First, we consider the case
\(1 < p <n\). By assumption, we have
\(q < \frac{np}{n-p}\). Using the Poincaré-type inequality, Lemma
2.2 to differential forms
\(T(H(u))\)
$$ \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B} \bigr| ^{np/(n-p)}\,dx \biggr)^{(n-p)/np} \leq C_{1} \biggl( \int_{ B} \bigl|d\bigl(T\bigl(H(u)\bigr) \bigr)\bigr|^{p} \,dx \biggr)^{1/p}, $$
(2.10)
we find that
$$ \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr| ^{q}\,dx \biggr)^{1 / q} \leq C_{2} \biggl( \int_{B} \bigl|d\bigl(T\bigl(H(u)\bigr) \bigr)\bigr|^{p} \,dx \biggr)^{1/p}. $$
(2.11)
It is well known that, for any differential form
u,
\(d(T(u))=u_{B}\) and
\(\|u_{B} \|_{p, B} \leq C_{3} \|u \|_{p, B}\). Hence,
$$ \biggl( \int_{ B} \bigl|d\bigl(T\bigl(H(u)\bigr) \bigr)\bigr|^{p} \,dx \biggr)^{1/p} \leq C_{4} \biggl( \int _{ B} \bigl|H(u)\bigr|^{p} \,dx \biggr)^{1/p}. $$
(2.12)
Note that
$$\bigl\| \Delta G(u) \bigr\| _{p, B} = \bigl\| \bigl(d^{\star}d + d d^{\star}\bigr) G(u) \bigr\| _{p, B} \leq C_{5} \|u \|_{p, B}. $$
We have
$$\begin{aligned} \bigl\| H(u)\bigr\| _{p, B} &= \bigl\| u - \Delta G(u) \bigr\| _{p, B} \\ &\leq\|u \|_{p, B} + \bigl\| \Delta G(u) \bigr\| _{p, B} \\ &\leq C_{6} \|u\|_{p, B}. \end{aligned}$$
(2.13)
Combining (
2.11), (
2.12), and (
2.13), we obtain
$$ \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr| ^{q}\,dx \biggr)^{1 / q} \leq C_{7} \biggl( \int_{ B} |u|^{p} \,dx \biggr)^{1/p} $$
(2.14)
for
\(1 < p <n\). Next, for the case of
\(p \geq n\), since the
\(L^{p}\)-norm of
\(|T(H(u))- (T(H(u)))_{B}|\) increases with
p and
\(\frac{np}{n-p} \to \infty\) as
\(p \to n\). Then, there exists
\(1 < p_{0} <n\) such that
\(q < \frac{np_{0} }{ n-p_{0}} \). Hence, it follows that
$$\begin{aligned} & \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B}\bigr| ^{q}\,dx \biggr)^{1 / q} \\ &\quad\leq\biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B} \bigr| ^{n{p_{0}}/(n-{p_{0}} )}\,dx \biggr)^{(n-p_{0})/np_{0}} \\ & \quad\leq C_{8} \biggl( \int_{ B} \bigl|d\bigl(T \bigl(H(u)\bigr)\bigr)\bigr|^{p_{0}}\,dx \biggr)^{1/p_{0}} \\ &\quad\leq C_{9} \biggl( \int_{ B} \bigl|d\bigl(T \bigl(H(u)\bigr)\bigr)\bigr|^{p}\,dx \biggr)^{1/p} \\ &\quad\leq C_{10} \biggl( \int_{ B} |u|^{p}\,dx \biggr)^{1/p}. \end{aligned}$$
(2.15)
Hence, from (
2.14) and (
2.15), we obtain
$$ \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr| ^{q}\,dx \biggr)^{1 / q} \leq C_{10} \biggl( \int_{B} |u|^{p} \,dx \biggr)^{1/p} $$
(2.16)
for any
\(p>1\). Using the Hölder inequality with
\(1= \frac{q}{ n+q}+ \frac{n }{{n+q}}\), we obtain
$$\begin{aligned} &\int_{B} \varphi\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B} \bigr| \bigr)\,dx \\ &\quad= \int_{B} \frac{ \varphi(|T(H(u))- (T(H(u)))_{B} | ) }{ |T(H(u))- (T(H(u)))_{B}|^{\frac{nq}{{n+q}}}} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{\frac{nq}{{n+q}}}\,dx \\ &\quad\leq\biggl( \int_{B} \frac{{\varphi (|T(H(u))-(T(H(u)))_{B}|)}^{\frac{n+q}{q}} }{|T(H(u))-(T(H(u)))_{B}|^{n}}\,dx \biggr)^{\frac{q}{{n+q}}}\\ &\qquad{}\times{ \biggl( \int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}}. \end{aligned}$$
Applying Lemma
2.3 and noticing
