Proof. Let $g:=(g_1,g_2)\in L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dxd\sigma(k)))\oplus_{\infty} L_{\infty}\\ (L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})$. Next, we show that

\begin{equation*}\left\|\Psi(g)\right\|_{\mathrm{bmo}_{A}^c({{{\mathbb R}}^d},\,{\mathcal M})}\lesssim\|g\|_ {L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dxd\sigma(k)))\oplus_{\infty} L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}.\end{equation*}

To do this, we only need to prove that, for any $B\in\mathfrak{B}_1$ and some constant $C_B$,

(3.13)\begin{align} &\left\|\left( \unicode{x02A0F}_{\!\!\!\!B}|\Psi(g)(z)-C_{B}|^2\,dz\right)^{\frac12}\right\|_{{\mathcal M}}\nonumber\\ &\quad\lesssim\|g\|_ {L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dxd\sigma(k)))\oplus_{\infty} L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}, \end{align}

and for any $B\in\mathfrak{B}_2$,

(3.14)\begin{align} &\left\|\left( \unicode{x02A0F}_{\!\!\!\!B}|\Psi(g)(z)|^2\,dz\right)^{\frac12}\right\|_{{\mathcal M}}\nonumber\\ &\quad\lesssim\|g\|_ {L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dxd\sigma(k)))\oplus_{\infty} L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}. \end{align}

Let $x_0+B_{k_0}\in\mathfrak{B}$ with $x_0\in{{{\mathbb R}}^d}$ and $k_0\in{\mathbb Z}\backslash{\mathbb N}$. Then we can write

\begin{align*} g_1(x,\,y,\,k)&=g_1(x,\,y,\,k)\mathbf{1}_{x_0+B_{k_0+4\sigma}}(x)+g_1(x,\,y,\,k) \mathbf{1}_{(x_0+B_{k_0+4\sigma})^{\complement}}(x)\\ &=:g_{11}(x,\,y,\,k)+g_{12}(x,\,y,\,k). \end{align*}

Set

\begin{equation*}C_{x_0+B_{k_0}}:=\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}|\det A|^{-\frac{k}{2}}\int_{x+B_k} g_{12}(x,\,y,\,k)\unicode{x02A0F}_{\!\!\!\!x_0+B_{k_0}}\varphi_k(y-z)\,dzdydx.\end{equation*}

Then

\begin{align*} & \unicode{x02A0F}_{\!\!\!\!x_0+B_{k_0}}\left|\Psi(g)(z)-C_{x_0+B_{k_0}}\right|^2\,dz\\ &\quad \leq 2|\det A|^{-k_0}\int_{x_0+B_{k_0}}\left|\int_{(x_0+B_{k_0+4\sigma})^{\complement}}\sum_{k=-\infty}^{-1}|\det A|^{-\frac{k}{2}}\right.\\ &\qquad\left.\times\int_{x+B_k}g_{12}(x,\,y,\,k) \left[\varphi_k(y-z)-Q_{x_0+B_{k_0}}\right]\,dydx\right|^2\,dz\\ &\qquad +2|\det A|^{-k_0}\int_{x_0+B_{k_0}}\left|\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}|\det A|^{-\frac{k}{2}}\right.\\ &\qquad\left.\times\int_{x+B_k}g_{11}(x,\,y,\,k) \varphi_k(y-z)\,dydx\right|^2\,dz\\ &\qquad +\unicode{x02A0F}_{\!\!\!\!x_0+B_{k_0}}\left|\int_{{{{\mathbb R}}^d}}\overline{\phi(x-z)}g_2(x)\,dx\right|^2\,dz\\ &=:\mathrm{II}_1+\mathrm{II}_2+\mathrm{II}_3, \end{align*}

where $Q_{x_0+B_{k_0}}:=\unicode{x02A0F}_{x_0+B_{k_0}}\varphi_k(y-z)\,dz$.

To deal with $\mathrm{II}_1$ and $\mathrm{II}_2$, by using the duality theory of noncommutative $L_p$-spaces, the Cauchy–Schwarz inequality, the fact that $L_2(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})=\mathbf{h}_2^{A,c}({{{\mathbb R}}^d},{\mathcal M})$, (2.1), and an estimate similar to that in [Reference Wang40, Lemma 3.5], we obtain

\begin{align*} \left\|\mathrm{II}_2\right\|_{{\mathcal M}} &\lesssim \|g_1\|_{L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L^c_2(\widetilde{\Gamma}_x, \,dyd\sigma(k)))}^2 \ \ \mathrm{and}\\ \quad\left\|\mathrm{II}_{1}\right\|_{{\mathcal M}}&\lesssim\|g_1\| _{L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L^c_2(\widetilde{\Gamma}_x,\,dyd\sigma(k))}^2. \end{align*}

For the last term $\mathrm{II}_3$, by (2.1), we obtain

\begin{align*} \left\|\mathrm{II}_3\right\|_{{\mathcal M}}&=\left\|\unicode{x02A0F}_{\!\!\!\!x_0+B_{k_0}}\left|\int_{{{{\mathbb R}}^d}}\overline{\phi(x-z)}g_2(x) \,dx\right|^2\,dz\right\|_{{\mathcal M}} \lesssim\left\|g_2\right\|_{L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2, \end{align*}

which, together with the estimates of $\mathrm{II}_1$ and $\mathrm{II}_2$, implies (3.13).

Next, we prove (3.14). Let $x_0+B_{k_0}\in\mathfrak{B}$ with $x_0\in{{{\mathbb R}}^d}$ and $k_0\in{\mathbb N}$. Then we can write

\begin{align*} &\unicode{x02A0F}_{\!\!\!\!x_0+B_{k_0}}\left|\Psi(g)(z)\right|^2\,dz\\ &\quad\leq 2|\det A|^{-k_0}\int_{x_0+B_{k_0}}\left|\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}|\det A|^{-\frac{k}{2}}\right.\\ &\qquad\left.\times\int_{x+B_k}g_1(x,\,y,\,k) \varphi_k(y-z)\,dydx\right|^2\,dz\\ &\qquad +2\unicode{x02A0F}_{\!\!\!\!x_0+B_{k_0}}\left|\int_{{{{\mathbb R}}^d}}\overline{\phi(x-z)}g_2(x)\,dx\right|^2\,dz\\ \quad =&:\mathrm{III}_1+\mathrm{III}_2, \end{align*}

To deal with $\mathrm{III}_1$, set

\begin{equation*}g_1(x,\,y,\,k):=g_1(x,\,y,\,k)\mathbf{1}_{x_0+B_{k_0+2\sigma}}(x)+g_1(x,\,y,\,k) \mathbf{1}_{(x_0+B_{k_0+2\sigma})^{\complement}}(x).\end{equation*}

From this and a similar argument of $\mathrm{II}_1$ and $\mathrm{II}_2$, we can deduce that, for $x_0\in{{{\mathbb R}}^d}$, $k_0\in{\mathbb N}$,

\begin{align*} &\left\|\mathrm{III}_{1}\right\|_{{\mathcal M}}\thicksim |\det A|^{-k_0}\\ &\qquad\times\left\|\int_{x_0+B_{k_0}}\left|\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}|\det A|^{-\frac{k}{2}}\int_{x+B_k}g_1(x,\,y,\,k) \varphi_k(y-z)\,dydx\right|^2\,dz\right\|_{{\mathcal M}}\\ &\quad\lesssim\|g_1\| _{L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L^c_2(\widetilde{\Gamma}_x,\,dyd\sigma(k))}^2. \end{align*}

For the term $\mathrm{III}_2$, from (2.1), we can conclude that, for $x_0\in{{{\mathbb R}}^d}$, $k_0\in{\mathbb N}$,

\begin{align*} \left\|\mathrm{III}_2\right\|_{{\mathcal M}}&=\left\|\unicode{x02A0F}_{\!\!\!\!x_0+B_{k_0}}\left|\int_{{{{\mathbb R}}^d}}\overline{\phi(x-z)}g_2(x) \,dx\right|^2\,dz\right\|_{{\mathcal M}}\\ &\lesssim\unicode{x02A0F}_{\!\!\!\!x_0+B_{k_0}}\int_{{{{\mathbb R}}^d}}\left|\overline{\phi(x-z)}\right|^2\,dxdz \left\|\int_{{{{\mathbb R}}^d}}\left|g_2(x)\right|^2\,dx\right\|_{{\mathcal M}}\\ &\lesssim\left\|\int_{{{{\mathbb R}}^d}}\left|g_2(x) \right|^2\,dx\right\|_{{\mathcal M}} \lesssim\left\|g_2\right\|_{L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2. \end{align*}

The above estimates imply (3.14). Therefore, we complete the proof of Lemma 3.2.

Proof. From the Fubini theorem and (2.9), we conclude that, for any $S_{{\mathcal M}}$-valued simple function $g$,

\begin{align*} &\int_{{{{\mathbb R}}^d}}\Psi\Phi(f)(z)g^*(z)\,dz\\ &\quad =\int_{{{{\mathbb R}}^d}}\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}\int_{x+B_k}|\det A|^{-k}f\ast\varphi_k(y) \overline{\varphi_k(y-z)}dydxg^*(z)\,dz\\ &\qquad+\int_{{{{\mathbb R}}^d}}\int_{{{{\mathbb R}}^d}}f*\phi(x)\overline{\phi(x-z)}g(z)^*\,dxdz\\ &\quad=\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}\int_{x+B_k}|\det A|^{-k}f\ast\varphi_k(y)(g\ast\varphi_k(y))^*\,dydx\\ &\qquad+\int_{{{{\mathbb R}}^d}}\phi*f(x)(\phi*g)^*(x)\,dx =\int_{{{{\mathbb R}}^d}}f(z)g^*(z)\,dz. \end{align*}

Therefore, $\Psi\Phi(f)=f.$

Proof. Set $ p=\frac q2 \gt 1. $ For simplicity, for $k\in{\mathbb Z}\backslash{\mathbb N}$, define

\begin{equation*} A_k(f)(x) := \unicode{x02A0F}_{\!\!\!\!x+B_k}|f(y)-f_{x+B_k}|^2\,dy, \end{equation*}

and, for $k\in{\mathbb N}$, define

\begin{equation*} D_k(f)(x) := \unicode{x02A0F}_{\!\!\!\!x+B_k}|f(y)|^2\,dy. \end{equation*}

Then the norm in the definition of $L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$ can be written as

\begin{equation*} \begin{aligned} \|f\|_{L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})} = \left[ \left\| \sup_{k\in{\mathbb Z}\setminus{\mathbb N}}^{\ \ \ \ \ +}A_k(f) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^p + \left\| \sup_{k\in{\mathbb N}}^{\ \ \ \ \ +}D_k(f) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^p \right]^{\frac 1q}. \end{aligned} \end{equation*}

By the definition of $L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}};\ell_{\infty})$, it is easy to check that this expression defines a norm.

Next, we prove a useful embedding estimate. By (2.5), if $i\leq j$, then $ B_i+B_j\subset B_{j+{\sigma}}. $ Thus, for arbitrary $i,j\in{\mathbb Z}$, $ B_i+B_j\subset B_{\max\{i,j\}+{\sigma}}. $ In particular, the quasi-triangle inequality for $\rho$ gives

\begin{equation*} \rho(y)\lesssim 1+\rho(x) \qquad\text{and}\qquad \rho(x)\lesssim 1+\rho(y), \qquad x\in y+B_1. \end{equation*}

Consequently,

\begin{equation*} \frac1{1+\rho(y)^2} \lesssim \frac1{1+\rho(x)^2}, \qquad x\in y+B_1. \end{equation*}

Averaging over $y+B_1$, we obtain

\begin{equation*} \frac1{1+\rho(y)^2} \lesssim \unicode{x02A0F}_{\!\!\!\!y-B_1}\frac1{1+\rho(x)^2}\,dx . \end{equation*}

Therefore,

\begin{equation*} \begin{aligned} \int_{{{{\mathbb R}}^d}}|f(y)|^2\frac1{1+\rho(y)^2}\,dy &\lesssim \int_{{{{\mathbb R}}^d}}|f(y)|^2 \unicode{x02A0F}_{\!\!\!\!y-B_1}\frac1{1+\rho(x)^2}\,dx\,dy\\ &=\int_{{{{\mathbb R}}^d}} \unicode{x02A0F}_{\!\!\!\!x+B_1}|f(y)|^2\,dy \frac1{1+\rho(x)^2}\,dx\\ & =\int_{{{{\mathbb R}}^d}}D_1(f)(x)\frac1{1+\rho(x)^2}\,dx . \end{aligned} \end{equation*}

Hence

\begin{equation*} \begin{aligned} \|f\|_{L_q({\mathcal M};L_2^c({{{\mathbb R}}^d},\frac1{1+\rho(x)^2}\,dx))}^2 &= \left\| \int_{{{{\mathbb R}}^d}}|f(y)|^2\frac1{1+\rho(y)^2}\,dy \right\|_{L_p({\mathcal M})}\\ &\lesssim \left\| \int_{{{{\mathbb R}}^d}}D_1(f)(x)\frac1{1+\rho(x)^2}\,dx \right\|_{L_p({\mathcal M})}. \end{aligned} \end{equation*}

