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Inequalities for integral operators in Hölder–Morrey spaces on differential forms
Journal of Inequalities and Applications volume 2023, Article number: 71 (2023)
Abstract
The Hölder–Morrey spaces \(\Lambda _{\kappa}^{p,\tau}(\Omega ,\wedge ^{l})\) are proposed in this paper. The imbedding inequalities for homotopy operator are derived in Hölder–Morrey spaces on differential forms. The Hölder continuity for Riesz potential with envelope function is deduced. As application, some composite theorems, which are associated with conjugate A-harmonic equations on differential forms, are given.
1 Introduction
The purpose of this paper is to study the imbedding theory for homotopy operator in Hölder–Morrey spaces on differential forms. We first give a new definition for the Hölder–Morrey spaces \(\Lambda _{\kappa}^{p,\tau}(\Omega ,\wedge ^{l})\) that is based on the classical Hölder theory and Sobolev spaces \(W^{p,\kappa}(\Omega ,\wedge ^{l})\) on differential forms. The theory of Hölder continuity of order \(\kappa \in (0,1]\) has been widely used in various fields such as partial differential equations, harmonic analysis, and basic geometrical theory, see [1–3].
The concept of Morrey space was proposed in 1938, and many scholars have conducted in-depth research on it. Morrey space theory was extended to generalized Morrey spaces, weighted Morrey spaces, variable exponent Morrey spaces, weak Morrey spaces, and so on, see [4–7]; with the deepening of research, to composite spaces of Morrey spaces such as Morrey–Herz spaces and Orlicz–Morrey spaces, see [8, 9]. The concept of Hölder–Morrey spaces we present is in Sect. 2. Furthermore, the theory of envelope functions is introduced in the same section, and we give the definition for a class of admissible envelope functions. Envelope functions have been widely studied in super-lattice band structure, medical science, and nonlinear periodic structure, see [10–12]. In the present paper, the real envelope function \(\lambda :\mathbb{R}^{n}\rightarrow \mathbb{R}\) satisfies the condition of locally Hölder-continuous in \(L^{p,\tau}\)-norm.
Homotopy operator T is a key tool used in the decomposition theorem and Poincaré-type inequality on differential forms, see [13, 14] for more details. Homotopy operator T on differential forms is a linear mapping from \(\wedge ^{l}(\mathbb{R}^{n})\) to \(\wedge ^{l-1}(\mathbb{R}^{n})\). In fact, by using the Poincaré lemma in [15], we obtain
for a closed form u on Ω. The primary properties for homotopy operator T on differential forms can be found in [15].
In this paper, we introduce the definition of Hölder–Morrey spaces and envelope functions on differential forms in Sect. 2. The imbedding inequality for homotopy operator T and Poincaré-type inequality are given in Sect. 3. In Sect. 4, the Hölder continuity for Riesz potential operator with envelope functions is derived. Finally, some estimates that are closely related to homotopy operator applied to the solutions of conjugate A-harmonic equations on differential forms are obtained.
2 Preliminary
Before stating our main results, we introduce some notations and basic theory for differential forms.
Let \(\Omega \subset \mathbb{R}^{n}\) be a bounded, convex domain, \(n\geq 2\) and an l-form u be a locally integrable function on Ω with values in \(\wedge ^{l}(\mathbb{R}^{n})\). If \(u(x)\in \wedge ^{l}(\mathbb{R}^{n})\), then the value of \(u(x)\) at the vectors \(\xi _{1},\ldots ,\xi _{l}\in \mathbb{R}^{n}\) is denoted by \(u(x)(\xi _{1},\ldots ,\xi _{l})=u(x;\xi _{1},\ldots ,\xi _{l})\). For more details on differential forms, see [16].
