$\sum_{i=k+1}^{k+m}\frac{1}{2^{l}}\sum_{j=0}^{2^{l}-1}\left\|\pi\left(v^{-j2^{il}}\right)P_{(i+1)l}f(a)-P_{il}f(a)\right\|_{X}^{q}=\sum_{i=k+1}^{k+m}\frac{1}{2^{l}}\sum_{j=0}^{2^{l}-1}\left\|P_{(i+1)l}f(a)-\pi\left(v^{j2^{il}}\right)P_{il}f(a)\right\|_{X}^{q}\leq(2K_{q}(X))^{q}.$

Thus

$\displaystyle|S|(2K_{q}(X))^{q}$
$\displaystyle\geq\sum_{i=k+1}^{k+m}\frac{1}{2^{l}}\sum_{j=0}^{2^{l}-1}\sum_{a\in
S}\left\|\pi\left(v^{-j2^{il}}\right)P_{(i+1)l}f(a)-P_{il}f(a)\right\|_{X}^{q}$
$\displaystyle\geq m\min_{\begin{subarray}{c}k+1\leq i\leq k+m\\\ 1\leq j\leq
2^{l}-1\end{subarray}}\sum_{a\in
S}\left\|\pi\left(v^{-j2^{il}}\right)P_{(i+1)l}f(a)-P_{il}f(a)\right\|_{X}^{q},$

and so there exist $i\in[k+1,k+m]$, $j\in[0,2^{l}-1]$ such that

$\max_{a\in
S}\left\|\pi\left(v^{-j2^{il}}\right)P_{(i+1)l}f(a)-P_{il}f(a)\right\|_{X}\lesssim_{\Gamma}\frac{K_{q}(X)}{m^{1/q}}.$

As $v^{\lfloor n^{\rho}\rfloor}=a_{1}\cdots a_{b}$ for some $a_{i}\in S$,
where $b=O_{\Gamma}(n)$, we have by the cocycle identity

$f(v^{\lfloor n^{\rho}\rfloor})=\sum_{i=1}^{b}\pi(a_{1}\cdots
a_{i-1})f(a_{i}).$

Since $P_{r}$, $r\geq 0$, and $\pi(g)$, $g\in\Gamma$, commute, and $v$ and
$g\in\Gamma$ commute, and $\pi(g)$ is an isometry for $g\in\Gamma$, we have

$\displaystyle\left\|\pi\left(v^{-j2^{il}}\right)P_{(i+1)l}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)-P_{il}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}\leq\sum_{i=1}^{b}\left\|\pi\left(v^{-j2^{il}}\right)P_{(i+1)l}f\left(a_{i}\right)-P_{il}f\left(a_{i}\right)\right\|_{X}\lesssim_{\Gamma}\frac{K_{q}(X)n}{m^{1/q}},$

as claimed. ∎

######  Lemma 44 (Analogue of [ANT13, Lemma 4.2]).

For every $m,n\in\mathbb{N}$,

$\left\|P_{m}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}\lesssim_{\Gamma}\frac{n^{(\rho^{2}+\rho-1)/(2\rho-1)}}{2^{m(\rho-1)/(2\rho-1)}}.$

###### Proof.

If $\frac{n^{\rho}}{2^{m}}\geq\rho-1$ then we have the obvious bound

$\left\|P_{m}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}\leq\left\|f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}\leq d_{W}\left(e_{\Gamma},v^{\lfloor
n^{\rho}\rfloor}\right)\lesssim_{\Gamma}n\left(\frac{n^{\rho}}{2^{m}}\right)^{(\rho-1)/(2\rho-1)}.$

Thus we assume $\frac{n^{\rho}}{2^{m}}<\rho-1$.

Because

$P_{m}-\pi(v^{k})P_{m}=\frac{1}{2^{m}}\sum_{j=0}^{k-1}\pi(v)^{j}-\frac{1}{2^{m}}\sum_{j=2^{m}}^{2^{m}+k-1}\pi(v)^{j},$

we have

$\left\|P_{m}-\pi(v^{k})P_{m}\right\|\leq\frac{2k}{2^{m}}.$

By the cocycle identity,

$f\left(v^{\lfloor k^{\rho}\rfloor\lfloor
n^{\rho}\rfloor}\right)=\sum_{j=0}^{\lfloor
k^{\rho}\rfloor-1}\pi\left(v^{j\lfloor
n^{\rho}\rfloor}\right)f\left(v^{\lfloor n^{\rho}\rfloor}\right)$

Because $f$ is $1$-Lipschitz, $d_{W}\left(e_{\Gamma},v^{\lfloor
k^{\rho}\rfloor\lfloor n^{\rho}\rfloor}\right)\lesssim_{\Gamma}kn$, and
$\|P_{m}\|\leq 1$,

$\displaystyle kn$ $\displaystyle\gtrsim_{\Gamma}\left\|P_{m}f\left(v^{\lfloor
k^{\rho}\rfloor\lfloor n^{\rho}\rfloor}\right)\right\|_{X}=\left\|\lfloor
k^{\rho}\rfloor P_{m}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)-\sum_{j=0}^{\lfloor
k^{\rho}\rfloor-1}\left(P_{m}-\pi\left(v^{j\lfloor
n^{\rho}\rfloor}\right)P_{m}\right)f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}$ $\displaystyle\geq\lfloor
k^{\rho}\rfloor\left\|P_{m}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}-\sum_{j=0}^{\lfloor
k^{\rho}\rfloor-1}\left\|P_{m}-\pi\left(v^{j\lfloor
n^{\rho}\rfloor}\right)P_{m}\right\|\cdot\left\|f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}$ $\displaystyle\geq\lfloor
k^{\rho}\rfloor\left\|P_{m}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}-\sum_{j=0}^{\lfloor
k^{\rho}\rfloor-1}\frac{2j\lfloor
n^{\rho}\rfloor}{2^{m}}d_{W}\left(e_{\Gamma},v^{\lfloor
n^{\rho}\rfloor}\right)$

and rearranging terms,

$\left\|P_{m}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}\lesssim_{\Gamma}\frac{n}{k^{\rho-1}}+\frac{n^{\rho+1}k^{\rho}}{2^{m}}.$

Since this is true for all $k$, choosing
$k=\left\lceil\left(\frac{(\rho-1)2^{m}}{\rho
n^{\rho}}\right)^{1/(2\rho-1)}\right\rceil$ gives the stated bound. Indeed, by
$\frac{n^{\rho}}{2^{m}}<\rho-1$ we have
$k\asymp\left(\frac{(\rho-1)2^{m}}{\rho n^{\rho}}\right)^{1/(2\rho-1)}$ and
plugging in gives the desired bound. ∎

######  Lemma 45 (Analogue of [ANT13, Lemma 4.3]).

For every $m,n\in\mathbb{N}$,

$\left\|f\left(v^{\lfloor n^{\rho}\rfloor}\right)-P_{m}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}\lesssim_{\Gamma}2^{m/(\rho+1)}n^{1/(\rho+1)}.$

###### Proof.

If we define $\tilde{f}:\Gamma\to X$ by

$\tilde{f}(h)\coloneqq f(h)-P_{m}f(h)=(I-P_{m})f(h),$

then $\tilde{f}\in Z^{1}(\pi)$. Let $k\geq 1$ be an integer to be determined
later. If we set

$w\coloneqq-\frac{1}{k}\sum_{j=0}^{k-1}\tilde{f}(v^{j}),$

then

$\|w\|_{X}\lesssim_{\Gamma}\frac{1}{k}\sum_{j=0}^{k-1}j^{1/\rho}\lesssim
k^{1/\rho}.$

For every $h\in\Gamma$ we have the following identity:

$\displaystyle\begin{aligned}
-\pi(h)w+\tilde{f}(h)&=\frac{1}{k}\sum_{j=0}^{k-1}\left(\pi(h)\tilde{f}(v^{j})+\tilde{f}(h)\right)=\frac{1}{k}\sum_{j=0}^{k-1}\tilde{f}(hv^{j})=\frac{1}{k}\sum_{j=0}^{k-1}\tilde{f}(v^{j}h)\\\
&=\frac{1}{k}\sum_{j=0}^{k-1}\left(\pi(v^{j})\tilde{f}(h)+\tilde{f}(v^{j})\right)=\frac{1}{k}\sum_{j=0}^{k-1}\pi(v^{j})\tilde{f}(h)-w.\end{aligned}$
(45)

But

$\displaystyle\frac{1}{k}\sum_{j=0}^{k-1}\pi(v^{j})\tilde{f}(h)=\frac{1}{k}\sum_{j=0}^{k-1}\left(\pi(v^{j})f(h)-\frac{1}{2^{m}}\sum_{i=0}^{2^{m}-1}\pi(v^{j+i})f(h)\right)=\frac{1}{2^{m}}\sum_{i=0}^{2^{m}-1}\left(\frac{1}{k}\sum_{j=0}^{k-1}\pi(v^{j})-\frac{1}{k}\sum_{j=i}^{i+k-1}\pi(v^{j})\right)f(h),$

so that

$\left\|\frac{1}{k}\sum_{j=0}^{k-1}\pi(v^{j})\tilde{f}(h)\right\|_{X}\leq\frac{d_{W}(h,e_{\Gamma})}{2^{m}}\sum_{i=0}^{2^{m}-1}\frac{2i}{k}\leq\frac{2^{m}}{k}d_{W}(h,e_{\Gamma}).$

Because of (45), $\tilde{f}$ is close to a coboundary in the following sense:

$\left\|\tilde{f}(h)-(\pi(h)w-w)\right\|\leq\frac{2^{m}}{k}d_{W}(h,e_{\Gamma}).$

Writing $v^{\lfloor n^{\rho}\rfloor}=a_{1}\cdots a_{b}$ for some $a_{i}\in S$,
where $b=O_{\Gamma}(n)$, we have

$\displaystyle\left\|\tilde{f}\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}$
$\displaystyle=\left\|\sum_{i=1}^{b}\pi(a_{1}\cdots
a_{i-1})\tilde{f}(a_{i})\right\|_{X}\lesssim_{\Gamma}\left\|\sum_{i=1}^{b}\pi(a_{1}\cdots
a_{i-1})\left(\pi(a_{i})w-w\right)\right\|_{X}+\frac{n2^{m}}{k}$
$\displaystyle=\left\|\pi\left(v^{\lfloor
n^{\rho}\rfloor}\right)w-w\right\|_{X}+\frac{n2^{m}}{k}\lesssim_{\Gamma}k^{1/\rho}+\frac{n2^{m}}{k}.$

With $k=\lceil n^{\frac{\rho}{\rho+1}}2^{\frac{m\rho}{\rho+1}}\rceil$, we have
the stated bound. ∎

We now prove theorem 9.

