An interesting problem in Riemannian geometry is to study the relationship between the curvature and the topology of a Riemannian manifold. It is well-known that the space of harmonic $ r $-forms is isomorphic to its $ r $-th de Rham cohomology group for compact manifolds. When the manifold is non-compact, it is natural to study the square integrable harmonic forms. So it is an interesting problem in geometry and topology to find sufficient conditions on a complete manifold such that we can obtain vanishing theorems of harmonic forms.

Let $ (M,g,\mathcal{F}) $ be a $ (n+m) $-dimensional complete foliated Riemannian manifold with a foliation $ \mathcal{F} $ of codimention $ m $ and a bundle-like metric $ g $ with respect to $ \mathcal{F} $. Let $ d_{B} $ be the restriction of the exterior differential operator $ d $ on the basic forms $ \Omega_{B}^{r}(\mathcal{F}) $. Then the formal dual operator of $ d_{B} $ is defined by

$ \begin{equation*} \delta_{B}=(-1)^{m(r+1)+1}\star_{B}(d_{B}-\kappa_{B}\wedge)\star_{B}, \end{equation*} $

where $ \star_{B} $ is the basic Hodge star operator and $ \kappa_{B} $ is the basic part of the mean curvature form $ \kappa $. Then the basic Laplician operator $ \Delta_{B} $ acting on the space of basic $ r $-forms $ \Omega_{B}^{r}(\mathcal{F}) $ is given by

$ \begin{equation*} \Delta_{B}=d_{B}\delta_{B}+\delta_{B}d_{B}. \end{equation*} $

A basic $ r $-form $ \omega\in \Omega_{B}^r $ is said to be harmonic if $ \Delta_{B}\omega=0 $. It is well-known that $ \omega $ is harmonic if and only if $ d_{B}\omega=0 $ and $ \delta_{B}\omega=0 $ when $ M $ is compact. Hence, for any $ p\geq 2 $, we say a basic $ r $-form $ \omega\in \Omega_{B}^r(\mathcal{F}) $ is a basic $ p $-harmonic $ r $-form if it satisfies the following properties:

$ \begin{equation*} \begin{cases} d_{B}\omega=0,\\ \delta_{B}(|\omega|^{p-2}\omega)=0. \end{cases} \end{equation*} $

It is easy to see that when $ p=2 $ and $ M $ is compact, a basic $ p $-harmonic $ r $-form is exactly a basic harmonic $ r $-form.

It is worth noting that the main tools to study the spaces of harmonic $ r $-forms are the Bochner–Weizenböck type formulas. For basic harmonic forms, some vanishing theorems are obtained on compact foliated Riemannian manifolds. For example, M. Min-Oo et al. ([2]) proved that if the transversal curvature operator of $ \mathcal{F} $ is positive definite, there is no non-trivial basic harmonic $ r $-forms on a closed foliated Riemannian manifold. Recently, Jung–Liu ([3]) studied the basic $ r $-forms on a complete foliated Riemannian manifold. Under the assumption that the mean curvature form is bounded and co-closed, they obtained some vanishing theorems on complete foliated Riemannian manifolds.

For $ p $-harmonic $ 1 $-forms, Zhang ([4]) proved that if $ M $ is a complete non-compact Riemannian manifold with non-negative Ricci curvature, there is non-trivial $ L^q $ $ p $-harmonic $ 1 $-form for $ p>1 $ and $ 0<q<\infty $. Later, inspired by Zhang's result, Chang–Guo–Sung ([5]) extended Zhang's result and obtained the compactness for any bounded set of $ p $-harmonic $ 1 $-forms. In 2017, Dung ([1]) obtained some vanishing theorems for $ L^{p} $ $ p $-harmonic $ r $-forms on complete non-compact sub-manifolds of Euclidean space. For further details, the readers can refer to [6–8] and the references therein.

Motivated by these results, in this paper, our aim is to study basic $ L^p $ $ p $-harmonic $ r $-forms on a complete foliated Riemannian manifold with nonnegative transversal curvature operator. As usual, we define the space of the basic $ p $-harmonic $ r $-forms on $ M $ by

$ \begin{equation*} \mathcal{H}^{r,p}_{B}(L^p(M))=\{\omega\in \Omega_{B}^r(\mathcal{F})| d_{B}\omega=0,\delta_{B}(|\omega|^{p-2}\omega)=0,\displaystyle{\int}_{M}|\omega|^p\mu_M<\infty\}. \end{equation*} $

In fact, we establish the following vanishing theorem.