Arbitrary l-form \(u:\Omega \rightarrow \wedge ^{l}(\mathbb{R}^{n})\) can be denoted as \(u(x)=\sum_{I}u_{I}(x)\,dx_{I}\), and each \(u_{I}\) is the coefficient function. Moreover, u is called a differential form, the coefficient functions \(u_{I}\) are differentiable, see [17, 18]. The operator \(\star :\wedge ^{l}(\mathbb{R}^{n})\rightarrow \wedge ^{n-l}( \mathbb{R}^{n})\) is the Hodge-star operator, and the linear operator \(d:D'(\Omega ,\wedge ^{l})\rightarrow{D'(\Omega ,\wedge ^{l+1})}\), \(0\leq l\leq n-1\), is called the exterior differential. \(d^{\star}: D'(\Omega ,\wedge ^{l+1})\rightarrow{D^{\prime }(\Omega , \wedge ^{l})}\) denotes the Hodge codifferential operator, which is the formal adjoint of d.
Denote by \(L^{p}(\Omega ,\wedge ^{l})\) the space of differential l-forms satisfying \(\int _{\Omega}|u_{I}|^{p} <\infty \) with the norm
Definition 2.1
(Hölder space)
If \(1< p<\infty \) and \(\kappa \in (0,1]\). Let \(\Lambda ^{p,\kappa}(\Omega ,\wedge ^{l})\) denote the Hölder space of all differential l-forms u whose norm
The Hölder space on differential forms will reduce to the classical type when \(l=0\).
Clearly, Sobolev spaces \(W^{p,\kappa}(\Omega ,\wedge ^{l})\) in [19] could replace Hölder spaces in what follows, but we selected Hölder spaces because they are slightly more elementary.
The following definition is contributed in Sect. 4.
Definition 2.2
(Envelope function)
A real envelope function is a function \(\lambda :\mathbb{R}^{n}\rightarrow \mathbb{R}\) such that for every \(1< p<\infty \) there exists \(\kappa \in (0,1)\) such that \(\lambda \in \Lambda ^{p,\kappa}(\mathbb{R}^{n})\). Thus, the space of all envelope functions is
In this paper, we consider that each envelope function \(\lambda \in \Lambda \) satisfies the condition: there exists a constant \(C_{1}\geq 1\) such that
Remark 1
Any function in the Schwartz class is an envelope function, and the product of two envelope functions \(\lambda _{1},\lambda _{2}\in \Lambda \) is again an envelope function.
Let \(u\in L_{\mathrm{loc}}^{1}(\Omega ,\wedge ^{l})\) be a differential l-form on Ω. For \(1\leq p<\infty \), \(0<\tau \leq n\), and \(r>0\), we denote the Morrey space on differential forms by \(L^{p,\tau}(\Omega ,\wedge ^{l})\) with the norm
where \(B(x,r)\subset \Omega \) is an open ball centered at x of radius r.
Easily we see that \(L^{p,\tau}(\Omega ,\wedge ^{l})\) is an expansion of \(L^{p}(\Omega ,\wedge ^{l})\) in the sense that \(L^{p,n}(\Omega ,\wedge ^{l})=L^{p}(\Omega ,\wedge ^{l})\). If \(l=0\), then the Morrey space on differential forms reduces to the classical Morrey space, which was first introduced by C. Morrey in [20]. In recent years, Morrey spaces and generalized Morrey spaces have received much investigation. The boundedness for Hardy–Littlewood maximal function and singular integrals in Morrey spaces were given in [21–23].
Definition 2.3
(Hölder–Morrey space)
Assume \(1\leq p<\infty \), \(0<\tau \leq n\), and \(\kappa \in (0,1]\). Let \(\Lambda ^{p,\tau}_{\kappa}(\Omega ,\wedge ^{l})\) denote the Hölder–Morrey space of all differential l-forms u whose norm
We choose \(l=0\) and \(\tau =n\), then \(\Lambda ^{p,\tau}_{\kappa}(\mathbb{R}^{n},\wedge ^{l})\) coincides with \(\Lambda ^{p,\kappa}(\mathbb{R}^{n})\). Furthermore, the space
denotes all envelope functions with Morrey norms. Recall some properties of BMO spaces and integral average on differential forms. For \(b\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\), let
where
Define \({\mathrm{BMO}}(\mathbb{R}^{n})=\{b\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n}): \|b\|_{ \ast}<\infty \}\). Analogically, we show integral average on differential forms. The following lemma appeared in [24].