By the Minkowski integral inequality and the Hölder inequality, we obtain

\begin{equation*} \begin{aligned} \left\| \int_{{{{\mathbb R}}^d}}D_1(f)(x)\frac1{1+\rho(x)^2}\,dx \right\|_{L_p({\mathcal M})}& \leq \int_{{{{\mathbb R}}^d}} \|D_1(f)(x)\|_{L_p({\mathcal M})}\frac1{1+\rho(x)^2}\,dx\\ &\leq \left\|\frac1{1+\rho(\cdot)^2}\right\|_{L_{p'}({{{\mathbb R}}^d})} \|D_1(f)\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}. \end{aligned} \end{equation*}

Here $p'=p/(p-1)$. Moreover, $\frac1{1+\rho(\cdot)^2}\in L_{p'}({{{\mathbb R}}^d})$. Indeed, since $\rho(x)=|\det A|^j$ on $B_{j+1}\backslash B_j$, we have

\begin{equation*} \begin{aligned} \int_{{{{\mathbb R}}^d}}\left[\frac1{1+\rho(x)^2}\right]^{p'}\,dx &\leq |B_0| + \sum_{j=0}^{\infty} \int_{B_{j+1}\setminus B_j} \left[\frac{1}{1+\rho(x)^2}\right]^{p'}dx\\ &\lesssim 1+ \sum_{j=0}^{\infty} |\det A|^j|\det A|^{-2p'j} \lt \infty, \end{aligned} \end{equation*}

because $2p' \gt 1$. Consequently,

\begin{equation*} \begin{aligned} \|f\|_{L_q({\mathcal M};L_2^c({{{\mathbb R}}^d},\frac1{1+\rho(x)^2}\,dx))} &\lesssim \|D_1(f)\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^{\frac12}\\ &\leq \left\| \sup_{k\in{\mathbb N}}^{\ \ \ \ \ +}D_k(f) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^{\frac12} \leq \|f\|_{L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})}. \end{aligned} \end{equation*}

Thus

\begin{equation*} L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M}) \hookrightarrow L_q({\mathcal M};L_2^c({{{\mathbb R}}^d},\frac1{1+\rho(x)^2}\,dx)) \end{equation*}

continuously.

Now let $\{f_n\}_{n\geq1}$ be a Cauchy sequence in $L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$. By the embedding estimate above, we know that $\{f_n\}$ is Cauchy in $ L_q({\mathcal M};L_2^c({{{\mathbb R}}^d},\frac1{1+\rho(x)^2}\,dx)). $ Since this space is complete, there exists $ f\in L_q({\mathcal M};L_2^c({{{\mathbb R}}^d},\frac1{1+\rho(x)^2}\,dx)) $ such that $ f_n\to f$ in $L_q({\mathcal M};L_2^c({{{\mathbb R}}^d},\frac1{1+\rho(x)^2}\,dx)). $ Since $\frac1{1+\rho(\cdot)^2}$ is bounded from below on every bounded subset of ${{{\mathbb R}}^d}$, this convergence implies local convergence in the column $L_q$-space.

Next, we claim that $f_n\to f$ in $L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$ as $n\rightarrow\infty$. Let $\varepsilon \gt 0$. Since $\{f_n\}$ is Cauchy in $L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$, there exists $N\in{\mathbb N}$ such that

\begin{equation*} \|f_n-f_m\|_{L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})} \lt \varepsilon, \qquad m,n\geq N. \end{equation*}

Fix $n\geq N$ and write $ h_m=f_n-f_m, $ $ h=f_n-f. $ Then

\begin{equation*} h_m\to h \quad\text{in}\quad L_q({\mathcal M};L_2^c({{{\mathbb R}}^d},\frac1{1+\rho(x)^2}\,dx)), \end{equation*}

and hence locally in the column $L_q$-space.

Now we justify the convergence of the local oscillation terms. For $k\in{\mathbb Z}$, define

\begin{equation*} T_k u(x):=\unicode{x02A0F}_{\!\!\!\!x+B_k}u(y)\,dy. \end{equation*}

Then

\begin{equation*} D_k(u)=T_k(|u|^2), \qquad A_k(u)=T_k(|u|^2)-|T_ku|^2. \end{equation*}

Let $E\subset{{{\mathbb R}}^d}$ be bounded and fix $k\in{\mathbb Z}$. Put $ K=E+B_k. $ Since $h_m\to h$ locally in the column $L_q$-space, we have

\begin{equation*} \|h_m-h\|_{L_q({\mathcal M};L_2^c(K))}\to0. \end{equation*}

For $u\in L_q({\mathcal M};L_2^c(K))$ and $x\in E$, since $x+B_k\subset K$, we have

\begin{equation*} D_k(u)(x) = \unicode{x02A0F}_{\!\!\!\!x+B_k}|u(y)|^2\,dy \leq |B_k|^{-1}\int_K |u(y)|^2\,dy. \end{equation*}

Thus

\begin{equation*} \|D_k(u)\|_{L_p(L_\infty(E)\overline{\otimes}{\mathcal M})} \leq |E|^{\frac1p}|B_k|^{-1}\|u\|_{L_q({\mathcal M};L_2^c(K))}^2. \end{equation*}

Moreover, for $v,w\in L_q({\mathcal M};L_2^c(K))$, the Cauchy–Schwarz inequality gives

\begin{align*} &\left\| x\mapsto \unicode{x02A0F}_{\!\!\!\!x+B_k}v(y)^*w(y)\,dy \right\|_{L_p(L_\infty(E)\overline{\otimes}{\mathcal M})}\\ &\quad\leq |E|^{\frac1p}|B_k|^{-1} \|v\|_{L_q({\mathcal M};L_2^c(K))}\|w\|_{L_q({\mathcal M};L_2^c(K))}. \end{align*}

Let $ v_m=h_m-h. $ Since $v_m\to0$ in $L_q({\mathcal M};L_2^c(K))$ and $\{h_m\}$ is bounded in this local column norm, we get

\begin{equation*} \begin{aligned} &D_k(h_m)-D_k(h)\\ &\quad=\unicode{x02A0F}_{\!\!\!\!x+B_k}\big(v_m(y)^*h_m(y)+h(y)^*v_m(y)\big)\,dy \longrightarrow0 \ \mathrm{in} \ L_p(L_\infty(E)\overline{\otimes}{\mathcal M}). \end{aligned} \end{equation*}

Next, by Kadison’s inequality for the positive unital averaging map $T_k$, we have

\begin{equation*} |T_ku(x)|^2\leq D_k(u)(x), \qquad x\in E. \end{equation*}

Consequently,

\begin{equation*} \|T_ku\|_{L_q(L_\infty(E)\overline{\otimes}{\mathcal M})} \leq |E|^{\frac1q}|B_k|^{-\frac12}\|u\|_{L_q({\mathcal M};L_2^c(K))}. \end{equation*}

Therefore, $ T_kh_m\to T_kh$ in $L_q(L_\infty(E)\overline{\otimes}{\mathcal M}). $ Since

\begin{equation*} |T_kh_m|^2-|T_kh|^2 = (T_kv_m)^*T_kh_m+(T_kh)^*T_kv_m, \end{equation*}

the noncommutative Hölder inequality gives $ |T_kh_m|^2\to |T_kh|^2$ in $ L_p(L_\infty(E)\overline{\otimes}{\mathcal M}). $ Using $ A_k(u)=D_k(u)-|T_ku|^2, $ we conclude that, for every fixed $k\in{\mathbb Z}$,

\begin{equation*} D_k(h_m)\to D_k(h), \qquad A_k(h_m)\to A_k(h) \end{equation*}

in $L_p(L_\infty(E)\overline{\otimes}{\mathcal M})$, hence locally in measure.

Now we use the Fatou property of noncommutative vector-valued $L_p$-spaces. Let $F\subset{\mathbb Z}\backslash{\mathbb N}$ be finite and let $E_j=B_j$, $j\in{\mathbb N}$. Since $A_k(h_m)\to A_k(h)$ locally in measure, the Fatou property of the finite-index vector-valued space $ L_p(L_\infty(E_j)\overline{\otimes}{\mathcal M};\ell_\infty^F) $ gives

\begin{equation*} \left\| \sup_{k\in F}^{\ \ \ \ \ +}A_k(h) \right\|_{L_p(L_\infty(E_j)\overline{\otimes}{\mathcal M})} \leq \liminf_{m\to\infty} \left\| \sup_{k\in F}^{\ \ \ \ \ +}A_k(h_m) \right\|_{L_p(L_\infty(E_j)\overline{\otimes}{\mathcal M})}. \end{equation*}

Since restriction to $E_j$ is contractive,

\begin{equation*} \left\| \sup_{k\in F}^{\ \ \ \ \ +}A_k(h_m) \right\|_{L_p(L_\infty(E_j)\overline{\otimes}{\mathcal M})} \leq \left\| \sup_{k\in{\mathbb Z}\setminus{\mathbb N}}^{\ \ \ \ \ +}A_k(h_m) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}. \end{equation*}

Letting $j\to\infty$ and using again the Fatou property, we obtain

\begin{equation*} \left\| \sup_{k\in F}^{\ \ \ \ \ +}A_k(h) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \leq \liminf_{m\to\infty} \left\| \sup_{k\in{\mathbb Z}\backslash{\mathbb N}}^{\ \ \ \ \ +}A_k(h_m) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}. \end{equation*}

Taking the supremum over all finite subsets $F\subset{\mathbb Z}\backslash{\mathbb N}$ yields

\begin{equation*} \left\| \sup_{k\in{\mathbb Z}\backslash{\mathbb N}}^{\ \ \ \ \ +}A_k(h) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \leq \liminf_{m\to\infty} \left\| \sup_{k\in{\mathbb Z}\backslash{\mathbb N}}^{\ \ \ \ \ +}A_k(h_m) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}. \end{equation*}

The same argument applied to the family $\{D_k\}_{k\in{\mathbb N}}$ gives

\begin{equation*} \left\| \sup_{k\in{\mathbb N}}^{\ \ \ \ \ +}D_k(h) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \leq \liminf_{m\to\infty} \left\| \sup_{k\in{\mathbb N}}^{\ \ \ \ \ +}D_k(h_m) \right\|_{L_p(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}. \end{equation*}

Consequently,

\begin{equation*} \|h\|_{L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})} \leq \liminf_{m\to\infty} \|h_m\|_{L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})} \leq \varepsilon. \end{equation*}

That is,

\begin{equation*} \|f_n-f\|_{L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})} \leq \varepsilon, \qquad n\geq N. \end{equation*}

Thus

\begin{equation*} f_n\to f \quad\text{in}\quad L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M}). \end{equation*}

In particular, since $f_n\in L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$ and $f=f_n-(f_n-f)$ for any fixed $n\geq N$, we obtain $ f\in L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M}). $ This proves that $L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$ is a Banach space.

Proof. Let $g\in L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$. For $k\in{\mathbb N}$ and $x\in{\mathbb R}^d$, put $ D_k(g)(x) = \unicode{x02A0F}_{x+B_k}|g(z)|^2\,dz . $ From Lemma 2.16 and [Reference Wang40, Lemma 4.3], we can deduce that

\begin{align*} &\left\| \sup_{k\in\mathbb Z\backslash{\mathbb N}}^{\ \ +} \frac1{|x+B_k|} \int_{(y,i)\in T(x+B_k)} |g*\varphi_i(y)|^2\,dy\,d\sigma(i) \right\|_{L_{\frac q2}(L_{\infty}({\mathbb R}^d)\overline{\otimes}{\mathcal M})}\\ &\lesssim \|g\|_{L_q\mathrm{mo}_A^c(\mathbb R^d,\mathcal M)}^2 . \end{align*}

It remains to estimate the low-frequency term $g*\phi$. Since $\phi\in\mathcal S(\mathbb R^d)$, for every sufficiently large $N$, $ |\phi(u)|\lesssim (1+\rho(u))^{-N}. $ By the Cauchy–Schwarz inequality, we obtain

\begin{equation*} |g*\phi(x)|^2 \le \|\phi\|_{L_1(\mathbb R^d)} \int_{\mathbb R^d}|\phi(x-y)|\,|g(y)|^2\,dy . \end{equation*}

Decomposing $\mathbb R^d$ into the annuli $x+B_0$ and $x+B_{j+1}\backslash B_j$, $j\ge0$, we get

\begin{equation*} \begin{aligned} |g*\phi(x)|^2 \lesssim \sum_{j=0}^{\infty} |\det A|^{-Nj} \int_{x+B_{j+1}}|g(y)|^2\,dy \lesssim \sum_{j=0}^{\infty} |\det A|^{-(N-1)j} D_{j+1}(g)(x). \end{aligned} \end{equation*}

Thus,

\begin{equation*} \begin{aligned} \|g*\phi\|_{L_q(L_{\infty}({\mathbb R}^d)\overline{\otimes}{\mathcal M})}^2 &=\left\||g*\phi|^2\right\|_{L_{\frac q2}(L_{\infty}({\mathbb R}^d)\overline{\otimes}{\mathcal M})}\\ &\lesssim \sum_{j=0}^{\infty} |\det A|^{-(N-1)j} \left\|D_{j+1}(g)\right\|_{L_{\frac q2}(L_{\infty}({\mathbb R}^d)\overline{\otimes}{\mathcal M})} \\ &\lesssim \left\| \sup_{m\in{\mathbb N}}^{\ \ +} D_m(g) \right\|_{L_{\frac q2}(L_{\infty}({\mathbb R}^d)\overline{\otimes}{\mathcal M})} \le \|g\|_{L_q\mathrm{mo}_A^c(\mathbb R^d,\mathcal M)}^2 . \end{aligned} \end{equation*}

Combining the two estimates proves the lemma.

