the algorithms of differential forms, for \(i_{1}< i_{2}<\cdots <i_{k}\) and \(j_{1}< j_{2}<\cdots <j_{k}\), we obtain
where \(e_{1},e_{2},\ldots ,e_{n}\) are orthogonal basis for the tangent space of \(\mathbb{R}^{n}\). So, for a differential form
with \(I=(i_{1},i_{2},\ldots ,i_{l})\), \(1\leq {i_{1}}<{i_{2}}<\cdots <{i_{l}}\leq n\), we have
where \({e}_{I}=(e_{i_{1}},e_{i_{2}},\ldots ,e_{i_{l}})\). It is also a differential form for the image of the multilinear Calderón–Zygmund operator on a differential form. So, there exist \(a_{I}, I=(i_{1},i_{2},\ldots ,i_{l})\), \(1\leq {i_{1}}<{i_{2}}<\cdots <{i_{l}}\leq n\) satisfying
If we write \(\mathrm{d}{y}=\mathrm{d}y_{1}\cdots \mathrm{d}y_{m}\), then it follows that
Applying the Calderón–Zygmund decomposition to each function \(u^{(i)}_{I}\) at height \(\alpha =(\lambda \rho )^{1/m}\), we obtain a sequence of pairwise disjoint cubes \(\{Q_{i,k_{i}}\}_{k_{i}=1}^{\infty }\) and a decomposition
such that, for all \(i = 1, \ldots , m\),
\((a)\) \(|g^{(i)}_{I}|\leq C_{1}\alpha \),
\((b)\) \(\operatorname{supp}(b^{(i,k_{i})}_{I})\subset Q_{i,k_{i}}\), \(\int _{Q_{i,k_{i}}} b^{(i,k_{i})}_{I}(x)\,\mathrm{d}x=0\), and
\((c)\) \(\sum_{k_{i}}|Q_{i,k_{i}}|\leq \frac{C_{1}}{\alpha }\int _{ \mathbb{R}^{n}}|u^{(i)}_{I}(x)|\,dx\).
Applying the Calderón–Zygmund decomposition above to the operator \(\mathcal{L}\), we have
where \(h^{(i)}_{I}\in \{b^{(i)}_{I},g^{(i)}_{I}\}\). Then, by properties \((a)\) and \((c)\), one has
Using (5) and the weak \(-(q_{1},q_{2},\ldots ,q_{m},q) \) boundedness to the first item of the right-hand side of inequality (4), we get
Now we begin to estimate the measure of the following set which appeared in (4):
where \(h^{(i_{r})}_{I}=b^{(i_{r})}_{I}\), \(1 \le r \le t\), and \(h^{(i_{t+j})}_{I}=g^{(i_{t+j})}_{I}\), \(1\le j\le m-t\), \(1 \le t \le m\). For the sake of simplicity, there is no harm in the setting \(b^{(i_{r})}_{I}\) appearing with superscript \(1,\ldots ,t\). We set that \(\Omega =\bigcup _{r=1}^{t}\Omega _{r}=\bigcup _{r=1}^{t}\bigcup _{k_{r}} \tilde{Q}_{r,k_{r}}\) with \(\tilde{Q}_{r,k_{r}}\) is a concentric and double diameter cube to \(Q_{r,k_{r}}\). Then we obtain
Using properties \((a)\), \((b)\) and the Chebyshev inequality, we have
where \(c_{1,k_{1}}\) is the center of cube \(\tilde{Q}_{1,k_{1}}\). We represent the difference set \((2^{\tau _{r}+1}Q_{r,k_{r}})\setminus (2^{\tau _{r}}Q_{r,k_{r}})\) by \(H^{\tau _{r}}_{r,k_{r}}\) for \(\tau _{r}=1,2,\ldots \) , and \(r=1,\ldots ,t\). Then the difference set of \(\mathbb{R}^{n}\) and Ω satisfies
Note that, for the arbitrary \(x\in \bigcap^{t}_{r=1} (H^{\tau _{r}}_{r,k_{r}} )\) and \(y_{r}\in Q_{r,k_{r}}\), we have
and
where \(\tilde{l}(Q_{r,k_{r}})\) denotes the diameter of the cube \(\tilde{Q}_{r,k_{r}}\). Hence, it follows that
Bringing formula (10) into (8) and combining with inequality (9), we get
Owing to the fact that the sets \(H^{\tau _{r}}_{r,k_{r}}\) and \(\tau _{r}=1,\ldots ,\infty \) are disjoint, we know
Similarly, we have
Combining (8), (11), (12), and (13), we obtain
Choose \(\rho =\frac{1}{A+B}\), then inequality (6) shows that
On the other hand, property \((c)\) implies that
where
And it should be noted that
Finally, combining (15), (16), and (17), we have
This ends the proof of Theorem 1. □
4 Poincaré-type inequalities
In this section, we establish the Poincaré-type inequalities for multilinear Calderón–Zygmund operators on differential forms in the local and global domain. So we first define the operator on differential l-form in a local domain as follows:
for \(x\notin \bigcap ^{m}_{i=1}\operatorname{supp} u^{(i)}_{I}\), \(B\subset \Omega \), and the kernel function \(K(y_{1},\ldots ,y_{m})\) satisfies
for \(|x-x'|\le \frac{1}{2}\max_{1\le j\le m}|x-y_{j}|\).