Lemma 2.4
Let Ω be a bounded, convex domain in \(\mathbb{R}^{n}\). To each \(z\in \Omega \), there corresponds a linear operator \(K_{z}:C^{\infty}(\Omega ,\wedge ^{l})\rightarrow C^{\infty}(\Omega , \wedge ^{l-1})\) defined by
and the decomposition
holds at any point z in Ω.
Assume that \(\varphi \in C_{0}^{\infty}(\Omega )\) is a test function and satisfies \(\int _{\Omega}\varphi (z)\,dz=1\). We define \(T : L^{p}(\Omega ,\wedge ^{l})\rightarrow L^{p}(\Omega ,\wedge ^{l-1})\) is a homotopy operator by averaging \(K_{z}\) in Ω:
It is obvious that
Then we denote the integral average of u over Ω by \(u_{\Omega}=d(Tu)\) for \(l=1,2, \ldots , n\). If \(l=0\), then the differential form u reduces to the function type and integral average \(u_{\Omega}={\frac{1}{|\Omega |}}\int _{\Omega }u(y)\,dy\). Clearly, we see that
From Poincaré lemma in [15], \(u_{\Omega}\) is a closed form. Suppose that u is a differential l-form in \(L_{\mathrm{loc}}^{1}(\Omega ,\wedge ^{l})\), \(l=0,1,2,\ldots,n\), and for \(1< p<\infty \), \(0<\tau \leq n\). Let \({\mathrm{BMO}}^{p,\tau}(\Omega ,\wedge ^{l})\) be a space on differential form with the finite norm
where \(u_{B}\) coincides with (2.3). If \(l=0\) and \(\tau =0\), then \({\mathrm{BMO}}^{p,\tau}(\Omega ,\wedge ^{l})\) spaces reduce to the classical BMO spaces.
3 Imbedding inequalities for homotopy operator
In this section, we are concerned with the imbedding inequality for homotopy operator. The following lemma appeared in [13, 14].
Lemma 3.1
Let u be a differential l-form on Ω, \(l=0,1,\ldots ,n\). Let \(T : L^{p}(\Omega ,\wedge ^{l})\rightarrow L^{p}(\Omega ,\wedge ^{l-1})\) be a homotopy operator and \(\Omega \subset \mathbb{R}^{n}\) be a bounded, convex domain. Then there exists a constant C, independent of u, such that
where \(B:=B(x,r)\subset \Omega \) is an open ball centered at x of radius r.
Now, we give the imbedding inequality for homotopy operator on differential forms.
Theorem 3.2
Let \(u\in \Lambda _{\kappa}^{p,\tau}(\Omega ,\wedge ^{l})\) be a differential l-form on Ω, \(l=1,2,\ldots ,n\), \(0<\tau \leq n-p\), and \(\kappa \in (0,1]\). Let also \(T:L^{p}(\Omega ,\wedge ^{l})\rightarrow L^{p}(\Omega ,\wedge ^{l-1})\) be a homotopy operator. Then there exists a constant C, independent of u, such that
Proof
For arbitrary \(y\in \Omega \), we have
By using Lemma 3.1, we get the first part of the right-hand side for (3.1)
For any \(\xi _{1},\ldots ,\xi _{l-1}\in \mathbb{R}^{n}\), we denote \(\boldsymbol{\xi}=(\xi _{1},\ldots ,\xi _{l-1})\) for notational simplicity. Then we deduce that
Note
where \(\boldsymbol{\theta}=(\theta _{1},\ldots ,\theta _{l-1})\) and each \(\theta _{i}\) for \(i=1,2,\ldots ,l-1\) is a 0-vector in \(\mathbb{R}^{n}\). For (3.2), we only consider the condition for \(l=1\). In fact, if \(l>1\), then the determinant of \((x-y,\boldsymbol{\theta})=0\). For \(l=1\) and each \(u_{i}\) is a locally integrable function, we obtain
For \(\mathcal{T}_{2}\), we have
Combining (3.2) and (3.3), we obtain
Collecting the above facts, we deduce that
The proof of Theorem 3.2 has been completed. □
Next, we show the Poincaré-type inequality in Hölder–Morrey spaces on differential forms.