###### Proof of Theorem 9.

By [NP11, Theorem 9.1], by amenability of $\Gamma$, if $X$ is $q$-uniformly
convex and $f:\Gamma\to X$ is $1$-Lipschitz, then there exists a Banach space
$(Y,\|\cdot\|_{Y})$ that is also $q$-uniformly convex with
$K_{q}(Y)=K_{q}(X)$, an action $\pi$ of $\Gamma$ on $Y$ by linear isometric
automorphisms, and a $1$-cocycle $F:\Gamma\to Y$ such that
$\omega_{F}=\omega_{f}$, where $F:\Gamma\to Y$ is a $1$-cocycle if
$F(xy)=\pi(x)F(y)+F(x)$ for all $x,y\in\Gamma$. Thus, we may assume without
loss of generality that $f\in Z^{1}(\pi)$ for some action $\pi$ of $\Gamma$ on
$X$ by linear isometric automorphisms.

Let $C_{1}<C_{2}$ be constants depending on $\Gamma$ such that

$C_{1}n\leq d_{W}\left(v^{\lfloor n^{\rho}\rfloor},e_{\Gamma}\right)\leq
C_{2}n,\quad n\in\mathbb{N}.$

We may assume $t$ is sufficiently large so that if $m$ is the largest integer
such that

$\frac{2C_{1}}{C_{2}}2^{\frac{m}{\rho}}m^{\frac{m(\rho^{2}+2\rho-2)}{p\rho(\rho-1)}}\leq\sqrt{t},$
(46)

then

$\frac{p(\rho-1)}{\rho^{2}+2\rho-2}\leq\frac{1}{2}\log_{2}m.$ (47)

Given $m$, let $k$ be the smallest integer such that

$m^{\frac{\rho+1}{\rho p}+\frac{(\rho^{2}+2\rho-2)(k+1)}{p\rho(\rho-1)}}\geq
t,$ (48)

and define

$\ell\coloneqq\left\lceil\frac{\rho^{2}+2\rho-2}{p(\rho-1)}\log_{2}m\right\rceil.$
(49)

Using Lemma 43, we may find integers $i\in[k+1,k+m]$ and $j\in[0,2^{\ell}-1]$
such that for all $n\in\mathbb{N}$,

$\left\|\pi\left(v^{-j2^{il}}\right)P_{(i+1)l}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)-P_{il}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}\lesssim_{\Gamma}\frac{K_{q}(X)n}{m^{1/p}}.$
(50)

Finally, define

$n\coloneqq\left\lceil\frac{1}{C_{1}}m^{\frac{\rho+1}{\rho
p}}2^{\frac{i\ell}{\rho}}\right\rceil,$ (51)

Expanding, we have

$f\left(v^{\lfloor
n^{\rho}\rfloor}\right)=\pi\left(v^{-j2^{i\ell}}\right)P_{(i+1)\ell}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)+\left(P_{i\ell}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)-\pi\left(v^{-j2^{i\ell}}\right)P_{(i+1)\ell}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right)+\left(f\left(v^{\lfloor
n^{\rho}\rfloor}\right)-P_{i\ell}f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right).$

Thus by Lemmas 44 and 45 and inequality (50), we obtain:

$\omega_{f}\left(d_{W}\left(v^{\lfloor
n^{\rho}\rfloor},e_{\Gamma}\right)\right)\leq\left\|f\left(v^{\lfloor
n^{\rho}\rfloor}\right)\right\|_{X}\lesssim_{\Gamma}\frac{n^{(\rho^{2}+\rho-1)/(2\rho-1)}}{2^{(i+1)l(\rho-1)/(2\rho-1)}}+\frac{K_{q}(X)n}{m^{1/p}}+2^{il/(\rho+1)}n^{1/(\rho+1)}\\\
\stackrel{{\scriptstyle\eqref{eq:def l}\wedge\eqref{eq:def
n}}}{{\lesssim_{\Gamma}}}\frac{K_{q}(X)n}{m^{1/p}}.$ (52)

We compute

$d_{W}\left(v^{\lfloor n^{\rho}\rfloor},e_{\Gamma}\right)\geq
C_{1}n\stackrel{{\scriptstyle\eqref{eq:def n}}}{{\geq}}m^{\frac{\rho+1}{\rho
p}}2^{\frac{i\ell}{\rho}}\geq m^{\frac{\rho+1}{\rho
p}}2^{\frac{(k+1)\ell}{\rho}}\stackrel{{\scriptstyle\eqref{eq:def
l}}}{{\geq}}m^{\frac{\rho+1}{\rho
p}+\frac{(\rho^{2}+2\rho-2)(k+1)}{p\rho(\rho-1)}}\stackrel{{\scriptstyle\eqref{eq:def
k}}}{{\geq}}t$

and

$\displaystyle d_{W}\left(v^{\lfloor n^{\rho}\rfloor},e_{\Gamma}\right)\leq
C_{2}n\stackrel{{\scriptstyle\eqref{eq:def
n}}}{{\leq}}\frac{2C_{2}}{C_{1}}m^{\frac{\rho+1}{\rho
p}}2^{\frac{i\ell}{\rho}}\leq\frac{2C_{2}}{C_{1}}m^{\frac{\rho+1}{\rho
p}}2^{\frac{(k+m)\ell}{\rho}}\stackrel{{\scriptstyle\eqref{eq:def
l}}}{{\leq}}\frac{2C_{1}}{C_{2}}2^{\frac{k+m}{\rho}}m^{\frac{\rho+1}{\rho
p}+\frac{k(\rho^{2}+2\rho-2)}{p\rho(\rho-1)}}m^{\frac{m(\rho^{2}+2\rho-2)}{p\rho(\rho-1)}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:def
k}}}{{<}}\frac{2C_{1}}{C_{2}}2^{\frac{k+m}{\rho}}tm^{\frac{m(\rho^{2}+2\rho-2)}{p\rho(\rho-1)}}\stackrel{{\scriptstyle\eqref{eq:def
k}}}{{\leq}}\frac{2C_{1}}{C_{2}}2^{\frac{p(\rho-1)}{\rho^{2}+2\rho-2}\log_{m}t}2^{\frac{m}{\rho}}tm^{\frac{m(\rho^{2}+2\rho-2)}{p\rho(\rho-1)}}\stackrel{{\scriptstyle\eqref{eq:def
m}}}{{\leq}}t^{3/2}2^{\frac{p(\rho-1)}{\rho^{2}+2\rho-2}\log_{m}t}\stackrel{{\scriptstyle\eqref{eq:prop
m}}}{{\leq}}t^{2}.$

Therefore $t\leq d_{W}\left(v^{\lfloor n^{\rho}\rfloor},e_{\Gamma}\right)\leq
t^{2}$. The definition of $m$ implies $m\gtrsim_{\Gamma}\min\left\\{\log
n,\frac{p\log n}{\log\log n}\right\\}$, and thus by (52), we have

$\frac{\omega_{f}\left(d_{W}\left(v^{\lfloor
n^{\rho}\rfloor},e_{\Gamma}\right)\right)}{n}\lesssim
K_{q}(X)\left(\frac{\log\log n}{\log n}\right)^{1/p}.$

The proof of Theorem 9 is complete. ∎

##  6\. Derivation of Theorem 24 from Dorronsoro’s Theorem 22

In this section, we prove Theorem 24, which proves Theorem 17 since it is a
special case. This section follows Section 7 of [FO20a] closely.