Next, we give the Poincaré-type inequality for the multilinear Calderón–Zygmund operator \({\widetilde{\mathcal{L}}}\) on a differential form in the local domain.
Theorem 2
Let \(\widetilde{\mathcal{L}}\) be a multilinear Calderón–Zygmund operator defined by (19), \(B\subset \Omega \), the kernel function K satisfies condition (2) in Definition 1, and \(u^{(1)},\ldots ,u^{(m)} \in L^{p}_{loc}(\Omega ,\Lambda ^{l})\) satisfy A-harmonic equation (1), then for \(\theta ={1+\frac{2}{n}-\frac{m-1}{p}}\), we have
Proof
Applying the decomposition theorem for a differential form and Lemma 1, we get
For convenience, we write
Then,
In terms of the conditions of kernel function K, we know
It follows from (23) and (22) that
Further, we get
Apply Hölder’s inequality to the innermost layer integral of the right-hand side of the inequality above with Hölder index satisfying \(1=\frac{n(m-1)-\varepsilon }{n(m-1)}+\frac{\varepsilon }{n(m-1)}\), \(0<\varepsilon <1\). For the convenience of writing, we write \(\frac{1}{\zeta }=\frac{n(m-1)-\varepsilon }{n(m-1)}\), \(\frac{1}{\eta }= \frac{\varepsilon }{n(m-1)}\), then it follows that
Similarly, applying Hölder’s inequalities \((m-2)\) times with the indexes satisfying \(1=\frac{1}{{\zeta }}+\frac{1}{{\eta }}\), we have
Let \(\chi _{B}\) be a characteristic function as
then we get
Applying Hölder’s inequalities again with the indexes satisfying \(1=\frac{1}{p}+\frac{1}{q}\), we obtain
The integral in the second bracket above can be simplified by the basic inequality as follows:
Here, \(B'=B(y_{0}')\) and \(B=B(y_{0})\) with \(y_{0}'=y_{0}-x\), \(r_{1}\) is the distance from 0 to \(B'\). And then, by Fubini’s theorem, we have
Combining (24), (25), (26), and (27), we have
Applying Lemma 2 to \(\|u^{(i)}_{I}\|_{{\eta },B}\), \(1 \le i\le m-1\), we get
for \(\sigma >1\) with \(\sigma B\subset \Omega \). Combining (21), (28), (29), we obtain
□
Finally, we give the Poincaré-type inequality for the multilinear Calderón–Zygmund operator \({\widetilde{\mathcal{L}}}\) in a bounded convex domain. We need the following covering lemma.
Lemma 3
([21])
There exists a cover \({\mathcal{V}} = \{B_{i}\}\) for any bounded subset Ω in an n-dimensional Euclidean space \(\mathbb{R}^{n}\) satisfying
for a constant \(N>1\). And if \(B_{i} \cap B_{j} \neq \emptyset \), then there exists a cube Q included in \(B_{i} \cap B_{j}\) with \(B_{i} \cup B_{j} \subset NR\), and the cube Q does not have to be an element in the set family \(\mathcal{V}\).
Theorem 3
Let \(\widetilde{\mathcal{L}}\) be a multilinear Calderón–Zygmund operator defined by (19), \(\Omega _{1}\subset \subset \Omega \), kernel function K satisfies condition (2) in Definition 1, \(u^{(1)},\ldots ,u^{(m)} \in L^{p}_{loc}(\Omega ,\Lambda ^{l})\) satisfy A-harmonic equation (1) and \(2p+(p+1-m)n>0\), then
Proof
By Lemma 3 and Theorem 2, we have
where N is the constant in Lemma 3. Next, from the decomposition theorem of homotopy operator, we get
which completes the proof of Theorem 3. □
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This work was supported by the Fundamental Research Funds for the Central Universities (Grant no.2572019BC06) and the Natural Science Foundation for the Universities in Jiangsu Province (Grant no.20KJB110008).
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All authors put their efforts together into the research and writing of this manuscript. XL carried out the proofs of all research results in this manuscript and wrote its draft. YX and JN proposed the study, participated in its design, and revised its final version. All authors read and approved the final manuscript.
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Li, X., Xing, Y. & Niu, J. Weak and strong type estimates for multilinear Calderón–Zygmund operators on differential forms. J Inequal Appl 2021, 92 (2021). https://doi.org/10.1186/s13660-021-02628-5
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DOI: https://doi.org/10.1186/s13660-021-02628-5