Theorem 3.3
Let \(u\in \Lambda _{\kappa}^{p,\tau}(\Omega ,\wedge ^{l})\) be a differential l-form and \(du\in \Lambda _{\kappa}^{p,\tau}(\Omega ,\wedge ^{l+1})\), \(l=1,2,\ldots ,n-1\), \(0<\tau \leq n-p\). Then there exists a constant C, independent of u, such that
where B is any ball in Ω.
Proof
Using Lemma 3.1, we get
Note that
From Lemma 2.4, we have
and
where \(u'(x):\mathbb{R}^{n}\rightarrow \wedge ^{l}(\mathbb{R}^{n})\) is a derivative mapping. Moreover, \(u'(x)\xi _{i}\) is an l-linear antisymmetric function on \(\mathbb{R}^{n}\times \cdots \times \mathbb{R}^{n}\) for each \(\xi _{i}\in \mathbb{R}^{n}\). We first estimate \(\mathcal{K}_{1}-\mathcal{K}_{3}\):
Similarly, for \(\mathcal{K}_{2}-\mathcal{K}_{4}\), we obtain
Combining (3.5) and (3.6), we deduce that
Finally, we obtain
The proof of Theorem 3.3 has been completed. □
Remark 2
Replacing \(u\in L^{p}_{\mathrm{loc}}(\Omega ,\wedge ^{l})\) with the solution for A-harmonic equation on differential forms in [25, 26]
we can get the high order Poincaré-type estimate with Hölder norm by using the reverse Hölder inequality
for \(\sigma B\subset \Omega \) with \(\sigma >1\).
Next, we show the vectorial differential forms
where each \(u_{i}\) is a differential \(l_{i}\) form for \(i=1,2,\ldots ,k\) and all \(l_{i}\) satisfy the condition \(1\leq l_{1}\leq \cdots \leq l_{k}\leq n\), \(\sum_{1}^{k}l_{i}\leq n\). We denote the spaces of all vectorial differential forms by \(\mathcal{L}^{P}(\Omega ,\wedge ^{l_{1}}\times \cdots \times \wedge ^{l_{k}})\) with the norm
where \(P=(p_{1},p_{2},\ldots ,p_{k})\) and each \(p_{i}\) belongs to \((1,\infty )\). Analogically, we denote the vectorial Hölder–Morrey space by \(\Lambda _{\kappa}^{P,\tau}(\Omega ,\wedge ^{l_{1}}\times \cdots \times \wedge ^{l_{k}})\) with the finite norm
Then we can obtain the following corollary.
Corollary 3.4
Suppose that \(U=(u^{1},u^{2},\ldots ,u^{k})\) is a vectorial differential form on Ω. Each \(u^{i}\) belongs to \(\Lambda ^{p_{i},\tau}_{\kappa}(\Omega ,\wedge ^{l_{i}})\) with \(P=(p_{1},\ldots ,p_{k})\) and each \(du^{i}\) belongs to \(\Lambda ^{q_{i},\tau}_{\kappa}(\Omega ,\wedge ^{l_{i}+1})\) with \(Q=(q_{1},\ldots ,q_{k})\), \(l=l_{1}+\cdots +l_{k}\). \(P=(p_{1},\ldots ,p_{k})\) and \(Q=(q_{1},\ldots ,q_{k})\) satisfying \(\max \{1,np_{i}/(n+p_{i})\}\leq q_{i}\leq p_{i}\) for \(i=1,2,\ldots ,k\). Let \(T^{k}=T\times \cdots \times T\) be a k-order vectorial homotopy operator and \(T^{k}U=(Tu^{1},\ldots ,Tu^{k})\). Then
(i) There exists a constant \(C>0\) independent of u such that
where \(\tau +P=:(p_{1}+\tau ,\ldots ,p_{k}+\tau )\).