Theorem 1.1    Let $ (M,g,\mathcal{F}) $ be a foliated Riemannian manifold with all leaves compact. Assume that the basic part of the mean curvature form is bounded and co-closed. If the transversal curvature operator of $ \mathcal{F} $ is nonnegative and positive at least one point, any basic $ L^p $ $ p $-harmonic $ r $-form is trivial.

We should remark that our main theorem generalized Jun–Liu's result ([3, Main Theorem]) when $ p=2 $.

The rest of this paper is organized as follows. In Section 2, we recall some preliminary knowledge about foliated Riemannian manifolds and some useful lemmas. In Section 3, we will give a detailed proof of our main Theorem 1.1.

Let $ (M,g,\mathcal{F}) $ be a $ (n+m) $-dimensional complete foliated Riemannian manifold with a foliation $ \mathcal{F} $ of co-dimension $ m $ and a bundle-like metric $ g $ with respect to $ \mathcal{F} $. Let $ TM $ be the tangent bundle of $ M $, $ T\mathcal{F} $ its integrable subbundle given by $ \mathcal{F} $, and $ Q=TM/T\mathcal{F} $ the corresponding normal bundle of $ \mathcal{F} $. Then we have an exact sequence of vector bundles

$ \begin{equation*} 0\to T\mathcal{F}\to TM\overset{\pi}{\rightarrow } Q\to 0, \end{equation*} $

where $ \pi: TM\to Q $ is a projection. Then we have a bundle map $ \sigma: Q\to T\mathcal{F}^{\perp}\subset TM $ satisfying $ \pi\circ \sigma={\rm Id} $.

Let $ g_Q $ be the holonomy invariant metric on $ Q $ induced by $ g $. That is, for any vector fields $ X\in \Gamma(T\mathcal{F}) $, we have $ \mathcal{L}_Xg_{Q}=0 $, where $ \mathcal{L}_{X} $ is the transverse Lie derivative. Let $ \nabla=\nabla^Q $ be the transverse Levi-Civita connection on the normal bundle $ Q $ ([9, 10]). In fact, for any $ s\in \Gamma(Q) $ and $ Y_s=\sigma(s)\in \Gamma(T\mathcal{F}^{\perp}) $, the connection $ \nabla $ is defined by

$ \begin{equation*} \nabla_Xs=\begin{cases} \pi([X,Y_s]), &\text{for}\quad X\in \Gamma(T\mathcal{F}),\\ \pi(\nabla^M_XY_s), &\text{for}\quad X\in \Gamma(T\mathcal{F}^{\perp}), \end{cases} \end{equation*} $

where $ \nabla^M $ is the Levi-Civita connection with respect to the Riemannian metric $ g $ on $ M $. Then the transverse curvature tensor $ R^Q $ of $ \nabla $ is given by

$ \begin{equation*} R^Q(X,Y)s=\nabla_X\nabla_Ys-\nabla_Y\nabla_Xs-\nabla_{[X,Y]}s, \end{equation*} $

for any $ X,Y\in \Gamma(TM) $ and $ s\in \Gamma(Q) $. Let $ {\rm Ric}^Q $ be the transversal Ricci operator of $ \mathcal{F} $ with respect to the transversal Levi-Civita connection $ \nabla $.

A differential form $ \omega\in \Omega^r(M) $ is called basic, if for any $ X\in \Gamma(T\mathcal{F}) $, we have

$ \begin{equation*} \iota_{X}\omega=0,\quad \iota_{X}d\omega=0, \end{equation*} $

where $ \iota_X $ is the interior product with respect to $ X $. In a local chart $ (x^1,\cdots, x^n;y^1,\cdots, y^m) $ of $ \mathcal{F} $, a basic $ r $-form $ \omega $ can be written as

$ \begin{equation*} \omega=\sum\limits_{i_1<\cdots<i_r}\omega_{i_1\cdots i_r}(y^1,\cdots,y^m) dy^{i_1}\wedge \cdots \wedge dy^{i_r}, \end{equation*} $

where the coefficient functions $ \omega_{i_1\cdots i_r} $ are independent of $ x $. We denote $ \Omega_{B}^r(\mathcal{F}) $ by the set of all basic $ r $-forms on $ M $. Then we have $ \Omega^r(M)=\Omega_{B}^r(\mathcal{F})\oplus \Omega^r_{B}(\mathcal{F})^{\perp} $.

