1 Introduction

Let (M, g) be a compact Riemannian manifold of dimension n with a compatible almost complex structure. Denote by \({\mathcal {J}}_g\) the space of smooth almost complex structures compatible with g, i.e., \(g(J\cdot , J\cdot )=g(\cdot , \cdot )\). Consider the following functional, for all \(m\in {\mathbb {N}}^+\), \(J\in {\mathcal {J}}_{g}\),

$$\begin{aligned} {\mathcal {E}}_{m}(J) = \int _M \big |\varDelta ^{\frac{m}{2}} J \big |^2 dV :=\left\{ \begin{aligned}&\int _{M}|\nabla \varDelta ^{k-1}J|^{2}\,dV, \quad m=2k-1, \\&\int _{M}|\varDelta ^{k}J|^{2}\,dV, \qquad m=2k, \\ \end{aligned} \right. \end{aligned}$$
(1)

where \(\nabla \) and \(\varDelta \) are Levi-Civita connection and Laplace-Beltrami operator on (M, g), respectively, and dV denotes the volume element of (M, g). We call the critical points of functional \({\mathcal {E}}_{m}(J)\) m-harmonic almost complex structures. These objects are tensor-valued version of polyharmonic maps which have attracted quite some attention in recent years. When \(m=1\), the critical points of functional \({\mathcal {E}}_{m}(J)\) are also called harmonic almost complex structures introduced by Wood [17] in 1990s. We refer the reader to the recent survey [3] for the background and results in this subject. The first author have studied the existence and regularity of harmonic almost complex structures [7] from the point of view of geometric analysis. In this paper, we focus on the case of polyharmonic almost complex structures with \(m\ge 2\). Recall the definition of the Sobolev spaces of almost complex structures.

Definition 1

Suppose \((M^n,g)\) be an almost Hermitian manifold with compatible almost complex structures in \({\mathcal {J}}_g\). We define \(W^{k,p}({\mathcal {J}}_g)\) to be the closed subspace of \(W^{k,p}(T^*M \otimes TM)\) consisting of those sections \(J \in W^{k,p}(T^*M \otimes TM)\), which satisfy \(J^2=-id,\quad g(J\cdot , J\cdot )=g(\cdot , \cdot )\) almost everywhere.

Now, we state our main results.

  • Theorem 1 There always exists an energy-minimizer of \({\mathcal {E}}_{m}(J)\) in \(W^{m,2}({\mathcal {J}}_g)\).

  • Theorem 2 Suppose \(J\in W^{m,2}({\mathcal {J}}_{g})\) is a weakly m-harmonic almost complex structure on \((M^{2m}, g)\) with \(m\in \{2, 3\}\). Then J is Hölder-continuous.

For semilinear elliptic systems with critical growth nonlinearities, the most essential step towards the smoothness is to prove the Hölder continuity, such as the systems for (poly)harmonic maps, see for example [2, 5, 10, 15, 16] and references therein. It is well-known that a semilinear elliptic system with critical growth nonlinearities and at critical dimension might be singular [4, 9]. For weakly harmonic map, it can be even singular everywhere [13] when the dimension is three and above. The smooth regularity starts with Helein’s seminal result [10] for harmonic maps in dimension two where the special (algebraic) structure of the system plays a substantial role. New proofs and understanding of Helein’s seminal results can be found [1, 14]. The methods can be generalized to fourth order elliptic system in dimension four [2, 11]. General smooth regularity for biharmonic maps and polyharmonic maps have been obtained by [16] and [5], respectively.

We shall briefly compare our results with the results in the theory of (poly)harmonic maps. Theorem 1 is a standard practice in calculus of variations, while the main point is that the absolute energy-minimizer is not trivial due to its tensor-valued nature. The main result is to prove the Hölder regularity in Theorem 2 and our method is motivated by the work in [2] and [5]. In [2], the authors explore a special divergence structure of the biharmonic system into the spheres and our elliptic system shares some similarities. On the other hand, the tensor-valued nature makes our arguments much more complicated, mainly due to the fact that matrix multiplication is not commutative. We certainly believe that this divergence structure should hold for all weakly polyharmonic almost complex structures but we do not find a systematic way to argue that. Instead we only show that the elliptic system for polyharmonic almost complex structures has a desired divergence structure when \(m=2, 3\) by brutal computations. Given this divergence structure, our argument for Hölder regularity is quite different from the method used in [2], but more like a generalization of [5]. We use extension of maps (almost complex structures) instead of solving boundary value problem. Our methods are very general and work for all dimensions. A main difficulty is that the background metric is not necessarily Euclidean, while most results in the setting of polyharmonic maps (see [2, 5, 16] etc) only consider the Euclidean case. Even though the methods for semilinear system are expected to work similarly, the non-Euclidean background metric really leads to complicated computations and presentations. Once the Hölder regularity is assured, the proof of smoothness follows the strategy in [5].

The paper is organized as follows. In Sect. 2, we collect some facts for Lorentz spaces and Green’s functions. In Sect. 3, we establish the existence of the energy-minimizers and derive the Euler-Lagrange equations. Moreover we show that a weak limit of a sequence of weakly m-harmonic almost complex structures in \(W^{m,2}\) is still m-harmonic. In Sect. 4, we prove decay estimates for a class of semilinear elliptic equations in critical dimension and obtain the Hölder regularity of weakly m-harmonic almost complex structures on \((M^{2m},g)\) for \(m=2,3\). In Sect. 5, we generalize the higher regularity results of Gastel and Scheven [5] to prove smoothness of weakly m-harmonic almost complex structures. Appendix derive the divergence structures in detail for m-harmonic almost complex structures when \(m=2, 3\).

2 Preliminaries

In this section, we gather some facts that will be used later. First of all, let us denote by \(G(x)=c_m \ln |x|\) the fundamental solution for \(\varDelta ^m\) on \({\mathbb {R}}^{2m}\), where \(c_m\) is a suitable constant only dependent of m. We have the following lemma,

Lemma 1

Suppose \(k \in [1,2m]\) is a positive integer and \(p,q \in (1,\infty )\) satisfy

$$\begin{aligned} 1+\frac{1}{p}= \frac{k}{2m}+\frac{1}{q}. \end{aligned}$$

If \(f\in L^{q}({\mathbb {R}}^{2m})\), then we have

$$\begin{aligned} \bigg \Vert \int _{{\mathbb {R}}^{2m}} \nabla ^k G(x-y) f(y) dy \bigg \Vert _{L^p\left( {\mathbb {R}}^{2m}\right) } \le C \Vert f\Vert _{L^q\left( {\mathbb {R}}^{2m}\right) } \end{aligned}$$
(2)

where C is a positive constant only dependent of m, k, q.

Proof

Since \(\nabla ^{2m}G\) is a Calderón-Zygmund kernel, (2) holds for \(k=2m\) and all \(p=q\in (1,\infty )\). For \(k=1,\cdots ,2m-1\), we have

$$\begin{aligned} \nabla ^k G \in L^{\frac{2m}{k}, \infty }\left( {\mathbb {R}}^{2m}\right) \end{aligned}$$

where \(L^{\frac{2m}{k}, \infty }({\mathbb {R}}^{2m})\) is a Lorentz space. By the convolution inequality for Lorentz spaces (cf. [12] Theorem 2.6), we deduce that, for \(s\le p\)

$$\begin{aligned} \bigg \Vert \int _{{\mathbb {R}}^{2m}} \nabla ^k G(x-y) f(y) dy \bigg \Vert _{L^{p,p}\left( {\mathbb {R}}^{2m}\right) }&\le C \Vert \nabla ^k G\Vert _{L^{\frac{2m}{k},\infty }\left( {\mathbb {R}}^{2m}\right) } \Vert f\Vert _{L^{q,s}\left( {\mathbb {R}}^{2m}\right) } \end{aligned}$$

The fact that \(k \in [1,2m-1]\) implies \(\frac{1}{p} < \frac{1}{q}\). Thus, we can choose \(s=q\). Moreover, there holds that \(L^p({\mathbb {R}}^{2m})=L^{p,p}({\mathbb {R}}^{2m})\) for all \(p\in (1,\infty )\) (cf. [18] Lemma 1.8.10), which implies (2). \(\square \)

For more details about Lorentz spaces, we refer the readers to [12, 18]. We also need the following standard fact about the elliptic operator \(\varDelta ^m\).

Lemma 2

Let \(B_1\) be the unit ball of \({\mathbb {R}}^n\). Suppose \(v(x)\in W^{m,2}(B_1) \cap L^\infty \) and \(f\in L^{\infty }(B_1)\). If v(x) satisfies \(\varDelta ^m v(x) =f(x)\) in distributional sense, then

$$\begin{aligned} \Vert v(x)\Vert _{L^{\infty }\left( B_{\frac{1}{2}}\right) } \le C \left( \Vert v(x)\Vert _{L^1(B_1)} +\Vert f(x) \Vert _{L^\infty (B_1)} \right) , \end{aligned}$$
(3)

where C is a positive constant only dependent of n

3 Existence of Energy-Minimizer and the Euler-Lagrange Equation

In this section, we establish the existence of the energy-minimizers of \({\mathcal {E}}_{m}(J)\), derive its Euler-Lagrange equation and define the weak solutions. Moreover we prove that a weak limit of a sequence of weakly m-harmonic almost complex structures with bounded \(W^{m,2}\) norm is still m-harmonic.

Theorem 1

There always exists an energy-minimizer of \({\mathcal {E}}_{m}(J)\) in \(W^{m,2}({\mathcal {J}}_g)\).

Proof

The proof is standard in calculus of variations. We include the details for completeness. Take a minimizing sequence \(J_k \in W^{m,2}({\mathcal {J}}_g)\) such that

$$\begin{aligned} \inf \limits _{J \in W^{m,2}} {\mathcal {E}}_{m}(J) = \lim _{k \rightarrow \infty } {\mathcal {E}}_{m}(J_k). \end{aligned}$$

Note that by interpolation inequality and integration by parts,

$$\begin{aligned} \Vert J \Vert ^2_{W^{m,2}} \le C \left( \sum _{|\alpha |=m}\Vert \nabla ^\alpha J \Vert ^2_{L^2} + \Vert J\Vert ^2_{L^\infty } \right) \le C \big ( {\mathcal {E}}_{m}(J) + 1 \big ), \,\, \forall J \in W^{m,2}({\mathcal {J}}_g). \end{aligned}$$

Hence, the sequence \(\{J_k\}\) is bounded in \(W^{m,2}\). This implies that there exists a subsequence, still denoted by \(J_k\), and \(J_0 \in W^{m,2}\), such that \(J_k\) converges weakly to \(J_0\) in \(W^{m,2}\) and \({\mathcal {E}}_{m}(J_0) \le \varliminf _{k \rightarrow \infty } {\mathcal {E}}_{m}(J_k).\) Moreover, \(J_k\) converges strongly to \(J_0\) in \(W^{m-1,2}\) and hence \(J_0 \in {\mathcal {J}}_{g}\). It follows that \(J_0\) is an energy-minimizer of the functional \({\mathcal {E}}_{m}(J)\). \(\square \)

Denote by \(T_{q}^{p}(M)\) the set of all (p, q) tensor fields on (M, g). There is a natural inner product on \(T_{q}^{p}(M)\) induced by g, denoted by \(\left\langle , \right\rangle \). In local coordinate \(\{x^{i}\}_{i=1}^{n}\), \(A\in T_{q}^{p}(M)\) can be expressed by

$$\begin{aligned} A=A_{i_1 \cdots i_q}^{j_1 \cdots j_p} \frac{\partial }{\partial x^{j_1}} \otimes \cdots \otimes \frac{\partial }{\partial x^{j_p}} \otimes dx^{i_1} \otimes \cdots \otimes dx^{i_q}. \end{aligned}$$

The inner product of \(A,B \in T^p_q(M)\) is given by

$$\begin{aligned} \left\langle A,B\right\rangle =A_{i_1 \cdots i_q}^{k_1 \cdots k_p} B_{j_1 \cdots j_q}^{l_1 \cdots l_p} g^{i_1j_1} \cdots g^{i_q j_q} g_{k_1 l_1} \cdots g_{k_p l_p}, \end{aligned}$$

where \(g=g_{ij} dx^i \otimes dx^j\) and \((g^{ij})\) is the inverse of \((g_{ij})\). For \(A\in T_1^1(M)\), define the adjoint operator \(A^*\) of A by

$$\begin{aligned} g(X,A^{*}Y) := g(AX,Y),\quad \forall X,Y\in {\mathfrak {X}}(M). \end{aligned}$$

where \({\mathfrak {X}}(M)\) is the set of all smooth vector fields on (M, g). In local coordinates, if \(A=A_{i}^{j}\partial _{x^{j}}\otimes dx^{i}\) we have \((A^*)_i^j=A_k^l g^{kj} g_{li}\).