\(A(t)\) is a concave function, we obtain
$$\begin{aligned} &\int_{B} \varphi\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr| \bigr)\,dx \\ &\quad\leq\biggl(\int_{B} K\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr|^{q}\bigr)\,dx \biggr)^{\frac{q}{n+q}} { \biggl( \int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad\leq\biggl(\int_{B} A\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr|^{q}\bigr)\,dx \biggr)^{\frac{q}{n+q}} { \biggl( \int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad\leq A^{\frac{q}{n+q}} \biggl(\int_{B} \bigl(\bigl|T \bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr|^{q} \bigr)\,dx \biggr) { \biggl( \int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad\leq C_{1}(n, q) K^{\frac{q}{n +q}} \biggl(\int _{B} \bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B} \bigr|^{q}\bigr)\,dx \biggr) \\ &\qquad{}\times{\biggl( \int _{B} \bigl|T\bigl(H(u)\bigr)-\bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad= C_{1}(n, q) \frac{\varphi( (\int_{B} (|T(H(u))- (T(H(u)))_{B} |^{q})\,dx )^{1/q} )}{ (\int_{B} (|T(H(u))- (T(H(u)))_{B} |^{q})\,dx )^{\frac{n}{ n+q} }} \\ &\qquad{}\times{ \biggl( \int _{B} \bigl|T\bigl(H(u)\bigr)-\bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n }{{n+q} }} \\ & \quad= C_{1}(n, q) \varphi\biggl( \biggl(\int_{B} \bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr|^{q}\bigr)\,dx \biggr)^{1/q} \biggr). \end{aligned}$$
(2.17)
Note that
φ is increasing and satisfies
\(\Delta_{2}\)-condition, substituting (
2.16) into (
2.17) gives
$$ \int_{B} \varphi\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B} \bigr| \bigr)\,dx \leq C_{11} \varphi \biggl( \biggl( \int_{ B} |u|^{p} \,dx \biggr)^{1/p} \biggr). $$
(2.18)
Let
\(h(t)= \int_{0}^{t} \frac{\varphi(s) }{ s}\,ds\). From (
2.1) we know that
\(\varphi(t)/t^{q}\) is decreasing with
t, thus,
$$h(t) = \int_{0}^{t} \frac{\varphi(s) }{ s}\,ds = \int _{0}^{t} \frac{\varphi(s) }{ s^{q}}s^{q-1}\,ds \geq \varphi(t)/t^{q} \frac{1}{q} s^{q} \bigg|_{0}^{t} = \frac{1}{q} \varphi(t). $$
Similarly, using the fact that
\(\varphi(t)/t^{p}\) is increasing with
t, we have
\(h(t) \leq\frac{1}{p} \varphi(t)\). Therefore,
$$ \frac{1}{q} \varphi(t) \leq h(t) \leq\frac{1}{p} \varphi(t). $$
(2.19)
Let
\(g(t)=h(t^{1/p})\), then
\(( h(t^{1/p}) )' = \frac{1}{p} \frac{\varphi(t^{1/p})}{t}\) is increasing. Hence,
g is a convex function. From the definitions of
g and
h and using Jensen’s inequality to
g, we have
$$ h \biggl( \biggl(\int_{B} |u|^{p} \,dx \biggr)^{1/p} \biggr) = g \biggl(\int_{B} |u|^{p} \,dx \biggr) \leq\int_{B} g \bigl(|u|^{p}\bigr)\,dx = \int_{B} h\bigl(|u|\bigr)\,dx. $$
(2.20)
Combining (
2.18), (
2.19), and (
2.20), we have
$$\begin{aligned} &\int_{B} \varphi\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B} \bigr| \bigr)\,dx \\ &\quad \leq C_{11} \varphi\biggl( \biggl( \int_{ B} |u|^{p} \,dx \biggr)^{1/p} \biggr) \\ &\quad\leq C_{12} h \biggl( \biggl(\int_{B} |u|^{p} \,dx \biggr)^{1/p} \biggr) \\ &\quad\leq C_{13} \int_{B} h\bigl(|u|\bigr)\,dx \\ &\quad\leq C_{14} \int_{B} \varphi\bigl(|u|\bigr)\,dx, \end{aligned}$$
which indicates (
2.9) holds. We have completed the proof of Theorem
2.6. □