Using an argument similar to the Vitali covering lemma, we may find a
collection $\mathcal{B}_{r}$ of balls $B$ of radius $r$ whose union covers
$G$, and such that the concentric balls $\hat{B}$ of radius $(n+1)r$ have
bounded overlap. Then

$\displaystyle\int_{G}\left(\frac{1}{r^{\alpha}}\left|\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}f(x(\delta_{r}(v))^{j})\right|\right)^{p}dx$
$\displaystyle\leq\sum_{B\in{\mathcal{B}}_{r}}\int_{B}\left(\frac{1}{r^{\alpha}}\left|\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}f(x(\delta_{r}(v))^{j})\right|\right)^{p}dx$
$\displaystyle\lesssim_{p}\sum_{j=0}^{n}\binom{n}{j}^{p}\sum_{B\in{\mathcal{B}}_{r}}\int_{B}\left(\frac{1}{r^{\alpha}}\left|(f(x(\delta_{r}(v))^{j})-A^{\lfloor\alpha\rfloor}_{\hat{B}}f(x(\delta_{r}(v))^{j})\right|\right)^{p}dx$
$\displaystyle\qquad\qquad+\int_{B}\left(\frac{1}{r^{\alpha}}\left|\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}A^{\lfloor\alpha\rfloor}_{\hat{B}}f(x(\delta_{r}(v))^{j})\right|\right)^{p}dx$
$\displaystyle\lesssim_{n,p}\sum_{B\in{\mathcal{B}}_{r}}\int_{\hat{B}}\left(\frac{1}{r^{\alpha}}\left|f(x)-A^{\lfloor\alpha\rfloor}_{\hat{B}}f(x)\right|\right)^{p}dx+0$
$\displaystyle=\sum_{B\in{\mathcal{B}}_{r}}\left[\frac{1}{r^{\alpha}}\beta_{f,\lfloor\alpha\rfloor,p}(\hat{B})\right]^{p}|\hat{B}|\lesssim_{G,\alpha}\int_{G}\left[\frac{1}{r^{\alpha}}\cdot\beta_{f,\lfloor\alpha\rfloor,p}(B_{2(n+1)r}(x))\right]^{p}dx.$

We have used in the penultimate inequality the fact that since
$A^{\lfloor\alpha\rfloor}_{\hat{B}}f$ is a polynomial of weighted degree at
most $\lfloor\alpha\rfloor$, and since $\delta_{r}(v)\in
V_{\lfloor\alpha/n\rfloor+1}\oplus\cdots\oplus V_{s}$,
$A^{\lfloor\alpha\rfloor}_{\hat{B}}f(x(\delta_{r}(v))^{j})$ is a polynomial in
$j$ of degree at most $n-1$, and hence

$\Delta_{\delta_{r}(v)}^{n}f(x)=\sum_{j=0}^{n}(-1)^{n-j}\binom{n}{j}A^{\lfloor\alpha\rfloor}_{\hat{B}}f(x(\delta_{r}(v))^{j})=0.$

In the last inequality we used Corollary 47. Now, the inequality (9) follows
from Minkowski’s integral inequality and Theorem 22 with $q=p$:

$\displaystyle\left(\int_{0}^{\infty}\left[\int_{G}\left(\frac{1}{r^{\alpha}}\left|\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}f(x(\delta_{r}(v))^{j})\right|\right)^{p}dx\right]^{2/p}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\lesssim_{G,\alpha,n,p}\left(\int_{0}^{\infty}\left[\int_{G}\left[\frac{1}{r^{\alpha}}\beta_{f,\lfloor\alpha\rfloor,p}(B_{2(n+1)r}(x))\right]^{p}dx\right]^{2/p}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\leq\left(\int_{G}\left[\int_{0}^{\infty}\left[\frac{1}{r^{\alpha}}\beta_{f,\lfloor\alpha\rfloor,p}(B_{2(n+1)r}(x))\right]^{2}\frac{dr}{r}\right]^{p/2}dx\right)^{1/p}\lesssim_{G,\alpha,n,p}\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)}.$

The proof of Theorem 24 is complete.

##  7\. Proof of Dorronsoro’s Theorem 22 for Carnot groups

We need to prove two directions. In one direction, we assume $f\in
S^{p}_{\alpha}(G)$ and prove the $\lesssim$ direction of (8). In the other
direction, we assume $f\in L^{p}(G)$ with the left-hand side of (8) finite,
and prove $f\in S^{p}_{\alpha}(G)$ along with the $\gtrsim$ direction of (8).
Our proof will be based on Dorronsoro’s original proof of the
$G=\mathbb{R}^{n}$ case in [Dor85] and will borrow modifications inspired by
the proof of $G=\mathbb{H}^{k}$ case in [FO20a] to deal with general Carnot
groups.

We will first prove Theorem 22 for $q=1$, the $\lesssim$ direction in
subsection 7.2 and the $\gtrsim$ direction in subsection 7.3. Since the left-
hand side of (8) is minimized when $q=1$, this will finish the proof of the
$\gtrsim$ direction; for the $\lesssim$ direction, we will see in subsection
7.4 that the $\lesssim$ inequality for $q=1$ implies the $\lesssim$ inequality
for $1\leq q<\frac{\min\\{p,2\\}n_{h}}{n_{h}-\min\\{p,2\\}}$.

For the $\lesssim$ direction, we will first see by an approximation argument
that it is enough to look at smooth functions $f$ (subsubsection 7.2.1), for
which it turns out that we may as well approximate at all scales
simultaneously by Taylor polynomials (subsubsection 7.2.2). Then, for
$0<\alpha<1$, it will turn out that the desired inequality has already been
proven in [CRTN01, Theorem 5] (subsubsection 7.2.4), and for $\alpha>1$
nonintegral we may use an induction argument on $\alpha$ (subsubsection
7.2.5). We finish off the case of $\alpha\geq 1$ integral by an interpolation
argument (subsubsection 7.2.6), with the extra terms arising in this process
having been taken care of by a homogenization argument (subsubsection 7.2.3).

For the $\gtrsim$ direction, we assume $f\in L^{p}(G)$ and the finiteness of a
certain singular integral and need to derive $f\in S^{p}_{\alpha}(G)$. We
first see again that we may approximate by a fixed polynomial for all scales
(which we suspect to be the Taylor polynomial in the distributive sense but we
can only prove this up to first derivatives) in Proposition 54. For
$0<\alpha\leq 1$ we again use characterizations of the fractional Laplacian
given by [CRTN01, Theorem 5] for $0<\alpha<1$ and [DNM21, Theorem 1.4] for
$\alpha=1$ (subsubsection 7.3.1). We then prove the case $\alpha>1$ by
induction (subsubsection 7.3.2).

For simplicity, we define the $L^{1}$-beta numbers

$\beta_{f,d}(B_{r}(x))\coloneqq\beta_{f,d,1}(B_{r}(x)).$

If we define the function

$\mathfrak{G}_{\alpha}f(x)\coloneqq\left(\int_{0}^{\infty}\left(\frac{\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))}{r^{\alpha}}\right)^{2}\frac{dr}{r}\right)^{1/2},\quad
x\in G,$

then Theorem 22 states that

$\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}\asymp_{G,\alpha,p}\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)},\quad
f\in S^{p}_{\alpha}.$ (53)

Before we begin the proof, we briefly remark on coordinate and multi-index
notation on $G$. Recall the coordinate system
$x=\exp\left(\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}x_{r,i}X_{r,i}\right)$. A multi-
index $\gamma=((\gamma_{r,i})_{i=1}^{k_{r}})_{r=1}^{s}$ is a multi-index on
$\sum_{r=1}^{s}k_{r}$ entries, and we denote

$|\gamma|\coloneqq\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}r|\gamma_{r,i}|\in\mathbb{N},\quad\gamma!\coloneqq\prod_{r=1}^{s}\prod_{i=1}^{k_{r}}\gamma_{r,i}!,\quad
x^{\gamma}\coloneqq\prod_{r=1}^{s}\prod_{i=1}^{k_{r}}x_{r,i}^{\gamma_{r,i}}\in\mathbb{R}.$

The latter should not be confused with the previous notation
$x^{t}=\exp\left(\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}tx_{r,i}X_{r,i}\right)$ for
$t\in\mathbb{R}$.

We now define the weighted degree of linear combinations of ‘polynomial
differential operators’, that is, linear combinations of differential
operators of the form

$x^{\gamma_{1}}\left(\frac{\partial}{\partial x}\right)^{\gamma_{2}},$

by assigning weight $|\gamma_{2}|-|\gamma_{1}|$ to the above term. Then, a
polynomial differential operator of homogeneous weighted degree $d_{1}$ will
act on a polynomial of homogeneous weighted degree $d_{2}$ to produce a
polynomial of homogeneous weighted degree $d_{2}-d_{1}$. We observe that the
left-invariant differential operators $X_{r,i}$ is a polynomial differential
operator of homogeneous weighted degree $r$ for $r=1,\cdots,s$ and
$i=1,\cdots,k_{r}$.

Recall from subsection 1.1 that the polynomials of weighted degree $\leq r$
are left-invariant.

###  7.1. $L^{1}$-optimality of the $A^{d}_{x,r}f$’s

We begin the proof by discussing some basic properties of $A^{d}_{x,r}f$ and
$\beta_{f,d}(B_{r}(x))$.