Let $ \star_{B}: \Omega_{B}^r(\mathcal{F})\to \Omega_{B}^{m-r}(\mathcal{F}) $ be the basic Hodge star operator defined by

$ \begin{equation*} \star_{B}\omega=(-1)^{n(m-r)}\star (\omega\wedge \chi_{\mathcal{F}}), \forall \omega\in \Omega_{B}^{r}(\mathcal{F}), \end{equation*} $

where $ \star $ is the Hodge star operator with respect to $ g $ and $ \chi_{\mathcal{F}} $ is the characteristic form of $ \mathcal{F} $. Then we can easily check that, for any basic $ r $-forms $ \alpha,\beta \in \Omega_{B}^r(\mathcal{F}) $,

$ \begin{equation*} \alpha\wedge \star_{B}\beta=\beta\wedge \star_{B} \alpha,\quad \star_{B}^2\alpha=(-1)^{r(m-r)}\alpha. \end{equation*} $

Let $ \nu $ be the transversal volume form such that $ \star \nu=\chi_{\mathcal{F}} $, The point-wise inner product $ \langle\cdot,\cdot\rangle $ on $ \Omega_{B}^{r}(\mathcal{F}) $ is given by

$ \begin{equation*} \langle \alpha,\beta\rangle \nu=\alpha\wedge \star_{B}\beta, \end{equation*} $

where $ \alpha,\ \beta\in \Omega_{B}^{r}(\mathcal{F}). $ So the global inner product can be defined by

$ \begin{equation*} (\alpha,\beta)_{B}=\displaystyle{\int}_{M}\langle \alpha,\beta\rangle \mu_{M}=\displaystyle{\int}_{M}\alpha\wedge \star_{B}\beta \wedge \chi_{\mathcal{F}}, \end{equation*} $

where $ \mu_{M}=\nu\wedge \chi_{\mathcal{F}} $ is the volume form.

Let $ d_{B} $ be the restriction of $ d $ to the basic forms, i.e., $ d_{B}=d|_{\Omega_{B}^{*}(\mathcal{F})}. $ Then the operator $ \delta_{B}:\Omega_{B}^r(\mathcal{F})\to \Omega_{B}^{r-1}(\mathcal{F}) $ is defined by

$ \begin{equation*} \delta_{B}\omega=(-1)^{m(r+1)+1}\star_{B}(d_{B}-\kappa_{B}\wedge )\star_{B}\omega, \end{equation*} $

where $ \kappa $ is the mean curvature form of $ \mathcal{F} $ and $ \kappa_{B} $ is the basic part of $ \kappa $ ([11]). It is well that $ \delta_{B} $ is the formal adjoint of $ d_{B} $ with respect to the global inner product $ (\cdot, \cdot)_{B} $ ([12]). In general, $ \delta_{B} $ is not a restriction of $ \delta $ on $ \Omega_{B}^{r}(\mathcal{F}) $, i.e., $ \delta_{B}\neq \delta|_{\Omega_{B}^r(\mathcal{F})} $, where $ \delta $ is the formal adjoint of $ d $. But for any basic $ 1 $-form $ \omega $, we have $ \delta_{B}\omega=\delta\omega $. So the basic Laplacian $ \Delta_{B} $ acting on $ \Omega_{B}^{*}(\mathcal{F}) $ is defined by $ \Delta_{B}=d_{B}\delta_{B}+\delta_{B}d_{B}. $ It is well known that $ \Delta|_{\Omega_{B}^0(\mathcal{F})}=\Delta_{B}, $ where $ \delta $ is the Beltrami Laplacian of $ M $ ([13]).

In order to obtain the vanishing theorems for basic $ p $-harmonic $ r $-forms, we need the following lemmas.

Lemma 2.2 ([14, Lemma 3.2])    Let $ \mathcal{F} $ be a Riemannian foliation. Then the operators $ d_{B} $ and $ \delta_{B} $ on $ \Omega_{B}^{*}(\mathcal{F}) $ are given by

$ \begin{equation*} d_{B}=\sum\limits_{a=1}^m\theta^a\wedge \nabla_{E_a},\quad \delta_{B}=-\sum\limits_{a=1}^m\iota_{E_a}\nabla_{E_a}+\iota_{\kappa_{B}^{\sharp}}, \end{equation*} $

where $ \{E_a\}_{a=1}^m $ is a local orthonormal basic frame in $ Q $ and $ \{\theta^a\}_{a=1}^m $ its $ g_{Q} $-dual $ 1 $-form.

