ISOPERIMETRIC TYPE INEQUALITIES FOR DIFFERENTIAL FORMS ON MANIFOLDS

Flavia Giannetti * Antonia Passarelli di Napoli *

*Dipartimento di Matematica e Applicazioni ”R. Caccioppoli” Università di Napoli “Federico II”, via Cintia - 80126 Napoli

e-mail:giannett@unina.it antonia.passarelli@unina.it

ABSTRACT. Let X be a smooth oriented Riemannian n-manifold without boundary and (Φ, Ψ) ∈ L^{p}(∧^{l}X) × L^{r}(∧^{n−l}X) , \frac{1}{p} + \frac{1}{r} = 1 + \frac{1}{n} , be a pair of closed differential forms. We prove an isoperimetric type inequality for such differential forms under suitable assumptions. As an application we derive Hölder continuity for solutions of Hodge systems.

AMS Classifications. 58A10; 58A14 ; 58J99.

Key words. Wedge Products; Isoperimetric Inequality; Hölder continuity.

1 Introduction

The familiar geometric form of the isoperimetric inequality reads as

n^{n−1}ω_{n−1}|U|^{n−1} ⩽ |∂U|^{n}

where |U| stands for the volume of a region U ⊂ IR^{n}, while |∂U| is its (n−1)- dimensional surface area. Now, if f : R → U is a smooth diffeomorphism of a “regular” domain Q ⊂ IR^{n} onto U then |U| = ∫_{Q} J (x, f)dx, where J (x, f) is the Jacobian determinant of Df, while |∂U| is dominated by ∫_{∂Q} |Df|^{n−1}dx or ∫_{∂Q} |D^{♯}f|dx. Thus we obtain the following integral form of the isoperimetric inequalities

| ∫_{Q} J (x, f)dx| ⩽ C(n)( ∫_{∂Q} |Df|^{n−1}dx)^{\frac{n}{n−1}}  (1)

and

| ∫_{Q} J (x, f)dx| ⩽ C(n)( ∫_{∂Q} |D^{♯}f|dx)^{\frac{n}{n−1}} ,  (2)

valid also for all smooth mappings f : Q → IR^{n}, not necessarily diffeomorphisms. Reshetnyak [13] established sharp Hölder continuity for mappings of bounded distortion extending ideas of Morrey [7]. This required the isoperimetric inequality (2) for mappings in the Sobolev class W^{1,n}. This inequality has later been explored in nonlinear elasticity [8],[9]. Let us note that given a mapping f : IR^{n} → IR^{n} in W^{1,n}, we have two vector fields associated with f. The first one is curl free while the second is divergence free. These are, respectively

E = ∇f^{1}  and  B = (B^{1}, …, B^{n})

where

B^{j} = (−1)^{j+1} \frac{∂(f^{2}, …, f^{n})}{∂(x^{1}, …, x^{j−1}, x^{j+1}, …, x^{n})}

Their scalar product ⟨B, E⟩, which equals exactly the Jacobian determinant of Df, is integrable for f ∈ W^{1,n}(IR^{n}).

Now, any pair (B, E) of vector fields with div B=0 and curl E=0 will be referred to as div-curl couple. By analogy to the previous discussion the scalar product J = ⟨B, E⟩ will be called the Jacobian of (B, E).

Recently [1] an isoperimetric inequality of type (1) for div-curl couples (B, E) ∈ L^{p} × L^{q}(IR^{n} × IR^{n}) has been proved, under the assumption that p, q are Hölder conjugate exponents. This makes their Jacobian integrable.

As a consequence the authors derived a Hölder continuity property of p-harmonic functions.

Actually, local integrability of the Jacobian of an orientation preserving mapping f was established under somewhat weaker assumption, i.e. f ∈

W^{1, \frac{n^{2}}{n+1}} (IR^{n}) and

lim inf_{t→∞} t^{\frac{n+1}{n}} ∫_{|Df|>t} |Df(x)|^{\frac{n^{2}}{n+1}} dx = 0 .

Later [11] an isoperimetric inequality of the type (2) for such mappings f has also been proved.