It is well-known that $A^{d}_{x,r}f$ is the optimal
$L^{2}(B_{r}(x))$-approximation of $f$ in ${\mathcal{A}}_{d}$. It turns out
that it is also an optimal $L^{1}(B_{r}(x))$-approximation up to constants. To
see this, we first observe that for all $d\geq 0$, there exists an integral
formula for the coefficients of $A^{d}_{x,r}f$, given by the Gram–Schmidt
process. For example in $d=1$,

$A^{1}_{x,r}f(xy)=\langle
f\rangle_{B_{r}(x)}+\sum_{j=1}^{k}\frac{\fint_{B_{r}(x)}f(z)(z_{1,j}-x_{1,j})dz}{\fint_{B_{r}(x)}(z_{1,j}-x_{1,j})^{2}dz}y_{j},\quad
y\in G.$ (54)

More generally, for each $d\geq 0$, there exists a family
$\\{p^{d}_{\gamma}\\}_{|\gamma|\leq d}$ of polynomials such that

$A^{d}_{0,1}f(y)=\sum_{|\gamma|\leq
d}\left(\int_{B_{1}}fp^{d}_{\gamma}\right)y^{\gamma},\quad y\in G,$

and by rescaling we have for $x\in G$ and $r>0$ that

$A^{d}_{x,r}f(xy)=\sum_{|\gamma|\leq
d}r^{-n_{h}-|\gamma|}\left(\int_{B_{r}(x)}f(z)p^{d}_{\gamma}(\delta_{1/r}(x^{-1}z))dz\right)y^{\gamma}.$

Therefore

$\|A^{d}_{x,r}f\|_{L^{\infty}(B_{r}(x))}\lesssim_{G,d}\fint_{B_{r}(x)}|f(y)|dy.$
(55)

In other words, $A^{d}_{x,r}$ is a linear projection of $L^{1}(B_{r}(x))$ onto
$\left.\mathcal{A}_{d}\right|_{B_{r}(x)}$ which is bounded in the $L^{1}\to
L^{\infty}$ norm.

We now prove the $L^{1}$-optimality of the $A^{d}_{x,r}f$’s.

######  Lemma 46.

For $x\in G$, $r>0$, $f\in L^{1}_{\mathrm{loc}}(B_{r}(x))$, and
$d\in\mathbb{Z}_{\geq 0}$,

$\fint_{B_{r}(x)}|f(y)-A^{d}_{x,r}f(y)|dy\lesssim_{G,d}\fint_{B_{r}(x)}|f(y)-A(y)|dy,\quad
A\in\mathcal{A}_{d}.$

###### Proof.

From (55),

$\fint_{B_{r}(x)}|f-A^{d}_{x,r}f|\leq\fint_{B_{r}(x)}|f-A|+\fint_{B_{r}(x)}|A^{d}_{x,r}(f-A)|\lesssim_{G,d}\fint_{B_{r}(x)}|f-A|.$

∎

It follows that $\beta$ possesses a weak monotonicity property.

######  Corollary 47.

Let $x,y\in G$, $0<r<s<\infty$, $d\in\mathbb{Z}_{\geq 0}$ be such that
$B_{r}(x)\subset B_{s}(y)$. Then

$\beta_{f,d}(B_{r}(x))\lesssim_{G,d}\left(\frac{s}{r}\right)^{n_{h}}\beta_{f,d}(B_{s}(y)).$

By Corollary 47, we have that

$\beta_{f,d}(x,r)\lesssim_{G,d}\beta_{f,d}(x,s)\lesssim_{G,d}\beta_{f,d}(x,2r),\quad
r\leq s\leq 2r.$

Thus $\mathfrak{G}_{\alpha}f(x)$ is comparable to the series

$\left(\sum_{i=-\infty}^{\infty}\left[2^{-i\alpha}\beta_{f,d}(x,2^{i})\right]^{2}\right)^{1/2}.$

One can also see that, if $X$ is a left-invariant differential operator of
weighted order $n$, then because $A^{d}_{x,r}f$ is a polynomial of weighted
degree $d$,

$\|XA^{d}_{x,r}f\|_{L^{\infty}(B_{r}(x))}\lesssim_{G,n,d,X}r^{-n}\|A^{d}_{x,r}f\|_{L^{\infty}(B_{r}(x))}\lesssim_{G,d}r^{-n}\fint_{B_{r}(x)}|f|.$
(56)

###  7.2. Proof of the $\lesssim$ direction, $q=1$

####  7.2.1. Approximation by smooth compactly supported functions

It is enough to prove the $\lesssim$ statement of Theorem 22 for $f$ in the
space $\mathcal{D}$ of smooth compactly supported functions on $G$. Indeed, by
[Fol75, Theorem 4.5], $\mathcal{D}$ is dense in $S^{p}_{\alpha}$ for all
$1<p<\infty$ and $\alpha\geq 0$. Given $f\in S^{p}_{\alpha}$, choose a
sequence $\\{f_{j}\\}_{j=1}^{\infty}\subset\mathcal{D}$ that converges to $f$
in $S^{p}_{\alpha}$. Then $f_{j}\to f$ in $L^{p}(G)$, so by the contraction
property (55) we have

$\beta_{f_{j},\lfloor{\alpha}\rfloor}(B_{r}(x))\to\beta_{f,\lfloor{\alpha}\rfloor}(B_{r}(x))\mbox{
as }j\to\infty,\quad x\in G,~{}r>0.$

Now, by two applications of Fatou’s lemma, it follows that

$\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}\leq\liminf_{j\to\infty}\|\mathfrak{G}_{\alpha}f_{j}\|_{L^{p}(G)}\lesssim_{G,\alpha,p}\liminf_{j\to\infty}\|(-\Delta_{p})^{\alpha/2}f_{j}\|_{L^{p}(G)}=\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)}.$

This completes the proof of our claim.

####  7.2.2. Taylor polynomials

Recall the coordinate system on $G$ where each $y\in G$ is uniquely expressed
as $y=\exp\left(\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}y_{r,i}X_{r,i}\right)$,
$y_{r,i}\in\mathbb{R}$. Let $f\in\mathcal{D}$ and $x\in G$. The well-known
Taylor formula tells us that for $n\in\mathbb{Z}_{\geq 0}$,

$f(xy)=f(x)+\sum_{j=1}^{n}\frac{1}{j!}\left.\frac{d^{j}}{dt^{j}}\right|_{t=0}f(xy^{t})+\frac{1}{n!}\int_{0}^{1}(1-t)^{n}\frac{d^{n+1}}{dt^{n+1}}f(xy^{t})dt.$

As $y^{t}$ is the one-parameter subgroup generated by
$\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}y_{r,i}X_{r,i}$, we have

$f(xy)=f(x)+\sum_{j=1}^{n}\frac{1}{j!}\left(\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}y_{r,i}X_{r,i}\right)^{j}f(x)+\frac{1}{n!}\int_{0}^{1}(1-t)^{n}\left(\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}y_{r,i}X_{r,i}\right)^{n+1}f(xy^{t})dt.$

For each multi-index $\gamma$, if we denote by
$\operatorname{Sym}(X^{\gamma})$ the ‘coefficient operator’ of $y^{\gamma}$ in
$\frac{\gamma!}{j!}\left(\sum_{r=1}^{s}\sum_{j=1}^{k_{r}}y_{r,j}X_{r,j}\right)^{|\gamma|}$
where $j=\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}\gamma_{r,i}$, then the above
expression can be rewritten as

$f(xy)=\sum_{\begin{subarray}{c}\gamma\mathrm{~{}multi-index}\\\
\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}\gamma_{r,i}\leq
n\end{subarray}}\frac{1}{\gamma!}\left[\operatorname{Sym}(X^{\gamma})f(x)\right]y^{\gamma}+(n+1)\int_{0}^{1}\sum_{\begin{subarray}{c}\gamma\mathrm{~{}multi-
index}\\\
\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}\gamma_{r,i}=n+1\end{subarray}}\frac{1}{\gamma!}\left[\operatorname{Sym}(X^{\gamma})f(xy^{t})\right]y^{\gamma}dt.$

By isolating the terms with multi-index $\gamma$ such that $|\gamma|\leq n$,
we see that there exist for each multi-index $\gamma$ with $|\gamma|\geq n+1$
and $\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}\gamma_{r,i}\leq n+1$ a function
$q_{x,\gamma}f\in\mathcal{D}$ such that

$f(xy)=\underbrace{\sum_{|\gamma|\leq
n}\frac{1}{\gamma!}\left[\operatorname{Sym}(X^{\gamma})f(x)\right]y^{\gamma}}_{\eqqcolon
T^{n}_{x}f(xy)}+\underbrace{\sum_{\begin{subarray}{c}|\gamma|\geq n+1\\\
\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}\gamma_{r,i}\leq
n+1\end{subarray}}q_{x,\gamma}f(y)y^{\gamma}}_{=o(d_{G}(y,e_{G})^{n})\mathrm{~{}as~{}}d_{G}(y,e_{G})\to
0},\quad y\in G,$

where we denote the first sum to be the _Taylor polynomial of $f$ of degree
$n$ at $x\in G$_, while we observe the second term to be
$o(d_{G}(y,e_{G})^{n})$ as $d_{G}(y,e_{G})\to 0$ because
$d_{G}(y,e_{G})\asymp_{G}\sum_{r=1}^{s}\sum_{i=1}^{k_{r}}|y_{r,i}|^{1/r}$.

We may observe that if $Y$ is a left-invariant homogeneous differential
operator of degree $d\leq n$, then

$Yf(xy)=\underbrace{\sum_{|\gamma|\leq
n}\frac{1}{\gamma!}\left[\operatorname{Sym}(X^{\gamma})f(x)\right]Yy^{\gamma}}_{=T^{n-d}_{x}Yf(xy)}+\underbrace{\sum_{\begin{subarray}{c}|\gamma|\geq
n+1\\\ \sum_{r=1}^{s}\sum_{i=1}^{k_{r}}\gamma_{r,i}\leq
n+1\end{subarray}}Y(q_{x,\gamma}f(y)y^{\gamma})}_{=o(d_{G}(y,e_{G})^{n-d})\mathrm{~{}as~{}}d_{G}(y,e_{G})\to
0},\quad y\in G,$

with the second sum being $o(d_{G}(y,e_{G})^{n-d})$ and the first sum being a
polynomial of weighted degree at most $n-d$ and hence is the Taylor polynomial
of $Yf$ of degree $n-d$ at $x\in G$. This proves that

$YT^{n}_{x}f=T^{n-d}_{x}Yf.$

In particular, we have for the left-invariant horizontal vector fields that
with $d\geq 1$,

$X_{i}T^{d}_{x}f=T^{d-1}_{x}X_{i}f,\quad i=1,\cdots,k.$

We may define the $\beta$-numbers and $\mathfrak{G}$-function using the Taylor
polynomial instead.