Proposition 2.1    For any basic form $ \phi\in \Omega_{B}^r(\mathcal{F}) $, the operator $ \delta_{B} $ can be also written as

$ \begin{equation*} \delta_{B}\phi=(-1)^{m(r+1)+1}\star_{B}d_{B}\star_{B}\phi+\iota_{\kappa_{B}^{\sharp}}\phi. \end{equation*} $

Furthermore, for any function $ f\in C^{\infty}(M) $, we have

$ \begin{equation*} \delta_{B}(f\phi)=f\delta_{B}\phi+(-1)^{m(r+1)+1}\star_{B}(d_{B}f\wedge \star_{B}\phi). \end{equation*} $

Proof    By direct calculation, we have

$ \begin{equation*} \begin{split} \delta_{B}(f\phi)=&(-1)^{m(r+1)+1}\star_{B}d_{B}\star_{B}(f\phi)+f\iota_{\kappa_{B}^{\sharp}}\phi\\ =&(-1)^{m(r+1)+1}\star_{B}d_{B}(f\star_{B}\phi)+f\iota_{\kappa_{B}^{\sharp}}\phi\\ =&(-1)^{m(r+1)+1}\star_{B}(d_{B}f\wedge \star_{B}\phi+fd_{B}\star_{B}\phi)+f\iota_{\kappa_{B}^{\sharp}}\phi\\ =&f\delta_{B}\phi+(-1)^{m(r+1)+1}\star_{B}(d_{B}f\wedge \star_{B}\phi). \end{split} \end{equation*} $

Lemma 2.3 On a Riemannian foliation $ \mathcal{F} $, we have that for any $ \phi\in \Omega_{B}^r(\mathcal{F}) $,

$ \begin{equation*} \Delta_{B}\phi=\nabla_{{\rm tr}}^{*}\nabla_{{\rm tr}}\phi+F(\phi)+A_{\kappa_{B}^{\sharp}}\phi, \end{equation*} $

where $ A_{\kappa_{B}^{\sharp}}\phi=L_{\kappa_{B}^{\sharp}}\phi-\nabla_{\kappa_{B}^{\sharp}}\phi $, $ L_{\kappa_{B}^{\sharp}}=d_{B}\iota(\kappa_{B}^{\sharp})+\iota(\kappa_{B}^{\sharp}) d_{B} $, and $ F $ is the transverse curvature operator locally given by

$ \begin{equation*} F(\phi)=\sum\limits_{a,b=1}^m\theta^a\wedge \iota_{E_b}R^Q(E_b,E_a)\phi. \end{equation*} $

Furthermore, since $ \frac{1}{2}\Delta_{B}|\phi|^2=\langle \nabla^{*}_{{\rm tr}}\nabla_{{\rm tr}}\phi,\phi\rangle-|\nabla_{\rm tr}\phi|^2 $, we obtain that for any $ \phi\in \Omega_{B}^{r}(\mathcal{F}) $,

$ \begin{equation*} \frac{1}{2}\Delta_{B}|\phi|^2=\langle \Delta_{B}\phi,\phi\rangle-|\nabla_{\rm tr} \phi|^2-\langle F(\phi),\phi\rangle-\langle A_{\kappa_{B}^{\sharp}}\phi,\phi\rangle. \end{equation*} $

Remark 1     The above operator $ A_{\kappa_{B}^{\sharp}}: \Omega_{B}^{r}(\mathcal{F})\rightarrow \Omega_{B}^{r}(\mathcal{F}) $ is $ C^{\infty}(M) $ linear. That means for every $ f\in C^{\infty}(M) $, it holds $ A_{\kappa_{B}^{\sharp}}(f\phi)=fA_{\kappa_{B}^{\sharp}}(\phi). $

Proof

$ \begin{equation*} \begin{split} A_{\kappa_{B}^{\sharp}}(f\phi)=d_{B}\iota(\kappa_{B}^{\sharp})(f\phi)+\iota(\kappa_{B}^{\sharp}) d_{B}(f\phi)-\nabla_{\kappa_{B}^{\sharp}}(f\phi) =:I_1+I_2-I_3 \end{split}. \end{equation*} $