Proof. For any $f\in L_p\mathrm{mo}_A^{c}({{{\mathbb R}}^d},\,{\mathcal M})$, we know that

\begin{align*} \|f\|_{L_p\mathrm{mo}_A^{c}({{{\mathbb R}}^d},\,{\mathcal M})}^p&=\left\| \sup^{\ \ \ \ +}_{k\in{\mathbb Z}\backslash{\mathbb N}}\unicode{x02A0F}_{\!\!\!\!x+B_k}\left|f(y)-f_{x+B_k}\right| ^2\,dy\right\|_{L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^{\frac p2}\\ &\quad+\left\|\sup^{\ \ \ \ +}_{k\in{\mathbb N}}\unicode{x02A0F}_{\!\!\!\!x+B_k}\left|f(y)\right| ^2\,dy\right\|_{L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^{\frac p2}. \end{align*}

Then, it suffices to show that for any $g:=(g_1,g_2)\in L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\, L_2^c(\widetilde{\Gamma}_x,\\ \,dyd\sigma(k))) \oplus_p L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})$,

(3.17)\begin{align} &\left\|\sup^{\ \ \ \ +}_{k\in\mathbb{Z}\backslash{\mathbb N}}\unicode{x02A0F}_{\!\!\!\!x+B_k} \left|\Psi(g)(z)-C_{x+B_k}\right|^2\,dz\right\|_{L_{\frac{p}{2}} (L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^{\frac p2}\nonumber\\ &\quad\lesssim \|g\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\, L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(k))) \oplus_p L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^p. \end{align}

and

(3.18)\begin{align} &\left\|\sup^{\ \ \ \ +}_{k\in{\mathbb N}}\unicode{x02A0F}_{\!\!\!\!x+B_k} \left|\Psi(g)(z)\right|^2\,dz\right\|_{L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} ^{\frac p2}\nonumber\\ &\quad\lesssim \|g\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\, L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(k))) \oplus_p L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^p, \end{align}

where $C_{x+B_k}$ is a constant depending on $x+B_k$. To show these inequalities, we write

\begin{align*} g_1(x,\,y,\,i)&=g_1(x,\,y,\,i)\mathbf{1}_{\omega+B_{k+4\sigma}}(x) +g_1(x,\,y,\,i) \mathbf{1}_{(\omega+B_{k+4\sigma})^{\complement}}(x)\\ &=:g_{11}(x,\,y,\,i)+g_{12}(x,\,y,\,i). \end{align*}

To estimate (3.17), we only need to show the $L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,\ell_\infty)$-norm of expressions

(3.19)\begin{align} \unicode{x02A0F}_{\!\!\!\!\omega+B_k} &\left|\int_{(\omega+B_{k+4\sigma})^{\complement}}\sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\right.\nonumber\\ &\left.\times \int_{x+B_i}g_{12}(x,\,y,\,i)\left[\varphi_i(y-z)-Q_{x+B_k}(y)\right]\,dydx\right|^2\,dz \end{align}

and

(3.20)\begin{align} \unicode{x02A0F}_{\!\!\!\!\omega+B_k}\left|\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\right. \left.\int_{x+B_i}g_{11}(x,\,y,\,i) \overline{\varphi_i(y-z)}\,dydx\right|^2\,dz \end{align}

are dominated by $\|g_1\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))}^2$, where $Q_{B}(y):=\unicode{x02A0F}_{\!\!B}\varphi_i(y-z)\,dz$ for $y\in{{{\mathbb R}}^d}$, and

(3.21)\begin{equation} \unicode{x02A0F}_{\!\!\!\!\omega+B_k}\left|\int_{{{{\mathbb R}}^d}}\overline{\phi(x-z)}g_2(x)\,dx\right|^2\,dz \end{equation}

are dominated by $\|g_2\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2$.

To estimate the $L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,\ell_\infty)$-norm of (3.19), choose $N_0 \gt \frac52$, it is easy to deduce that if $z\in \omega+B_{k}$, then

\begin{align*} &\left| \int_{(\omega+B_{k+4\sigma}) ^{\complement}} \sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\int_{x+B_i}g_{12}(x,\,y,\,i) \left[\varphi_i(y-z)-Q_{\omega+B_k}(y)\right]\,dydx\right|^2\\ &\quad\lesssim|\det A|^{-\frac k2}\int_{(\omega+B_{k+4\sigma})^{\complement}}[\rho(x-z)]^{\frac32} \unicode{x02A0F}_{\!\!\!\!\omega+B_k}\\ &\qquad\times\sum_{i=-\infty}^{-1}|\det A|^{-i}\int_{x+B_i} \left|\varphi_i(y-z)-\varphi_i(y-s)\right|^2 \,dyds\\ &\qquad\times\sum_{i=-\infty}^{-1}\int_{x+B_i}|g_{12}(x,\,y,\,i)|^2\,dydx\\ &\quad\lesssim \sum_{j=k}^{\infty}|\det A|^{-j}|\det A|^{(k-j)(N_0-\frac52)}\\ &\qquad\times\int_{(\omega+B_{j+4\sigma+1})\backslash (\omega+B_{j+4\sigma})} \sum_{i=-\infty}^{-1}\int_{x+B_i}|g_{12}(x,\,y,\,i)|^2\,dydx, \end{align*}

where the last inequality is a variant of the combination of Lemma 3.2 with a slight modification that the $L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))$-norm of $g$ is not taken outside. Next, we shall use a duality argument to estimate the $L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,\ell_\infty)$-norm of the right-hand side. To this end, we take a sequence of positive elements $\{a_k\}_{k\in\mathbb{Z}\backslash{\mathbb N}}\in L_{(\frac{p}{2})'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})$ such that $\|\sum_k a_k\|_{L_{(\frac{p}{2})'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}=1$. Then, we use Hölder’s inequality and the noncommutative Hardy–Littlewood maximal inequality to see that

\begin{align*} &\tau\int_{{\mathbb R}^d}\sum_{k=-\infty}^{-1} \sum_{j=k}^{\infty}|\det A|^{(k-j)(N_0-\frac52)}\unicode{x02A0F}_{\!\!\!\!\omega+B_{j+4\sigma+1}}\\ &\qquad\times\sum_{i=-\infty}^{-1}\int_{x+B_i}|g_{12}(x,\,y,\,i)|^2\,dydxa_k(\omega)d\omega\\ &\quad=\tau \int_{{\mathbb R}^d}\sum_{i=-\infty}^{-1}\int_{x+B_i}|g_{12}(x,\,y,\,i)|^2\,dy\sum_{j\in{\mathbb Z}} \unicode{x02A0F}_{\!\!\!\!\omega+B_{j+4\sigma+1}}\\ &\qquad\times\sum_{k=-\infty}^{\min\{j,-1\}}|\det A|^{(k-j)(N_0-\frac52)}a_k(\omega)d\omega dx\\ &\quad\leq \|g_1\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))}^2 \left\|\sum_{j\in{\mathbb Z}}\unicode{x02A0F}_{\!\!\!\!x+B_{j+4\sigma+1}}\right.\\ &\qquad\left.\times\sum_{k=-\infty}^{\min\{j,-1\}}|\det A|^{(k-j)(N_0-\frac52)}a_k(\omega)d\omega \right\|_{L_{(\frac{p}{2})'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\quad\lesssim \|g_1\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))}^2\\ &\qquad\times\left\|\sum_{k=-\infty}^{-1}\sum_{j=k}^{\infty}|\det A|^{(k-j) (N_0-\frac52)}a_k\right\|_{L_{(\frac{p}{2})'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \\ &\quad\lesssim \|g_1\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))}^2. \end{align*}

By taking the supremum in the above inequality over all $\{a_k\}_{k\in\mathbb{Z}\backslash{\mathbb N}}$ satisfying the stated properties, we conclude that the $L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,\ell_\infty)$-norm of (3.19) is dominated by $\|g_1\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))}^2$.

To estimate the $L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,\ell_\infty)$-norm of (3.20), we shall use the duality argument again. To this end, we take a sequence of positive elements $\{b_k\}_{k\in\mathbb{Z}\backslash{\mathbb N}}\in L_{(\frac{p}{2})'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})$ such that $\|\sum_{k\in{\mathbb Z}\backslash{\mathbb N}} b_k\|_{L_{(\frac{p}{2})'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}=1$. Note that

(3.22)\begin{align} &\tau\int_{{\mathbb R}^d}\sum_{k=-\infty}^{-1}\unicode{x02A0F}_{\!\!\!\!\omega+B_k} \left|\int_{{\mathbb R}^d}\sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\right.\nonumber\\ &\qquad\times\left.\int_{x+B_i}g_{11}(x,\,y,\,i) \overline{\varphi_i(y-z)}\,dydx \right|^2dz\,b_k(\omega)d\omega\nonumber\\ &\quad\leq\sum_{k=-\infty}^{-1} \int_{{{{\mathbb R}}^d}}|\det A|^{-k} \left\|\int_{\omega+B_{k+4\sigma+1}} \sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\right.\nonumber\\ &\left.\qquad\times\int_{x+B_i} g_1(x,\,y,\,i)b_k(\omega)^{\frac12} \overline{\varphi_i(y-\cdot)}\,dy dx\right\|_{L_2(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2\,d\omega.\end{align}

To estimate the $L_2(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})$-norm in the above expression, we use the duality again, Cauchy–Schwarz inequality to deduce that

(3.23)\begin{align} &\left\|\int_{\omega+B_{k+4\sigma+1}}\sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\int_{x+B_i}g_1(x,\,y,\,i)b_k(\omega)^{\frac12} \overline{\varphi_i(y-z)}\,dydx\right\| _{L_2(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2\nonumber\\ &\quad=\sup\limits_{\|h\|_{L_2(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}=1} \left|\tau\int_{{\mathbb R}^d}\int_{\omega+B_{k+4\sigma+1}}\sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\right.\nonumber\\ &\qquad\left.\times\int_{x+B_i} g_1(x,\,y,\,i)b_k(\omega)^{\frac12} \overline{\varphi_i(y-z)}\,dydxh(z)dz\right|^2 \nonumber\\ &\quad=\sup\limits_{\|h\|_{L_2(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})=1}} \left|\tau\int_{\omega+B_{k+4\sigma+1}}\sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\right.\nonumber\\ &\qquad\left.\times\int_{x+B_i} g_1(x,\,y,\,i)b_k(\omega)^{\frac12} (h*\varphi_i(y))^*\,dydx\right|^2\nonumber\\ &\quad\lesssim \tau\int_{\omega+B_{k+4\sigma+1}}\sum_{i=-\infty}^{-1}\int_{x+B_i}|g_1(x,\,y,\,i)|^2b_k(\omega)\, dydx.\end{align}

Substituting the inequality (3.23) into (3.22), we conclude that

\begin{align*} &\tau\int_{{\mathbb R}^d}\sum_{k=-\infty}^{-1}\unicode{x02A0F}_{\!\!\!\!\omega+B_k}\left|\int_{{\mathbb R}^d} \sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\right.\\ &\qquad\left.\times\int_{x+B_i}g_{11}(x,\,y,\,i) \overline{\varphi_i(y-z)}\,dydx\right|^2dz \,b_k(\omega)d\omega\nonumber\\ &\quad\lesssim \tau \int_{{\mathbb R}^d}\sum_{i=-\infty}^{-1}\int_{x+B_i}|g_{11}(x,\,y,\,i)|^2dyd\sigma(i) \sum_{k=-\infty}^{-1}\unicode{x02A0F}_{\!\!\!\!x+B_{k+4\sigma+1}}b_k(\omega)d\omega dx\\ &\quad\lesssim \|g_1\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))}^2 \left\|\sum_{k=-\infty}^{-1}\unicode{x02A0F}_{\!\!\!\!x+B_{k+4\sigma+1}} b_k(\omega)d\omega\right\|_{L_{(\frac{p}{2})'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\quad\lesssim \|g_1\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))}^2\left\|\sum_{k=-\infty}^{-1}b_k \right\|_{L_{(\frac{p}{2})'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\quad\lesssim \|g_1\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))}^2. \end{align*}

By taking the supremum in the above inequality over all $\{b_k\}_{k\in\mathbb{Z}\backslash{\mathbb N}}$ satisfying the stated properties, we conclude that the $L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,\ell_\infty)$-norm of (3.20) is dominated by $\|g_1\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(i)))}^2$.