Proposition 1

We have the following standard facts,

  1. 1.

    For all \(A,B\in T_{q}^{p}(M)\), there holds

    $$\begin{aligned} \int _{M}\left\langle \nabla A,\nabla B\right\rangle =-\int _{M}\left\langle A,\varDelta B\right\rangle . \end{aligned}$$
  2. 2.

    For all \(A\in T_{1}^{1}(M)\) and \(X \in {\mathfrak {X}}(M)\), there holds \(\left( \nabla _X A\right) ^{*}=\nabla _X (A^{*}).\)

  3. 3.

    For all \(A,B\in T_{1}^{1}(M)\), there holds \(\left\langle A,B\right\rangle =\left\langle A^{*},B^{*}\right\rangle .\)

  4. 4.

    For all \(A,B,C\in T_{1}^{1}(M)\), there holds \(\left\langle A,BC\right\rangle =\left\langle B^{*}A,C\right\rangle =\left\langle AC^{*},B\right\rangle .\)

For \(A, B \in T^1_1(M)\), AB is regarded as the composition of linear maps, i.e., \(AB\in T^1_1(M)\). In local coordinate we have \((AB)_i^j=A_s^jB_i^s.\) With these notations, we have

$$\begin{aligned} {\mathcal {J}}_{g}=\left\{ J\in T^1_1(M):\, J^2=-id, \, J^*+J=0\right\} . \end{aligned}$$
(4)

Let \(\{ J(t)\}_{t \in (-\delta ,\delta )}\) be a \(C^1\) curve in \({\mathcal {J}}_g\) with \(J(0)=J\). Let \(S=\frac{dJ}{dt} |_{t=0}\). Such S is called an admissible variational direction of J in \({\mathcal {J}}_{g}\). Denote \({\mathcal {S}}_J\) to be the set of all admissible variational directions of J.

Proposition 2

We have

$$\begin{aligned} {\mathcal {S}}_J=\left\{ S \in T_1^1(M): SJ+JS=0, S+S^*=0 \right\} . \end{aligned}$$

For any \(J\in {\mathcal {J}}_{g}\), define the operator \(\varPhi _J: T^1_1(M) \rightarrow S_J\) by

$$\begin{aligned} \varPhi _J(T)=\frac{1}{4} \bigg ( (T+JTJ)-(T+JTJ)^* \bigg ). \end{aligned}$$

On each fiber of \(T^1_1 (M)\), \(\varPhi _J\) is precisely the orthogonal projection onto \((S_J)_x\), satisfying that for all \(T\in T^1_1(M)\) and \(S\in {\mathcal {S}}_J\),

$$\begin{aligned} \left\langle T,S \right\rangle = \left\langle \varPhi _J(T), S\right\rangle . \end{aligned}$$

Proposition 3

The Euler-Lagrange equation of functional \({\mathcal {E}}_{m}(J)\) is

$$\begin{aligned} \left[ \varDelta ^m J, J\right] =0. \end{aligned}$$
(5)

Proof

Suppose \(J\in {\mathcal {J}}_{g}\) is a critical point of \({\mathcal {E}}_{m}(J)\). For any \(S\in {\mathcal {S}}_J\), we have

$$\begin{aligned} 0=\delta {\mathcal {E}}_{m}(J) =(-1)^m 2\int _M \left\langle \varDelta ^m J, S \right\rangle =(-1)^m 2\int _M \left\langle \varPhi _J\left( \varDelta ^m J\right) , S \right\rangle \end{aligned}$$

which implies \(\varPhi _J(\varDelta ^m J)=0.\) Equivalently we have

$$\begin{aligned} \left[ \varDelta ^m J, J\right] :=\varDelta ^m J J-J \varDelta ^m J=0. \end{aligned}$$

\(\square \)

An almost complex structure \(J\in W^{m,2}({\mathcal {J}}_{g})\) satisfying (5) in distributional sense is called weakly m-harmonic.

Proposition 4

A weakly m-harmonic almost complex structure J satisfies the following in distributional sense,

$$\begin{aligned} \varDelta ^m J=\sum _{s=0}^{m-1}(-1)^{m+1+s} \nabla ^s \cdot g_s \end{aligned}$$
(6)

where \(g_s =\sum C_{k_1,k_2,k_3}\, \nabla ^{k_1} J \nabla ^{k_2} J \nabla ^{k_3} J\) for nonnegative integers \(k_1+k_2+k_3=2m-s, \;k_i \in [0,m]\) and \(C_{k_1,k_2,k_3}\in {\mathbb {Z}}.\) That is, for any \(T \in T^1_1(M)\), there holds

  1. 1.

    when \(m=2k\), \(k\in {\mathbb {N}}^+\),

    $$\begin{aligned} \int _M \left\langle \varDelta ^k J , \varDelta ^k T \right\rangle +\sum _{s=0}^{m-1} \int _M \left\langle g_s, \nabla ^s T \right\rangle =0 \end{aligned}$$
    (7)
  2. 2.

    when \(m=2k-1\), \(k\in {\mathbb {N}}^+\),

    $$\begin{aligned} \int _M \left\langle \nabla \varDelta ^{k-1} J , \nabla \varDelta ^{k-1} T \right\rangle +\sum _{s=0}^{m-1} \int _M \left\langle g_s, \nabla ^s T \right\rangle =0 \end{aligned}$$
    (8)

For simplicity, we will give the exact meaning of \(\nabla ^s\) in the proof.

Proof

We focus on the case \(m=2k\) since the case \(m=2k-1\) is similar. Suppose J is weakly m-harmonic. Then for any \(S\in {\mathcal {S}}_J\), we have

$$\begin{aligned} 0=2\int _M \left\langle \varDelta ^k J, \varDelta ^k S \right\rangle . \end{aligned}$$

Taking \(S=\varPhi _J(T)\) for any \(T\in T^1_1(M)\),

$$\begin{aligned} 0&=2\int _M \left\langle \varDelta ^k J, \varDelta ^k \varPhi _J(T)\right\rangle \nonumber \\&= \frac{1}{2} \int _M \bigg \langle \varDelta ^k J \, ,\, \varDelta ^k \big ((T+JTJ)-(T+JTJ)^* \big ) \bigg \rangle \nonumber \\&=\frac{1}{2} \int _M \bigg \langle \varDelta ^k J, \varDelta ^k \big (T+JTJ \big ) \bigg \rangle -\bigg \langle \varDelta ^k J^*, \varDelta ^k \big (T+JTJ \big ) \bigg \rangle \nonumber \\&=\int _M \bigg \langle \varDelta ^k J, \varDelta ^k \big (T+JTJ \big ) \bigg \rangle \end{aligned}$$
(9)
$$\begin{aligned}&=\int _M \left\langle \varDelta ^k J, \varDelta ^k T \right\rangle +\left\langle \varDelta ^k J, J\varDelta ^k TJ \right\rangle +\left\langle \varDelta ^k J, R_1 \right\rangle \nonumber \\&=\int _M \left\langle \varDelta ^k J, \varDelta ^k T \right\rangle +\left\langle J \varDelta ^k J J, \varDelta ^k T \right\rangle +\left\langle \varDelta ^k J, R_1 \right\rangle \nonumber \\&=\int _M \left\langle \varDelta ^k J, \varDelta ^k T \right\rangle -\left\langle \big (\varDelta ^k J J+R_2 \big ) J, \varDelta ^k T \right\rangle +\left\langle \varDelta ^k J, R_1 \right\rangle \nonumber \\&=\int _M 2\left\langle \varDelta ^k J, \varDelta ^k T \right\rangle -\left\langle R_2 J, \varDelta ^k T \right\rangle +\left\langle \varDelta ^k J, R_1 \right\rangle \nonumber \\&=\int _M 2\left\langle \varDelta ^k J, \varDelta ^k T \right\rangle +\left\langle \nabla (R_2 J), \nabla \varDelta ^{k-1} T \right\rangle +\left\langle \varDelta ^k J, R_1 \right\rangle \end{aligned}$$
(10)

where

$$\begin{aligned} R_1&= \varDelta ^k (JTJ)-J \varDelta ^k T J, \\ R_2&=\varDelta ^k (JJ) - \varDelta ^k J J- J\varDelta ^k J= - \varDelta ^k J J- J\varDelta ^k J \end{aligned}$$

We describe the terms of \(R_1\) and \(R_2\) by taking a local orthonormal fields \(\{ e_i\}_{i=1}^n\) as follows,

$$\begin{aligned} R_1&=\varDelta ^k (JTJ) -J \varDelta ^k T J \\&=\nabla _{i_1}^2 \cdots \nabla _{i_k}^2 (JTJ) -J \varDelta ^k T J \\&=\sum _{\begin{array}{c} \alpha<, \beta<, \gamma< \\ k_1+k_2+k_3=m\\ k_2\le m-1 \end{array}} \nabla _{i_{\alpha _1}} \cdots \nabla _{i_{\alpha _{k_1}}} J \, \nabla _{i_{\beta _1}} \cdots \nabla _{i_{\beta _{k_2}}} T\, \nabla _{i_{\gamma _1}} \cdots \nabla _{i_{\gamma _{k_3}}} J \\ R_2&=\varDelta ^k (JJ)-\varDelta ^k J J-J\varDelta ^k J \\&=\sum _{\begin{array}{c} \alpha<, \beta <, \\ k_1+k_2=m\\ 1\le k_1, k_2\le m-1 \end{array}} \nabla _{i_{\alpha _1}} \cdots \nabla _{i_{\alpha _{k_1}}} J \, \nabla _{i_{\beta _1}} \cdots \nabla _{i_{\beta _{k_2}}} J \end{aligned}$$

where the symbol \(\alpha<\) means \(1\le \alpha _1\le \cdots \le \alpha _{k_1}\le k\) and we write

$$\begin{aligned} \nabla ^{k_1} = \sum _{\alpha <} \nabla _{i_{\alpha _1}} \cdots \nabla _{i_{\alpha _{k_1}}} . \end{aligned}$$

Then, we can rewrite \(R_1\) and \(R_2\) in the following,

$$\begin{aligned} R_1&=\sum _{\begin{array}{c} k_1+k_2+k_3=m \\ k_2\le m-1 \end{array}} \nabla ^{k_1} J \,\nabla ^{k_2} T\,\nabla ^{k_3} J, \\ R_2&=\sum _{\begin{array}{c} k_1+k_2=m\\ 1\le k_1, k_2\le m-1 \end{array}} \nabla ^{k_1} J \,\nabla ^{k_2} J. \end{aligned}$$

Substituting the above into (10), we get (7) as follows,

$$\begin{aligned} 0&=\int _M 2\left\langle \varDelta ^k J, \varDelta ^k T \right\rangle +\int _M\sum _{\begin{array}{c} k_1+k_2=m\\ 1\le k_1, k_2\le m-1 \end{array}} \left\langle \nabla \left( \nabla ^{k_1} J \,\nabla ^{k_2} J J\right) , \nabla \varDelta ^{k-1} T \right\rangle \\&\quad +\int _M \sum _{\begin{array}{c} k_1+k_2+k_3=m \\ k_2\le m-1 \end{array}} \left\langle \varDelta ^k J, \nabla ^{k_1} J \,\nabla ^{k_2} T\,\nabla ^{k_3} J \right\rangle \\&=\int _M 2\left\langle \varDelta ^k J, \varDelta ^k T \right\rangle +\int _M\sum _{\begin{array}{c} k_1+k_2=m\\ 1\le k_1, k_2\le m-1 \end{array}} \left\langle \nabla \left( \nabla ^{k_1} J \,\nabla ^{k_2} J J\right) , \nabla \varDelta ^{k-1} T \right\rangle \\&\quad + \int _M\sum _{\begin{array}{c} k_1+k_2+k_3=m \\ k_2\le m-1 \end{array}} \left\langle \nabla ^{k_1} J \varDelta ^k J \,\nabla ^{k_3} J, \nabla ^{k_2} T \right\rangle , \end{aligned}$$

\(\square \)

Proposition 5

A weak limit of a sequence of weakly m-harmonic almost complex structures with uniformly bounded \(W^{m,2}\) norm is still m-harmonic.