By direct calculation, we have

$ \begin{equation*} \begin{split} I_1:=&d_{B}\iota(\kappa_{B}^{\sharp})(f\phi)=d_B(f\iota(\kappa_{B}^{\sharp})\phi)=d_Bf\wedge [\iota(\kappa_{B}^{\sharp})\phi]+fd_B\iota(\kappa_{B}^{\sharp})\phi, \\ I_2:=&\iota(\kappa_{B}^{\sharp}) d_{B}(f\phi)=\iota(\kappa_{B}^{\sharp})(d_Bf\wedge\phi+fd_B\phi)\\ =&\kappa_{B}^{\sharp}(f)\varphi-d_Bf\wedge [\iota(\kappa_{B}^{\sharp})\phi]+f\iota(\kappa_{B}^{\sharp})d_B\phi\\ I_3:=&\nabla_{\kappa_{B}^{\sharp}}(f\phi)=\kappa_{B}^{\sharp}(f)\phi+f\nabla_{\kappa_{B}^{\sharp}}\phi. \end{split} \end{equation*} $

And then

$ \begin{equation*} I_1+I_2-I_3=fd_B\iota(\kappa_{B}^{\sharp})\phi+f\iota(\kappa_{B}^{\sharp})d_B\phi-f\nabla_{\kappa_{B}^{\sharp}}\phi=fA_{\kappa_{B}^{\sharp}}(\phi). \end{equation*} $

Lemma 2.4 ([15, Lemma 3.4])    For any $ d_{B} $-closed $ r $-form $ \phi $ and $ f\in C^{\infty}(M) $, we have

$ \begin{equation} |d_{B}(f\phi)|=|d_{B}f\wedge \phi|\leq |d_{B}f|\cdot |\phi|. \end{equation} $

(2.1)

Lemma 2.5 ([15, Lemma 3.5])    Let $ \phi $ be a basic $ r $-form and $ \kappa_{B}^{\sharp} $ the dual vector field of $ \kappa_{B} $.Then we have

$ \begin{equation*} |\iota_{\kappa_{B}^\sharp}\phi|\leq |\kappa_{B}^{\sharp}||\phi|\leq |\kappa_{B}||\phi|. \end{equation*} $

Proof    Let $ \omega $ be any basic $ p $-harmonic $ r $-form on $ M $, $ 1\leq r\leq m-1 $. Then we have

$ \begin{equation*} \begin{cases} d_B\omega=0,\\ \delta_{B}(|\omega|^{p-2}\omega)=0. \end{cases} \end{equation*} $

Applying the Bochner formula to the form $ |\omega|^{p-2}\omega $ and Lemma 1 we obtain

$ \begin{equation*} \begin{split} \frac{1}{2}\Delta_{B}|\omega|^{2p-2}= & \langle \Delta_{B}(|\omega|^{p-2}\omega),|\omega|^{p-2}\omega\rangle-|\nabla_{\rm tr}(|\omega|^{p-2}\omega)|^2\\ &-\langle F(|\omega|^{p-2}\omega),|\omega|^{p-2}\omega\rangle-|\omega|^{2p-4}\langle A_{\kappa_{B}^{\sharp}}(\omega),\omega\rangle. \end{split} \end{equation*} $

On the other hand, it holds

$ \begin{equation*} \frac{1}{2}\Delta_{B}|\omega|^{2p-2}= \frac{1}{2}\Delta_{B}(|\omega|^{p-1})^2=|\omega|^{p-1}\Delta_{B}|\omega|^{p-1}-|d_{B}|\omega|^{p-1}|^2. \end{equation*} $

By the first Kato's inequality, we have $ |\nabla_{\rm tr}(|\omega|^{p-2}\omega)|\geq |d_{B}|\omega|^{p-1}|. $ Then, we have

$ \begin{equation} \begin{split} |\omega|\Delta_{B}|\omega|^{p-1}+|\omega|^{p-2}\langle F(\omega),\omega\rangle\leq &\langle \Delta_{B}(|\omega|^{p-2}\omega),\omega\rangle-|\omega|^{p-2}\langle A_{\kappa_{B}^{\sharp}}(\omega),\omega\rangle \end{split}. \end{equation} $

(3.1)

Let $ B_l=\{y\in M|\rho(y)\leq l\} $, where $ \rho(y) $ is the distance between leaves through $ y $ and a fixed point. Then we can define a basic function $ \varphi_l $ on $ M $ satisfying the following properties:

$ \begin{equation} \begin{cases} 0\leq \varphi_l(y)\leq 1, &\text{for any} \quad y\in M,\\ \text{ supp}\;\varphi_l\subset B_{2l},\\ \varphi_l(y)=1, &\text{for any} \quad y\in B_l, \end{cases} \end{equation} $

(3.2)

and

$ \begin{equation} \lim\limits_{l\to +\infty}\varphi_l=1,\quad |d_{B}\varphi_l|\leq \frac{C_1}{l}, \end{equation} $

(3.3)

where $ C_1 $ is a positive constant independent of $ l $. ([16]) In the following, we will denote $ \varphi_l $ by $ \varphi $ without confusion.