It remains to estimate (3.21). Put

\begin{equation*} A_k(\omega)=\unicode{x02A0F}_{\!\!\!\!\omega+B_k}\left|\int_{{{{\mathbb R}}^d}}\overline{\phi(x-z)}g_2(x)\,dx\right|^2\,dz , \quad k\in{\mathbb Z}\backslash{\mathbb N} . \end{equation*}

By the Cauchy–Schwarz inequality, we obtain

\begin{equation*} \left|\int_{{{{\mathbb R}}^d}}\overline{\phi(x-z)}g_2(x)\,dx\right|^2 \le \|\phi\|_{L_1({{{\mathbb R}}^d})}\int_{{{{\mathbb R}}^d}}|\phi(x-z)|\,|g_2(x)|^2\,dx . \end{equation*}

Since $k\in{\mathbb Z}\backslash{\mathbb N}$, we have $B_k\subset B_0$. Define

\begin{equation*} \eta(u)=\sup_{v\in B_0}|\phi(u-v)| . \end{equation*}

Then $\eta\in L_1({{{\mathbb R}}^d})$, and

\begin{equation*} \unicode{x02A0F}_{\!\!\!\!\omega+B_k}|\phi(x-z)|\,dz \le \eta(x-\omega). \end{equation*}

Hence, for all $k\in{\mathbb Z}\backslash{\mathbb N}$,

\begin{equation*} A_k(\omega) \le \|\phi\|_{L_1({{{\mathbb R}}^d})} \int_{{{{\mathbb R}}^d}}\eta(x-\omega)|g_2(x)|^2\,dx =:\|\phi\|_{L_1({{{\mathbb R}}^d})} G(\omega). \end{equation*}

Therefore, using the positive domination in $L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\ell_\infty)$,

\begin{align*} \left\|\sup_{k\in{\mathbb Z}\backslash{\mathbb N}}^{\ \ \ \ +}A_k \right\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} &\le \|\phi\|_{L_1({{{\mathbb R}}^d})}\|G\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\le \|\eta\|_{L_1({{{\mathbb R}}^d})} \left\|\left|g_2\right|^2\right\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\lesssim \|g_2\|_{L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2. \end{align*}

Consequently,

\begin{equation*} \left\| \sup_{k\in{\mathbb Z}\backslash{\mathbb N}}^{\ \ \ \ +} \unicode{x02A0F}_{\!\!\!\!\omega+B_k} \left| \int_{{{{\mathbb R}}^d}}\overline{\phi(x-z)}g_2(x)\,dx \right|^2dz \right\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \lesssim \|g_2\|_{L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2. \end{equation*}

This finishes the proof of (3.17).

Next, we prove (3.18). Write

\begin{equation*} \Psi(g)=\Psi_1(g_1)+\Psi_2(g_2), \end{equation*}

where

\begin{align*} \Psi_1(g_1)(z)&:=\int_{\mathbb R^d}\sum_{i=-\infty}^{-1}|\det A|^{-\frac i2}\int_{x+B_i}\overline{\varphi_i(y-z)}g_1(x,y,i)\,dy\,dx\ \ \mathrm{and}\\ \Psi_2(g_2)(z)&:=\int_{\mathbb R^d}\overline{\phi(x-z)}g_2(x)\,dx . \end{align*}

Since $ |\Psi(g)|^2 \lesssim |\Psi_1(g_1)|^2+|\Psi_2(g_2)|^2, $ it is enough to estimate the two terms separately.

First we deal with the term containing $g_1$. Let $ \{a_k\}_{k\in{\mathbb N}} \subset L_{(\frac p2)'}(L_\infty(\mathbb R^d)\overline\otimes\mathcal M) $ be a sequence of positive elements such that $ \|\sum_{k\in{\mathbb N}}a_k\|_{L_{(\frac p2)'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\le 1. $ By the duality of $L_{\frac p2}(L_\infty(\mathbb R^d)\overline\otimes\mathcal M;\ell_\infty)$, it suffices to estimate

\begin{equation*} \Lambda = \tau\int_{\mathbb R^d} \sum_{k\in{\mathbb N}} \unicode{x02A0F}_{\!\!\!\!\omega+B_k} |\Psi_1(g_1)(z)|^2\,dz\, a_k(\omega)\,d\omega . \end{equation*}

Arguing as in the proof of (3.17), namely using $L_2$-duality, the Cauchy–Schwarz inequality, the $L_2$-boundedness of the local square function, and the noncommutative Hardy–Littlewood maximal inequality, we obtain

\begin{equation*} \begin{aligned} \Lambda&\lesssim\tau\int_{\mathbb R^d}\sum_{i=-\infty}^{-1}\int_{x+B_i}|g_1(x,y,i)|^2\,dy\sum_{k\in{\mathbb N}} \unicode{x02A0F}_{\!\!\!\!x+B_{k+C_\sigma}}a_k(\omega)\,d\omega\,dx \\ &\le\left\|\sum_{i=-\infty}^{-1}\int_{x+B_i}|g_1(x,y,i)|^2\,dy\right\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\quad\times\left\|\sum_{k\in{\mathbb N}}\unicode{x02A0F}_{\!\!\!\!x+B_{k+C_\sigma}}a_k(\omega)\,d\omega \right\|_{L_{(\frac p2)'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \\ &\lesssim\|g_1\|_{L_p(L_\infty(\mathbb R^d)\overline\otimes\mathcal M; L_2^c(\widetilde\Gamma_x))}^2\left\|\sum_{k\in{\mathbb N}}a_k\right\|_{L_{(\frac p2)'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \\ &\lesssim\|g_1\|_{L_p(L_\infty(\mathbb R^d)\overline\otimes\mathcal M; L_2^c(\widetilde\Gamma_x))}^2 . \end{aligned} \end{equation*}

Taking the supremum over all such positive sequences $\{a_k\}$, we get

\begin{equation*} \left\| \sup_{k\in{\mathbb N}}^{\ \ \ \ +} \unicode{x02A0F}_{\!\!\!\!\omega+B_k}|\Psi_1(g_1)(z)|^2\,dz \right\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \lesssim \|g_1\|_{L_p(L_\infty(\mathbb R^d)\overline\otimes\mathcal M; L_2^c(\widetilde\Gamma_x))}^2 . \end{equation*}

It remains to estimate the term containing $g_2$. Set

\begin{equation*} F_2(z)=\Psi_2(g_2)(z)=\widetilde{\phi}*g_2(z), \qquad \widetilde{\phi}(z)=\overline{\phi(-z)}. \end{equation*}

By the anisotropic noncommutative Hardy–Littlewood maximal inequality, we obtain

\begin{align*} &\left\| \sup_{k\in{\mathbb N}}^{\ \ \ \ +} \unicode{x02A0F}_{\!\!\!\!\omega+B_k}|F_2(z)|^2\,dz \right\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\quad\lesssim \left\||F_2|^2\right\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} = \|F_2\|_{L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2 . \end{align*}

Since $\phi\in\mathcal S(\mathbb R^d)\subset L_1(\mathbb R^d)$, it follows that

\begin{equation*} \|F_2\|_{L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \le \|\phi\|_{L_1({{{\mathbb R}}^d})}\|g_2\|_{L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}. \end{equation*}

Consequently,

\begin{equation*} \left\| \sup_{k\in{\mathbb N}}^{\ \ \ \ +} \unicode{x02A0F}_{\!\!\!\!\omega+B_k}|\Psi_2(g_2)(z)|^2\,dz \right\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \lesssim \|g_2\|_{L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2 . \end{equation*}

Combining the preceding two estimates gives

\begin{align*} &\left\| \sup_{k\in{\mathbb N}}^{\ \ \ \ +} \unicode{x02A0F}_{\!\!\!\!\omega+B_k}|\Psi(g)(z)|^2\,dz \right\|_{L_{\frac p2}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^{\frac12}\\ &\quad\lesssim \|g\|_{ L_p(L_\infty(\mathbb R^d)\overline\otimes\mathcal M; L_2^c(\widetilde\Gamma_x)) \oplus_p L_p(L_\infty(\mathbb R^d)\overline\otimes\mathcal M)} . \end{align*}

This proves (3.18).

Proof of Theorem 3.10

Let $p\in[1,\,2)$ and $p'$ be the conjugate index of $p$. For convenience, we divide the proof into the following two steps:

Step 1. In this step, we will show

\begin{equation*}L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})\subset\left(\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})\right)^{\ast}.\end{equation*}

Since $S_{{\mathcal M}}$-valued simple functions are dense in $ \mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})$, by approximation, we only need to show that, for any $S_{{\mathcal M}}$-valued simple function $f$, $g\in L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$,

\begin{equation*}\left|{\mathcal L}_g(f)\right|\leq\|f\|_{ \mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})}\|g\|_{L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})}.\end{equation*}

Let $f$ be a $S_{{\mathcal M}}$-valued simple function and $g\in L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$. Then, by (2.2) and the Fubini theorem, we get

(3.24)\begin{align} \left|{\mathcal L}_g(f)\right|&=\left|\tau\otimes\int_{{{{\mathbb R}}^d}}f(x)g^{*}(x)\,dx\right|\nonumber\\ &=C\left|\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}|\det A|^{-i}\int_{B_{\rho}(x,\, \frac{|\det A|^i}{2})}f\ast\varphi_i(y)(g\ast\varphi_i(y))^{*}\,dydx\right.\nonumber\\ &\quad+\left.\tau\otimes\int_{{{{\mathbb R}}^d}}f*\phi(x)(g*\phi(x))^* \,dx\right|\nonumber\\ &=:\mathrm{I}+\mathrm{II}. \end{align}

For the term $\mathrm{II}$, by the Hölder inequality and Lemmas 2.16 and 3.9, we obtain

\begin{align*} &\left|\tau\otimes\int_{{{{\mathbb R}}^d}}f*\phi(x)(g*\phi(x))^* \,dx\right|\\ & \quad\leq\|f*\phi\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\|g*\phi\|_{L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\\ &\quad\lesssim\|f*\phi\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\|g\|_{L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})}. \end{align*}

The estimate of the term $\mathrm{I}$ is harder. To deal with it, for $x\in{{{\mathbb R}}^d}$ and $i\in{\mathbb Z}\backslash{\mathbb N}$, define

\begin{align*} \mathcal{S}_{A,\varphi}^c(f)(x,\,i):=\left(\sum_{k= i+\sigma}^{-1}|\det A| ^{-k}\int_{B_{\rho}(x,\,|\det A|^{k-\sigma}-\frac{|\det A|^i}{2})}|f\ast\varphi_k(y)|^2\,dy\right)^{\frac12} \end{align*}

and

\begin{align*} \widetilde{\mathcal{S}}_{A,\varphi}^c(f)(x,\,i):=\left(\sum_{k= i+\sigma}^{-1}|\det A|^{-k}\int_{B_{\rho}(x,\,\frac{|\det A|^{k-\sigma}}{2})}|f\ast\varphi_k(y)|^2\,dy\right)^{\frac12}. \end{align*}

From the above two definitions, we see that $\mathcal{S}_{A,\varphi}^c(f)(x,\,i)$ and $\widetilde{\mathcal{S}}_{A,\varphi}^c(f)(x,\,i)$ are decreasing in $i$. Moreover, it is easy to obtain that, for any $(x,\,i)\in{{{\mathbb R}}^d} \times{\mathbb Z}\backslash{\mathbb N}$,

\begin{align*} &\widetilde{\mathcal{S}}_{A,\varphi}^c(f)(x,\,-\infty)\leq \mathcal{S}_{A,\varphi}^c(f)(x), \ \ \mathcal{S}_{A,\varphi}^c(f)(x,\,0)=0 \ \ \mathrm{and}\\ &\widetilde{\mathcal{S}}_{A,\varphi}^c(f)(x,\,i)\leq \mathcal{S}_{A,\varphi}^c(f)(x,\,i).\end{align*}

In what follows, we denote $\mathcal{S}_{A,\varphi}^c(f)(x,\,i)$ and $\widetilde{{\mathcal S}}_{A,\varphi}^c(f)(x,\,i)$ simply by $\mathcal{S}(x,\,i)$ and $\widetilde{\mathcal{S}}(x,\,i)$, respectively. By approximation, we can assume that $\tau$ is finite and $\mathcal{S}(x,\,i)$, $\widetilde{\mathcal{S}}(x,\,i)$ are invertible in ${\mathcal M}$ for any $(x,\,i)\in{{{\mathbb R}}^d} \times{\mathbb Z}\backslash{\mathbb N}$. In fact, whenever negative powers of square functions occur, we first replace ${\mathcal S}_{A,\varphi}^c$ and $\widetilde {{\mathcal S}}_{A,\varphi}^c$ by

\begin{equation*} {\mathcal S}_\varepsilon=\left[({\mathcal S}_{A,\varphi}^c)^2(x,i)+\varepsilon I\right]^{\frac 12} \ \ \mathrm{and} \ \ \widetilde {{\mathcal S}}_\varepsilon=\left[(\widetilde {{\mathcal S}}_{A,\varphi}^c)^2(x,i)+\varepsilon I\right]^{\frac 12}. \end{equation*}

These operators are strictly positive, so all negative powers are defined by functional calculus. The required estimates are first proved uniformly in $\varepsilon \gt 0$, and then we let $\varepsilon\downarrow0$, using the normality of the trace.