$\tilde{\beta}_{f,d}(B_{r}(x))\coloneqq\fint_{B_{r}(x)}|f-T^{\lfloor\alpha\rfloor}_{x,r}f|,$

and

$\tilde{\mathfrak{G}}_{\alpha}f(x)\coloneqq\left(\int_{0}^{\infty}\left(\frac{1}{r^{\alpha}}\tilde{\beta}_{f,\lfloor\alpha\rfloor}(B_{r}(x))\right)^{2}\frac{dr}{r}\right)^{1/2},\quad
x\in G.$

######  Proposition 48.

If $f\in\mathcal{D}$ and $\alpha$ is nonintegral, then
$\mathfrak{G}_{\alpha}f(x)\asymp_{G,\alpha}\tilde{\mathfrak{G}}_{\alpha}f(x)$
for all $x\in G$.

###### Proof.

That
$\mathfrak{G}_{\alpha}f(x)\lesssim_{G,\alpha}\tilde{\mathfrak{G}}_{\alpha}f(x)$
follows from Lemma 46. Denote $d=\lfloor\alpha\rfloor$ and

$A^{d}_{x,r}f(xy)=\sum_{|\gamma|\leq d}f_{\gamma}(x,r)y^{\gamma},$

then

$f_{\gamma}(x,r)=\frac{1}{\gamma!}\operatorname{Sym}(X^{\gamma})A^{d}_{x,r}f(x).$

Also denote

$T^{d}_{x}f(xy)=\sum_{|\gamma|\leq d}f_{\gamma}(x)y^{\gamma},$

then, in the case $f\in\mathcal{D}$, $f_{\gamma}(x,r)\to f_{\gamma}(x)$ as
$r\to 0$ by Taylor expansion. Indeed, recalling that
$f(xy)=T^{d}_{x}f(xy)+O_{f}(d_{G}(y,e_{G})^{d+1})$,

$\|A^{d}_{x,r}f-T^{d}_{x}f\|_{L^{\infty}(B_{r}(x))}=\|A^{d}_{x,r}[f-T^{d}_{x}f]\|_{L^{\infty}(B_{r}(x))}\stackrel{{\scriptstyle\eqref{claim}}}{{\lesssim}}_{G,d}\fint_{B_{r}(x)}|f-T^{d}_{x}f|=O_{G,d,f}(r^{d+1})\quad\mathrm{as~{}}r\to
0.$

By the above expansions for $A^{d}_{x,r}f$ and $T^{d}_{x}f$, we have

$\|\sum_{|\gamma|\leq
d}(f_{\gamma}(x,r)-f_{\gamma}(x))r^{|\gamma|}y^{\gamma}\|_{L^{\infty}(B_{1})}=\|\sum_{|\gamma|\leq
d}(f_{\gamma}(x,r)-f_{\gamma}(x))y^{\gamma}\|_{L^{\infty}(B_{r})}=O_{G,d,f}(r^{d+1}).$

But because the space $\mathcal{A}^{d}$ of polynomials of weighted degree
$\leq d$ normed by $L^{\infty}(B_{1})$ is finite dimensional, all norms are
equivalent, and hence

$\max_{|\gamma|\leq
d}r^{|\gamma|}|f_{\gamma}(x,r)-f_{\gamma}(x)|=O_{G,d,f}(r^{d+1}).$

This completes the proof of the convergence $f_{\gamma}(x,r)\to f_{\gamma}(x)$
as $r\to 0$.

With this, we provide a different bound on $|f_{\gamma}(x,r)-f_{\gamma}(x)|$
using the $\beta$ numbers. For each $x\in G$,

$\displaystyle\begin{aligned}
|f_{\gamma}(x,r)-f_{\gamma}(x)|&\leq\sum_{i=-\infty}^{0}|f_{\gamma}(x,2^{i}r)-f_{\gamma}(x,2^{i-1}r)|\\\
&=\frac{1}{\gamma!}\sum_{i=-\infty}^{0}\left|\operatorname{Sym}(X^{\gamma})(A^{d}_{x,2^{i}r}f-A^{d}_{x,2^{i-1}r}f)(x)\right|\\\
&\stackrel{{\scriptstyle\eqref{polydiff}}}{{\lesssim_{G,|\gamma|}}}\sum_{i=-\infty}^{0}2^{-i|\gamma|}r^{-|\gamma|}\fint_{B_{2^{i-1}r}(x)}|f-A^{d}_{x,2^{i}r}f|\\\
&=\
\sum_{i=-\infty}^{0}2^{-i|\gamma|}r^{-|\gamma|}\beta_{f,d}(B_{2^{i}r}(x))\\\
&\stackrel{{\scriptstyle\mathrm{Corollary}~{}\ref{monotoneBeta}}}{{\lesssim_{G,d}}}\int_{0}^{2r}\beta_{f,d}(B_{u}(x))u^{-|\gamma|}\frac{du}{u}.\end{aligned}$
(57)

(With the Cauchy–Schwartz inequality, we may prove that
$|f_{\gamma}(x,r)-f_{\gamma}(x)|\lesssim_{G,\alpha}r^{\alpha-|\gamma|}\mathfrak{G}_{\alpha}f(x)$
for $|\gamma|<\alpha$; see Proposition 54 for the computation. This is weaker
than the $O_{G,d,f}(r^{d+1-|\gamma|})$ bound we get using Taylor
approximation. This is because the Taylor approximation argument leverages on
the fact that $f$ has $(d+1)$ and higher derivatives, whereas the
$\mathfrak{G}_{\alpha}f$ (in principle) only measures the “$\alpha$-th
derivative” of $f$ at $x$; however, the latter approach is more natural since
we have to work with $\mathfrak{G}_{\alpha}f$.)

Now we bound

$\displaystyle\tilde{\beta}_{f,d}(B_{r}(x))$
$\displaystyle=\fint_{B_{r}(x)}|f-T^{d}_{x}f|$
$\displaystyle\leq\fint_{B_{r}(x)}|f-A^{d}_{x,r}f|+\fint_{B_{r}(x)}|T^{d}_{x}f-A^{d}_{x,r}f|$
$\displaystyle\leq\beta_{f,d}(B_{r}(x))+C\sum_{|\gamma|\leq
d}|f_{\gamma}(x,r)-f_{\gamma}(x)|r^{|\gamma|}$
$\displaystyle\leq\beta_{f,d}(B_{r}(x))+C\sum_{i=0}^{d}r^{i}\int_{0}^{2r}\beta_{f,d}(B_{u}(x))u^{-i}\frac{du}{u},$

where $C$ is a constant depending on $G$ and $d$. By plugging into the
definition of $\tilde{\mathfrak{G}}$ and using the triangle inequality,

$\displaystyle\tilde{\mathfrak{G}}_{\alpha}f(x)\leq\mathfrak{G}_{\alpha}f(x)+C\sum_{i=0}^{d}\left(\int_{0}^{\infty}\left[r^{i-\alpha}\int_{0}^{2r}\beta_{f,d}(B_{u}(x))u^{-i}\frac{du}{u}\right]^{2}\frac{dr}{r}\right)^{1/2}.$

But by Hardy’s inequality (23), we have

$\displaystyle\tilde{\mathfrak{G}}_{\alpha}f(x)$
$\displaystyle\leq\mathfrak{G}_{\alpha}f(x)+C\sum_{i=0}^{d}\frac{1}{\alpha-i}\left(\int_{0}^{\infty}\left[\frac{\beta_{f,d}(B_{r}(x))}{r^{\alpha}}\right]^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\lesssim_{G,\alpha}\mathfrak{G}_{\alpha}f(x).$

This is where we use the fact that $\alpha$ is nonintegral. ∎

####  7.2.3. Inhomogeneous estimates to homogeneous estimates

We provide one more reduction in the proof of Theorem 22. It is enough to
prove the weaker estimate

$\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}\lesssim_{G,p,\alpha}\|f\|_{p,\alpha}=\|f\|_{L^{p}(G)}+\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)},\quad
f\in\mathcal{D}.$

(In fact, the inequality Dorronsoro proves in [Dor85, Theorem 2] is

$\|f\|_{L^{p}(G)}+\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}\asymp_{\alpha,p}\|f\|_{p,\alpha}$

when $G=\mathbb{R}^{n}$.) This is because a dimensional analysis reveals that
the above inequality is inhomogeneous, so by exploiting scaling properties we
may remove the inhomogeneous term $\|f\|_{L^{p}(G)}$.