Proof

Let J be weakly m-harmonic. Then for any \(T\in T^1_1(M)\), there holds

  1. 1.

    when \(m=2k\), \(k\in {\mathbb {N}}^+\)

    $$\begin{aligned} \int _M \left\langle \varDelta ^k J, \left[ J, \varDelta ^k T\right] \right\rangle +\sum _{\begin{array}{c} k_1+k_2=m \\ 1\le k_1, k_2 \le m-1 \end{array}} \left\langle \varDelta ^k J, \left[ \nabla ^{k_1} J, \nabla ^{k_2}T\right] \right\rangle =0, \end{aligned}$$
    (11)
  2. 2.

    when \(m=2k-1\), \(k\in {\mathbb {N}}^+\)

    $$\begin{aligned} \int _M \left\langle \nabla \varDelta ^{k-1} J, \left[ J, \nabla \varDelta ^{k-1} T\right] \right\rangle +\sum _{\begin{array}{c} k_1+k_2=m \\ 1\le k_1, k_2 \le m-1 \end{array}} \left\langle \nabla \varDelta ^{k-1} J, \left[ \nabla ^{k_1} J, \nabla ^{k_2}T\right] \right\rangle =0. \end{aligned}$$

We only prove the case \(m=2k\). Recall (9) holds for all \(T\in T^1_1(M)\),

$$\begin{aligned} \int _M \left\langle \varDelta ^k J, \varDelta ^k \big (T+JTJ \big ) \right\rangle =0 \end{aligned}$$

By replacing T by JT, we derive

$$\begin{aligned} \int _M \left\langle \varDelta ^k J, \varDelta ^k \big (JT-TJ \big ) \right\rangle =\int _M \left\langle \varDelta ^k J, \varDelta ^k [J,T] \right\rangle =0. \end{aligned}$$

This implies (11) since

$$\begin{aligned} \varDelta ^k [J,T]&=\left[ \varDelta ^k J, T\right] +\left[ J,\varDelta ^k T\right] +\sum _{\begin{array}{c} k_1+k_2=m \\ 1\le k_1, k_2 \le m-1 \end{array}} \left[ \nabla ^{k_1}J, \nabla ^{k_2}T\right] ,\\ \left\langle \varDelta ^k J, \left[ \varDelta ^k J, T\right] \right\rangle&=\left\langle \varDelta ^k J, \varDelta ^k J\, T \right\rangle -\left\langle \varDelta ^k J, T \,\varDelta ^k J \right\rangle \\&=\left\langle \left( \varDelta ^k J\right) ^*\,\varDelta ^k J, T \right\rangle -\left\langle \varDelta ^k J\,\left( \varDelta ^k J\right) ^*, T \right\rangle \\&=\left\langle \varDelta ^k J^*\,\varDelta ^k J, T \right\rangle -\left\langle \varDelta ^k J\,\varDelta ^k J^*, T \right\rangle =0. \end{aligned}$$

Now, suppose \(\{J_l\}\) is a sequence of weakly m-harmonic almost complex structures in \(W^{m,2}\) such that \(J_l \rightharpoonup J_0 \quad \text{ in }\,\, W^{m,2} \quad \text{ and } \quad \sup _l \Vert J_l\Vert _{W^{m,2}} <\infty .\) By Rellich-Kondrachov theorem, we know that \(J_l\) converges to \(J_0\) in \(W^{m-1,2}\). Hence \(J_0 \in W^{m,2}({\mathcal {J}}_{g})\). Since \(J_l \rightharpoonup J_0\) in \(W^{m,2}\), we have

$$\begin{aligned} \lim _{l\rightarrow \infty } \int _M \left\langle \varDelta ^k J_l, \left[ J_0, \varDelta ^k T\right] \right\rangle = \int _M \left\langle \varDelta ^k J_0, \left[ J_0, \varDelta ^k T\right] \right\rangle . \end{aligned}$$
(12)

Since \(J_l\) converges to \(J_0\) in \(W^{m-1,2}\), \(\sup _l \Vert J_l\Vert _{W^{m,2}} <\infty \) and

$$\begin{aligned} \left| \int _M \left\langle \varDelta ^k J_l, \left[ J_l-J_0, \varDelta ^k T\right] \right\rangle \right| \le \Vert \varDelta ^k J_l\Vert _{L^2} \Vert J_l-J_0\Vert _{L^2} \Vert \varDelta ^k T\Vert _{L^\infty }, \end{aligned}$$

we have \(\lim _{l\rightarrow \infty } \int _M \left\langle \varDelta ^k J_l, [J_l-J_0, \varDelta ^k T]\right\rangle = 0.\) With (12) this implies

$$\begin{aligned} \lim _{l\rightarrow \infty } \int _M \left\langle \varDelta ^k J_l, \left[ J_l, \varDelta ^k T\right] \right\rangle =\int _M \left\langle \varDelta ^k J_0, \left[ J_0, \varDelta ^k T\right] \right\rangle . \end{aligned}$$

Similarly we conclude that, for all \(k_1+k_2=m\) and \(1\le k_1, k_2 \le m-1\),

$$\begin{aligned} \lim _{l\rightarrow \infty } \int _M \left\langle \varDelta ^k J_l, \left[ \nabla ^{k_1} J_l, \nabla ^{k_2}T\right] \right\rangle =\int _M \left\langle \varDelta ^k J_0, \left[ \nabla ^{k_1} J_0, \nabla ^{k_2}T\right] \right\rangle . \end{aligned}$$

Hence \(J_0\) is weakly m-harmonic and this completes the proof. \(\square \)

4 Decay Estimates and Hölder Regularity

In this section, we establish decay estimates for a class of semilinear elliptic equations in critical dimension and deduce the Hölder regularity of \(W^{m,2}\) m-harmonic almost complex structure on \((M^{2m},g)\) for \(m=2,3\). For simplicity, we use C to denote a uniform positive constant.

4.1 Decay Estimates for \(W^{2,2}\) Biharmonic Almost Complex Structure on \({\mathbb {R}}^4\)

First, we consider decay estimates for biharmonic almost complex structure defined on \(B_1\) in \({\mathbb {R}}^4\) as a special case. The presentation is much clearer and more streamlined for this case and the main ideas are essentially the same. Consider the biharmonic almost complex structure equation,

$$\begin{aligned} \varDelta ^2 J = J \bigg ( \nabla \varDelta J \nabla J + \nabla J \nabla \varDelta J+ \varDelta J \varDelta J + \varDelta ( \nabla J )^2\bigg ) \end{aligned}$$

where \(J : B_1 \subset {\mathbb {R}}^4 \rightarrow M_4({\mathbb {R}})\) (the set of all \(4\times 4\) real matrices) satisfies

$$\begin{aligned} J^2=- id, \quad J+J^T=0 \end{aligned}$$

Proposition 6 asserts that the biharmonic almost complex structure equation admit a good divergence form. That is, for any given constant matrix \(\lambda _0\), biharmonic almost complex structure J satisfies

$$\begin{aligned} \varDelta ^2 J = T_{\lambda _0} \end{aligned}$$
(13)

where \(T_{\lambda _0}\) is a linear combination of the following terms

$$\begin{aligned} \nabla ^{\alpha }\bigg ((J-\lambda _{0})*\nabla ^{\beta }J*\nabla ^{\gamma }J\bigg ),\qquad \lambda _0*\nabla ^{\alpha }\bigg ((J-\lambda _{0})*\nabla ^{\delta }J\bigg ), \end{aligned}$$

where \(\alpha ,\beta ,\gamma ,\delta \) are multi-indices such that \(1\le |\alpha | \le 3\), \(0\le |\beta |,|\gamma |,|\delta |\le 2\), \(|\alpha |+|\beta |+|\gamma |=4\) and \(|\alpha |+|\delta |=4\). The notation \(A*B\) means the composition of terms A and B, such as AB and BA. Then we have the following,

Lemma 3

Suppose \(J \in W^{2,2}(B_1, M_4({\mathbb {R}}))\) is a weakly biharmonic almost complex structure on unit ball \(B_1 \subset {\mathbb {R}}^4\). Then, given any \(\tau \in (0,1)\), there exists \(\epsilon _0>0\) and \(\theta _0 \in (0, \frac{1}{2})\) such that if

$$\begin{aligned} E(J,1):=\left( \int _{B_1} |\nabla J|^4 \right) ^\frac{1}{4} + \left( \int _{B_1} |\nabla ^2 J|^2 \right) ^\frac{1}{2} \le \epsilon _0, \end{aligned}$$

then we have

$$\begin{aligned} D_{p_0}(J,\theta _0) \le \theta _0^\tau D_{p_0}(J,1), \end{aligned}$$
(14)

where \(p_0=\frac{8}{3}\) and \(D_{p}(J,r):= \bigg (r^{p-4} \int _{B_r} |\nabla u|^p \bigg )^{\frac{1}{p}}.\)

Proof

We extend J to \({\widetilde{J}}\in W^{2,2}({\mathbb {R}}^4, M_4({\mathbb {R}})) \cap L^\infty \) such that

$$\begin{aligned}&{\widetilde{J}}|_{B_1}= J, \quad {\widetilde{J}}|_{{\mathbb {R}}^{4} \setminus B_2}= \lambda _0:=\frac{1}{|B_1|}\int _{B_1} J \nonumber \\&\quad \Vert \nabla {\widetilde{J}}\Vert _{L^{p_0}({\mathbb {R}}^{4})} \le C\, \Vert \nabla J\Vert _{L^{p_0}(B_1)} \end{aligned}$$
(15)
$$\begin{aligned}&E({\widetilde{J}},\infty ) \le C\, E(J,1), \end{aligned}$$
(16)

By the standard extension to \(J-\lambda _0\) in \(B_1\), there exists a function \({\widetilde{J}}-\lambda _0\) on \({\mathbb {R}}^{4}\) with compact support contained in \(B_2\) and satisfying

$$\begin{aligned} \Vert {\widetilde{J}}-\lambda _0 \Vert _{L^\infty \left( {\mathbb {R}}^{4}\right) }&\le C \Vert J -\lambda _0\Vert _{L^\infty (B_1)}, \end{aligned}$$
(17)
$$\begin{aligned} \Vert {\widetilde{J}} -\lambda _0\Vert _{W^{1,p_0}\left( {\mathbb {R}}^{4}\right) }&\le C \Vert J -\lambda _0 \Vert _{W^{1,p_0}(B_1)} \end{aligned}$$
(18)
$$\begin{aligned} \Vert {\widetilde{J}} -\lambda _0 \Vert _{W^{2,2}\left( {\mathbb {R}}^{4}\right) }&\le C \Vert J- \lambda _0\Vert _{W^{2,2}(B_1)}. \end{aligned}$$
(19)

Since \({\widetilde{J}}-\lambda _0\) has a compact support, (17) implies \({\widetilde{J}}-\lambda _0 \in L^q({\mathbb {R}}^{4})\) for all \(q\in [1,\infty ]\). By Poincáre inequality (15) follows from (18). We obtain (16) by Poincáre inequality, Sobolev inequality and (19),

$$\begin{aligned} E({\widetilde{J}},\infty )&= E({\widetilde{J}}-\lambda _0,\infty ) \\&\le C \Vert \nabla ^2 {\widetilde{J}} \Vert _{L^2\left( {\mathbb {R}}^{4}\right) } \le C \Vert J- \lambda _0\Vert _{W^{2,2}(B_1)} \le C \Vert \nabla J\Vert _{W^{1,2}(B_1)} \le C E(J,1) \end{aligned}$$

Note that \({\widetilde{J}}\) may not be almost complex structure outside \(B_1\). Now let \(G(x)=c \ln |x|\) be the fundamental solution for \(\varDelta ^2\) on \({\mathbb {R}}^{4}\), where c is a constant. Then \(\nabla ^{4} G\) is a Calderón-Zygmund kernel. Define

$$\begin{aligned} \omega (x)&=\int _{{\mathbb {R}}^{4}} G(x-y)\, {\widetilde{T}}_{\lambda _0} (y) dy =\sum _{\alpha , \beta , \gamma } \omega _{\alpha ,\beta ,\gamma } + \sum _{\alpha , \delta } \omega _{\alpha ,\delta }, \end{aligned}$$

where

$$\begin{aligned} \omega _{\alpha ,\beta ,\gamma }&= \int _{{\mathbb {R}}^4} \nabla ^{\alpha }G(x-y) \bigg ( \big ({\widetilde{J}}(y)-\lambda _0 \big ) *\nabla ^\beta {\widetilde{J}}(y) *\nabla ^\gamma {\widetilde{J}}(y) \bigg ) dy, \\ \omega _{\alpha ,\delta }&= \int _{{\mathbb {R}}^4} \nabla ^{\alpha }G(x-y) \bigg ( \lambda _0 *\big ({\widetilde{J}}(y)-\lambda _0 \big ) *\nabla ^\delta {\widetilde{J}}(y) \bigg ) dy, \end{aligned}$$

and \({\widetilde{T}}_{\lambda _0}\) is defined by replacing J by \({\widetilde{J}}\) in \(T_{\lambda _0}\) (see (13)) , and \(\alpha ,\beta ,\gamma ,\delta \) are multi-indices such that \(1\le |\alpha | \le 3\), \(0\le |\beta |,|\gamma |,|\delta |\le 2\), \(|\alpha |+|\beta |+|\gamma |=4\) and \(|\alpha |+|\delta |=4\). We claim that for \(E(J,1)\le 1\), there holds

$$\begin{aligned} \Vert \nabla \omega \Vert _{L^{p_0}(B_1)} \le C E(J,1) \Vert \nabla J \Vert _{L^{p_0}(B_1)}. \end{aligned}$$
(20)