(ii) There exists a test form \(\eta \in C^{\infty}_{0}(\Omega ,\wedge ^{n-l})\) that satisfies \(\|\eta \|_{\infty}\leq 1\) and \(\|d\eta \|_{\infty}\leq 1\) with \(l=\sum_{1}^{k}l_{i}\) such that
4 Hölder continuity with envelope function
For \(0<\alpha <n\) and a differential l-form u on Ω, we define Riesz potential operator \(I_{\alpha }u\) of order α by
The operator \(I_{\alpha }u\) will reduce to function type if \(l=0\). The boundedness and Sobolev-type imbedding inequality in classic Morrey spaces for Riesz potential operator were given by Adams and Xiao in [27, 28]. The Hölder continuity for \(I_{\alpha}\) with envelope functions is as follows.
Theorem 4.1
Suppose that \(1< p< q<\infty \), \(\tau \in (\alpha +\kappa -1,\alpha +\kappa )\) and \(\tau \in (\alpha p-np/q,\alpha q-nq/p)\), \(h=|x-z|\). Let \(u\in L^{1}_{\mathrm{loc}}(\Omega ,\wedge ^{l})\) be a differential l-form and \(\lambda (\cdot )\in \Lambda ^{\tau}\) be an envelope function. Then there exists a constant \(C>0\) such that
for all differential l-forms \(u\in L^{p}_{\mathrm{loc}}(\Omega ,\wedge ^{l})\) with \(\|u\|_{L^{p,\tau}(\Omega ,\wedge ^{l})}\leq 1\).
Proof
Since \(\lambda (\cdot )\in \Lambda ^{\tau}\) and each \(\lambda (\cdot )\) on \(\mathbb{R}^{n}\) satisfies condition (2.1), we get
where C is a constant independent of λ. Assume that each \(u_{I}\) for \(u=\sum_{I}u_{I}\,dx_{I}\) is nonnegative on Ω and \(\|u\|_{L^{p,\tau}(\Omega ,\wedge ^{l})}\leq 1\). Let \(B(x,h)\subset \Omega \) be an open ball centered at x of radius h. Then we have
We first estimate \(\mathcal{S}_{1}\)
Hence
where \(\alpha +\kappa -\tau >0\). For \(\mathcal{S}_{12}\),
where \(\alpha -\tau /q-n/p>0\). Similar to \(\mathcal{S}_{1}\), we omit the calculation of \(\mathcal{S}_{2}\). For \(\mathcal{S}_{3}\), we have
since \(\alpha -\tau /p-n/q<0\). Next, we estimate \(\mathcal{S}_{4}\):
Then
for \(\alpha +\kappa -\tau -1<0\) and
for \(\alpha -\tau /p-n/q-1<0\).
Let \(B(x_{0},r_{0})\supset \Omega \) centered at \(x_{0}\) and radius of \(r_{0}\). Then \(r_{0}\) is a finite constant independent of x. Combining all the above facts, we deduce that
where C is a constant independent of x and z. The proof of Theorem 4.1 has been completed. □
The inequality for composite operator with Morrey norms on differential forms is as follows.
Theorem 4.2
Let \(u\in \Lambda ^{p,\tau}_{\kappa}(\Omega ,\wedge ^{l})\) be a differential l-form and \(du\in \Lambda ^{p,\tau}_{\kappa}(\Omega ,\wedge ^{l+1})\), \(l=0,1,\ldots ,n-1\), \(h=|x-z|\), let \(T:L^{p}(\Omega ,\wedge ^{l})\rightarrow L^{p}(\Omega ,\wedge ^{l-1})\) be a homotopy operator and \(\lambda (\cdot )\in \Lambda ^{\tau}\). Then there exists a constant C, independent of u, such that
where B is any ball in Ω.