######  Lemma 49.

For $1<p<\infty$ and $\alpha>0$, if we have the estimate,

$\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}\leq
C_{G,p,\alpha}\|f\|_{p,\alpha},\quad f\in\mathcal{D},$ (58)

then with the same constant $C_{G,p,\alpha}$,

$\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}\leq
C_{G,p,\alpha}\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)},\quad f\in\mathcal{D}.$
(59)

###### Proof.

Given $f\in\mathcal{D}$, set $f_{s}\coloneqq f\circ\delta_{s}\in\mathcal{D}$,
for $s>0$, where $\delta_{s}$ is the Carnot group dilation. It is easy to see
that

$\begin{cases}\|f_{s}\|_{L^{p}({G})}=s^{-n_{h}/p}\|f\|_{L^{p}({G})},\\\
\|\mathfrak{G}_{\alpha}f_{s}\|_{L^{p}(G)}=s^{\alpha-
n_{h}/p}\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)},\end{cases}\quad s>0.$

Also, it is morally clear that we should have

$\|(-\Delta_{p})^{\alpha/2}f_{s}\|_{L^{p}(G)}=s^{\alpha-
n_{h}/p}\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)},\quad s>0,$

and one may indeed verify this using the definition of
$(-\Delta_{p})^{\alpha/2}$; see the proof of [FO20a, Lemma 2.6] for details.

We now obtain (59) for $f$ from (58) for $f_{s}$ and taking $s\to\infty$. ∎

####  7.2.4. The case $0<\alpha<1$

We now begin proving Theorem 22.

Fix $1<p<\infty$ and $0<\alpha<1$. Note that we have

$\tilde{\mathfrak{G}}_{\alpha}f(x)=\left(\int_{0}^{\infty}\left[\frac{1}{r^{\alpha}}\fint_{B_{r}}|f(xy)-f(x)|dy\right]^{2}\frac{dr}{r}\right)^{1/2},\quad
f\in\mathcal{D},~{}x\in G,$

then by Proposition 48

$\mathfrak{G}_{\alpha}f(x)\asymp_{G,\alpha}\tilde{\mathfrak{G}}_{\alpha}f(x),\quad
x\in G.$

But by [CRTN01, Theorem 5],

$\|\tilde{\mathfrak{G}}_{\alpha}f\|_{L^{p}(G)}\asymp_{G,\alpha,p}\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)},\quad
f\in\mathcal{D}.$

Therefore $\lesssim$ of Theorem 22 follows in this case. Note that we haven’t
proven both directions of the inequalities yet due to the restriction
$f\in\mathcal{D}$.

####  7.2.5. The case $\alpha$ nonintegral, $\alpha>1$

We will reduce to the case $0<\alpha<1$.

For induction, we need the following result. Although we are currently working
with $f\in\mathcal{D}$, we state the proposition below for $f\in
S_{\alpha}^{p}(G)$, because we will need it later again in the proof of the
$\gtrsim$ direction of Dorronsoro’s theorem.

######  Proposition 50 ([Fol75, Theorem 4.10]).

Let $1<p<\infty$ and $\alpha>1$, and let $f\in L^{p}(G)$. Then $f\in
S^{p}_{\alpha}(G)$ if and only if $f\in S^{p}_{\alpha-1}(G)$ and the
distributional derivatives $X_{i}f\in S^{p}_{\alpha-1}(G)$ for $i=1,\cdots,k$,
in which case

$\sum_{j=1}^{k}\|(-\Delta_{p})^{(\alpha-1)/2}X_{j}f\|_{L^{p}(G)}\asymp_{G,\alpha,p}\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)},\quad
f\in S_{\alpha}^{p}(G).$ (60)

######  Remark 51.

To be precise, Theorem 4.10 of [Fol75] states the inhomogeneous estimate

$\displaystyle\|f\|_{L^{p}(G)}+\|(-\Delta_{p})^{(\alpha-1)/2}f\|_{L^{p}(G)}+\sum_{j=1}^{k}\|X_{j}f\|_{L^{p}(G)}+\|(-\Delta_{p})^{(\alpha-1)/2}X_{j}f\|_{L^{p}(G)}$
$\displaystyle\asymp_{G,\alpha,p}\|f\|_{L^{p}(G)}+\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)},\quad
f\in S_{\alpha}^{p}(G).$

But now (60) easily follows using the homogenization argument of subsubsection
7.2.3.

The following is an analogue of [Dor85, Theorem 5] and [FO20a, Proposition
4.2].

######  Proposition 52 (Analogue of [Dor85, Theorem 5] and [FO20a, Proposition
4.2]).

Let $1<p<\infty$ and $\alpha>0$. Then,

$\|\mathfrak{G}_{\alpha+1}f\|_{L^{p}(G)}\lesssim_{G,\alpha}\sum_{j=1}^{k}\|\mathfrak{G}_{\alpha}(X_{j}f)\|_{L^{p}(G)},\qquad
f\in\mathcal{D}.$

###### Proof.

Let $d=\lfloor\alpha\rfloor$. Define a function
$\tilde{A}^{d+1}_{x,r}f\in{\mathcal{A}}_{d+1}$ for $x\in G$ and $r>0$ as

$\tilde{A}^{d+1}_{x,r}f(xy)\coloneqq T^{d+1}_{x}f(xy)-\langle
T^{d+1}_{x}f\rangle_{B_{r}(x)}+\langle f\rangle_{B_{r}(x)}.$

We choose $C\geq 1$ depending on $G$ such that the weak 1-Poincaré inequality
due to Jerison [Jer86] holds:

$\fint_{B_{s}(y)}|g-\langle
g\rangle_{B_{s}(y)}|\lesssim_{G}s\fint_{B_{Cs}(y)}|\nabla g|,\quad y\in
G,~{}s>0,~{}g\in\mathcal{D}.$

As $\langle f-\tilde{A}^{d+1}_{x,r}f\rangle_{B_{r}(x)}=0$, we have

$\displaystyle\beta_{f,d+1}(B_{r}(x))$
$\displaystyle\stackrel{{\scriptstyle\mathrm{Lemma}~{}\ref{L1-approx}}}{{\lesssim_{G,d}}}\fint_{B_{r}(x)}|f-\tilde{A}^{d+1}_{x,r}f|\stackrel{{\scriptstyle\mathrm{weak~{}1-Poincar\acute{e}}}}{{\lesssim_{G}}}r\fint_{B_{Cr}(0)}|\nabla
f(xy)-\nabla T^{d+1}_{x}f(xy)|dy$
$\displaystyle\asymp_{G}r\sum_{j=1}^{k}\fint_{B_{Cr}(0)}|X_{j}f(xy)-X_{j}T^{d+1}_{x}f(xy)|=r\sum_{j=1}^{k}\tilde{\beta}_{X_{j}f,d}(B_{Cr}(x)).$

because $X_{j}T^{d+1}_{x}f=T^{d}_{x}X_{j}f$. Now, by definition,

$\displaystyle\mathfrak{G}_{\alpha+1}f(x)$
$\displaystyle=\left(\int_{0}^{\infty}\left[\frac{1}{r^{\alpha+1}}\beta_{f,d+1}(B_{r}(x))\right]^{2}\frac{dr}{r}\right)^{1/2}\lesssim_{G,d}\sum_{j=1}^{k}\left(\int_{0}^{\infty}\left[\frac{1}{r^{\alpha}}\tilde{\beta}_{X_{j}f,d}(B_{Cr}(x))\right]^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle=C^{\alpha}\sum_{j=1}^{k}\tilde{\mathfrak{G}}_{\alpha}(X_{j}f)(x),\quad
x\in G.$

By Proposition 48, the proof is complete. ∎

If we suppose $\lesssim$ of Theorem 22 for $a<d$ nonintegral,
$d\in\mathbb{N}$, then from the above two Propositions, for $d<\alpha<d+1$ and
$f\in\mathcal{D}$,

$\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}\lesssim_{G,\alpha}\sum_{j=1}^{k}\|\mathfrak{G}_{\alpha-1}(X_{j}f)\|_{L^{p}(G)}\lesssim_{G,\alpha,p}\sum_{j=1}^{k}\|(-\Delta_{p})^{(\alpha-1)/2}X_{j}f\|_{L^{p}(G)}\asymp_{G,\alpha,p}\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)}.$

This completes the proof of Theorem 22 for $\alpha$ nonintegral.

####  7.2.6. Interpolation and the case $\alpha$ integral

The case $\alpha=d\in\mathbb{Z}$ is proven by complex interpolation. This
subsubsection follows Section 5 of [FO20a] closely.

For $1<p<\infty$ and $0<\alpha_{0}<\alpha_{1}<\infty$, the pair
$(S^{p}_{\alpha_{0}},S^{p}_{\alpha_{1}})$ of Banach spaces is an interpolation
pair in the sense of Calderón, as they embed continuously in the space
$\mathcal{S}^{\prime}(G)$ of tempered distributions on $G$. Thus, we may
define the complex interpolation space
$[S^{p}_{\alpha_{0}},S^{p}_{\alpha_{1}}]_{\theta}$, $\theta\in(0,1)$.

The following lemma asserts that we have a continuous embedding

$S^{p}_{(1-\theta)\alpha_{0}+\theta\alpha_{1}}\hookrightarrow[S^{p}_{\alpha_{0}},S^{p}_{\alpha_{1}}]_{\theta}.$

######  Lemma 53 ([FO20a, Lemma 5.1]).

Let $1<p<\infty$, $0<\alpha_{0}<\alpha_{1}<\infty$, and $\theta\in(0,1)$. Then

$\|f\|_{[S^{p}_{\alpha_{0}},S^{p}_{\alpha_{1}}]_{\theta}}\lesssim_{G,\alpha_{0},\alpha_{1},p}\|f\|_{p,(1-\theta)\alpha_{0}+\theta\alpha_{1}}.$

The proof is exactly the same as in [FO20a, Lemma 5.1].