Then multiplying both sides of inequality (3.1) by $ \varphi^2 $ and integrating over $ M $, we obtain

$ \begin{equation} \begin{split} &\displaystyle{\int}_{M}\varphi^2|\omega|\Delta_{B}|\omega|^{p-1}\mu_{M}+\displaystyle{\int}_{M}|\omega|^{p-2}\langle F(\omega),\varphi^2\omega\rangle \mu_{M}\\ \leq &\displaystyle{\int}_{M}\langle \Delta_{B}(|\omega|^{p-2}\omega),\varphi^2\omega\rangle\mu_{M}-\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}\langle A_{\kappa_{B}^{\sharp}}(\omega),\omega\rangle\mu_{M}. \end{split} \end{equation} $

(3.4)

In order to express the following estimate explicitly, we set

$ \begin{equation*} \begin{split} L_1:= &\displaystyle{\int}_{M}\varphi^2|\omega|\Delta_{B}|\omega|^{p-1}\mu_{M},\\ R_1:=&\displaystyle{\int}_{M}\langle \Delta_{B}(|\omega|^{p-2}\omega),\varphi^2\omega\rangle\mu_{M},\\ R_2:=&\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}\langle A_{\kappa_{B}^{\sharp}}(\omega),\omega\rangle\mu_{M}. \end{split} \end{equation*} $

Step 1: We compute the above three terms separately.

Firstly, we deal with $ L_1: $

$ \begin{equation*} \begin{split} L_1=&\displaystyle{\int}_{M}\langle d_{B}(\varphi^2|\omega|),d_{B}|\omega|^{p-1}\rangle \mu_{M}\\ =&2(p-1)\displaystyle{\int}_{M}\varphi |\omega|^{p-1}\langle d_{B}\varphi,d_{B}|\omega|\rangle\mu_{M}+(p-1)\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}\\ \geq & -2(p-1)\displaystyle{\int}_{M}\varphi |\omega|^{p-1}| d_{B}\varphi|\cdot \big|d_{B}|\omega|\big|\mu_{M}+(p-1)\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}. \end{split} \end{equation*} $

By Young's inequality, we have

$ \begin{equation} \begin{split} &\varphi |\omega|^{p-1}|d_{B}\varphi|\cdot |d_{B}|\omega||=|\omega|^{p-2}(\varphi|d_{B}|\omega||)\cdot (|d_{B}\varphi||\omega|)\\ \leq & |\omega|^{p-2}\left( \frac{\epsilon_1}{2}\varphi^2|d_{B}|\omega||^2+\frac{1}{2\epsilon_1}|d_{B}\varphi|^2|\omega|^{2}\right), \end{split} \end{equation} $

(3.5)

where $ \epsilon_1>0 $ is a constant. Then we have

$ \begin{equation} \begin{split} L_1\geq&-(p-1)\epsilon_1\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}-\frac{p-1}{\epsilon_1}\displaystyle{\int}_{M}|d_{B}\varphi|^2|\omega|^{p}\mu_{M}\\ &+(p-1)\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}\\ =&(p-1)(1-\epsilon_{1})\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}-\frac{p-1}{\epsilon_{1}}\displaystyle{\int}_{M}|d_{B}\varphi|^2|\omega|^{p}\mu_{M}. \end{split} \end{equation} $

(3.6)

Then we compute $ R_1: $

$ \begin{equation} \begin{split} R_1=&\displaystyle{\int}_{M}\langle d_{B}(|\omega|^{p-2}\omega),d_{B}(\varphi^2\omega)\rangle\mu_{M}\\ =&2(p-2)\displaystyle{\int}_M|\omega|^{p-3}\varphi\langle d_{B}|\omega|\wedge \omega, d_{B}\varphi\wedge \omega\rangle\mu_{M}\\ \leq &2(p-2)\displaystyle{\int}_{M}\varphi|\omega|^{p-3}|d_{B}|\omega|\wedge \omega||d_{B}\varphi\wedge\omega|\mu_{M}\\ \leq &2(p-2)\displaystyle{\int}_{M}\varphi |\omega|^{p-1}| d_{B}\varphi|\cdot \big|d_{B}|\omega|\big|\mu_{M}\\ \leq&(p-2)\epsilon_1\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}+\frac{p-2}{\epsilon_1}\displaystyle{\int}_{M}|d_{B}\varphi|^2|\omega|^{p}\mu_{M}, \end{split} \end{equation} $

(3.7)

where we used the Young's inequality (3.5) again and $ \epsilon_1 $ is a positive constant.