In this spirit one might try to obtain an isoperimetric type inequality for a wedge product of closed differential forms on a smooth Riemannian manifold analogous to that in [1]. However, attempts to weaken the assumptions on the integrability of the forms fail when the wedge product doesn’t obey the rule of integration by parts. Actually, this problem translates into the question how the distributional wedge product relates to the pointwise product.

In this paper we were able to identify these two quantities. Then we proved an isoperimetric type inequality under assumptions similar to those in [3].

Let X be a smooth oriented Riemannian n-manifold without boundary. A pair of closed differential forms Φ ∈ L^{p}(∧^{l}X)∩ ker d and Ψ ∈ L^{r}(∧^{n−l}X)∩ ker d, where 1 < p, r < ∞ satisfy Sobolev’s relation \frac{1}{p} + \frac{1}{r} = 1 + \frac{1}{n}, will be called an admissible pair if Φ ∧ Ψ ≥ 0^{1} and

lim inf_{t→∞} t^{\frac{1}{n}} ∫_{H>t} H(x)dx = 0  (3)

where H = |Φ|^{p} + |Ψ|^{r}.

The reader may wish to notice that this equation is automatic if H ∈ L^{\frac{n+1}{n}} (X). However, there are many interesting cases in which (3) holds for H ∉ L^{\frac{n+1}{n}} (X)

Our main result is contained in the following

Theorem 1.1. Let (Φ, Ψ) be an admissible pair. Given x ∈ X, for almost every ball B = B(x, ρ) in X we have

∫_{B} Φ ∧ Ψ << ( ∫_{∂B} |Φ|^{p}dH^{n−1})^{\frac{1}{p}} ( ∫_{∂B} |Ψ|^{s}dH^{n−1})^{\frac{1}{s}}  (4)

provided 1 < s = \frac{p(n−1)}{np−n+1} and 0 < ρ ≤ R_{X}, where R_{X} depends only on X.

The symbol << means that inequality (4) holds with a constant depending on X. As a first consequence we obtain an isoperimetric inequality of the type (1) when the target manifold has a nontrivial cohomology H^{l}(Y), for some 1 < l < n − 1. Secondly, as in [1], we derive a Hölder continuity property for solutions of Hodge systems.

1Throughout this text we fix an orientation of X, that is a nonvanishing n-form dx ∈ C^{∞}(∧^{n}X). We then say that an n-form Θ on X is nonnegative if Θ = λdx for some nonnegative function λ.

2 Preliminaries

2.1 Differential forms

Our ambient space is an oriented compact (without boundary) smooth Riemannian manifold X of dimension n > 2. Fix an atlas A consisting of a finite number of coordinate charts (Ω_{i}, ϑ_{i}), where ϑ_{i} : Ω_{i} → IR^{n} is a C^{∞}- diffeomorphism of the open region Ω_{i} ⊂ X onto IR^{n} and choose a C^{∞}- partition of unity {χ_{i} ∈ C_{0}^{∞}(Ω_{i}) : ∑ χ_{i} = 1} subordinate to A. Introduce a radius R = R_{X} such that all concentric balls B(x, ρ) ⊂ B(x, 3ρ) ⊂ X, with ρ ≤ R lay in the same coordinate region Ω_{i} of the fixed atlas A. Such balls share many properties of the Euclidean balls.

Let us denote by C^{∞}(∧^{l}X), 1 ≤ l ≤ n = dimX, the space of smooth l-forms on X. Consider the exterior derivative

d : C^{∞}(∧^{l}X) → C^{∞}(∧^{l+1}X)

and its formal adjoint

d^{∗} = (−1)^{nl+1} ∗ d∗ : C^{∞}(∧^{l+1}X) → C^{∞}(∧^{l}X)

where ∗ : C^{∞}(∧^{l}X) → C^{∞}(∧^{n−l}X) denotes the Hodge star duality operator. The duality between d and d^{∗} is emphasized in the integration by parts

∫_{X}⟨dφ, ψ⟩ = ∫_{X}⟨φ, d^{∗}ψ⟩

for φ ∈ C^{∞}(∧^{l}X) and ψ ∈ C^{∞}(∧^{l+1}X).