From (3.24) and the Cauchy–Schwarz inequality, we conclude that

\begin{align*} \mathrm{I}&\leq C\left(\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}{\mathcal S}^{p-2}(x,\,i)|\det A|^{-i}\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})}|f \ast\varphi_i(y)|^2\,dydx\right)^{\frac12}\\ &\ \ \ \ \times\left(\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}{\mathcal S}^{2-p}(x,\,i)|\det A|^{-i}\int_{B_{\rho} (x,\,\frac{|\det A|^i}{2})}|g\ast\varphi_i(y)|^2\,dydx\right)^{\frac12}\\ &=:\mathrm{I}_1\cdot\mathrm{I}_2. \end{align*}

Fix $x\in{{{\mathbb R}}^d}$ and $i\in{\mathbb Z}\backslash{\mathbb N}$, we have $\widetilde{\mathcal{S}}(x,\,i)\leq \mathcal{S}(x,\,i)$. Thus we can deduce that $\mathcal{S}^{-1}(x,\,i)\leq\widetilde{\mathcal{S}}^{-1}(x,\,i)$ with $(x,\,i)\in{{{\mathbb R}}^d} \times{\mathbb Z}\backslash {\mathbb N}$. Furthermore, for the term $\mathrm{I}_1$, by the fact that $0\leq2-p \lt 1$, and the property of $\tau$, we have

\begin{align*} (\mathrm{I}_1)^2&\leq\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\widetilde{{\mathcal S}}^{p-2}(x,\,i)|\det A|^{-i}\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})}|f \ast\varphi_i(y)|^2\,dydx\\ &\thicksim\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\widetilde{{\mathcal S}}^{p-2}(x,\,i) \left[\widetilde{{\mathcal S}}^2(x,\,i)-\widetilde{{\mathcal S}}^2(x,\,i+1)\right]\,dx\\ &=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\left[\widetilde{{\mathcal S}}^p(x,\,i) -\widetilde{{\mathcal S}}^{p-1}(x,\,i)\widetilde{{\mathcal S}}(x,\,i+1)+ \widetilde{{\mathcal S}}^{p-1}(x,\,i)\widetilde{{\mathcal S}}(x,\,i+1)\right.\nonumber\\ &\quad-\left.\widetilde{{\mathcal S}}^{p-2}(x,\,i) \widetilde{{\mathcal S}}^2(x,\,i+1)\right]\,dx\\ &=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\widetilde{{\mathcal S}}^{p-1}(x,\,i) \left[\widetilde{{\mathcal S}}(x,\,i)-\widetilde{{\mathcal S}}(x,\,i+1)\right]\,dx\\ &\ \ \ \ \ + \tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\widetilde{{\mathcal S}}^{p-2}(x,\,i)\widetilde{{\mathcal S}}(x,\,i+1) \left[\widetilde{{\mathcal S}}(x,\,i) -\widetilde{{\mathcal S}}(x,\,i+1)\right]\,dx\\ &=:\mathrm{I}_{11}+\mathrm{I}_{12}. \end{align*}

By the fact that $\widetilde{{\mathcal S}}(x,\,i)\leq\widetilde{{\mathcal S}}(x,\,-\infty)\leq {\mathcal S}_{A,\varphi}^c(f)(x)$ and $\widetilde{{\mathcal S}}(x,\,0)=0$ with $x\in{{{\mathbb R}}^d}$, and $0\leq p-1 \lt 1$, we have

\begin{align*} \mathrm{I}_{11}&=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\widetilde{{\mathcal S}}^{p-1}(x,\,i)\left[\widetilde{{\mathcal S}}(x,\,i) -\widetilde{{\mathcal S}}(x,\,i+1)\right]\,dx\\ &\leq\tau\otimes\int_{{{{\mathbb R}}^d}}\widetilde{{\mathcal S}}^{p-1}(x,\,-\infty)\sum_{i=-\infty}^{-1}\left[\widetilde{{\mathcal S}}(x,\,i) -\widetilde{{\mathcal S}}(x,\,i+1)\right]\,dx\\ &\leq\tau\otimes\int_{{{{\mathbb R}}^d}}\widetilde{{\mathcal S}}^{p-1}(x,\,-\infty)\widetilde{{\mathcal S}}(x,\,-\infty)\,dx \leq\left\|{\mathcal S}_{A,\varphi}^c(f)\right\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}^p. \end{align*}

For the term $\mathrm{I}_{12}$, by the noncommutative Hölder inequality and the property of $\tau$, we obtain

(3.25)\begin{align} \mathrm{I}_{12}&=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\left[\widetilde{{\mathcal S}}(x,\,i)\right]^{p-2} \widetilde{{\mathcal S}}(x,\,i+1)\left[\widetilde{{\mathcal S}}(x,\,i)-\widetilde{{\mathcal S}}(x,\,i+1)\right]\,dx\nonumber\\ &=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}} \left[\widetilde{{\mathcal S}}(x,\,i)\right]^{p-2} \widetilde{{\mathcal S}}(x,\,i+1)\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\nonumber\\ &\ \ \ \ \times\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{p-1}{2}} \left[\widetilde{{\mathcal S}}(x,\,i)-\widetilde{{\mathcal S}}(x,\,i+1)\right] \left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{p-1}{2}}\,dx\nonumber\\ &\leq\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\left\|\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right] ^{\frac{1-p}{2}}\right.\nonumber\\ &\quad\times\left.\left[\widetilde{{\mathcal S}}(x,\,i)\right]^{p-2} \widetilde{{\mathcal S}}(x,\,i+1)\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right] ^{\frac{1-p}{2}}\right\|_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}}\nonumber\\ &\ \ \ \ \times\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{p-1}{2}} \left[\widetilde{{\mathcal S}}(x,\,i)-\widetilde{{\mathcal S}}(x,\,i+1)\right] \left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{p-1}{2}}\,dx.\end{align}

Using the fact that $\widetilde{{\mathcal S}}(x,\,i+1)\leq \widetilde{{\mathcal S}}(x,\,i)$ with $x\in{{{\mathbb R}}^d}$ and $i\in{\mathbb Z}\backslash{\mathbb N}$, we have

\begin{align*} &\left\|\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\left[\widetilde{{\mathcal S}}(x,\,i)\right]^{p-2} \widetilde{{\mathcal S}}(x,\,i+1)\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}} \right\|_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}}\\ &\quad=\left\|\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\left[\widetilde{{\mathcal S}}(x,\,i)\right] ^{\frac{p-1}{2}} \left[\widetilde{{\mathcal S}}(x,\,i)\right]^{\frac{p-3}{2}}\left[\widetilde{{\mathcal S}}(x,\,i+1)\right]^{\frac{3-p}{2}}\right.\\ &\qquad\times\left.\left[\widetilde{{\mathcal S}}(x,\,i+1)\right]^{\frac{p-1}{2}} \left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\right\|_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}}\\ &\quad\leq\left\|\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\left[\widetilde{{\mathcal S}}(x,\,i)\right] ^{\frac{p-1}{2}}\right\|_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}}\\ &\qquad\times\left\|\left[\widetilde{{\mathcal S}}(x,\,i)\right]^{\frac{p-3}{2}}\left[\widetilde{{\mathcal S}}(x,\,i+1)\right]^{\frac{3-p}{2}} \right\|_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}}\\ &\qquad\times\left\|\left[\widetilde{{\mathcal S}}(x,\,i+1)\right]^{\frac{p-1}{2}} \left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\right\|_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}} \leq1. \end{align*}

Indeed, each of the three factors on the right-hand side is a contraction. Here we only consider the first one:

\begin{align*} &\left\|\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\left[\widetilde{{\mathcal S}}(x,\,i)\right] ^{\frac{p-1}{2}}\right\|_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}}\\ &\quad=\left\|\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\left[\widetilde{{\mathcal S}}(x,\,i)\right] ^{\frac{p-1}{2}}\left\{\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\left[\widetilde{{\mathcal S}}(x,\,i)\right] ^{\frac{p-1}{2}}\right\}^*\right\|_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}}^{\frac12}\\ &\quad=\left\|\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}}\left[\widetilde{{\mathcal S}}(x,\,i)\right] ^{p-1}\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{\frac{1-p}{2}} \right\|_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}}^{\frac12}\leq1. \end{align*}

The last equality is deduced by $\widetilde{{\mathcal S}}(x,\,i)\leq\widetilde{{\mathcal S}}(x,\,-\infty)$ with $x\in{{{\mathbb R}}^d}$ and $i\in{\mathbb Z}\backslash{\mathbb N}$. From this and (3.25), we deduce that

\begin{align*} \mathrm{I}_{12}&\leq\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i=-\infty}^{-1}\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{p-1} \left[\widetilde{{\mathcal S}}(x,\,i)-\widetilde{{\mathcal S}}(x,\,i+1)\right]\,dx\\ &=\tau\otimes\int_{{{{\mathbb R}}^d}}\left[\widetilde{{\mathcal S}}(x,\,-\infty)\right]^{p}\,dx \leq\left\|{\mathcal S}_{A,\varphi}^c(f)\right\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}^p, \end{align*}

which, together with the estimate $\mathrm{I}_{11}$, further implies that

\begin{equation*}(\mathrm{I}_{1})^2\lesssim\left\|\mathcal{S}_{A,\varphi}^c(f)\right\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}^p.\end{equation*}

Now, we deal with the term $\mathrm{I}_{2}$. In what follows, let $\beta$ be the minimum integer such that $2\leq |\det A|^{\beta}$ and $2\beta \gt \sigma$, and $\alpha$ be as in Lemma 3.12. Therefore,

(3.26)\begin{align} (\mathrm{I}_{2})^2&\leq C\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{j=-\infty}^{-\alpha-2\beta-2}\sum_{i=j+\alpha+2\beta } ^{j+\alpha+2\beta+1}\left[\mathcal{S}(x,\,i)\right]^{2-p}|\det A|^{-i}\nonumber\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})}|g\ast\varphi_i(y)|^2\,dydx\nonumber\\ &=C\tau\sum_{m\in{\mathbb Z}^d}\sum_{j=-\infty}^{-\alpha-2\beta-2}\int_{Q_m^j} \sum_{i=j+\alpha+2\beta} ^{j+\alpha+2\beta+1}\left[\mathcal{S}(x,\,i)\right]^{2-p}|\det A|^{-i}\nonumber\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})}|g\ast\varphi_i(y)|^2\,dydx,\end{align}

where $Q_m^j\in\mathcal{Q}$ and $\mathcal{Q}$ is as in (3.16).

Let

\begin{align*}\mathbb{S}(x,\,j):&=\left(\sum_{k= j+\alpha+2\beta}^{-1}|\det A|^{-k}\right.\\ &\quad\times\left.\int_{B_{\rho}(x_{Q_m^j},\,|\det A|^k-|\det A|^{j+\alpha+\sigma})} |f\ast\varphi_k(y)|^2\,dy\right)^{1/2},\ \ \ x\in Q_m^j.\end{align*}

It follows from Lemma 3.12 that

\begin{align*} &B_{\rho}(x,\,|\det A|^{k-\sigma}-{|\det A|^i}/{2})\subset B_{\rho}(x_{Q_m^j},\,|\det A|^k\\ &\quad-|\det A|^{j+\alpha+\sigma}),\ \ \ \mathrm{for}\ \mathrm{any}\ x\in Q_m^j, \ i\geq j+\alpha+2\beta. \end{align*}

From this and the definitions of $\mathcal{S}(x,\,i)$ and $\mathbb{S}(x,\,j)$, we conclude that

\begin{align*} {\mathcal S}(x,\,i)\leq \mathbb{S}(x,\,j),\ \ \ \mathrm{for}\ \mathrm{any}\ x\in Q_m^j \ \mathrm{and}\ i\geq j+\alpha+2\beta. \end{align*}

Then, by (3.26), we find that

(3.27)\begin{align} (\mathrm{I}_{2})^2&\leq C\tau\sum_{m\in{\mathbb Z}^d}\sum_{j=-\infty}^{-\alpha-2\beta-2}\int_{Q_m^j}\left[\mathbb{S}(x,\,j)\right]^{2-p} \sum_{i=j+\alpha+2\beta} ^{j+\alpha+2\beta+1}|\det A|^{-i}\nonumber\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})} |g\ast\varphi_i(y)|^2\,dydx\nonumber\\ &=C\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{j=-\infty}^{-\alpha-2\beta-2}\left[\mathbb{S}(x,\,j)\right]^{2-p} \sum_{i=j+\alpha+2\beta} ^{j+\alpha+2\beta+1}|\det A|^{-i}\nonumber\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})} |g\ast\varphi_i(y)|^2\,dydx.\end{align}