We will prove the above inequality term by term. By Lemma 1, we have

$$\begin{aligned} \Vert \nabla \omega _{\alpha ,\beta , \gamma } \Vert _{L^{p_0}\left( {\mathbb {R}}^4\right) }&\le C \bigg \Vert |{\widetilde{J}}-\lambda _0|\, |\nabla ^\beta {\widetilde{J}}| \,|\nabla ^\gamma {\widetilde{J}}| \bigg \Vert _{L^{q_0}\left( {\mathbb {R}}^4\right) } \\&\le C \Vert {\widetilde{J}}-\lambda _0 \Vert _{L^{q_1}\left( {\mathbb {R}}^4\right) } \Vert \nabla ^\beta {\widetilde{J}} \Vert _{L^{\frac{4}{|\beta |}}\left( {\mathbb {R}}^4\right) } \Vert \nabla ^\gamma {\widetilde{J}}\Vert _{L^{\frac{4}{|\gamma |}}\left( {\mathbb {R}}^4\right) } \\&\le C \Vert \nabla {\widetilde{J}} \Vert _{L^{p_0}\left( {\mathbb {R}}^4\right) } E({\widetilde{J}},\infty )^{N_{\beta ,\gamma }} \\&\le C \Vert \nabla J \Vert _{L^{p_0}(B_1)} E(J,1)^{N_{\beta ,\gamma }} \end{aligned}$$

where we let \(\frac{4}{s}:=\infty \) for \(s=0\), \(N_{\beta ,\gamma }\) stands for the number of non-zero elements in \(\{\beta ,\gamma \}\) and \(q_0, q_1 \in (1,\infty )\) satisfy

$$\begin{aligned} \frac{1}{p_0}+1 = \frac{|\alpha |+1}{4} + \frac{1}{q_0}, \qquad \frac{1}{q_0} = \frac{1}{q_1} +\frac{|\beta |}{4}+ \frac{|\gamma |}{4}. \end{aligned}$$

Since \(|\alpha |+|\beta |+|\gamma |=4\) and \(1\le |\alpha | \le 3\), we know that \(1\le N_{\beta ,\gamma }\le 2\) and hence such \(q_0\) and \(q_1\) exist. If \(E(J,1)\le 1\), there holds

$$\begin{aligned} \Vert \nabla \omega _{\alpha ,\beta , \gamma } \Vert _{L^{p_0}(B_1)} \le \Vert \nabla \omega _{\alpha ,\beta , \gamma } \Vert _{L^{p_0}\left( {\mathbb {R}}^4\right) } \le C \Vert \nabla J \Vert _{L^{p_0}(B_1)} E(J,1). \end{aligned}$$
(21)

By a similar argument, we also have

$$\begin{aligned} \Vert \nabla \omega _{\alpha ,\delta } \Vert _{L^{p_0}(B_1)} \le \Vert \nabla \omega _{\alpha ,\delta } \Vert _{L^{p_0}\left( {\mathbb {R}}^4\right) } \le C \Vert \nabla J \Vert _{L^{p_0}(B_1)} E(J,1). \end{aligned}$$
(22)

Combining (21) and (22), we deduce (20).

Finally, we turn to proving (14). Let \(v(x):=J(x)-\omega (x)\), then we know v(x) is biharmonic on unit ball \(B_1\), i.e., \(\varDelta ^2 v(x)=0\). Since \(\nabla v\) is also biharmonic, it follows from Lemma 2 (or see Lemma 6.2 in [5]) that there holds

$$\begin{aligned} \Vert \nabla v \Vert _{L^\infty \left( B_{\frac{1}{2}}\right) } \le C \Vert \nabla v \Vert _{L^1(B_1)}. \end{aligned}$$

Hence, for any \(\theta \in (0,\frac{1}{2})\) and \(E(u,1)\le 1\), there holds

$$\begin{aligned} D_{p_0}(J,\theta )&= \theta ^{1-\frac{4}{p_0}} \Vert \nabla J\Vert _{L^{p_0}(B_\theta )} \\&\le \theta ^{1-\frac{4}{p_0}} \left( \Vert \nabla \omega (x)\Vert _{L^{p_0}(B_\theta )}+ \Vert \nabla v(x)\Vert _{L^{p_0}(B_\theta )} \right) \\&\le C \theta ^{1-\frac{4}{p_0}} \left( \Vert \nabla \omega (x)\Vert _{L^{p_0}(B_\theta )} + \theta ^{\frac{4}{p_0}}\Vert \nabla v(x)\Vert _{L^{\infty }(B_\theta )} \right) \\&\le C \theta ^{1-\frac{4}{p_0}} \left( \Vert \nabla \omega (x)\Vert _{L^{p_0}(B_\theta )} + \theta ^{\frac{4}{p_0}}\Vert \nabla v(x)\Vert _{L^{p_0}(B_1)} \right) \\&\le C \theta ^{1-\frac{4}{p_0}} \left( \Vert \nabla \omega (x)\Vert _{L^{p_0}(B_\theta )} + \theta ^{\frac{4}{p_0}}\Vert \nabla \omega (x)\Vert _{L^{p_0}(B_1)} \right. \\&\left. \quad +\theta ^{\frac{4}{p_0}}\Vert \nabla J(x)\Vert _{L^{p_0}(B_1)} \right) \\&\le C \theta ^{1-\frac{4}{p_0}} \left( \Vert \nabla \omega (x)\Vert _{L^{p_0}(B_\theta )} +\theta ^{\frac{4}{p_0}}\Vert \nabla J(x)\Vert _{L^{p_0}(B_1)} \right) \\&\le C \left( \theta ^{1-\frac{4}{p_0}}E(J,1) \Vert \nabla J(x)\Vert _{L^{p_0}(B_\theta )} +\theta \Vert \nabla J(x)\Vert _{L^{p_0}(B_1)} \right) \\&\le C \left( \theta ^{1-\frac{4}{p_0}} E(J,1) +\theta \right) D_{p_0}(J,1). \end{aligned}$$

Thus, for any given \(\tau \in (0,1)\), by choosing \(\theta =\theta _0\) and \(\epsilon _0\) sufficiently small, we obtain (14) for \(E(J,1)\le \epsilon _0\). \(\square \)

4.2 Decay Estimates for a Class of Semilinear Elliptic Equations

Consider the following semilinear elliptic equation for \(u: B_1 \subset {\mathbb {R}}^n\rightarrow {\mathbb {R}}^K\),

$$\begin{aligned} \varDelta ^m u = \varPsi \left( x, \nabla u, \cdots , \nabla ^{2m-1} u\right) \end{aligned}$$
(23)

where \(\varPsi : {\mathbb {R}}^n \times {\mathbb {R}}^{nK}\times \cdots \times {\mathbb {R}}^{n^{2m-1}K} \rightarrow {\mathbb {R}}^K\) is smooth and \(B_1\) is the unit ball in \({\mathbb {R}}^n\) centered at origin. We can generalize the results in Sect,. 4.1 to (23) which admit a good divergence structure specified in the following,

Definition 2

We say that the equation (23) admits a good divergence form if for any fixed constant vector \(\lambda _0 \in {\mathbb {R}}^K\), \(\varPsi \) can be decomposed into \(\varPsi _H+\varPsi _L\), the highest order term \(\varPsi _H\) and the lower order term \(\varPsi _L\), which satisfy the following properties:

  1. 1.

    \(\varPsi _H \) is a linear combination of the following terms

    $$\begin{aligned} \nabla ^\alpha ((u-\lambda _0) * h_{\alpha , \beta }), \quad \text{ with } |h_{\alpha , \beta }| \le C \prod _{i=1}^{s} \big |\nabla ^{\beta _i} u \big |, \end{aligned}$$
    (24)

    where \(\alpha , \beta _i\) are multi-indices and \(\beta =(\beta _1, \cdots , \beta _s)\) such that

    $$\begin{aligned}&|\alpha |+\sum _{i=1}^{s} |\beta _i|= 2m, \end{aligned}$$
    (25)
    $$\begin{aligned}&|\beta _i| \le m, \quad i=1, \cdots , s, \,\, s \in {\mathbb {N}}^+, \end{aligned}$$
    (26)
    $$\begin{aligned}&1 \le \sum _{i=1}^s|\beta _i| \le 2m-1, \end{aligned}$$
    (27)
  2. 2.

    \(\varPsi _L\) is a linear combination of the following three types of terms

    $$\begin{aligned} \begin{aligned}&\nabla ^{\alpha }(a_{\alpha ,\gamma }(x)*\ell _{\alpha ,\gamma }), \quad \text{ with }\,\,\, |\ell _{\alpha ,\gamma }| \le C \prod _{i=1}^{s} \big |\nabla ^{\gamma _i} u \big |, \\&b_t(x)*\big (u(x)-\lambda _0 \big )*\ell _{0,t}, \quad \text{ with }\,\,\, |\ell _{0,t}| \le C |u|^t, \quad t \in {\mathbb {N}}, \\&c(x), \end{aligned} \end{aligned}$$
    (28)

    where \(\gamma =(\gamma _1, \cdots , \gamma _s)\), \(a_{\alpha ,\gamma }(x), b_t(x), c(x) \in C^{2m}(\overline{B_1}, {\mathbb {R}}^K)\) and

    $$\begin{aligned}&|\alpha |+\sum _{i=1}^{s} |\gamma _i|\le 2m-1, \end{aligned}$$
    (29)
    $$\begin{aligned}&|\gamma _i| \le m, \quad i=1, \cdots , s, \,\, s \in {\mathbb {N}}^+, \end{aligned}$$
    (30)
    $$\begin{aligned}&\sum _{i=1}^{s} |\gamma _i| \ge 1. \end{aligned}$$
    (31)

Remark 1

  1. 1.

    The condition (26) and (30) are natural for us to define the weak solution to (23) for \(u \in W^{m,2}\).

  2. 2.

    The condition (27) plays an important role in proving the Hölder continuity of u in critical dimension \(n=2m\) under the structure (24) of \(\varPsi _H\).

  3. 3.

    A trivial verification shows that the terms in the form

    $$\begin{aligned} g(x)*\nabla ^{\alpha _1}u * \cdots * \nabla ^{\alpha _t}u \quad \text{ for } \,\, g(x) \in C^{4m}(\overline{B_1}, {\mathbb {R}}^K), \,\, \sum _i |\alpha _i| \le 2m-1 \end{aligned}$$

    can always be rewritten as a linear combination of terms (28).

For any ball \(B_r\) of radius r centered at origin in \({\mathbb {R}}^n\), any \(p>1\), and \(q_l\in (1,\infty )\) given by \(\frac{1}{q_l}=\frac{1}{2}-\frac{m-l}{n}\) for \(l=1,\cdots , m\) and \(n\ge 2m\), denote

$$\begin{aligned} E(u,r)&=\sum _{l=1}^m \left( r^{lq_l-n}\int _{B_r} |\nabla ^l u|^{q_l} \right) ^{\frac{1}{q_l}}, \end{aligned}$$
(32)
$$\begin{aligned} D_p(u,r)&=\left( r^{p-n} \int _{B_r} |\nabla u|^p \right) ^{\frac{1}{p}}. \end{aligned}$$
(33)

Lemma 4

Suppose \(n=2m\) and \(u \in W^{m,2}(B_1, {\mathbb {R}}^K) \cap L^\infty \) satisfies (23) in distributional sense. If (23) admits a good divergence form and \(\Vert u\Vert _{L^\infty (B_1)} \le {\mathcal {B}}<\infty \), then, given any \(\tau \in (0,1)\), there exists \(\epsilon _0>0\) and \(\theta _0 \in (0, \frac{1}{2})\), which are only dependent of \(\tau , {\mathcal {B}}, m\), such that if \(E(u,1)\le \epsilon _0,\) then we have

$$\begin{aligned} D_{p_0}(u,\theta _0)\le \theta _0^{\tau } \big (D_{p_0}(u,1)+ \varLambda ), \end{aligned}$$
(34)

where \(p_0=\frac{4m}{3} \in (1,2m)\) and

$$\begin{aligned} \varLambda :=\sum _{\alpha , \gamma } \Vert a_{\alpha , \gamma }(x)\Vert _{L^\infty (B_1)} + \sum _t \Vert b_t(x) \Vert _{L^\infty (B_1)}+\Vert \nabla c(x) \Vert _{L^\infty (B_1)} \end{aligned}$$

where \(a_{\alpha , \gamma }(x), b_t(x), c(x)\) are from (28) in lower order terms \(\varPsi _L\) of (23).