Next, we define, for $\alpha>0$, the Banach space $H_{\alpha}$ of functions
$F:B_{1}\times(0,\infty)\to\mathbb{R}$ with norm

$\|F\|_{H_{\alpha}}\coloneqq\left(\int_{0}^{\infty}\left[\frac{1}{r^{\alpha}}\fint_{B_{1}}|F(y,r)|dy\right]^{2}\frac{dr}{r}\right)^{1/2}<\infty.$

Then, for $1<p<\infty$, we may define the function space
$L^{p}(G,H_{\alpha})$. Now, for $0<\alpha_{0}<\alpha_{1}<\infty$,

$(L^{p}(G,H_{\alpha_{0}}),L^{p}(G,H_{\alpha_{1}}))$

is a compatible couple, as both embed continuously into
$L_{\mathrm{loc}}^{1}(G\times B_{1}\times(0,\infty))$. As in [Dor85, FO20a],
we apply the results of [BL12, p. 107, 121] so that for $\theta\in(0,1)$,

$[L^{p}(G,H_{\alpha_{0}}),L^{p}(G,H_{\alpha_{1}})]_{\theta}=L^{p}(G,[H_{\alpha_{0}},H_{\alpha_{1}}]_{\theta})=L^{p}(G,H_{(1-\theta)\alpha_{0}+\theta\alpha_{1}}).$

Now, for $d-1<\alpha_{0}<d<\alpha_{1}<d+1$ (recall $d=\alpha$) and
$\theta\in(0,1)$, consider the linear map
$T:S^{p}_{\alpha_{0}}+S^{p}_{\alpha_{1}}\to
L^{p}(G,H_{\alpha_{0}})+L^{p}(G,H_{\alpha_{1}})$ given by

$f\mapsto Tf(x,r,y)=f(x\delta_{r}(y))-A^{d}_{x,r}(x\delta_{r}(y)).$

By Lemma 46 and Theorem 22 for $\alpha\in(d-1,d)\cup(d,d+1)$, we have

$\|Tf\|_{L^{p}(G,H_{\alpha})}\lesssim_{G,d}\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}\lesssim_{G,\alpha_{0},\alpha_{1},p}\|f\|_{p,\alpha},\quad
f\in S^{p}_{\alpha},~{}\alpha=\alpha_{0},\alpha_{1},$

i.e., $T:S^{p}_{\alpha_{0}}\to L^{p}(G,H_{\alpha_{0}})$ and
$T:S^{p}_{\alpha_{1}}\to L^{p}(G,H_{\alpha_{1}})$ are bounded. By complex
interpolation,

$T:[S^{p}_{\alpha_{0}},S^{p}_{\alpha_{1}}]_{\theta}\to[L^{p}(G,H_{\alpha_{0}}),L^{p}(G,H_{\alpha_{1}})]_{\theta}=L^{p}(G,H_{(1-\theta)\alpha_{0}+\theta\alpha_{1}})$

is bounded.

Now set $\alpha_{0}=d-1/2$, $\alpha_{1}=d+1/2$, and $\theta=1/2$. Then

$\|\mathfrak{G}_{d}f\|_{L^{p}(G)}=\|Tf\|_{L^{p}(G,H_{d})}\lesssim_{G,d}\|f\|_{[S^{p}_{d-1/2},S^{p}_{d+1/2}]_{1/2}}\stackrel{{\scriptstyle\mathrm{Lemma~{}}\ref{lem:interpolation}}}{{\lesssim_{G,d,p}}}\|f\|_{p,d},\quad
f\in\mathcal{D}.$

This completes the proof of Theorem 22 for $\alpha=d\in\mathbb{Z}$.

###  7.3. Proof of the $\gtrsim$ direction

Now, given a $f\in L^{p}(G)$ and $\alpha>0$ with $\mathfrak{G}_{\alpha}f\in
L^{p}(G)$, we need to prove that $f\in S^{p}_{\alpha}(G)$ and that the
$\gtrsim$ direction of (8) holds.

We first prove the following preparatory result.

######  Proposition 54.

Let $f\in L^{p}(G)$ and $\alpha>0$ with $\mathfrak{G}_{\alpha}f\in L^{p}(G)$.
Denote

$A^{\lfloor\alpha\rfloor}_{x,r}f(xy)=\sum_{|\gamma|\leq\lfloor\alpha\rfloor}f_{\gamma}(x,r)y^{\gamma},$

Then

  1. (1)

For each multi-index $\gamma$ with $|\gamma|<\alpha$,
$|f_{\gamma}(x,r)-f_{\gamma}(x,s)|\lesssim_{G,\alpha}r^{\alpha-|\gamma|}\mathfrak{G}_{\alpha}f(x)$
for $x\in G$ and $0<s<r$, and
$|f_{\gamma}(x,r)|\lesssim_{G,\alpha}\mathfrak{G}_{\alpha}f(x)+Mf(x)$ for
$x\in G$ and $0<r<1$, where $M$ denotes the Hardy–Littlewood maximal operator.
In particular, $f_{\gamma}(\cdot,r)\in L^{p}(G)$ with
$\|f_{\gamma}(\cdot,r)\|_{L^{p}(G)}\lesssim_{G,\alpha}\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}+\|f\|_{L^{p}(G)}$.

  2. (2)

For each multi-index $\gamma$ with $|\gamma|<\alpha$, there exists
$f_{\gamma}\in L^{p}(G)$ such that $f_{\gamma}(\cdot,r)\to f_{\gamma}(\cdot)$
as $r\to 0$, satisfying

$|f_{\gamma}(x,r)-f_{\gamma}(x)|\lesssim_{G,|\gamma|}\int_{0}^{2r}\beta_{f,d}(B_{u}(x))u^{-|\gamma|}\frac{du}{u}\lesssim_{G,\alpha}r^{\alpha-|\gamma|}\mathfrak{G}_{\alpha}f(x)$

for a.e. $x\in G$ and $r>0$.

  3. (3)

If $\alpha$ is nonintegral, then

$\left(\int_{0}^{\infty}\left(\frac{1}{r^{\alpha}}\fint_{B_{r}}|f(xy)-\sum_{|\gamma|\leq\lfloor\alpha\rfloor}f_{\gamma}(x)y^{\gamma}|dy\right)^{2}\frac{dr}{r}\right)^{1/2}\asymp_{G,\alpha}\mathfrak{G}_{\alpha}f(x),\quad\mathrm{a.e.~{}}x\in
G,$

  4. (4)

If $\alpha$ is integral and $|\gamma|=\alpha$, then
$|f_{\gamma}(x,r)|\lesssim_{G,\alpha}\log(2/r)\mathfrak{G}_{\alpha}f(x)+Mf(x)$,
$0<r<1$. Thus, $f_{\gamma}(\cdot,r)\in L^{p}(G)$ with
$\|f_{\gamma}(\cdot,r)\|_{L^{p}(G)}\lesssim_{G,\alpha}\log(2/r)\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}+\|f\|_{L^{p}(G)}$.

  5. (5)

If $0<\alpha<1$, then $f_{0}(x)=f(x)$ for a.e. $x\in G$.

###### Proof.

  1. (1)

By repeating the computation (57) made in the proof of Proposition 48, it
follows that

$|f_{\gamma}(x,r)-f_{\gamma}(x,s)|\lesssim_{G,|\gamma|}\int_{s}^{2r}\beta_{f,d}(B_{u}(x))u^{-|\gamma|}\frac{du}{u},\quad
x\in G,~{}0<s<r.$

In particular, by Cauchy–Schwartz,

$|f_{\gamma}(x,r)-f_{\gamma}(x,s)|\lesssim_{G,|\gamma|}\left(\int_{s}^{2r}\left[\frac{\beta_{f,d}(B_{u}(x))}{u^{\alpha}}\right]^{2}\frac{du}{u}\right)^{1/2}\left(\int_{s}^{2r}(u^{\alpha-|\gamma|})^{2}\frac{du}{u}\right)^{1/2}\lesssim_{\alpha}r^{\alpha-|\gamma|}\mathfrak{G}_{\alpha}f(x),\quad
x\in G,~{}0<s<r.$

Also, for $0<r<1$,
$|f_{\gamma}(x,r)|\leq|f_{\gamma}(x,r)-f_{\gamma}(x,1)|+|f_{\gamma}(x,1)|\lesssim_{G,\alpha}\mathfrak{G}_{\alpha}f(x)+Mf(x)$,
where the latter inequality used the fact that $f_{\gamma}(x,1)$ is given as
the integral of $f$ multiplied by a polynomial determined by $\gamma$ and
$\lfloor\alpha\rfloor$. In particular, this implies that
$f_{\gamma}(\cdot,r)\in L^{p}$.

  2. (2)

By (1) and the fact that $\mathfrak{G}_{\alpha}f\in L^{p}(G)$,
$\\{f_{\gamma}(\cdot,r)\\}_{r>0}$ is a Cauchy sequence in $L^{p}$ as $r\to 0$.
Thus, there exists $f_{\gamma}(\cdot)\in L^{p}(G)$ so that
$f_{\gamma}(\cdot,r)\to f_{\gamma}(\cdot)$ in $L^{p}(G)$ as $r\to 0$.