We denote by L^{p}(∧^{l}X) the space of all measurables l-forms ω for which

||ω||_{p} = ( ∫_{X} |ω|^{p})^{\frac{1}{p}} < +∞ for 1 ≤ p < +∞

and

||ω||_{∞} = ess sup |ω| for p = ∞

In order to define the Sobolev space W^{1,p}(∧^{l}X), 1 ≤ p ≤ +∞, we decompose an l-form ω as ω = ∑ ω_{i} where ω_{i} = χ_{i}ω and consider the pullback of ω_{i} via the mapping ϑ_{i}^{−1} : IR^{n} → Ω_{i}

ω_{i}^{♯}(x) = ∑_{I} ω_{i}^{I}(x)dx^{I} x ∈ θ_{i}(Ω_{i}) ⊂ IR^{n}

The sum is taken over all ordered l-tuples I = (i_{1}, …, i_{l}) and dx^{I} = dx_{i_{1}} ∧ … ∧ dx_{i_{l}}. If each function ω_{i}^{I} has generalized partial derivative ∂ω_{i}^{I} ∈ L^{p} then the Sobolev norm of ω is defined by

||ω||_{1,p} = ∑_{i} ||ω_{i}||_{1,p}

A differential form φ ∈ L^{p}(∧^{l}X) is said to be closed (in the sense of distributions), in writing dφ = 0, if

∫_{X}⟨φ, d^{∗}ψ⟩ = 0

for every test form ψ ∈ C^{∞}(∧^{l+1}X). Similarly we define the coclosed forms, d^{∗}ψ = 0. Finally a form h ∈ L^{p}(∧^{l}X) which is both closed and coclosed is called a harmonic field of degree l. Note that the space of all harmonic fields of degree l is finite dimensional and consists of C^{∞}-smooth forms.

We shall use the following Hodge decomposition of a differential form ω ∈ L^{p}(∧^{l}X) (see [6]):

Theorem 2.1. For 1 < p < ∞, and 1 ≤ l ≤ n every ω ∈ L^{p}(∧^{l}X) can be decomposed as

ω = dα + d^{∗}β + h

where α, β are in the Sobolev classes W^{1,p}(∧^{l−1}X) , W^{1,p}(∧^{l+1}X) respectively and h is a harmonic field of degree l.

2.2 Weakly differentiable mappings

Our applications will be concerned with weakly differentiable mappings f : X → Y where X and Y are oriented smooth Riemannian manifolds of dimension n ≥ 2 without boundary. In order to define the Sobolev class W^{1,p}(X, Y) we appeal to the well known Theorem of J. Nash ([10]) which ensures that Y can be C^{∞}-isometrically imbedded in the euclidean space IR^{n}.

Namely, we say that f : X → IR^{n} lies in W^{1,p}(X, Y) if each coordinate function f^{i} : X → IR belongs to the usual Sobolev space W^{1,p}(X) and f(x) ∈ Y for almost every x ∈ X.

To every C^{∞}-smooth l-form ω in the target space there corresponds its pullback f^{♯}ω via f. The pointwise estimate

|f^{♯}ω| << |Df|^{l}

relates the degree of integrability of f^{♯}ω to the one of Df. Here and in what follows the symbol << means that the inequality holds with a constant depending on the manifold X. Note that the commutation rule

f^{♯} ∘ d = d ∘ f^{♯}

holds provided f : X → Y belongs to C^{∞}.

Given a function h ∈ L^{p}(X) define the maximal function of h as

M_{p}h(x) = sup \{(∫_{B} |h(x)|^{p}dx)^{\frac{1}{p}}; x ∈ B ⊂ X\}

where the supremun runs over all balls B ⊂ X containing x. As in the Euclidean setting, the following weak type estimate holds (see [4])

Proposition 2.2. For every h ∈ L^{p}(X) we have

∫_{M_{p}h>2t} dx << \frac{1}{t^{p}} ∫_{|h|>t} |h(x)|^{p}dx  (5)

for all t > 0.

As a consequence of the weak type estimate at (5) we obtain

Corollary 2.3. Let {h_{j}} converges to h in L^{p}(X), then

lim_{j→∞} ∫_{M_{p}h_{j}>λ} dx = ∫_{M_{p}h>λ} dx  (6)

for every λ > 0. Moreover {M_{p}h_{j}} contains a subsequence converging to M_{p}h almost everywhere.