Proof

For simplicity, we denote by C a positive constant only dependent of \(\tau ,{\mathcal {B}},m\). Following the similar argument in the proof of Lemma 3, we can extend u to \({\widetilde{u}} \in W^{m,2}({\mathbb {R}}^{2m}, {\mathbb {R}}^K) \cap L^\infty \) such that

$$\begin{aligned}&{\widetilde{u}}|_{B_1}=u, \quad {\widetilde{u}}|_{{\mathbb {R}}^{2m} \setminus B_2}= \lambda _0:=\frac{1}{|B_1|}\int _{B_1}u \nonumber \\&\Vert {\widetilde{u}}\Vert _{L^\infty ({\mathbb {R}}^{2m})} \le C\, \Vert u \Vert _{L^\infty (B_1)} \end{aligned}$$
(35)
$$\begin{aligned}&\Vert \nabla {\widetilde{u}}\Vert _{L^{p_0}({\mathbb {R}}^{2m})} \le C\, \Vert \nabla u\Vert _{L^{p_0}(B_1)} \end{aligned}$$
(36)
$$\begin{aligned}&E({\widetilde{u}},\infty ) \le C\, E(u,1), \end{aligned}$$
(37)

Of course, by a standard extension theorem to the functions \(a_{\alpha , \gamma }(x)\), \(b_t(x)\)  \(\in C^{2m}(\overline{B_1}, {\mathbb {R}}^K)\) from the lower order term \(\varPsi _L\), there exist the corresponding functions \({\widetilde{a}}_{\alpha , \gamma }(x)\), \({\widetilde{b}}_t(x)\in C_0^{2m}({\mathbb {R}}^{2m}, {\mathbb {R}}^K)\) such that

$$\begin{aligned}&{\widetilde{a}}_{\alpha , \gamma }(x)|_{B_1}=a_{\alpha , \gamma }(x), \quad {\widetilde{b}}_t (x)|_{B_1}=b_t(x), \\&{\widetilde{a}}_{\alpha , \gamma }(x)|_{{\mathbb {R}}^{2m} \setminus B_2}=0, \quad {\widetilde{b}}_t (x) |_{{\mathbb {R}}^{2m} \setminus B_2}=0, \\&\Vert {\widetilde{a}}_{\alpha , \gamma }(x)\Vert _{L^\infty \left( {\mathbb {R}}^{2m}\right) } \le C \Vert a_{\alpha , \gamma }(x) \Vert _{L^\infty (B_1)},\\&\Vert {\widetilde{b}}_t(x)\Vert _{L^\infty \left( {\mathbb {R}}^{2m}\right) } \le C \Vert b_t(x)\Vert _{L^\infty (B_1)}. \end{aligned}$$

Let \(G(x)=c_m \ln |x|\) be the fundamental solution for \(\varDelta ^m\) on \({\mathbb {R}}^{2m}\). Then \(\nabla ^{2m} G\) is a Calderón-Zygmund kernel. Let us define

$$\begin{aligned} \omega (x) =\sum _{\alpha , \beta } \omega _{\alpha ,\beta }(x) +\sum _{\alpha , \gamma } \omega _{\alpha ,\gamma }(x) +\sum _{t}\omega _{0,t}(x), \end{aligned}$$

where

$$\begin{aligned} \omega _{\alpha ,\beta }(x)&=\int _{{\mathbb {R}}^{2m}} \nabla ^{\alpha } G(x-y) \bigg ( \big ( {\widetilde{u}}(y)-\lambda _0 \big ) * {\widetilde{h}}_{\alpha ,\beta }(y) \bigg ) dy \\ \omega _{\alpha ,\gamma }(x)&=\int _{{\mathbb {R}}^{2m}} \nabla ^{\alpha } G(x-y) \bigg ( {\widetilde{a}}_{\alpha , \gamma }(y) * {\widetilde{\ell }}_{\alpha ,\gamma }(y) \bigg ) dy \\ \omega _{0,t}(x)&= \int _{{\mathbb {R}}^{2m}} G(x-y) \bigg ({\widetilde{b}}_t(y)*\big ({\widetilde{u}}(y)-\lambda _0 \big )*{\widetilde{\ell }}_{0,t}(y) \bigg ) dy. \end{aligned}$$

We claim that, for \(p_0=\frac{4m}{3}\) and \(E(u,1)\le 1\), there holds

$$\begin{aligned} \Vert \nabla \omega \Vert _{L^{p_0}(B_1)} \le C \bigg ( E(u,1) \Vert \nabla u \Vert _{L^{p_0}(B_1)} + E(u,1)\cdot \varLambda \bigg ). \end{aligned}$$
(38)

We will prove above inequality term by term. By Lemma 1, we have

$$\begin{aligned} \Vert \nabla \omega _{\alpha , \beta } \Vert _{L^{p_0}(B_1)} \le \Vert \nabla \omega _{\alpha , \beta } \Vert _{L^{p_0}({\mathbb {R}}^{2m})}&\le C \bigg \Vert \big | {\widetilde{u}}-\lambda _0 \big | \cdot \big |{\widetilde{h}}_{\alpha ,\beta } \big | \bigg \Vert _{L^{q_{\alpha , \beta }}\left( R^{2m}\right) } \\&\le C \Vert {\widetilde{u}}-\lambda _0 \Vert _{L^{4m}\left( {\mathbb {R}}^{2m}\right) } \prod _{i=1}^s \Vert \nabla ^{\beta _i} {\widetilde{u}} \Vert _{L^{\frac{2m}{|\beta _i|}}\left( {\mathbb {R}}^{2m}\right) } \\&\le C \Vert \nabla {\widetilde{u}}\Vert _{L^{p_0}\left( {\mathbb {R}}^{2m}\right) } \prod _{i=1}^s \Vert \nabla ^{\beta _i} {\widetilde{u}} \Vert _{L^{\frac{2m}{|\beta _i|}}\left( {\mathbb {R}}^{2m}\right) }\\&\le C \Vert \nabla u\Vert _{L^{p_0}(B_1)} E(u,1)^{n_{\beta }} \Vert u \Vert _{L^\infty (B_1)}^{s-n_\beta } \nonumber \\&\le C {\mathcal {B}}^{s-n_\beta } \Vert \nabla u\Vert _{L^{p_0}(B_1)} E(u,1)^{n_{\beta }} \end{aligned}$$

where \(\frac{2m}{|\beta _i|}:=\infty \) for \(|\beta _i|=0\), \(n_\beta =\big |\{\beta _i: \beta _i \ne 0 \}\big |\ge 1\), and

$$\begin{aligned} \frac{1}{q_{\alpha , \beta }} =\frac{1}{2m} \left( \sum _{i=1}^s |\beta _i|+\frac{1}{2} \right) =1+\frac{3}{4m}-\frac{|\alpha |+1}{2m}. \end{aligned}$$

Note that (27) implies \(q_{\alpha , \beta }\in (1,\infty )\). Hence, if \(E(u,1)\le 1\), there holds

$$\begin{aligned} \Vert \nabla \omega _{\alpha , \beta } \Vert _{L^{p_0}(B_1)} \le C E(u,1) \Vert \nabla u\Vert _{L^{p_0}(B_1)}. \end{aligned}$$
(39)

Similarly, by Lemma 1, we obtain that

$$\begin{aligned} \Vert \nabla \omega _{\alpha , \gamma } \Vert _{L^{p_0}(B_1)} \le C\Vert \nabla \omega _{\alpha , \gamma } \Vert _{L^{4m}\left( {\mathbb {R}}^{2m}\right) }&\le C \bigg \Vert \big | {\widetilde{a}}_{\alpha , \gamma } \big | \cdot \big |{\widetilde{\ell }}_{\alpha ,\gamma } \big | \bigg \Vert _{L^{q_{\alpha , \gamma }}\left( {\mathbb {R}}^{2m}\right) } \\&\le C \big \Vert {\widetilde{a}}_{\alpha , \gamma } \big \Vert _{L^{q_1}({\mathbb {R}}^{2m})} \prod _{i=1}^s \big \Vert \nabla ^{\gamma _i} {\widetilde{u}} \big \Vert _{L^{\frac{2m}{|\gamma _i|}}\left( {\mathbb {R}}^{2m}\right) } \\&\le C \cdot \varLambda \cdot E(u,1)^{n_\gamma } \Vert u \Vert _{L^\infty (B_1)}^{s-n_\gamma } \end{aligned}$$

where \(\frac{2m}{|\gamma _i|}:=\infty \) for \(|\gamma _i|=0\), \(n_\gamma =\big |\{\gamma _i: \gamma _i \ne 0 \}\big |\ge 1\) due to (31), and

$$\begin{aligned} \frac{1}{q_{\alpha , \gamma }}=\frac{1}{4m}+1-\frac{|\alpha |+1}{2m}, \qquad \frac{1}{q_1}&=\frac{1}{4m}+1-\frac{1}{2m} \left( |\alpha |+\sum _{i=1}^s |\gamma _i|+1\right) . \end{aligned}$$

Note that (29) implies \(q_{\alpha ,\gamma }, q_1 \in (1,\infty )\). Hence, if \(E(u,1)\le 1\), there holds

$$\begin{aligned} \Vert \nabla \omega _{\alpha , \gamma } \Vert _{L^{p_0}(B_1)} \le C \cdot \varLambda \cdot E(u,1) \end{aligned}$$
(40)

Similar argument applies to terms \(\omega _{0,t}\) and yields

$$\begin{aligned} \Vert \nabla \omega _{0, t} \Vert _{L^{p_0}(B_1)} \le C \cdot \varLambda \cdot E(u,1). \end{aligned}$$
(41)

Combining (39), (40) and (41) gives (38). In the last we prove (34). Denote \(v(x):=u(x)-\omega (x)\), then v(x) satisfies \(\varDelta ^m v(x)=c(x)\) on \(B_1\) in distributional sense. By Lemma 2, we have

$$\begin{aligned} \Vert \nabla v(x)\Vert _{L^{\infty }\left( B_{\frac{1}{2}}\right) } \le C \big ( \Vert \nabla v(x)\Vert _{L^1(B_1)} +\Vert \nabla c(x) \Vert _{L^\infty (B_1)} \big ). \end{aligned}$$

Hence, for any \(\theta \in (0,\frac{1}{2})\) and \(E(u,1) \le 1\), there holds

$$\begin{aligned} D_{p_0}(u,\theta )&= \theta ^{1-\frac{2m}{p_0}} \Vert \nabla u \Vert _{L^{p_0}(B_\theta )} \\&\le \theta ^{1-\frac{2m}{p_0}} \Vert \nabla v \Vert _{L^{p_0}(B_\theta )} +\theta ^{1-\frac{2m}{p_0}} \Vert \nabla \omega \Vert _{L^{p_0}(B_\theta )}\\&\le C \theta \Vert \nabla v \Vert _{L^{\infty }(B_\theta )} +\theta ^{1-\frac{2m}{p_0}} \Vert \nabla \omega \Vert _{L^{p_0}(B_1)} \\&\le C \theta \big ( \Vert \nabla v \Vert _{L^{p_0}(B_1)}+\Vert \nabla c(x) \Vert _{L^\infty (B_1)} \big ) +\theta ^{1-\frac{2m}{p_0}} \Vert \nabla \omega \Vert _{L^{p_0}(B_1)} \\&\le C \theta \big ( \Vert \nabla u \Vert _{L^{p_0}(B_1)}+\Vert \nabla \omega \Vert _{L^{p_0}(B_1)}+ \varLambda \big ) +\theta ^{1-\frac{2m}{p_0}} \Vert \nabla \omega \Vert _{L^{p_0}(B_1)} \\&\le C \bigg ( \theta \big (\Vert \nabla u \Vert _{L^{p_0}(B_1)}+ \varLambda \big ) +\theta ^{1-\frac{2m}{p_0}} \Vert \nabla \omega \Vert _{L^{p_0}(B_1)} \bigg )\\&\le C \bigg ( \theta \big (\Vert \nabla u \Vert _{L^{p_0}(B_1)}+ \varLambda \big ) +\theta ^{1-\frac{2m}{p_0}} E(u,1) \big (\Vert \nabla u \Vert _{L^{p_0}(B_1)} + \varLambda \big ) \bigg )\\&\le C \bigg (\theta + \theta ^{1-\frac{2m}{p_0}} E(u,1)\bigg ) \big (D_{p_0}(u,1) + \varLambda \big ). \end{aligned}$$

Thus, for any given \(\tau \in (0,1)\), by choosing \(\theta =\theta _0\) and \(\epsilon _0\) sufficiently small, we obtain (34) for \(E(u,1)\le \epsilon _0\). \(\square \)

4.3 Hölder Regularity for m-Harmonic Almost Complex Structure

In this subsection, we prove the Hölder regularity using the decay estimates above,

Theorem 2

Suppose \(J\in W^{m,2}({\mathcal {J}}_{g})\) is a weakly m-harmonic almost complex structure on \((M^{2m}, g)\) with \(m\in \{2, 3\}\). Then J is Hölder-continuous.