Substituting (3.6) and (3.7) into (3.4), we have

$ \begin{equation} \begin{split} &A\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}+\displaystyle{\int}_{M}|\omega|^{p-2}\langle F(\omega),\varphi^2\omega\rangle \mu_{M}\\ \leq &B\displaystyle{\int}_{M}|d_{B}\varphi|^2|\omega|^{p}\mu_{M}-\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}\langle A_{\kappa_{B}^{\sharp}}(\omega),\omega\rangle\mu_{M}, \end{split} \end{equation} $

(3.8)

where $ A:=A(\epsilon_1)=(p-1)-(2p-3)\epsilon_1 $ and $ B:=B(\epsilon_1)=\frac{2p-3}{\epsilon_1} $. Since $ p-1>0 $, we can choose proper $ \epsilon_1>0 $ such that $ A>0 $.

At last, we calculate $ R_2: $

$ \begin{equation*} \begin{split} R_2:=&\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}\langle A_{\kappa_{B}^{\sharp}}(\omega),\omega\rangle\mu_{M}\\ =& \displaystyle{\int}_{M}\langle d_B (\iota_{\kappa_{B}^{\sharp}}\omega),(\varphi^2|\omega|^{p-2}\omega) \rangle\mu_{M}-\displaystyle{\int}_M\varphi^2|\omega|^{p-2}\langle \nabla_{\kappa_{B}^{\sharp}}\omega, \omega\rangle\mu_M \\ =:&R_{2.1}-R_{2.2}. \end{split} \end{equation*} $

From the Proposition 2.2, we have

$ \begin{equation} \begin{split} R_{2.1}=&\displaystyle{\int}_{M}\langle d_B \iota_{\kappa_{B}^{\sharp}}\omega,(\varphi^2|\omega|^{p-2}\omega) \rangle\mu_{M}\\ =&\displaystyle{\int}_{M}\langle \iota_{\kappa_{B}^{\sharp}}\omega,\delta_{B}(\varphi^2|\omega|^{p-2}\omega) \rangle\mu_{M}\\ =&\displaystyle{\int}_{M}\langle \iota_{\kappa_{B}^{\sharp}}\omega,\varphi^2\delta_{B}(|\omega|^{p-2}\omega)\rangle \mu_{M} +\displaystyle{\int}_{M}\langle \iota_{\kappa_{B}^{\sharp}}\omega,(-1)^{m(r+1)+1}\star_B\left(2\varphi d_{B}\varphi \wedge\star_B(|\omega|^{p-2}\omega)\right)\rangle\mu_{M}\\ =&\displaystyle{\int}_{M}\langle \iota_{\kappa_{B}^{\sharp}}\omega,(-1)^{m(r+1)+1}\star_B\left(2\varphi d_{B}\varphi \wedge\star_B(|\omega|^{p-2}\omega)\right)\rangle\mu_{M}. \end{split} \end{equation} $

(3.9)

So

$ \begin{equation} \begin{split} |R_{2.1}|\leq &2\displaystyle{\int}_{M}\left(\varphi|\iota_{\kappa_{B}^{\sharp}}\omega|\cdot|d_{B}\varphi\wedge\star_B\omega)|\right)|\omega|^{p-2}\mu_{M}\\ \leq &2\displaystyle{\int}_{M}(\varphi |\kappa_{B}||\omega|\cdot |d_{B}\varphi||\omega|)|\omega|^{p-2}\mu_{M}\\ =&2\displaystyle{\int}_{M}\varphi |\kappa_{B}||d_{B}\varphi||\omega|^p\mu_{M}\\ \leq & 2\sup\limits_{M}|\kappa_B|\displaystyle{\int}_{M}\varphi |d_{B}\varphi||\omega|^p\mu_{M}, \end{split} \end{equation} $

(3.10)

where $ \sup\limits_{M}|\kappa_B| $ is the supremum of the norm of $ \kappa_B $.