Proof
Assume that \(l\geq 1\), arbitrary \(x_{0}\in \Omega \), and \(B(x_{0},r)\subset \Omega \) is any ball centered at \(x_{0}\) and radius of r. Then we have
\(|\lambda (x)I_{\alpha}(du(x))-\lambda (z)I_{\alpha}(du(z))|\) is expressed by
Assume that each \({\frac{\partial u_{I}(y)}{\partial x_{k}}}=v_{I}(x)\) is a coefficient function of differential \((l+1)\) form du. By using Theorem 4.1, we obtain (4.1) holds. □
5 Homotopy operator and A-harmonic equations
As applications for homotopy operator on differential forms, we study the solutions for nonhomogeneous A-harmonic equation on differential forms
where \(A:\Omega \times \wedge ^{l}(\mathbb{R}^{n})\rightarrow \wedge ^{l}( \mathbb{R}^{n})\) and g, h are differential l-forms on Ω. To study the property of (5.1), we need the following conditions for operator A:
and
where arbitrary \(x\in \Omega \), \(\zeta \in \wedge ^{l}(\mathbb{R}^{n})\) and a is a positive constant. In the present paper, we choose \(g=h=0\) in (5.1), then we get the nonhomogeneous conjugate A-harmonic equation
If u, v is a pair of solutions for equation (5.2), then we say u and v are a pair of conjugate A-harmonic tensors. Furthermore, u was called homogeneous A-harmonic tensor if u is a solution for homogeneous A-harmonic equation
The following lemma appeared in [24].
Lemma 5.1
Let u and v be a pair of solutions for equation (5.1) on Ω. If \(g\in L^{p}(B,\wedge ^{l})\) and \(h\in L^{q}(B,\wedge ^{l})\), then \(du\in L^{p}(B,\wedge ^{l})\) if and only if \(d^{\star }v\in L^{q}(B,\wedge ^{l})\). Moreover, we have the following inequalities:
for \(C_{1}\) and \(C_{2}\) are two constants independent of u and v.
Now, we show the Caccioppoli-type estimate for homotopy operator with \(L^{p,\tau}(\Omega ,\wedge ^{l})\)-norms.
Theorem 5.2
Assume that \(1<\tau ,\upsilon <n\) and \(1< p<\infty \). Let u and v be a pair of conjugate A-harmonic tensors on Ω and \(T:L^{p}(\Omega ,\wedge ^{l})\rightarrow L^{p}(\Omega ,\wedge ^{l-1})\) be a homotopy operator. Then there exist a constant \(C>0\), independent of u and v, such that
where c is any closed form on Ω and \(\upsilon =\tau -p+q\).
Proof
From Theorem 2.9 in [29], we have the following Caccioppoli inequality in \(L^{p}\)-spaces:
for a homogeneous A-harmonic tensor u, B with σB in Ω and all closed forms c. Since u and v are a pair of conjugate A-harmonic tensors on Ω with \(g=h=0\), by using Lemma 5.1, we have
Hence
Clearly, \(d^{\star}=(-1)^{nl+1}\star d\star u\) and \(\star :\wedge ^{l}\rightarrow \wedge ^{n-l}\) is an isometric isomorphism mapping. Combining all the facts, we obtain
for all B with \(\sigma B\subset \Omega \) and \(\upsilon =\tau -p+q\). The proof of Theorem 5.2 has been completed. □
Next, we give the Poincaré-type estimate for homotopy operator with \(L^{p,\tau}(\Omega ,\wedge ^{l})\)-norms.
Theorem 5.3
Assume that \(1<\tau ,\upsilon <n\) and \(1< p<\infty \). Let u and v be a pair of conjugate A-harmonic tensors on Ω and \(T:L^{p}(\Omega ,\wedge ^{l})\rightarrow L^{p}(\Omega ,\wedge ^{l-1})\) be a homotopy operator. Then there exists a constant \(C>0\), independent of u and v, such that
where c is any closed form on Ω and \(\upsilon =\tau +p\).
Proof
Let u and v be a pair of conjugate A-harmonic tensors on Ω and \(g=h=0\). Using the second relation of Lemma 5.1, we have
By using Lemma 3.1, we have
for \(\tau =\upsilon +p\) and any ball B in Ω. Collecting these facts, we obtain
The proof of Theorem 5.3 has been completed. □
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This paper is supported by the Fundamental Research Funds for the Central Universities (2572022BC05) and National Natural Science Foundation of China (12001088).
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Li, X., Wang, J. & Pan, N. Inequalities for integral operators in Hölder–Morrey spaces on differential forms. J Inequal Appl 2023, 71 (2023). https://doi.org/10.1186/s13660-023-02977-3
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DOI: https://doi.org/10.1186/s13660-023-02977-3
Keywords
- Hölder–Morrey spaces
- Homotopy operator
- Riesz potential
- Differential forms