Since the Hölder regularity is a local property, we work on local coordinates on \((M^n, g)\). Let \(B_1\) be the unit ball of \({\mathbb {R}}^n\) and write g as a smooth metric on \(B_1\). First we consider the Euclidean case with \(g=g_0=\sum _i dx^i \otimes dx^i\) on \(B_1\). The general case is a small perturbation of the Euclidean case.

4.3.1 The Euclidean Case \((B_1, g_0)\)

In this case, an almost complex structure J on \(B_1\) can be regarded as a function in \(W^{m,2}(B_1, M_n({\mathbb {R}}))\) such that \(J^2=-id\) and \(J^T+J=0\), where \(M_n({\mathbb {R}})\) is the set of all real \(n\times n\) matrices and \(J^T\) is the transpose of matrix J. The inner product of \(A, B \in T^1_1(B_1)\) reads \(\langle A, B\rangle =\sum _{i,j=1}^n A^j_i B^j_i\) for \(A=A_i^j dx^i \otimes \frac{\partial }{\partial x^j}\) and \(B=B_i^j dx^i \otimes \frac{\partial }{\partial x^j}\). Thus, the inner product of (1, 1) tensor fields on \(B_1\) can be viewed as the inner product of two vectors in Euclidean space \({\mathbb {R}}^{n^2}\).

First we need to write the Euler-Lagrange equation in a good divergence form in the sense of Definition 2.

Lemma 5

Suppose J is a \(W^{m,2}\) weakly m-harmonic almost structure on \((B_1,g_0)\), \(m=2, 3\). Then J satisfies the following in distributional sense,

$$\begin{aligned} \varDelta ^m J = \varPsi \left( J, \nabla J, \cdots , \nabla ^{2m-1}J\right) \end{aligned}$$
(42)

where \(\varPsi \) can be rewritten as a linear combination of the following terms, for any fixed constant matrix \(\lambda _0\in M_n({\mathbb {R}})\),

$$\begin{aligned} \nabla ^{\alpha }*\bigg ((J-\lambda _{0})*\nabla ^{\beta }J*\nabla ^{\gamma }J\bigg ) \quad or\quad \lambda _0*\nabla ^{\alpha }*\bigg ((J-\lambda _{0})*\nabla ^{\delta }J\bigg ), \end{aligned}$$

where \(\alpha ,\beta ,\gamma ,\delta \) are multi-indices such that \(1\le |\alpha | \le 2m-1\), \(0\le |\beta |,|\gamma |,|\delta |\le m\), \(|\alpha |+|\beta |+|\gamma |=2m\) and \(|\alpha |+|\delta |=2m\).

Lemma 5 will be proved in Appendix. Now we prove Theorem 2 for the Euclidean case \((B_1,g_0)\). First, we use the normalized energy E(J; x, r) defined by replacing u,\(B_r\) by J, \(B_r(x)\) respectively in (32). That is, due to \(n=2m\),

$$\begin{aligned} E(J;x,r)= \sum _{l=1}^{m} \bigg (\int _{B_r(x)} |\nabla ^l J|^{\frac{2m}{l}} \bigg )^{\frac{l}{2m}}. \end{aligned}$$

For any fixed \(R_0 \in (0,1)\), we have that for every \(\epsilon _0>0\), there exists \(r_0 \in (0, 1-R_0)\) such that

$$\begin{aligned} \sup _{x \in {\overline{B}}_{R_0}}E(J;x,r_0) < \epsilon _0. \end{aligned}$$
(43)

For \(x_0 \in {\overline{B}}_{R_0}\), \(J_{x_0, r_0}(x):=J(x_0+r_0x)\) is also a \(W^{m,2}\) m-harmonic almost structure on \((B_1,g_0)\) with

$$\begin{aligned} E(J_{x_0,r_0}; 0,1)=E(J; x_0,r_0)<\epsilon _0. \end{aligned}$$

By Lemma 5, \(J_{x_0,r_0}\) admits a good divergence form (see Definition 2) with \(\varPsi _L=0\). Then it follows from Lemma 4 that by choosing suitable \(\epsilon _0>0\) in (43), there exists \(\theta _0 \in (0,\frac{1}{2})\) and \(p_0=\frac{4m}{3}\) such that

$$\begin{aligned}&D_{p_0}(J;x_0,\theta _0 r_0)=D_{p_0}(J_{x_0,r_0};0, \theta _0) \nonumber \\&\quad \le \sqrt{\theta _0} D_{p_0}(J_{x_0,r_0};0,1) =\sqrt{\theta _0} D_{p_0}(J;x_0,r_0). \end{aligned}$$

A standard iteration argument shows that there exists \(\alpha \in (0,1)\) such that

$$\begin{aligned} D_{p_0}(J;x_0, r) \le C r^\alpha , \quad \forall r \in (0,r_0). \end{aligned}$$

This, combined with the Morrey’s lemma, yields that \(J \in C^{0,\alpha }({\overline{B}}_{R_0})\), hence that \(J \in C^{0,\alpha }(B_1)\).

4.3.2 The General Case \((B_1, g)\)

In this subsection, we prove the Hölder regularity of the general case on \((B_1,g)\) by a perturbation method. We start by recalling the scaling invariance of the functional \({\mathcal {E}}_m(J)\) in critical dimension \(n=2m\). If \(g_{\lambda }:=\lambda ^2 g\) for some positive real number \(\lambda \), then \({\mathcal {E}}_m(J,g)= {\mathcal {E}}_{m}(J, g_{\lambda }),\) where \({\mathcal {E}}_{m}(J, g)=\int _M \big | \varDelta ^{\frac{m}{2}}_{g} J\big |^2 dV_{g}.\) It follows that if J is a weakly m-harmonic on (M, g), then J is also m-harmonic on \((M,g_{\lambda })\). If we take the geodesic normal coordinates on the unit geodesic ball centered at fixed point in \((M,g_{\lambda })\), then the metric \(g_{\lambda }\) in such local coordinates converges to the Euclidean metric in \(C^{\infty }(B_1)\) as \(\lambda \) goes to infinity. Hence, we can assume that, by a scaling if necessary, the metric g on \(B_1\) is sufficiently close to the Euclidean metric in the sense

$$\begin{aligned} |g_{ij}(x)-\delta _{ij}| + \sum _{k=1}^{2m} | D^k g_{ij}(x)| \le \delta _0, \quad \forall x\in B_1 \end{aligned}$$
(44)

where \(\delta _0\) is sufficiently small and will be determined later. Now we prove Theorem 2 in the general case \((B_1,g)\).

Firstly, we introduce an operator \({\mathfrak {m}}\) which maps a (1, 1) tensor field A on \((B_1,g)\) to a \(n\times n\) real matrix valued function,

$$\begin{aligned} A_{{\mathfrak {m}}}:={\mathfrak {m}}(A)=(A_i^j) \end{aligned}$$

where \(A=A_i^j dx^i \otimes \frac{\partial }{\partial x^j}\). In other words, A denotes tensor field and \(A_{{\mathfrak {m}}}\) denotes its coefficient matrix. Let us denote by \(\nabla \) the covariant derivative on \((B_1,g)\) and D the ordinary derivatives (i.e., \(D_k=\partial _k\)). Here it is necessary to emphasize the difference between the derivatives on tensor fields and matrix valued functions. For example, for \(A=A_i^j dx^i \otimes \partial _j\), we have

$$\begin{aligned} (\nabla _{\partial _k} A )_i^j=D_k A_i^j + A_i^s \varGamma _{k s}^j-A_s^j \varGamma _{ki}^s \end{aligned}$$

where \(\varGamma _{ij}^k\) denote the Christoffel symbols with respect to metric g. To simplify notation, we rewrite above equation as

$$\begin{aligned} \big ( \nabla _{\partial _k} A \big )_{{\mathfrak {m}}}=D_k A_{{\mathfrak {m}}}+ Dg *A_{{\mathfrak {m}}} \end{aligned}$$

where \(D_k A_{{\mathfrak {m}}}=D_k (A_i^j)=(D_k A_i^j)\). Similarly, there holds

$$\begin{aligned} \big ( \varDelta A \big )_{{\mathfrak {m}}}= \varDelta A_{{\mathfrak {m}}} + Dg *D A_{{\mathfrak {m}}} + \big ( D^2g + Dg *Dg) *A_{{\mathfrak {m}}}. \end{aligned}$$
(45)

Recall the m-harmonic almost complex structure equation (61), i.e.,

$$\begin{aligned} \varDelta ^m J = T(J, \nabla J, \cdots , \nabla ^{2m-1}J). \end{aligned}$$

We will reduce above equation to a perturbation form of the Euclidean case step by step. As a example, we show how to handle the term \(\varDelta ^m J\). Repeated application of (45) yields

$$\begin{aligned} (\varDelta ^m J )_{{\mathfrak {m}}}= \varDelta ^m J_{{\mathfrak {m}}} + L_1(D^ig, D^j J_{{\mathfrak {m}}}) \end{aligned}$$

where \(L_1\) stands for the lower order terms in the following form

$$\begin{aligned} L_1=\sum D^{i_1}g *\cdots *D^{i_s} g*D^j J_{{\mathfrak {m}}} \end{aligned}$$

with \(i_\mu \ge 1, \mu =1, \cdots , s\), \(j \ge 0\) and \(j+\sum _{\mu =1}^s i_{\mu }=2m\). Let \(\varDelta _0= \sum _{i=1}^{2m} \partial ^2_i\) be the standard Laplace operator on Euclidean space \({\mathbb {R}}^{2m}\). Recall that for any smooth function f,

$$\begin{aligned} \varDelta f =g^{ij} \frac{\partial ^2 f}{\partial x^i \partial x^j}- g^{ij}\varGamma ^s_{ij}\frac{\partial f}{\partial x^s} =:g^{ij} \frac{\partial ^2 f}{\partial x^i \partial x^j} + Dg *Df, \end{aligned}$$

where we omit the terms \(g^{ij}\) in the expression \(Dg*Df\) due to boundedness of g. Then we have

$$\begin{aligned} \big ( \varDelta ^m -\varDelta _0^m \big )J_{{\mathfrak {m}}} = P_1 + L_2 (D^i g, D^j J_{{\mathfrak {m}}}), \end{aligned}$$

where \(P_1\) stands for the perturbation term in the following form

$$\begin{aligned} P_1&= \big ( g^{i_1 j_1}\cdots g^{i_m j_m} - \delta ^{i_1 j_1}\cdots \delta ^{i_m j_m} \big ) D^{2m}_{i_1 j_1 \cdots i_m j_m} J_{{\mathfrak {m}}} \nonumber \\&=:\sum _{|\beta |=2m} a_{\beta }(x) *D^{\beta } J_{{\mathfrak {m}}}, \end{aligned}$$
(46)

and \(L_2\) also stands for the lower order terms and has the similar expression as \(L_1\). Hence, we obtain

$$\begin{aligned} \left( \varDelta ^m J \right) _{{\mathfrak {m}}}= \varDelta ^m_0 J_{{\mathfrak {m}}} + P_1+ {\widetilde{L}}_1\left( D^ig, D^j J_{{\mathfrak {m}}}\right) , \end{aligned}$$

where \({\widetilde{L}}_1=L_1+L_2\) has the following form

$$\begin{aligned} {\widetilde{L}}_1=\sum D^{i_1}g *\cdots *D^{i_s} g*D^j J_{{\mathfrak {m}}} \end{aligned}$$

with \(i_\mu \ge 1, \mu =1, \cdots , s\), \(j \ge 0\) and \((\sum _{\mu =1}^s i_{\mu }) +j=2m\). Similar arguments apply to the nonlinear terms and yield

$$\begin{aligned} T\left( J, \nabla J, \cdots , \nabla ^{2m-1} J\right) = T_s\left( J_{{\mathfrak {m}}}, D J_{{\mathfrak {m}}}, \cdots , D^{2m-1} J_{{\mathfrak {m}}}\right) + P_2 + {\widetilde{L}}_2 \end{aligned}$$