For convenience, let $\mathbb{W}(x,\,l):=\mathbb{S}^{2-p}(x,\,l)-\mathbb{S}^{2-p}(x,\,l+1)$ with $x\in{{{\mathbb R}}^d}$ and $l\in{\mathbb Z}\backslash{\mathbb N}$. Then, from (3.27) and the fact that $\mathbb{S}(x,\,l+1)$ is decreasing about $l$, and $\mathbb{W}(x,\,l)$ is constant on $Q_m^l$ for any $m\in{\mathbb Z}^d$, we deduce that

\begin{align*} (\mathrm{I}_{2})^2&\leq C\tau\otimes\int_{{\mathbb R}^d}\sum_{j=-\infty}^{-\alpha-2\beta-2}\sum_{l\geq j}\mathbb{W}(x,\,l) \sum_{i=j+\alpha+2\beta} ^{j+\alpha+2\beta+1}|\det A|^{-i}\nonumber\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})} |g\ast\varphi_i(y)|^2\,dydx\nonumber\\ &=C\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{l=-\infty}^{-\alpha-2\beta-2}\mathbb{W}(x,\,l)\sum_{j\leq l} \sum_{i=j+\alpha+2\beta} ^{j+\alpha+2\beta+1}|\det A|^{-i}\nonumber\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})} |g\ast\varphi_i(y)|^2\,dydx\nonumber\\ &\leq C\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{l=-\infty}^{-\alpha-2\beta-2}\mathbb{W}(x,\,l) \sum_{i=-\infty} ^{l+\alpha+2\beta+1}|\det A|^{-i}\nonumber\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})} |g\ast\varphi_i(y)|^2\,dydx\nonumber\\ &=C\tau\sum_{m\in{\mathbb Z}^d}\sum_{l=-\infty}^{-\alpha-2\beta-2}\mathbb{W}(x_{Q_m^l},\,l)\int_{Q_m^l} \sum_{i=-\infty} ^{l+\alpha+2\beta+1}|\det A|^{-i}\nonumber\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^i}{2})} |g\ast\varphi_i(y)|^2\,dydx\nonumber\\ &\leq C\tau\sum_{m\in{\mathbb Z}^d}\sum_{l=-\infty}^{-\alpha-2\beta-2}\mathbb{W}(x_{Q_m^l},\,l) \sum_{i=-\infty} ^{l+\alpha+2\beta+1}\nonumber\\ &\quad\times\int_{B_{\rho}(x_{Q_m^l},\,|\det A|^{l+\alpha+2\beta+1})} |g\ast\varphi_i(y)|^2\,dy. \end{align*}

From this, the fact that $\mathbb{W}(x,\,l)$ is a constant on the dilated cube $Q_m^l$ for any $m\in{\mathbb Z}^d$, (3.15) and Lemma 3.9, we conclude that

\begin{align*} (\mathrm{I}_{2})^2 &\leq C\tau\sum_{l=-\infty}^{-\alpha-2\beta-2}\sum_{m\in{\mathbb Z}^d}\int_{Q_m^l}\mathbb{W}(x,\,l)\,|\det A|^{-l-\alpha-2\beta-1} \nonumber\\ &\quad\times\sum_{i=-\infty} ^{l+\alpha+2\beta+1}\int_{B_{\rho}(x_{Q_m^l},\,|\det A|^{l+\alpha+2\beta+1})} |g\ast\varphi_i(y)|^2\,dydx\nonumber\\\\ &=C\tau\sum_{l=-\infty}^{-\alpha-2\beta-2}\int_{{{{\mathbb R}}^d}}\mathbb{W}(x,\,l)\,|\det A|^{-l-\alpha-2\beta-1}\nonumber\\ &\quad\times\sum_{i=-\infty} ^{l+\alpha+2\beta+1}\int_{B_{\rho}(x_{Q_m^l},\,|\det A|^{l+\alpha+2\beta+1})} |g\ast\varphi_i(y)|^2\,dydx\nonumber\\ &\leq C\left\|\sum_{l=-\infty}^{-\alpha-2\beta-2}\mathbb{W}(\cdot,\,l)\right\|_{L_{(\frac{p'}{2})'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} \left\|\sup_{l\in{\mathbb Z}, l\leq-\alpha-2\beta-2}^{\ \ \ \ \ +}|\det A|^{-l-\alpha-2\beta}\right.\nonumber\\ &\quad\times\left.\sum_{i=-\infty} ^{l+\alpha+2\beta+1}\int_{B_{\rho}(x_{Q_m^l},\,|\det A|^{l+\alpha+2\beta+1})} |g\ast\varphi_i(y)|^2\,dy\right\|_{L_{\frac{p'}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\leq C\left\|\mathbb{S}^{2-p}(\cdot,\,-\infty)\right\|_{L_{(\frac{p'}{2})'}(L_{\infty}({{{\mathbb R}}^d}) \overline{\otimes}{\mathcal M})} \|g\|_{L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},\,{\mathcal M})}^2\\ &\leq C\|f\|_{\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},\,{\mathcal M})} ^{2-p}\|g\|_{L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},\,{\mathcal M})}^2, \end{align*}

which, combined with the estimate of the term $\mathrm{I}_1$, implies that

\begin{align*} \left|{\mathcal L}_g(f)\right|&\leq C\left(\left\|{\mathcal S}_{A,\varphi}^c(f)\right\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}+ \|f*\phi\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\right)\|g\|_{L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})}\\ &=C \|f\|_{\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})}\|g\|_{L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})}, \end{align*}

this further implies the desired result.

Step 2. In this step, we will show

\begin{equation*}\left(\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})\right)^{\ast}\subset L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M}).\end{equation*}

Let ${\mathcal L}\in( \mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M}))^{\ast}$. We only need to demonstrate that there exists an operator-valued function $g\in L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$ such that

\begin{align*} {\mathcal L}(f)={\mathcal L}_g(f)=\tau\otimes\int_{{{{\mathbb R}}^d}}f(x)g^{\ast}(x)\,dx. \end{align*}

By the definition of the map $\Phi$ in Definition 3.1 and the Hahn-Banach theorem, we deduce that ${\mathcal L}$ can be extended to a bounded linear functional on $L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(k)))\!\oplus_p\! L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})$ of the same norm. Then, by

\begin{align*} &\left(L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(k)))\oplus_p L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})\right)^{\ast}\\ &\quad=L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x, \,dyd\sigma(k)))\oplus_{p'} L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}), \end{align*}

we can find that there exists $h=(h_1,\,h_2)\in L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(k))) \oplus_{p'} L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})$ such that

\begin{align*} \|h\|_{L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(k))) \oplus_{p'} L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})} =\|{\mathcal L}\|_{( \mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M}))^{\ast}} \end{align*}

and

\begin{align*} {\mathcal L}(f)&=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}\int_{x+B_k} \frac{f\ast\varphi_k(y)}{|\det A|^{\frac k2}}h_1^{\ast}(x,\,y,\,k)\,dydx +\tau\otimes\int_{{{{\mathbb R}}^d}}f*\phi(z)h_2^*(z)\,dz\\ &=\tau\otimes\int_{{{{\mathbb R}}^d}}f(z)(\Psi(h))^{\ast}(z)\,dz =\tau\otimes\int_{{{{\mathbb R}}^d}}f(z)g^{\ast}(z)\,dz, \end{align*}

where $g:=\Psi(h)$ and $\Psi$ is as in Definition 3.1. It follows from Lemmas 3.2 and 3.13 that, there exists $g\in L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$ such that

\begin{align*} \|g\|_{L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})} &=\|\Psi(h)\|_{L_{p'}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})}\\ &\leq C \|h\|_{L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M},\,L_2^c(\widetilde{\Gamma}_x, \,dyd\sigma(k)))\oplus_{p'} L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}. \end{align*}

This completes the proof of Theorem 3.10.

Proof. From Lemma 2.16, we only need to show the reverse inequality. By the proof of Theorem 3.10, we see that, if $d\mu_g= |g*\varphi_k(x)|^2\,dxd\sigma(k)$ is a ${\mathcal M}$-valued anisotropic Carleson measure on ${{{\mathbb R}}^d}\times\{{\mathbb Z}\backslash{\mathbb N}\}$, then $g$ defines a continuous linear functional on $S_{{\mathcal M}}$-valued simple functions by

\begin{align*} {\mathcal L}_g&=\tau\otimes\int_{{{{\mathbb R}}^d}}f(x)g^{*}(x)\,dx\ \ \ \mathrm{and} \ \ \ \left\|{\mathcal L}_g\right\|^2 &\lesssim \|d\mu_g\|_{\mathbf{C}}+\|g*\phi\|_{L_{\infty}(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2.\end{align*}

Therefore, by Theorem 3.10, we know that there exists $\widetilde{g}\in \mathrm{bmo}_A^c({{{\mathbb R}}^d},{\mathcal M})$ such that

\begin{align*} \|\widetilde{g}\|_{\mathrm{bmo}_A^c({{{\mathbb R}}^d},{\mathcal M})}^2=\left\|{\mathcal L}_g\right\|^2\lesssim \|d\mu_g\|_{\mathbf{C}}+ \|g*\phi\|_{L_{\infty}(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2 \end{align*}

and

\begin{align*} \tau\otimes\int_{{{{\mathbb R}}^d}}f(x)g^{*}(x)\,dx=\tau\otimes\int_{{{{\mathbb R}}^d}}f(x)\widetilde{g}^{*}(x)\,dx. \end{align*}

Therefore, $g=\widetilde{g}$ and $g\in \mathrm{bmo}_A^c({{{\mathbb R}}^d},{\mathcal M})$ with $\|g\|_{\mathrm{bmo}_A^c({{{\mathbb R}}^d},{\mathcal M})}^2\lesssim \|d\mu_g\|_{\mathbf{C}}+ \|g*\phi\|_{L_{\infty}(L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^2$.

Proof. From Lemma 3.9, we only need to show the last inequality. By the proof of Theorem 3.10, we know that, if $d\mu_g= |g*\varphi_i(y)|^2\,dyd\sigma(i)$ satisfies the Carleson condition on ${{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}$, then $g$ defines a continuous linear functional on $S_{{\mathcal M}}$-valued simple functions by

\begin{equation*}{\mathcal L}_g(f)=\tau\otimes\int_{{{{\mathbb R}}^d}}f(x)g^{*}(x)\,dx\end{equation*}

and

\begin{align*}\left\|{\mathcal L}_g\right\|^2 \lesssim&\left\|\sup_{k\in{\mathbb Z}\backslash{\mathbb N}}^{\ \ \ \ \ +}\frac{1}{|x+B_k|}\int_{(y,\,i)\in T(x+B_k)}|g\ast\varphi_i(y)|^2\,dyd\sigma(i)\right\|_{L_{\frac{q}{2}}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\\ &\quad+\left\|g*\phi\right\|_{L_{q} (L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}) }^2.\end{align*}

Therefore, by Theorem 3.10, we know that there exists $\widetilde{g}\in L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},\,{\mathcal M})$ such that

\begin{align*} &\|\widetilde{g}\|_{L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},\,{\mathcal M})}^2 \thicksim\left\|{\mathcal L}_g\right\|^2\\ &\quad\lesssim \left\|\sup_{k\in{\mathbb Z}\backslash{\mathbb N}}^{\ \ \ \ \ +}\frac{1}{|x+B_k|}\int_{(y,\,i)\in T(x+B_k)}|g\ast\varphi_i(y)|^2\,dyd\sigma(i)\right\|_{L_{\frac{q}{2}}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\\ &\qquad+\left\|g*\phi\right\|_{L_{q} (L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}) }^2 \end{align*}

and

\begin{align*} \tau\otimes\int_{{{{\mathbb R}}^d}}f(x)g^{*}(x)\,dx=\tau\otimes\int_{{{{\mathbb R}}^d}}f(x)\widetilde{g}^{*}(x)\,dx. \end{align*}

Therefore, $g=\widetilde{g}\in L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},\,{\mathcal M})$ and

\begin{align*} &\|g\|_{L_q\mathrm{mo}_A^c({{{\mathbb R}}^d},\,{\mathcal M})}^2\\ &\quad\lesssim \left\|\sup_{k\in{\mathbb Z}\backslash{\mathbb N}}^{\ \ \ \ \ +}\frac{1}{|x+B_k|}\int_{(y,\,i)\in T(x+B_k)}|g\ast\varphi_i(y)|^2\,dyd\sigma(i)\right\|_{L_{\frac{q}{2}}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\\ &\qquad+\left\|g*\phi\right\|_{L_{q} (L_\infty({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}) }^2.\end{align*}

This completes the proof of Corollary 3.15.