3 Weak wedge products

Our intention here is to identify the distributional wedge product of closed forms with their pointwise product. As a matter of fact, a significant part of this section comes from [4]. Consider closed differential forms Φ ∈ L^{p}(∧^{l}X)∩ ker d and Ψ ∈ L^{r}(∧^{k}X)∩ ker d, where 1 ≤ k, l < n, k + l = n and 1 < p, r < ∞. First assume that p is Hölder conjugate to r. Thus Φ ∧ Ψ is integrable and, therefore, defines a Schwartz distribution of order zero

(Φ ∧ Ψ)[η] = ∫_{X} η(Φ ∧ Ψ)  for η ∈ C^{∞}(X)  (7)

But we can do much better if we use the following Hodge decomposition, see Theorem 2.1,

Φ = dφ + θ  (8)

where the exact component dφ ∈ L^{p}(∧^{l}X) and the harmonic field θ ∈ C^{∞}(∧^{l}X). Then, using integration by parts, we obtain the following formula

(Φ ∧ Ψ)[η] = ∫_{X} η(dφ ∧ Ψ) + ∫_{X} η(θ ∧ Ψ)
= ∫_{X} η(θ ∧ Ψ) − ∫_{X} dη ∧ (φ ∧ Ψ)  (9)

This latter integral converges whenever φ ∧ Ψ is integrable. Assume now that 1 < p, r < ∞ are Sobolev conjugate exponents, that is

\frac{1}{p} + \frac{1}{r} = 1 + \frac{1}{n}

This condition forces that one of the exponents is less than n, for instance 1 < p < n. Then, by Sobolev imbedding Theorem, we find that φ ∈ L^{\frac{np}{n−p}} (∧^{l−1}X), where the exponent \frac{np}{n−p} is exactly Hölder conjugate to r so that φ ∧ Ψ ∈ L^{1}(∧^{n−1}X).

In this way we are justified to give the definition of the distributional wedge product.

Definition 3.1. Let Φ ∈ L^{p}(∧^{l}X)∩ ker d and Ψ ∈ L^{r}(∧^{n−l}X)∩ ker d, where 1 < p, r < ∞ are Sobolev conjugate exponents. The distribution Φ∧Ψ operates on a test function η ∈ C^{∞}(X) by the rule

(Φ ∧ Ψ)[η] = lim_{j→∞} ∫_{X} η(Φ_{j} ∧ Ψ_{j})  (10)

where Φ_{j} ∈ C^{∞}(∧^{l}X) and Ψ_{j} ∈ C^{∞}(∧^{n−l}X) are closed forms converging to Φ and Ψ in L^{p}(∧^{l}X) and L^{r}(∧^{n−l}X), respectively.

We shall use the following two Lemmas.

Lemma 3.1. Let Φ ∈ C^{∞}(∧^{l}X) and Ψ ∈ C^{∞}(∧^{n−l}X) be closed differential forms and let 1 < p, r < ∞ be Sobolev conjugate exponents. Then for every nonempty open set Ω ⊂ X and every test function η ∈ C^{∞}(X) we have

|∫_{Ω} η(Φ ∧ Ψ)| << ∥η∥_{C^{∞}(Ω)}∥M_{p}Φ∥_{L^{∞}(X\Ω)}∥M_{r}Ψ∥_{L^{∞}(X\Ω)}  (11)

Lemma 3.2. Let Φ ∈ C^{∞}(∧^{l}X) and Ψ ∈ C^{∞}(∧^{n−l}X) be closed differential forms, let 1 < p, r < ∞ be Sobolev conjugate exponents and let λ > 0. Then, setting H = |Φ|^{p} + |Ψ|^{r}, we have

|(Φ ∧ Ψ)[η] − ∫_{X\Ω} η(Φ ∧ Ψ)| << λ^{1+ \frac{1}{n}} |Ω| ∥η∥_{C^{∞}(X)}  (12)

where Ω = {x ∈ X : MH(x) > λ}.