Next, we estimate

$ \begin{equation*} \begin{split} R_{2.2}:=&\displaystyle{\int}_{M}\langle \nabla_{\kappa_{B}^{\sharp}}\omega,\varphi^2|\omega|^{p-2}\omega \rangle \mu_{M}\\ =&\frac{1}{2}\displaystyle{\int}_M\varphi^2|\omega|^{p-2}\kappa_{B}^{\sharp}(|\omega|^2)\mu_M\\ =&\frac{1}{p}\displaystyle{\int}_M\varphi^2\kappa_{B}^{\sharp}(|\omega|^p)\mu_M\\ =&\frac{1}{p}\displaystyle{\int}_M\kappa_{B}^{\sharp}(\varphi^2|\omega|^p)\mu_M-\frac{1}{p}\displaystyle{\int}_M|\omega|^p\kappa_{B}^{\sharp}(\varphi^2)\mu_M\\ =&\frac{1}{p}\displaystyle{\int}_M\langle d_B(\varphi^2|\omega|^p),\kappa_{B}\rangle\mu_M-\frac{2}{p}\displaystyle{\int}_M\varphi|\omega|^p\kappa_{B}^{\sharp}(\varphi)\mu_M. \end{split} \end{equation*} $

Since $ \kappa_{B} $ is coclosed, we have

$ \begin{equation} \begin{split} |R_{2.2}|=\left|-\frac{2}{p}\displaystyle{\int}_M\varphi|\omega|^p\kappa_{B}^{\sharp}(\varphi)\mu_M\right| \leq \frac{2}{p}\displaystyle{\int}_M |\omega|^{p}( \varphi|\kappa_{B}|)|d_B\varphi|\mu_M \leq \frac{2}{p}\sup\limits_{M}|\kappa_{B}|\displaystyle{\int}_{M}\varphi |d_{B}\varphi||\omega|^p\mu_{M}. \end{split} \end{equation} $

(3.11)

Combining (3.8), (3.10), (3.11), we have

$ \begin{equation} \begin{split} &A\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}+\displaystyle{\int}_{M}|\omega|^{2p-4}\langle F(\omega),\varphi^2\omega\rangle \mu_{M}\\ \leq &B\displaystyle{\int}_{M}|d_{B}\varphi|^2|\omega|^p\mu_{M}+\left(2+\frac{2}{p}\right)\sup\limits_{M}|\kappa_{B}|\displaystyle{\int}_{M}\varphi |d_{B}\varphi| |\omega|^p\mu_{M}. \end{split} \end{equation} $

(3.12)

Step 2: Apply estimate (3.2) and (3.3) to inequality (3.12).

From (3.2) and (3.3), we have

$ \begin{equation*} \displaystyle{\int}_{M}|d_{B}\varphi|^2|\omega|^{p}\mu_{M}\leq \frac{C_1^2}{l^2} \displaystyle{\int}_{M}|\omega|^{p}\mu_{M}, \end{equation*} $

and

$ \begin{equation*} \displaystyle{\int}_M \varphi |d_{B}\varphi| |\omega|^{p}\mu_M\leq \frac{C_1}{l}\displaystyle{\int}_M |\omega|^{p}\mu_M. \end{equation*} $

Therefore, we

$ \begin{equation*} A\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}+\displaystyle{\int}_{M}|\omega|^{p-2}\langle F(\omega),\varphi^2\omega\rangle\leq\left[\frac{BC_1^2}{l^2}+\frac{C_1}{l}\right]\displaystyle{\int}_{M}|\omega|^p\mu_{M}. \end{equation*} $

Letting $ l\to +\infty $, it holds that

$ \begin{equation*} \lim\limits_{l\to +\infty}\displaystyle{\int}_{M}\varphi^2|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}=\displaystyle{\int}_{M}|\omega|^{p-2}|d_{B}|\omega||^2\mu_{M}=0, \end{equation*} $

and

$ \begin{equation*} \lim\limits_{l\to \infty}\displaystyle{\int}_{M}|\omega|^{p-2}\langle F(\omega),\varphi^2\omega\rangle=\displaystyle{\int}_{M}|\omega|^{p-2}\langle F(\omega),\omega\rangle\mu_{M}=0. \end{equation*} $

If the transverse curvature operator $ F $ is nonnegative and positive at some point, $ |\omega|=0 $. This completes the proof.

Remark 2    We should remark that, by applying the same method as above, we can obtain the vanishing theorem for basic $ L^q (q\geq p) $ $ p $-harmonic $ r $-form when the basic part of mean curvature form of $ \mathcal{F} $ vanishes.