Proof of Theorem 4.1

By Theorem 3.10, we only need to show that $\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})\subset (\mathbf{h}_{p'}^{A,c}({{{\mathbb R}}^d},{\mathcal M}))^*$. Let $g\in\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})$. From the Hölder inequality, we conclude that, for any $f\in\mathbf{h}_{p'}^{A,c}({{{\mathbb R}}^d},{\mathcal M})$ with $\|f\|_{\mathbf{h}_{p'}^{A,c}({{{\mathbb R}}^d},{\mathcal M})}\leq1$,

\begin{align*} \left|\tau\otimes\int_{{{{\mathbb R}}^d}}f^{\ast}(x)g(x)\,dx\right| &\thicksim \left|\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{i\in{\mathbb Z}\backslash{\mathbb N}}|\det A|^{-i}\int_{x+B_i}g*\varphi_i(y)(f*\varphi_i(y))^{*}\,dydx\right|\\ &\ \ \ \ + \left|\tau\otimes\int_{{{{\mathbb R}}^d}}g*\phi(x)(f*\phi(x))^{*}\,dx\right|\\ &\lesssim \left[\tau\otimes\int_{{{{\mathbb R}}^d}}\left(\sum_{i\in{\mathbb Z}\backslash{\mathbb N}}|\det A|^{-i}\int_{x+B_i}\left|f*\varphi_i(y)\right|^2\,dy\right)^{\frac{p'}{2}}dx\right]^{\frac{1}{p'}}\\ &\ \ \ \times\left[\tau\otimes\int_{{{{\mathbb R}}^d}}\left(\sum_{i\in{\mathbb Z}\backslash{\mathbb N}}|\det A|^{-i}\int_{x+B_i}\left|g*\varphi_i(y)\right|^2\,dy\right)^{\frac{p}{2}}dx\right]^{\frac{1}{p}}\\ &\ \ \ \ +\left\|g*\phi\right\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\left\|f*\phi\right\|_{L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\\ &\thicksim\|f\|_{\mathbf{h}_{p'}^{A,c}({{{\mathbb R}}^d},{\mathcal M})}\|g\|_{\mathbf{h}_{p}^{A,c}({{{\mathbb R}}^d},{\mathcal M})} \lesssim\|g\|_{\mathbf{h}_{p}^{A,c}({{{\mathbb R}}^d},{\mathcal M})}, \end{align*}

which implies that $\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})\subset (\mathbf{h}_{p'}^{A,c}({{{\mathbb R}}^d},{\mathcal M}))^*$.

Next, we prove that $L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})\subset\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})$. We claim that, for any $p\in(2,\,\infty)$ and $g\in L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$, $g$ can induce a continuous linear functional ${\mathcal L}_g$ on $\mathbf{T}^{\mathrm{loc},c}_{p'}({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N},{\mathcal M})\oplus_{p'}L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$. In fact, for any $f=(f_1,\,f_2)\in\mathbf{T}^{\mathrm{loc},c}_{p'}({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N},\, {\mathcal M})\oplus_{p'}L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$, we define

(4.28)\begin{equation} {\mathcal L}_g(f):=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}f_1(y,\,k)(g\ast\varphi_k(y))^*\,dy +\tau\otimes\int_{{{{\mathbb R}}^d}}f_2(y)(g\ast\phi(y))^*\,dy. \end{equation}

Then, we can deduce that

(4.29)\begin{equation} \|g\|_{\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},\, {\mathcal M})}\lesssim\|{\mathcal L}_g\| \lesssim\|g\|_{L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})}. \end{equation}

To prove this, we define

\begin{equation*}\mathcal{A}^{\mathrm{loc},c}(f)(x,\,k):=\left(\sum_{i= k+\sigma}^{-1}|\det A|^{-i}\int_{B_{\rho}(x,\,|\det A|^{i-\sigma}-\frac{|\det A|^k}{2})}|f(y,\,i)|^2\,dy\right)^{\frac12}\end{equation*}

and

\begin{equation*}\widetilde{\mathcal{A}}^{\mathrm{loc},c}(f)(x,\,k):=\left(\sum_{i= k+\sigma}^{-1}|\det A|^{-i}\int_{B_{\rho}(x,\,\frac{|\det A|^{i-\sigma}}{2})}|f(y,\,i)|^2\,dy\right)^{\frac12}.\end{equation*}

By approximation, we assume that $\mathcal{A}^{\mathrm{loc},c}(f)(x,\,k)$ is invertible for any $(x,\,k)\in{{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}$. Then, by the Cauchy–Schwarz inequality, we have

\begin{align*} \left|{\mathcal L}_g(f)\right|^2&=\left|\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}f_1(y,\,k)(g\ast\varphi_k(y))^*\,dy +\tau\otimes\int_{{{{\mathbb R}}^d}}f_2(y)(g\ast\phi(y))^*\,dy \right|^2\\ &\leq C\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}\left[\mathcal{A}^{\mathrm{loc},c}(f_1)(x,\,k)\right]^{p'-2}|\det A|^{-k}\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^k}{2})} |f_1(y,\,k)|^2\,dydx\\ &\quad\times\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}\left[\mathcal{A}^{\mathrm{loc},c}(f_1)(x,\,k)\right]^{2-p'}|\det A|^{-k}\\ &\quad\times\int_{B_{\rho}(x,\,\frac{|\det A|^k}{2})} |g\ast\varphi_k(y)|^2\,dydx\\ &\quad+\left\|f_2\right\|_{L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}^2\left\|g\ast\phi\right\|_{L_{p}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}^2\\ &=:C(\mathrm{P}_1\cdot\mathrm{P}_2)+\left\|f_2\right\|_{L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}^2\left\|g\ast\phi\right\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}^2, \end{align*}

where

\begin{align*} \mathrm{P}_1=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}\left[\mathcal{A}^{\mathrm{loc},c}(f_1)(x,\,k)\right]^{p'-2}|\det A|^{-k} \int_{B_{\rho}(x,\,\frac{|\det A|^k}{2})} |f_1(y,\,k)|^2\,dydx \end{align*}

and

\begin{align*} \mathrm{P}_2=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}\left[\mathcal{A}^{\mathrm{loc},c}(f_1)(x,\,k)\right]^{2-p'}|\det A|^{-k} \int_{B_{\rho}(x,\,\frac{|\det A|^k}{2})} |g\ast\varphi_k(y)|^2\,dydx. \end{align*}

By estimates similar to those for $\mathrm{I}$ and $\mathrm{II}$ in the proof of Theorem 3.10, respectively, we conclude that

\begin{equation*} \mathrm{P}_1\lesssim\|f_1\|_{\mathbf{T}_{p'}^{\mathrm{loc},c}({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}, {\mathcal M})}^{p'} \ \ \ \mathrm{and} \ \ \ \mathrm{P}_2\lesssim\|f_1\|_{\mathbf{T}_{p'}^{\mathrm{loc},c}({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}, {\mathcal M})}^{2-p'}\|g\| _{L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})}^2.\end{equation*}

From this, we deduce that

(4.30)\begin{equation} \left|{\mathcal L}_g(f)\right|\lesssim\left(\|f_1\|_{\mathbf{T}_{p'}^{\mathrm{loc},c}({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}, {\mathcal M})}+\|f_2\|_{L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\right)\|g\|_{L_{p}\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})}. \end{equation}

Now we show that $\|g\|_{\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})}\lesssim\|{\mathcal L}_g\|$. By the definition of $\mathbf{T}_{p'}^{\mathrm{loc},c}({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}, {\mathcal M})$, we can identify $\mathbf{T}_{p'}^{\mathrm{loc},c}({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}, {\mathcal M})$ as a subspace of $L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}))$ via the following map:

\begin{align*} \widetilde{\Phi}:f(y,\,k)\mapsto |\det A|^{-\frac{k}{2}}f(y,\,k){\mathbf{1}}_{x+B_k}(y). \end{align*}

Therefore, by the Hahn-Banach theorem, we know that ${\mathcal L}_g$ can be extended to a continuous linear functional on $L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}))\oplus_{p'}L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$ of the same norm. Then, by the fact that

\begin{equation*}\left(L_{p'}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}))\right)^{*} =L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N})),\end{equation*}

we obtain that there exists $h=(h_1,h_2)\in L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}))\oplus_{p}L_{p}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$ such that

\begin{equation*}\|h\|_{L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}))\oplus_{p}L_{p}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})} =\|{\mathcal L}_g\|\end{equation*}

and, for any $f=(f_1,\,f_2)\in \mathbf{T}^{\mathrm{loc},c}_{p'}({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}, {\mathcal M})\oplus_{p'}L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$,

\begin{align*} {\mathcal L}_g(f)&=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}|\det A|^{-\frac k2}\int_{x+B_k}f_1(y,\,k)h_1^{*}(x,\,y,\,k)\,dydx\\ &\quad+\tau\otimes\int_{{{{\mathbb R}}^d}}f_2(x)h^*_2(x)\,dx\\ &=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}f_1(y,\,k)|\det A|^{-\frac k2}\int_{y+B_k}h_1^{*}(x,\,y,\,k)\,dxdy\\ &\quad+\tau\otimes\int_{{{{\mathbb R}}^d}}f_2(x)h^*_2(x)\,dx, \end{align*}

which, combined with (4.28), yields that

\begin{equation*} g*\varphi_k(y)=|\det A|^{-\frac k2}\int_{y+B_k}h_1(x,\,y,\,k)\,dx \ \ \ \mathrm{and} \ \ \ g*\phi(y)=h_2(y).\end{equation*}

Therefore, by (2.1), we have

(4.31)\begin{align} \|g\|_{\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})} &=\left[\tau\otimes\int_{{{{\mathbb R}}^d}}\left(\sum_{k=-\infty}^{-1}|\det A|^{-k}\int_{z+B_k}\left||\det A|^{-\frac {k}{2}}\right.\right.\right.\nonumber\\ &\quad\left.\left.\left.\times\int_{y+B_k}h_1(x,\,y,\,k)\,dx\right|^2\,dy\right)^{\frac p2}\,dz\right]^{\frac 1p} +\|h_2\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\nonumber\\ &\leq\left[\tau\otimes\int_{{{{\mathbb R}}^d}}\left(\sum_{k=-\infty}^{-1}|\det A|^{-k}\right.\right.\nonumber\\ &\quad\left.\left.\times\int_{z+B_k} \int_{y+B_k}\left|h_1(x,\,y,\,k)\right|^2\,dxdy\right)^{\frac p2}\,dz\right]^{\frac1p}+\|h_2\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}\nonumber\\ &\leq\left\|\sum_{k=-\infty}^{-1}|\det A|^{-k}\int_{{{{\mathbb R}}^d}} \int_{z+B_{k+\sigma}}\left|h_1(x,\,y,\,k)\right|^2\,dxdy\right\| _{L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}^{\frac12}\nonumber\\ &\quad+\|h_2\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}. \end{align}

By (3.15) and the Fubini theorem, we obtain that, for any positive $a\in L_{\frac{p}{p-2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})$ with $\|a\|_{L_{\frac{p}{p-2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\leq1$,

\begin{align*} &\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}|\det A|^{-k}\int_{{{{\mathbb R}}^d}} \int_{z+B_{k+\sigma}}\left|h_1(x,\,y,\,k)\right|^2\,dxdy\,a(z)\,dz\\ &\quad\quad=\tau\otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}\int_{{{{\mathbb R}}^d}} \left|h_1(x,\,y,\,k)\right|^2\,dy\,|\det A|^{-k}\int_{x+B_{k+\sigma}}a(z)\,dzdx\\ &\quad\quad\leq\left\|\sum_{k=-\infty}^{-1}\int_{{{{\mathbb R}}^d}} \left|h_1(x,\,y,\,k)\right|^2\,dy\right\|_{L_{\frac{p}{2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\quad\quad\times\left\|\sup_{k\in{\mathbb Z}\backslash{\mathbb N}}^{\ \ \ \ \ +}|\det A|^{-k}\int_{x+B_{k+\sigma}}a(z)\,dz\right\| _{L_{\frac{p}{p-2}}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})}\\ &\quad\quad\leq C\left\|h_1\right\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L^c_2({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}))}^2. \end{align*}

From this and (4.31), we conclude that

\begin{equation*}\|g\|_{\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})} \lesssim\left\|h_1\right\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L^c_2({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}))} +\left\|h_2\right\|_{L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})} \thicksim\left\|{\mathcal L}_g\right\|,\end{equation*}

which, combined with (4.30), further implies that claim (4.29) holds true.

Finally, to complete the proof of $\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})=L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$, we only need to show that the set of $S_{{\mathcal M}}$-valued simple functions is dense in $L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$. Indeed, from the proof of Theorem 3.10, we see that, for any $g\in L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$, there exists $h=(h_1,h_2)\in L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}; \,L_2^c({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N}))\oplus_{p}L_{p}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$ such that

\begin{align*} g&=\Psi(h)\ \mathrm{and} \ \|\Psi(h)\|_{L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})} \lesssim\|h\|_{L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N})) \oplus_{p}L_{p}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}.\end{align*}

Choosing a sequence of $S_{L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}}$-valued simple functions $\{h_1^j\}_{j\in{\mathbb N}}$ and a sequence of $S_{{\mathcal M}}$-valued simple functions $\{h^j_2\}_{j\in{\mathbb N}}$ such that

\begin{equation*}h_1^j\rightarrow h_1 \ \ \mathrm{in} \ \ L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c({{{\mathbb R}}^d}\times{\mathbb Z}\backslash{\mathbb N})), \ \mathrm{as} \ j\rightarrow\infty,\end{equation*}

and

\begin{equation*}h_2^j\rightarrow h_2 \ \ \mathrm{in} \ \ L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}), \ \mathrm{as} \ j\rightarrow\infty.\end{equation*}

Let $g^j=\Psi(h^j)$, where $h^j:=(h_1^j,h_2^j)$. Then we know that $g^j$ belongs to a dense test function class, and

\begin{equation*}g^j\rightarrow g\ \ \ \mathrm{in} \ \ L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M}),\ \ \mathrm{as} \ \ j\rightarrow\infty.\end{equation*}

This shows that the set of $S_{{\mathcal M}}$-valued simple functions is dense in $L_p\mathrm{mo}_A^c({{{\mathbb R}}^d},{\mathcal M})$ and hence completes the proof of Theorem 4.1.