Proof: The left hand side of the inequality (12) reduces to

| ∫_{Ω} η(Φ ∧ Ψ)|  (13)

Using Lemma 3.1 and the definition of Ω we get

| ∫_{Ω} η(Φ ∧ Ψ)| << |Ω| ∥η∥_{C^{∞}(X)}∥M_{p}Φ∥_{L^{∞}(X\Ω)}∥M_{r}Ψ∥_{L^{∞}(X\Ω)}
<< |Ω| ∥η∥_{C^{∞}(X)}∥MH∥L^{∞}(X\Ω)^{\frac{1}{p}}∥MH∥L^{∞}(X\Ω)^{\frac{1}{r}}
<< |Ω| ∥η∥_{C^{∞}(X)}λ^{\frac{1}{p}} λ^{\frac{1}{r}} = |Ω| ∥η∥_{C^{∞}(X)}λ^{1+ \frac{1}{n}}  (14)

Now, we are in position to prove Theorem 3.3. Let (Φ, Ψ) be an admissible pair of differential forms. Then there are measurable sets X_{1} ⊂ X_{2} … ⊂ X whose union is X and Φ ∧ Ψ is integrable over each X_{j}, such that

(Φ ∧ Ψ)[η] = lim_{j→∞} ∫_{X_{j}} η(Φ ∧ Ψ)  (15)

for every η ∈ C^{∞}(X).

Proof: Observe that Φ and Ψ satisfy inequality (12) for all but a countable set of the parameters λ > 0. Indeed, there exist sequences of closed forms Φ_{j} ∈ C^{∞}(∧^{l}X) and Ψ_{j} ∈ C^{∞}(∧^{n−l}X) converging to Φ and Ψ in L^{p}(∧^{l}X) and L^{r}(∧^{n−l}X), respectively. We need only to justify a passage to the limit in the inequalities

|(Φ_{j} ∧ Ψ_{j})[η] − ∫_{MH_{j}≤λ} η(Φ_{j} ∧ Ψ_{j})| << λ^{1+ \frac{1}{n}} ||η||_{C^{∞}(X)} ∫_{MH_{j}>λ} dx (16)

Letting j → ∞ the integrals in the right hand side converge to ∫_{MH>λ} dx, by Corollary 2.3. Of course, (Φ_{j} ∧ Ψ_{j})[η] converges to (Φ ∧ Ψ)[η] by the definition of the distributional wedge product.

Moreover, since MH_{j} converges to MH a.e. in X, the characteristic functions Ξ_{j} of the sets {x ∈ X : MH_{j} ≤ λ} converge almost everywhere to the characteristic function Ξ of the set {x ∈ X : MH ≤ λ}. Therefore if we write the integral in the left hand side as

∫_{X} η(Φ_{j} ∧ Ψ_{j})Ξ_{j}

and observe that the integrands are uniformly bounded by λ^{1+ \frac{1}{n}} ||η||_{C^{∞}(X)}, we can pass to the limit thank to the Dominated Convergence Theorem.

Let us consider a sequence t_{j} increasing to infinity such that

lim inf_{j→∞} tj^{\frac{1}{n}} ∫_{H>t_{j}} H(x)dx = 0  (17)

and define the sets X_{j} as

X_{j} = {x ∈ X : MH ≤ 2t_{j}}

We make use of the estimate in Lemma 3.2 with λ = 2t_{j}

|(Φ ∧ Ψ)[η] − ∫_{X_{j}} η(Φ ∧ Ψ)| << ∥η∥_{C^{∞}(X)}t_{j}^{1+ \frac{1}{n}} ∫_{MH>2t_{j}} dx

<< ∥η∥_{C^{∞}(X)}tj^{\frac{1}{n}} ∫_{H>t_{j}} H(x)dx (18)

The latter inequality follows by Proposition 2.2. Letting t_{j} go to ∞, we conclude using (17).

Using a monotone convergence argument, from previous Theorem we obtain

Theorem 3.4. Let (Φ, Ψ) be an admissible pair of differential forms. Then

(Φ ∧ Ψ)[η] = ∫_{X} η(Φ ∧ Ψ)  (19)

for every η ∈ C^{∞}(X).