where \(T_s\) admits a good divergence form as \(\varPsi \) in Lemma 5, \(P_2\) stands for the perturbation terms

$$\begin{aligned} P_2= \sum b_{ijk} *D^i \big ( (J_{{\mathfrak {m}}}-\lambda _0) *D^j J_{{\mathfrak {m}}} *D^k J_{{\mathfrak {m}}} \big ) \end{aligned}$$
(47)

where \(b_{ijk}\) consists of \(|g^{st}-\delta ^{st}|\), \(0\le j,k \le m\) and \(i+j+k=2m\), and \({\widetilde{L}}_2\) stands for the lower order terms in the following form

$$\begin{aligned} {\widetilde{L}}_2=\sum D^{i_1}g *\cdots *D^{i_s}g *D^j J_{{\mathfrak {m}}} *D^k J_{{\mathfrak {m}}} *D^l J_{{\mathfrak {m}}} \end{aligned}$$

with \(i_{\mu } \ge 1\) \(\mu =1, \cdots , s\), \(0\le j+k+l\le 2m-1\) and \(\big (\sum _{\mu =1}^s i_{\mu } \big )+ j+k+l=2m\). By the arguments above, we get the final reduced equation about \(J_{{\mathfrak {m}}}\)

$$\begin{aligned} \varDelta _0^m J_{{\mathfrak {m}}}= T_s + {\mathcal {P}} + {\mathcal {L}} \end{aligned}$$
(48)

where \({\mathcal {P}}=P_2 - P_1\) and \({\mathcal {L}}={\widetilde{L}}_2 - {\widetilde{L}}_1\). In other words, the nonlinear part of (48) consists of three types of terms: terms that admit a good divergence form, the perturbation terms and the lower order terms. Now recall the definition of E(u, r) and \(D_p(u,r)\) in (32) and (33) respectively. Then, we claim that for any given \(\tau \in (0,1)\), there exists \(\delta _0>0\), \(\epsilon _0>0\) and \(\theta _0 \in (0,\frac{1}{2})\) such that if the metric g satisfies (44) and \(E(J_{{\mathfrak {m}}},1)<\epsilon _0\), then we have

$$\begin{aligned} D_{p_0}(J_{{\mathfrak {m}}}, \theta _0) \le \theta _0^\tau \big (D_{p_0}(J_{{\mathfrak {m}}}, 1)+ \Vert Dg\Vert _{C^{2m-1}(B_1)} \big ). \end{aligned}$$
(49)

where \(p_0=\frac{4m}{3}\). The above claim is a direct consequence of Lemma 4 provided \({\mathcal {P}} \equiv 0\). Hence the key point is to prove that the inequality (38) in Lemma 4 still holds with additional nonlinear terms \({\mathcal {P}}\). We claim that, there holds

$$\begin{aligned} \Vert D \omega _{{\mathcal {P}}} \Vert _{L^{p_0}(B_1)} \le C \big ( \delta _0 \Vert D J_{{\mathfrak {m}}} \Vert _{L^{p_0}(B_1)} + E(J_{{\mathfrak {m}}},1) \Vert Dg \Vert _{C^{2m-1}(B_1)} \big ), \end{aligned}$$
(50)

where \(\omega _{{\mathcal {P}}}(x):= \int _{{\mathbb {R}}^{2m}} G(x-y) {\mathcal {P}}({\widetilde{J}}_{{\mathfrak {m}}})(y) dy.\) We now turn to proving (50). Since

$$\begin{aligned} a(x) D^{2m} J_{{\mathfrak {m}}}= D^{2m-1} \big ( a(x) DJ_{{\mathfrak {m}}} \big ) + \text{ Lower } \text{ order } \text{ terms, } \end{aligned}$$

and

$$\begin{aligned}&\bigg \Vert D\int _{{\mathbb {R}}^{2m}} D^{2m-1}G(x-y) {\widetilde{a}}(y) D {\widetilde{J}}_{{\mathfrak {m}}}(y) dy \bigg \Vert _{L^{p_0}\left( {\mathbb {R}}^{2m}\right) } \\&\quad = \bigg \Vert \int _{{\mathbb {R}}^{2m}} D^{2m}G(x-y) {\widetilde{a}}(y) D {\widetilde{J}}_{{\mathfrak {m}}}(y) dy \bigg \Vert _{L^{p_0}\left( {\mathbb {R}}^{2m}\right) } \\&\quad \le C \Vert {\widetilde{a}}(x) D {\widetilde{J}}_{{\mathfrak {m}}}(x) \Vert _{L^{p_0}\left( {\mathbb {R}}^{2m}\right) } \\&\quad \le C \Vert a(x)\Vert _{L^\infty (B_1)} \Vert D J_{{\mathfrak {m}}}(x) \Vert _{L^{p_0}(B_1)} \\&\quad \le C \,\delta _0 \, \Vert D J_{{\mathfrak {m}}}(x) \Vert _{L^{p_0}(B_1)}, \end{aligned}$$

it follows from estimates for lower order terms in Lemma 4 that (50) holds for the terms \(a(x) D^{2m} J_{{\mathfrak {m}}}\) in (46). In the same manner, (50) also holds for the terms in (47). Hence the decay estimate (49) holds and it implies \(J_{{\mathfrak {m}}} \in C^{0,\alpha }(B_1)\) for some \(\alpha \in (0,1)\).

5 Higher Regularity for m-Harmonic Almost Complex Structures

We state the higher regularity results for a class of semilinear elliptic equations as a generalization in [5]. The proof follows essentially Proposition 7.1 in [5].

Theorem 3

Suppose \(n\ge 2m\) and \(u\in C^{0,\mu } \cap W^{m,2}(B_1,{\mathbb {R}}^K)\) satisfies

$$\begin{aligned} \varDelta ^m u = \varPsi \left( x, \nabla u, \cdots , \nabla ^{2m-1} u\right) \end{aligned}$$
(51)

in distributional sense, where \(\varPsi \) can be divided into two parts: the highest order terms H and lower order terms L, i.e., \(\varPsi =H+L\), which admit the following structures:

$$\begin{aligned} H&=\sum _{k=0}^{m-1} \nabla ^k \cdot g_k, \text{ where } |g_k| \le C \sum _{l=1}^m |\nabla ^l u|^{\frac{2m-k}{l}}, \\ L&=\sum _{k=0}^{m-1} \nabla ^k \cdot {\widetilde{g}}_k, \text{ where } |{\widetilde{g}}_k| \le C \sum _{\gamma } \left( \prod _{i} |\nabla ^{\gamma _i}u| \right) \text{ for } \sum _{i}|\gamma _i| \le 2m-1-k. \end{aligned}$$

Then, \(u\in C^\infty (B_1, {\mathbb {R}}^K)\).

Proof

Gastel and Scheven in [5] proved the theorem in the case \(\varPsi =H\). According to the proof of Proposition 7.1 in [5], it suffices to prove the following two claims in the case \(\varPsi =L\):

  1. (1)
    $$\begin{aligned} \sup _{B_\rho (x)\subset B_R} \rho ^{2m-n-2\mu } \int _{B_\rho (x)}|\nabla ^m u|^2<\infty , \quad \forall \, \,0<R<1, \end{aligned}$$
    (52)
  2. (2)

    For every non-integer \(\nu :=[\nu ]+\sigma \in (0,m)\), if \(u\in C^{[\nu ],\sigma }(B_1, {\mathbb {R}}^K)\) and

    $$\begin{aligned} \sup _{B_\rho (x)\subset B_R} \rho ^{2m-n-2\nu } \int _{B_\rho (x)}|\nabla ^m u|^2<\infty , \quad \forall \,0<R<1, \end{aligned}$$
    (53)

    then we have that, for \(0\le k\le m-1\) and \(B_\rho (x) \subset B_R\), there holds

    $$\begin{aligned} \bigg ( \rho ^{2m-n}\int _{B_\rho (x)} |{\widetilde{g}}_k|^{\frac{2m}{2m-k}} \bigg )^{\frac{2m-k}{2m}} \le C \rho ^{\frac{m+1}{m}\nu } \end{aligned}$$
    (54)

Before proceeding to prove claims, we make some conventions: fix \(R\in (0,1)\), always assume \(B_\rho (x) \subset B_R\), and C stand for the positive constants only dependent of \(m,n,\Vert u\Vert _{C^{0,\mu }(B_R)}\).

We first prove the Claim (1) by standard integral estimates. Since \(u\in C^{0,\mu }(B_1)\), we have

$$\begin{aligned} \Vert u-{\overline{u}}\Vert _{L^\infty (B_\rho (x))} \le C [u]_{\mu ;B_R} \rho ^\mu \le C \rho ^\mu , \end{aligned}$$

where \({\overline{u}}=\frac{1}{|B_\rho (x)|}\int _{B_\rho (x)} u(y)\). To simplify the proof in the following estimate, we assume \(\Vert u-{\overline{u}}\Vert _{L^\infty (B_\rho (x))} \le 1\).

By Gagliardo-Nirenberg interpolation inequality, we have that, for \(1\le l\le m-1\), there holds

$$\begin{aligned} \bigg ( \rho ^{2m-n} \int _{B_\rho (x)}|\nabla ^l u|^{\frac{2m}{l}} \bigg )^{\frac{l}{2m}}&\le C \Vert u-{\overline{u}}\Vert _{L^\infty }^{1-\frac{l}{m}} \bigg ( \rho ^{2m-n} \int _{B_\rho (x)}|\nabla ^m u|^2 \bigg )^{\frac{l}{2m}} \nonumber \\&\quad +C \Vert u-{\overline{u}}\Vert _{L^\infty }. \end{aligned}$$
(55)

It follows that

$$\begin{aligned} \int _{B_\rho (x)} |\nabla ^l u|^{\frac{2m}{l}}&\le C \Vert u-{\overline{u}}\Vert _{L^\infty }^{\left( 1-\frac{l}{m}\right) \frac{2m}{l}} \int _{B_\rho (x)}|\nabla ^m u|^2 + C \rho ^{n-2m}\Vert u-{\overline{u}}\Vert _{L^\infty }^{\frac{2m}{l}} \nonumber \\&\le C \Vert u-{\overline{u}}\Vert _{L^\infty }^{\frac{2}{m}} \int _{B_\rho (x)}|\nabla ^m u|^2 + C \rho ^{n-2m}\Vert u-{\overline{u}}\Vert _{L^\infty }^{2}. \end{aligned}$$
(56)

On the other hand, by Hölder’s inequality, it follows from (55) that, for \(1\le l \le m-1\) and \(q \in [1,\frac{2m}{l}]\)

$$\begin{aligned} \begin{aligned} \bigg ( \rho ^{lq-n} \int _{B_\rho (x)}|\nabla ^l u|^{q} \bigg )^{\frac{1}{q}}&\le C \Vert u-{\overline{u}}\Vert _{L^\infty }^{1-\frac{l}{m}} \bigg ( \rho ^{2m-n} \int _{B_\rho (x)}|\nabla ^m u|^2 \bigg )^{\frac{l}{2m}} \\&\quad +C \Vert u-{\overline{u}}\Vert _{L^\infty }. \end{aligned} \end{aligned}$$
(57)

We choose a cut-off function \(\eta \in C^\infty _0(B_\rho (x),[0,1])\) such that

$$\begin{aligned} \eta |_{B_{\frac{\rho }{2}}(x)} \equiv 1 \quad \text{ and } \quad \Vert \nabla ^l \eta \Vert _{L^\infty } \le C \rho ^{-l}, \quad \forall \, l\in {\mathbb {N}}. \end{aligned}$$

Testing (51) with \(\eta ^{2m}(u-{\overline{u}})\), we compute

$$\begin{aligned} \int \eta ^{2m} |\nabla ^m u|^2 dy&\le C \sum _{k=0}^{m-1} \int |\nabla ^m u| \cdot |\nabla ^k(u-{\overline{u}})| \cdot |\nabla ^{m-k} \eta ^{2m}| \nonumber \\&\quad + C \sum _{k=0}^{m-1} \sum _{j=0}^{k} \int |\nabla ^j (u-{\overline{u}})| \cdot |\nabla ^{k-j} \eta ^{2m}| \cdot |{\widetilde{g}}_k| \nonumber \\&=: \sum _{k=0}^{m-1} I_k + \sum _{k=0}^{m-1} \sum _{j=0}^{k} II_{kj}. \end{aligned}$$