Proof. The case $p\in[2,\infty)$ can be deduced by Lemma 3.13 and Theorem 4.1. For the case $p\in(1,2)$, from Theorems 3.10 and 4.1, and the Hölder inequality, we conclude that, for any $h=(h_1,h_2)\in L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(k))) \oplus_{p}L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$,

\begin{align*} &\left\|\Psi(h)\right\|_{\mathbf{h}^{A,c}_p({{{\mathbb R}}^d},{\mathcal M})}\\ &\quad=\sup_{\|g\|_{L_{p'}\mathrm{mo}_{A}^c({{{\mathbb R}}^d},{\mathcal M})}\leq1}\left|\tau\otimes\int_{{{{\mathbb R}}^d}}\Psi(h)(z)g^*(z)\,dz\right|\\ &\quad=\sup_{\|g\|_{\mathbf{h}^{A,c}_{p'}({{{\mathbb R}}^d},{\mathcal M})}\leq1}\left|\tau\otimes\int_{{{{\mathbb R}}^d}}\Psi(h)(z)g^*(z)\,dz\right|\\ &\quad\leq\sup_{\|g\|_{\mathbf{h}^{A,c}_{p'}({{{\mathbb R}}^d},{\mathcal M})}\leq1}\left|\tau\otimes\int_{{{{\mathbb R}}^d}} \int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}|\det A|^{-\frac k2}\right.\\ &\qquad\times\left.\int_{x+B_k}h_1(x,\,y,\,k)\varphi_k(y-z)\,dydxg^*(z)\right|\\ &\qquad+\left|\tau\otimes\int_{{{{\mathbb R}}^d}}\int_{{{{\mathbb R}}^d}}h_2(x)\phi(x-z)\,g^*(z)\,dz\right|\\ &\quad=\sup_{\|g\|_{\mathbf{h}^{A,c}_{p'}({{{\mathbb R}}^d},{\mathcal M})}\leq1}\left|\tau \otimes\int_{{{{\mathbb R}}^d}}\sum_{k=-\infty}^{-1}|\det A|^{-\frac k2}\int_{x+B_k}h_1(x,\,y,\,k)(g*\varphi_k(y))^*\,dydx\right|\\ &\qquad+\left|\tau\otimes\int_{{{{\mathbb R}}^d}}h_2(x)(g*\phi(x))^*\,dx\right|\\ &\quad\leq\|h\|_{L_{p}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M};\,L_2^c(\widetilde{\Gamma}_x,\,dyd\sigma(k)))\oplus_p L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})}. \end{align*}

Proof. Since $L_{\infty}(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})\subset\mathrm{bmo}_A^c({{{\mathbb R}}^d},{\mathcal M})$, by taking the duality, we immediately have the embedding relationship $\mathbf{h}_1^{A,c}({{{\mathbb R}}^d},{\mathcal M})\subset L_1(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})$. From the above conclusions and the interpolation, we conclude that

\begin{equation*}\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})\subset L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}), \ \ \mathrm{if} \ \ p\in(1,\,2]\end{equation*}

and

\begin{equation*}L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})\subset\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M}), \ \ \mathrm{if} \ \ p\in(2,\,\infty).\end{equation*}

A similar result also holds true for the row space. Combining with Remark 5.1, we further have

\begin{equation*}\mathbf{H}_p^A({{{\mathbb R}}^d},{\mathcal M})\subset\mathbf{h}_p^A({{{\mathbb R}}^d},{\mathcal M})\subset L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}), \ \ \mathrm{if} \ \ p\in(1,\,2]\end{equation*}

and

\begin{equation*}L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})\subset\mathbf{h}_p^{A}({{{\mathbb R}}^d},{\mathcal M})\subset\mathbf{H}_p^A({{{\mathbb R}}^d},{\mathcal M}), \ \ \mathrm{if} \ \ p\in(2,\,\infty).\end{equation*}

From [Reference Wang40, Theorem 6.5], we know that

\begin{equation*}\mathbf{H}_p^A({{{\mathbb R}}^d},{\mathcal M})=L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M}).\end{equation*}

Therefore, we can further deduce that

\begin{equation*}\mathbf{h}_p^{A}({{{\mathbb R}}^d},{\mathcal M})=L_p(L_{\infty}({{{\mathbb R}}^d})\overline{\otimes}{\mathcal M})=\mathbf{H}_p^A({{{\mathbb R}}^d},{\mathcal M}), \ \ \mathrm{if} \ \ p\in(1,\,\infty).\end{equation*}

This finishes the proof of Theorem 5.2.

Proof of Theorem 5.3

Let $p\in(1,\,2)$ and $\frac1q=\frac{1-\theta}{p}+\theta$. Since the map $\Phi$ is an isomorphism from $\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M})$ to $L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}};L_2^c(\widetilde{\Gamma}_x,dyd\sigma(k)))\oplus_p L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$, we have

\begin{equation*}\left[\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},{\mathcal M}),\,\mathbf{h}^{A,c}_1({{{\mathbb R}}^d},{\mathcal M})\right]_{\theta} \subset\mathbf{h}_{q}^{A,c}({{{\mathbb R}}^d},{\mathcal M}).\end{equation*}

By the duality theory, we obtain

\begin{equation*}\mathbf{h}_{\frac{p'}{1-\theta}}^{A,c}({{{\mathbb R}}^d},{\mathcal M}) \subset\left[L_{p'}\mathrm{mo}^{c}_A({{{\mathbb R}}^d},{\mathcal M}),\,\mathrm{bmo}_A^c({{{\mathbb R}}^d},{\mathcal M})\right]_{\theta}.\end{equation*}

Therefore, we have

\begin{equation*} \left[\mathbf{h}_{p}^{A,c}({{{\mathbb R}}^d},{\mathcal M}),\, \mathbf{h}_{1}^{A,c}({{{\mathbb R}}^d},{\mathcal M})\right]_{\theta} =\mathbf{h}_{q}^{A,c}({{{\mathbb R}}^d},{\mathcal M}). \end{equation*}

Taking the duality for the above equality, we have

\begin{equation*} \left[\mathbf{h}_{p'}^{A,c}({{{\mathbb R}}^d},{\mathcal M}),\, \mathrm{bmo}_A^c({{{\mathbb R}}^d},{\mathcal M})\right]_{\theta} =\mathbf{h}_{\frac{p'}{1-\theta}}^{A,c}({{{\mathbb R}}^d},{\mathcal M}). \end{equation*}

For the case $1 \lt p_1,p_2 \lt \infty$, by the fact that, for $p\in(1,\,\infty)$, $\mathbf{h}_p^{A}({{{\mathbb R}}^d},{\mathcal M})$ is a complemented subspace of $L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}};L_2^c(\widetilde{\Gamma}_x,dyd\sigma(k)))\oplus_p L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$, we can deduce that, for any $1 \lt p_1,p_2 \lt \infty$,

\begin{equation*} \left[\mathbf{h}_{p_1}^{A,c}({{{\mathbb R}}^d},{\mathcal M}),\, \mathbf{h}_{p_2}^{A,c}({{{\mathbb R}}^d},{\mathcal M})\right]_{\theta} =\mathbf{h}_{p}^{A,c}({{{\mathbb R}}^d},{\mathcal M}), \end{equation*}

where $\frac{1}{p}=\frac{1-\theta}{p_1}+\frac{\theta}{p_2}$. Combining the above conclusions with Wolff’s interpolation theorem, we can obtain the desired result.

Proof of Theorem 5.4

Let $2 \lt q\leq p \lt \infty$. By an argument similar to the proof of Theorem 5.3, we have

\begin{equation*}\left[L_q\mathrm{mo}_A({{{\mathbb R}}^d},{\mathcal M}),\,\mathrm{bmo}_A({{{\mathbb R}}^d},{\mathcal M})\right]_{1-\frac{q}{p}} \subset L_{p}\mathrm{mo}_A({{{\mathbb R}}^d},{\mathcal M}),\end{equation*}

which, together with the duality theory, implies that

\begin{equation*}\left[\mathbf{h}_{q'}^{A}({{{\mathbb R}}^d},{\mathcal M}),\,\mathbf{h}_1^{A}({{{\mathbb R}}^d},{\mathcal M})\right]_{1-\frac{q}{p}} \supset \mathbf{h}_{p'}^A({{{\mathbb R}}^d},{\mathcal M})=L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}}).\end{equation*}

From Theorem 5.2 and the fact that $\mathbf{h}_1^A({{{\mathbb R}}^d},{\mathcal M})\subset L_1({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$, we conclude that

\begin{align*} &L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})\\ &\quad\subset\left[\mathbf{h}_{q'}^{A}({{{\mathbb R}}^d},{\mathcal M}),\,\mathbf{h}_1^{A}({{{\mathbb R}}^d},{\mathcal M})\right]_{1-\frac{q}{p}} =\left[L_{q'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}}),\,\mathbf{h}_1^{A}({{{\mathbb R}}^d},{\mathcal M})\right]_{1-\frac{q}{p}}\\ &\quad\subset \left[L_{q'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}}),\,L_1({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})\right]_{1-\frac{q}{p}} =L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}}). \end{align*}

Therefore, we have

\begin{equation*}L_{p'}({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})=\left[\mathbf{h}_{q'}^{A}({{{\mathbb R}}^d},{\mathcal M}),\,\mathbf{h}_1^{A}({{{\mathbb R}}^d},{\mathcal M})\right]_{1-\frac{q}{p}}.\end{equation*}

From this, the duality theory, Wolff’s interpolation theorem, and a similar argument, we can obtain our desired result.

Proof of Theorem 5.5

For any $1 \lt p,\,u,\,s \lt \infty$ and $\eta\in(0,\,1)$ such that

\begin{equation*}\frac{1}{p}=\frac{1-\eta}{u}+\frac{\eta}{s},\end{equation*}

by the fact that the space $\mathbf{h}_p^{A,c}$ is a complemented subspace of $L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}};L_2^c(\widetilde{\Gamma}_x,dyd\sigma(k)))\oplus_p L_p({L_{\infty}(\mathbb{R}^d)\overline{\otimes}\mathcal{M}})$ and a similar argument as above for the complex method, we obtain

(5.34)\begin{equation} \left[\mathbf{h}_u^{A,c}({{{\mathbb R}}^d},{\mathcal M}),\,\mathbf{h}_s^{A,c}({{{\mathbb R}}^d},{\mathcal M})\right]_{\eta,\,p} =\mathbf{h}_p^{A,c}({{{\mathbb R}}^d},\,{\mathcal M}). \end{equation}

Let $\theta:=\frac1p$, $0 \lt \theta_0 \lt \theta \lt \theta_1$. Then there exists a number $\eta\in(0,1)$ such that $\theta=(1-\eta)\theta_0+\eta\theta_1$. From this, [Reference Bergh and Löfström1, Theorem 4.7.2], Theorem 5.3 and (5.34), we deduce that

\begin{align*} &\left[\mathrm{bmo}_A^c({{{\mathbb R}}^d},{\mathcal M}),\,\mathbf{h}_1^{A,c}({{{\mathbb R}}^d},{\mathcal M})\right]_{\theta,\,p}\\ &\quad=\left[\left[\mathrm{bmo}^c_A({{{\mathbb R}}^d},{\mathcal M}),\,\mathbf{h}_1^{A,c}({{{\mathbb R}}^d},{\mathcal M})\right]_{\theta_0} ,\,\left[\mathrm{bmo}_A^c({{{\mathbb R}}^d},{\mathcal M}),\,\mathbf{h}_1^{A,c}({{{\mathbb R}}^d},{\mathcal M})\right]_{\theta_1}\right] _{\eta,\,p}\\ &\quad=\left[\mathbf{h}_{\frac{1}{\theta_0}}^{A,c}({{{\mathbb R}}^d},{\mathcal M}) ,\,\mathbf{h}_{\frac{1}{\theta_1}}^{A,c}({{{\mathbb R}}^d},{\mathcal M})\right] _{\eta,\,p} =\mathbf{h}_{p}^{A,c}({{{\mathbb R}}^d},{\mathcal M}), \end{align*}

with equivalent norms. ${\rm(ii)}$ can be proved in a similar way, the details are omitted here. Therefore, we finish the proof of Theorem 5.5.