Fix a nonnegative η ∈ C^{∞}(IR^{n}) supported in the closed unit ball and having integral 1. We shall denote by η_{t}, t > 0, the usual sequence of mollifiers, i.e. η_{t}(x) = t^{−n}η(t^{−1}x). The convolution between Φ ∧ Ψ and η_{t} is defined by the rule

(Φ ∧ Ψ) ⋆ η_{t}(x) = ∑_{Ω_{i}∈A} χ_{i}(x) ∫_{Ω_{i}} η_{t}(ϑ_{i}(x) − ϑ_{i}(z))J(z, ϑ_{i}(x))Φ ∧ Ψ(z) (20)

where {χ_{i}, i ∈ I} is the partition of unity subordinate to the fixed atlas A consisting of the coordinate charts (Ω_{i}, ϑ_{i}).

Note that (Φ ∧ Ψ) ⋆ η_{t} converges to Φ ∧ Ψ in the sense of distributions, as t → 0.

Then if (Φ, Ψ) is an admissible pair of differential forms we have

(Φ ∧ Ψ)(x) = lim_{t→0}(Φ ∧ Ψ) ⋆ η_{t}(x)  (21)

A simple limiting argument implies the following rule of integration by parts for admissible pairs of differential forms

∫_{X} η(Φ ∧ Ψ) = ∫_{X} dη ∧ (φ ∧ Ψ) + ∫_{X} η(h ∧ Ψ)  (22)

where Φ = dφ + h.

4 Isoperimetric type inequalities

In the Euclidean case it is possible to derive isoperimetric type inequalities from estimates of the div-curl product ⟨B, E⟩ under suitable assumptions on the degree of integrability of B and E (see [1]). In our context this idea translates in Theorem 1.1, whose proof is the following

Proof: Since B has trivial cohomology, then every closed form Φ is also exact on B, i.e. there exists a (l − 1)-form φ such that dφ = Φ. Moreover, from the assumption on s we infer that 1 < p < n − 1. Then there exists a closed (l − 1)-form φ_{0} such that φ − φ_{0} ∈ W^{1,p^{∗}}(Λ^{l−1}X) and the following Sobolev-Poincaré inequality holds

( ∫_{∂B} |φ − φ_{o}|^{p^{∗}}dH^{n−1})^{\frac{1}{p^{∗}}} << ( ∫_{∂B} |dφ|^{p}dH^{n−1})^{\frac{1}{p}}  (23)

where p^{∗} = \frac{(n−1)p}{n−1−p} and H^{n−1} denotes the (n − 1)-dimensional measure on ∂B.

By the rule of integration by parts at (22) and the assumption Ψ ∈ ker d we have

∫_{B} Φ ∧ Ψ = ∫_{B} d(φ − φ_{o}) ∧ Ψ
= ∫_{B} d[(φ − φ_{o})Ψ]
= ∫_{∂B}(φ − φ_{o})Ψ  (24)

Now, since \frac{1}{p^{∗}} + \frac{1}{s} = 1, by Hölder’s inequality we conclude

∫_{B} Φ ∧ Ψ << ∫_{∂B} |φ − φ_{o}| |Ψ| dH^{n−1}
<< ( ∫_{∂B} |φ − φ_{o}|^{p^{∗}}dH^{n−1})^{\frac{1}{p^{∗}}} ( ∫_{∂B} |Ψ|^{s}dH^{n−1})^{\frac{1}{s}}
<< ( ∫_{∂B} |Φ|^{p}dH^{n−1})^{\frac{1}{p}} ( ∫_{∂B} |Ψ|^{s}dH^{n−1})^{\frac{1}{s}}  (25)

One can easily deduce that Corollary 4.1. Let (Φ, Ψ) be an admissible pair. Given x ∈ X, for almost every ball B = B(x, ρ) in X we have

∫_{B} Φ ∧ Ψ << ( ∫_{∂B} (|Φ|^{p} + |Ψ|^{s})dH^{n−1})^{\frac{n}{n−1}}  (26)

provided 1 < s = \frac{p(n−1)}{np−n+1} and 0 < ρ ≤ R_{X}.