Let \(\epsilon _1, \epsilon _2\in (0,1)\) be constants to be chosen later. For \(I_0\), we obtain

$$\begin{aligned} I_0&\le C \int |\nabla ^m u| \cdot |u-{\overline{u}}| \cdot |\nabla ^{m} \eta ^{2m}| \nonumber \\&\le C \rho ^{\frac{n}{2}-m} \Vert u-{\overline{u}}\Vert _{L^\infty } \bigg (\int \eta ^{2m} |\nabla ^m u|^2\bigg )^{\frac{1}{2}} \nonumber \\&\le \epsilon _1 \int \eta ^{2m} |\nabla ^m u|^2 + C_{\epsilon _1} \rho ^{n-2m}\Vert u-{\overline{u}}\Vert _{L^\infty }^2 \nonumber \\&\le \epsilon _1 \int \eta ^{2m} |\nabla ^m u|^2 + C_{\epsilon _1} \rho ^{n-2m+2\mu }. \end{aligned}$$

For \(1\le k \le m-1\), we obtain

$$\begin{aligned} I_k&\le C \rho ^{k-m} \int |\nabla ^m u| \cdot |\nabla ^k u| \cdot \eta ^{m+k} \nonumber \\&\le C \rho ^{k-m} \bigg (\int \eta ^{2m} |\nabla ^m u|^2\bigg )^{\frac{1}{2}} \bigg (\int \eta ^{2k} |\nabla ^k u|^2\bigg )^{\frac{1}{2}} \nonumber \\&\le \epsilon _1 \int \eta ^{2m} |\nabla ^m u|^2 + C_{\epsilon _1} \rho ^{2k-2m} \int \eta ^{2k} |\nabla ^k u|^2 \nonumber \\&\le \epsilon _1 \int \eta ^{2m} |\nabla ^m u|^2 \nonumber \\&\quad + C_{\epsilon _1} \rho ^{2k-2m} \bigg ( \epsilon _2 \rho ^{2m-2k} \int _{B_\rho (x)}|\nabla ^m u|^2 + C_{\epsilon _2} \rho ^{n-2k}\Vert u-{\overline{u}}\Vert _{L^\infty }^2\bigg ) \nonumber \\&\le \epsilon _1 \int \eta ^{2m} |\nabla ^m u|^2 + \epsilon _2 C_{\epsilon _1} \int _{B_\rho (x)}|\nabla ^m u|^2 + C_{\epsilon _1}C_{\epsilon _2} \rho ^{n-2m}\Vert u-{\overline{u}}\Vert _{L^\infty }^2 \nonumber \\&\le C\epsilon _1 \int \eta ^{2m} |\nabla ^m u|^2 + \epsilon _2 C_{\epsilon _1} \int _{B_\rho (x)}|\nabla ^m u|^2 + C_{\epsilon _1}C_{\epsilon _2} \rho ^{n-2m+2\mu } \end{aligned}$$

where we use (57) with \(q=2\) and Young’s inequality in the fourth inequality. Next, we estimate \(II_{00}\) as follows

$$\begin{aligned} II_{00}&\le C \sum _\gamma \int |u-u| \cdot \eta ^{2m} \cdot \prod _{i}|\nabla ^{\gamma _i}u| \\&\le C \sum _\gamma \rho ^{\frac{n}{p_0}}\cdot \Vert u-{\overline{u}}\Vert _{L^\infty } \prod _i \Vert \eta ^{|\gamma _i|}\nabla ^{\gamma _i} u\Vert _{L^{\frac{2m}{|\gamma _i|}}} \\&\le C \Vert u-{\overline{u}}\Vert _{L^\infty } \bigg (\rho ^n + \sum _{\gamma _i \ne 0} \int \eta ^{2m} |\nabla ^{|\gamma _i|} u|^{\frac{2m}{|\gamma _i|}}\bigg ) \\&\le C \Vert u-{\overline{u}}\Vert _{L^\infty } \bigg (\rho ^n + \sum _{l=1}^m \int _{B_{\rho }(x)} |\nabla ^l u|^{\frac{2m}{l}}\bigg ) \\&\le C \Vert u-{\overline{u}}\Vert _{L^\infty } \int _{B_\rho (x)} |\nabla ^m u|^2 + C \rho ^n \Vert u-{\overline{u}}\Vert _{L^\infty } \\&\quad + C \bigg ( \Vert u-{\overline{u}}\Vert _{L^\infty }^{1+\frac{2}{m}} \int _{B_\rho (x)} |\nabla ^m u|^2 +\rho ^{n-2m} \Vert u-{\overline{u}}\Vert _{L^\infty }^{3} \bigg ) \\&\le C \rho ^{\mu } \int _{B_\rho (x)} |\nabla ^m u|^2 + C \rho ^{n-2m+2\mu } \end{aligned}$$

where we use (56) in the fifth inequality, and \(1=\frac{1}{p_0}+\frac{1}{2m}\sum _i |\gamma _i|.\) Note that, due to \(\sum _i |\gamma _i| \le 2m-1\), it follows that \(p_0 \in [1, 2m]\). Similar arguments apply to \(II_{k,0}\) and we obtain, for \(1\le k \le m-1\),

$$\begin{aligned} II_{k0} \le C \rho ^{\mu } \int _{B_\rho (x)} |\nabla ^m u|^2 + C \rho ^{n-2m+2\mu }. \end{aligned}$$

By Hölder’s inequality and Young’s inequality, we have that, for \(1\le k \le m-1\),

$$\begin{aligned} \int \eta ^{2m} |{\widetilde{g}}_k|^{\frac{2m}{2m-k}}&\le \epsilon _2 \int _{B_\rho (x)} |\nabla ^m u|^2 + C_{\epsilon _2} \bigg ( \rho ^n + \sum _{l=1}^{m-1} \int _{B_\rho (x)} |\nabla ^l u|^{\frac{2m}{l}} \bigg ) \nonumber \\&\le \left( \epsilon _2+C_{\epsilon _2} \rho ^{\frac{2\mu }{m}}\right) \int _{B_\rho (x)} |\nabla ^m u|^2 + C_{\epsilon _2} \rho ^{n-2m+2\mu }, \end{aligned}$$
(58)

where we apply (56) in second inequality. Now for \(1\le j <k\le m-1\), we have

$$\begin{aligned} II_{kj}&\le C \rho ^{j-k} \int |{\widetilde{g}}_k| \cdot |\nabla ^j u| \cdot \eta ^{2m+j-k} \\&\le C \rho ^{j-k} \Vert \eta ^j \nabla ^j u\Vert _{L^{\frac{2m}{k}}} \Vert \eta ^{2m-k} {\widetilde{g}}_k\Vert _{L^{\frac{2m}{2m-k}}} \\&\le C \rho ^{(j-k)\frac{2m}{k}} \int _{B_\rho (x)} |\nabla ^j u|^{\frac{2m}{k}} + C \int \eta ^{2m} |{\widetilde{g}}_k|^{\frac{2m}{2m-k}} \\&\le C \left( \epsilon _2+C_{\epsilon _2}\rho ^{\frac{2\mu }{m}}\right) \int _{B_\rho (x)} |\nabla ^m u|^2 + C_{\epsilon _2} \rho ^{n-2m+2\mu } \end{aligned}$$

where we use (57) with \(q=\frac{2m}{k}\) and (58) in the last inequality. Similarly, we obtain, for \(1\le k \le m-1\)

$$\begin{aligned} II_{kk}\le (\epsilon _2 + C_{\epsilon _2} \rho ^{\frac{2\mu }{m}}) \int _{B_\rho (x)} |\nabla ^m u|^2 + C_{\epsilon _2} \rho ^{n-2m+2\mu }. \end{aligned}$$

Combining above all estimates, we deduce that

$$\begin{aligned} \int \eta ^{2m} |\nabla ^m u|^2 dy&\le C\epsilon _1 \int \eta ^{2m} |\nabla ^m u|^2 + C \left( \epsilon _2+C_{\epsilon _2}\rho ^{\frac{2\mu }{m}}\right) \int _{B_\rho (x)} |\nabla ^m u|^2 \\&\quad + C_{\epsilon _1,\epsilon _2} \rho ^{n-2m+2\mu }. \end{aligned}$$

Thus, by choosing \(\epsilon _1, \epsilon _2, \rho _0\) small enough, we have that, for all \(\rho \le \rho _0\), there holds

$$\begin{aligned} \int _{B_{\frac{\rho }{2}}} |\nabla ^m u|^2 \le \varepsilon \int _{B_{\rho }} |\nabla ^m u|^2 + C \rho ^{n-2m+2\mu } \end{aligned}$$

where \(\varepsilon < 2^{2m-n-2\mu }\) is a fixed positive number. A standard iteration argument implies (52).

The task is now to prove Claim (2). Since \(u \in C^{[\nu ],\sigma }(B_1)\) with \(\nu =[\nu ]+\sigma \), we know that, there exists a Taylor polynomials \(P_x\) at the points x such that

$$\begin{aligned} \Vert u \Vert _{C^{[\nu ],\sigma }(B_R)}\le C <\infty ,\quad \Vert u-P_x \Vert _{L^\infty (B_\rho (x))}\le C \rho ^{\nu }. \end{aligned}$$

By Gagliardo-Nirenberg interpolation inequality and (53), we have that, for \(\nu < l \le m\), there holds

$$\begin{aligned}&\bigg ( \rho ^{2m-n} \int _{B_\rho (x)} |\nabla ^l u|^{\frac{2m}{l}} \bigg )^{\frac{l}{2m}} \\&\quad \le C \Vert u-P_x \Vert _{L^\infty (B_\rho (x))}^{1-\frac{l}{m}} \bigg ( \rho ^{2m-n} \int _{B_\rho (x)} |\nabla ^m u|^{2} \bigg )^{\frac{l}{2m}} +C \Vert u-P_x \Vert _{L^\infty (B_\rho (x))} \\&\quad \le C \rho ^\nu . \end{aligned}$$

Let us compute

$$\begin{aligned} \bigg ( \rho ^{2m-n}\int _{B_\rho (x)} |{\widetilde{g}}_k|^{\frac{2m}{2m-k}} \bigg )^{\frac{2m-k}{2m}}&\le C \rho ^{(2m-n) \frac{2m-k}{2m}}\cdot \rho ^{\frac{n}{q_0}} \prod _{|\gamma _i|>\nu } \big \Vert \nabla ^{|\gamma _i|} u \big \Vert _{L^{\frac{2m}{|\gamma _i|}}(B_{\rho }(x))} \\&\le C \rho ^{\tau } \end{aligned}$$

where

$$\begin{aligned}&\frac{1}{q_0}+ \frac{1}{2m}\sum _{|\gamma _i|>\nu } |\gamma _i|=\frac{2m-k}{2m}, \nonumber \\&\quad \tau =(2m-n) \cdot \frac{2m-k}{2m} + \frac{n}{q_0} + \sum _{|\gamma _i|>\nu } \left( \nu +\frac{n-2m}{2m}|\gamma _i| \right) . \end{aligned}$$
(59)

Combining above two identities yields

$$\begin{aligned} \tau&=(2m-n) \cdot \frac{2m-k}{2m}+ \frac{n}{q_0} + n_{\nu } \nu +(n-2m) \bigg ( \frac{2m-k}{2m}-\frac{1}{q_0} \bigg ) \\&=\frac{2m}{q_0}+n_\nu \nu . \end{aligned}$$

where \(n_{\nu }=\big |\{\gamma _i: |\gamma _i|>\nu \}\big |\).

We claim that

$$\begin{aligned} \tau \ge \frac{m+1}{m} \nu . \end{aligned}$$
(60)

which implies (54). Obviously, (60) holds for \(n_\nu \ge 2\). For \(n_\nu =0\), (59) implies \(\frac{1}{q_0}=\frac{2m-k}{2m}\). Hence, for \(\nu \in (0,m)\)

$$\begin{aligned} \tau =\frac{2m}{q_0}=2m-k\ge m+1\ge \frac{m+1}{m} \nu . \end{aligned}$$

For \(n_\nu =1\), (59) and the fact \(k+\sum _i |\gamma _i| \le 2m-1\) imply \(\frac{1}{q_0} \ge \frac{1}{2m}\). Hence, for \(\nu \in (0,m)\),

$$\begin{aligned} \tau =\frac{2m}{q_0}+ \nu \ge 1+ \nu \ge \frac{m+1}{m} \nu . \end{aligned}$$

Thus, the claim (60) is proved. \(\square \)

As a direct consequence, Theorem 3 implies the smoothness of weakly m-harmonic almost complex structures.

Corollary 1

Suppose \(n\ge 2m\) and \(J \in C^{0,\alpha } \cap W^{m,2}\) is a weakly m-harmonic almost complex structure on \((M^n,g)\). Then J is smooth.