Next we fix ω ∈ C^{∞}(∧^{n}Y) and a Sobolev mapping f ∈ W^{1, \frac{n^{2}}{n+1}} (X, Y). The pointwise Jacobian J_{ω}(x, f) is defined by the rule

J_{ω}(x, f)dx = f^{♯}ω

Suppose that the target manifold Y has one nontrivial cohomology H^{l}(Y), 1 < l < n − 1, then every n-form ω ∈ C^{∞}(∧^{n}Y) decomposes into wedge products of closed forms of degree smaller than n, namely

ω = ∑_{i=1}^{N} α_{i} ∧ β_{i}  degα_{i} > 1,  degβ_{i} > 1

These forms, once pulled back via the mapping f : X → Y, bring us to analogous decomposition of f^{♯}ω:

J_{ω}(x, f)dx = f^{♯}ω = ∑_{i=1}^{N} f^{♯}α_{i} ∧ f^{♯}β_{i}

see [4]. Therefore we obtain the following variant of Theorem 1.1 reminiscent to the classical isoperimetric inequality (1).

Corollary 4.2. Let f ∈ W^{1, \frac{n^{2}}{n+1}} (X, Y) be an orientation preserving mapping. If

lim inf_{t→∞} t^{\frac{n+1}{n}} ∫_{|Df|>t} |Df|^{\frac{n^{2}}{n+1}} = 0

then

∫_{B} J_{ω}(x, f)dx << ( ∫_{∂B} |Df|^{n−1}dH^{n−1})^{\frac{n}{n−1}}  (27)

for almost every ball B = B(x, ρ) in X with 0 < ρ < R_{X}.

Proof: We need only consider one wedge product ω = α ∧ β with α ∈ C^{∞}(∧^{l}X)∩ ker d and β ∈ C^{∞}(∧^{n−l}X)∩ ker d. Taking f^{♯}α and f^{♯}β for the closed forms Φ and Ψ in Theorem 1.1 and using the pointwise inequalities

|f^{♯}α| << |Df|^{l} |f^{♯}β| << |Df|^{n−l}

we get

∫_{B} J_{ω}(x, f)dx << ( ∫_{∂B} |f^{♯}α|^{p}dH^{n−1})^{\frac{1}{p}} ( ∫_{∂B} |f^{♯}β|^{s}dH^{n−1})^{\frac{1}{s}}
<< ( ∫_{∂B} |Df|^{pl}dH^{n−1})^{\frac{1}{p}} ( ∫_{∂B} |Df|^{s(n−l)}dH^{n−1})^{\frac{1}{s}}

In order to conclude it is sufficient to choose p = \frac{n−1}{l} and s = \frac{n−1}{n−l} in previous inequality. Note that this choice is legitimate thank to assumption 1 < l < n − 1.

Take n = 2l and denote by D^{l×l}f the matrix of all l × l-subdeterminants of Df. If H^{l}(Y) ≠ 0, the pointwise inequalities

|f^{♯}α| << |D^{l×l}f| |f^{♯}β| << |D^{l×l}f|

hold. The following estimate, reminiscent to the isoperimetric type inequality (2), can be obtained arguing as in the previous corollary.

Corollary 4.3. Let f ∈ W^{1, \frac{n^{2}}{n+1}} (X, Y) be an orientation preserving mapping. If

lim inf_{t→∞} t^{\frac{n+1}{n}} ∫_{|Df|>t} |Df|^{\frac{n^{2}}{n+1}} = 0

then

∫_{B} J_{ω}(x, f)dx << ( ∫_{∂B} |D^{\frac{n}{2} × \frac{n}{2}} f|^{\frac{2(n−1)}{n}} dH^{n−1})^{\frac{n}{n−1}}  (28)

for almost every ball B = B(x, ρ) in X with 0 < ρ < R_{X}.

5 Some applications

As an application of the isoperimetric type inequality proved in Theorem 1.1, we give a Morrey’s type estimate for the wedge product of some special closed forms. Namely we have the following

Theorem 5.1. Let (Φ, Ψ) be a pair of closed differential forms in L^{q}(∧^{l}X)× L^{q′}(∧^{n−l}X), where 1 < q, q′ < ∞ are Hölder conjugate exponents. If there exists a constant K > 1 such that

|Φ|^{q} + |Ψ|^{q′} ≤ K(Φ ∧ Ψ)  (29)

then

∫_{B(0,ρ)} Φ ∧ Ψ ≤ (\frac{ρ}{R})^{\frac{1}{cnK}} ∫_{B(0,R)} Φ ∧ Ψ  (30)

for all 0 < ρ < R = R_{X}.

Remark that inequality (29) means that the nonnegative function λ such that the n-form Φ ∧ Ψ = λdx satisfies |Φ|^{q} + |Ψ|^{q′} ≤ Kλ.

Proof: Observe that the exponents p = \frac{q(n−1)}{n} and s = \frac{q′(n−1)}{n} are related by the equality s = \frac{p(n−1)}{np−n+1}. The isoperimetric type inequality (26) combined with assumption (29) gives

∫_{B(0,ρ)} Φ ∧ Ψ ≤ c_{n}( ∫_{∂B(0,ρ)} (|Φ|^{p} + |Ψ|^{s})dH^{n−1})^{\frac{n}{n−1}}

≤ c_{n}K( ∫_{∂B(0,ρ)}(Φ ∧ Ψ)^{\frac{n−1}{n}} )^{\frac{n}{n−1}}

≤ c_{n}K( ∫_{∂B(0,ρ)} dx)^{\frac{1}{n−1}} ∫_{∂B(0,ρ)} Φ ∧ Ψ
≤ c_{n}Kρ ∫_{∂B(0,ρ)} Φ ∧ Ψ  (31)

Setting

h(ρ) = ∫_{B(0,ρ)} Φ ∧ Ψ

a routine calculation shows that

h(ρ) ≤ (\frac{ρ}{R})^{\frac{1}{cnK}} h(R)

for ρ ≤ R, which concludes the proof.

Now we deduce the following Hölder continuity result.

Theorem 5.2. Let (Φ, Ψ) be a pair of closed differential forms in L^{q}(∧^{l}X)× L^{q′}(∧^{n−l}X), where 1 < q, q′ < ∞ are Hölder conjugate exponents. If there exists a constant K > 1 such that

|Φ|^{q} + |Ψ|^{q′} ≤ K(Φ ∧ Ψ)  (32)

then the (l-1)-form φ such that dφ = Φ is Hölder continuous, provided K < \frac{C_{n}}{n−q}.

Proof Inequality (29) yields

|dφ|^{q} ≤ K(Φ ∧ Ψ)

Then Theorem 5.1 gives

∫_{B(0,ρ)} |dφ|^{q}dH^{n} ≤ c_{n}K ∫_{B(0,ρ)} Φ ∧ Ψ
≤ c_{n}K(\frac{ρ}{R})^{\frac{1}{cnK}} ∫_{B(0,R)} Φ ∧ Ψ
= cρ^{\frac{1}{cnK}}  (33)

for all 0 < ρ < R = R_{X}. Therefore φ is Hölder continuous of exponent γ = 1 + \frac{1}{c_{n}Kq} − \frac{n}{q}.

Now, following the lead of the classical q-harmonic operator, 1 < q < ∞, we consider a bundle map

H_{q} : ∧^{l}X → ∧^{l}X

such that

|H_{q}(ξ)| ≤ M|ξ|^{q−1}
⟨H_{q}(ξ), ξ⟩ ≥ M^{−1}|ξ|^{q}

for every l-form ξ on X, where M ≥ 1 is a constant. Consider a pair (Φ, Ψ) ∈ L^{q}(∧^{l}X) × L^{q′}(∧^{l}X) such that dΦ = 0, d^{∗}Ψ = 0 and coupled by

Ψ = H_{q}(Φ)  (34)

We refer to (34) as a Hodge system. Existence and uniqueness of solutions to (34) has been studied in [5], [6]. Applying the operator d^{∗} to (34) we obtain the second order elliptic equation

d^{∗}H_{q}(Φ) = 0

One can easily check that the l-form Φ and the (n − l)-form Ψ satisfy inequality (29). From Theorem 5.2 it follows that the (l − 1)-form φ such that dφ = Φ is Hölder continuous.

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