  3. (3)

The $\gtrsim$ follows from Lemma 47. For $\lesssim$, we bound

$\displaystyle\fint_{B_{r}}|f(xy)-\sum_{|\gamma|\leq\lfloor\alpha\rfloor}f_{\gamma}(x)y^{\gamma}|$
$\displaystyle\leq\fint_{B_{r}(x)}|f-A^{\lfloor\alpha\rfloor}_{x,r}f|+\fint_{B_{r}}|\sum_{|\gamma|\leq\lfloor\alpha\rfloor}f_{\gamma}(x)y^{\gamma}-A^{\lfloor\alpha\rfloor}_{x,r}f(xy)|dy$
$\displaystyle\lesssim_{G,\alpha}\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))+\sum_{|\gamma|\leq\lfloor\alpha\rfloor}|f_{\gamma}(x,r)-f_{\gamma}(x)|r^{|\gamma|}$
$\displaystyle\lesssim_{G,\alpha}\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))+\sum_{i=0}^{\lfloor\alpha\rfloor}r^{i}\int_{0}^{2r}\beta_{f,\lfloor\alpha\rfloor}(B_{u}(x))u^{-i}\frac{du}{u}.$

By plugging into the definition of $\tilde{\mathfrak{G}}$ and using the
triangle inequality,

$\displaystyle\left(\int_{0}^{\infty}\left(\frac{1}{r^{\alpha}}\fint_{B_{r}}|f(xy)-\sum_{|\gamma|\leq\lfloor\alpha\rfloor}f_{\gamma}(x)y^{\gamma}|dy\right)^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\lesssim_{G,\alpha}\mathfrak{G}_{\alpha}f(x)+\sum_{i=0}^{\lfloor\alpha\rfloor}\left(\int_{0}^{\infty}\left[r^{i-\alpha}\int_{0}^{2r}\beta_{f,\lfloor\alpha\rfloor}(B_{u}(x))u^{-i}\frac{du}{u}\right]^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\lesssim_{G,\alpha}\mathfrak{G}_{\alpha}f(x)+\sum_{i=0}^{\lfloor\alpha\rfloor}\frac{1}{\alpha-i}\left(\int_{0}^{\infty}\left[\frac{\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))}{r^{\alpha}}\right]^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\lesssim_{\alpha}\mathfrak{G}_{\alpha}f(x).$

where in the second inequality we used Hardy’s inequality (23) with
$\nu=\alpha-i+\frac{1}{2}$ and $p=2$ along with the fact that $\alpha$ is
nonintegral.

  4. (4)

By Cauchy–Schwartz,

$|f_{\gamma}(x,r)-f_{\gamma}(x,1)|\lesssim_{G,|\gamma|}\left(\int_{r}^{2}\left[\frac{\beta_{f,d}(B_{u}(x))}{u^{\alpha}}\right]^{2}\frac{du}{u}\right)^{1/2}\left(\int_{r}^{2}\frac{du}{u}\right)^{1/2}\lesssim_{\alpha}\log(2/r)\mathfrak{G}_{\alpha}f(x).$

  5. (5)

If $0<\alpha<1$, then $f_{0}(x,r)=\fint_{B_{r}(x)}f$. Taking $r\to 0$, the
claim follows by the Lebesgue differentiation theorem for doubling metric
measure spaces [HKST15, (3.4.10)].

∎

######  Remark 55.

It seems likely that for $|\gamma|<\alpha$ we have
$f_{\gamma}=\frac{1}{\gamma!}\operatorname{Sym}(X^{\gamma})f$, with the latter
derivative existing in the distributive sense. This is evident when
$0<\alpha<1$ and $\gamma=0$, and we will prove this holds when $\alpha>1$ and
$|\gamma|=1$ in Lemma 57. We will not prove this statement in full generality
since we do not need it in this paper.

####  7.3.1. The case $0<\alpha\leq 1$

The case $0<\alpha<1$ is a repetition and strengthening of subsubsection
7.2.4. If we denote

$\tilde{\mathfrak{G}}_{\alpha}f(x)=\left(\int_{0}^{\infty}\left[\frac{1}{r^{\alpha}}\fint_{B_{r}}|f(xy)-f(x)|dy\right]^{2}\frac{dr}{r}\right)^{1/2},\quad
x\in G,$

then by Proposition 54 (3) and (5)

$\mathfrak{G}_{\alpha}f(x)\asymp_{G,\alpha}\tilde{\mathfrak{G}}_{\alpha}f(x),\quad\mathrm{for~{}a.e.~{}}x\in
G.$

But by [CRTN01, Theorem 5],

$\|\tilde{\mathfrak{G}}_{\alpha}f\|_{L^{p}(G)}\asymp_{G,p}\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)}.$

For $\alpha=1$, it is known from [DNM21, Theorem 1.4] that for $0<\alpha<2$,
if we define

$S_{\alpha}f(x)=\left(\int_{0}^{\infty}\left[r^{-\alpha}\fint_{B_{r}(0)}|\Delta_{y}^{2,\mathrm{sym}}f(x)|dy\right]^{2}\frac{dr}{r}\right)^{1/2},$

where we denote the symmetric difference

$\Delta_{y}^{2,\mathrm{sym}}f(x)=f(xy)+f(xy^{-1})-2f(x),$

then

$\|f\|_{p,\alpha}\asymp_{G,p,\alpha}\|f\|_{L^{p}(G)}+\|S_{\alpha}f\|_{L^{p}(G)}.$

By the scaling argument of subsubsection 7.2.3,

$\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)}\asymp_{G,p,\alpha}\|S_{\alpha}f\|_{L^{p}(G)}.$

Let $f$ be a locally integrable function, $d\in\mathbb{N}$, $x=z_{0}\in G$ and
$r>0$. For any $y\in B_{r}(z_{0})$ and ball $B_{2^{-k}r}(z_{k})$ with $y\in
B_{2^{-k}r}(z_{k})\subset B_{r}(z_{0})$, let $B_{2^{-k}r}(z_{k})\subset
B_{2^{-k+1}r}(z_{k-1})\subset\cdots\subset B_{2^{-1}r}(z_{1})\subset B_{r}(x)$
be a sequence of balls. We have

$\displaystyle\fint_{B_{2^{-k}r}(z_{k})}|f-A_{z_{0},r}^{d}f|$
$\displaystyle\leq\fint_{B_{2^{-k}r}(z_{k})}|f-A_{z_{k},2^{-k}r}^{d}f|+\sum_{i=0}^{k-1}\fint_{B_{2^{-k}r}(z_{k})}|A_{z_{i},2^{-i}r}^{d}f-A_{z_{i+1},2^{-i-1}r}^{d}f|$
$\displaystyle\leq\fint_{B_{2^{-k}r}(z_{k})}|f-A_{z_{k},2^{-k}r}^{d}f|+\sum_{i=0}^{k-1}\left\|A_{z_{i+1},2^{-i-1}r}^{d}(f-A_{z_{i},2^{-i}r}^{d}f)\right\|_{L^{\infty}(B_{2^{-i-1}r}(z_{i+1}))}$
$\displaystyle\stackrel{{\scriptstyle\eqref{claim}}}{{\lesssim_{G,d}}}\fint_{B_{2^{-k}r}(z_{k})}|f-A_{z_{k},2^{-k}r}^{d}f|+\sum_{i=0}^{k-1}\fint_{B_{2^{-i-1}r}(z_{i+1})}|f-A_{z_{i},2^{-i}r}^{d}f|$
$\displaystyle\lesssim_{G}\sum_{i=0}^{k}\fint_{B_{2^{-i}r}(z_{i})}|f-A^{d}_{z_{i},2^{-i}r}f|\stackrel{{\scriptstyle\mathrm{Corollary~{}}\ref{monotoneBeta}}}{{\lesssim_{G,d}}}\int_{0}^{2r}\beta_{f,d}(B_{s}(y))\frac{ds}{s}$

so by Lebesgue’s differentiation theorem for doubling metric measure spaces,
for a.e. $y\in B_{r}(x)$,

$|f(y)-A_{x,r}^{d}f(y)|=\lim_{B_{2^{-k}r}(z_{k})\to
y}\fint_{B_{2^{-k}r}(z_{k})}|f-A_{z_{0},r}^{d}f|\lesssim_{G,d}\int_{0}^{2r}\beta_{f,d}(B_{s}(y))\frac{ds}{s}.$

As the operator $\Delta_{h}^{2,\mathrm{sym}}$ annihilates polynomials of
weighted degree $1$, we obtain for a.e. $x\in G$

$\displaystyle\fint_{B_{r}}|\Delta_{h}^{2,\mathrm{sym}}f(x)|dh=\fint_{B_{r}}|\Delta_{h}^{2,\mathrm{sym}}(f-A_{x,2r}^{1}f)(x)|dh$
$\displaystyle=\fint_{B_{r}}|(f-A_{x,2r}^{1}f)(xh)+(f-A_{x,2r}^{1}f)(xh^{-1})-2(f-A_{x,2r}^{1}f)(x)|dh$
$\displaystyle\lesssim_{G,d}\fint_{B_{r}}\left(\int_{0}^{2r}\beta_{f,d}(B_{s}(xh))\frac{ds}{s}+\int_{0}^{2r}\beta_{f,d}(B_{s}(xh^{-1}))\frac{ds}{s}+2\int_{0}^{2r}\beta_{f,d}(B_{s}(x))\frac{ds}{s}\right)dh$
$\displaystyle\lesssim\int_{0}^{2r}M\beta_{f,1}(B_{s}(\cdot))(x)\frac{ds}{s}.$

By Hardy’s inequality (23) with $\nu=\frac{3}{2}$ and $p=2$, we obtain

$S_{1}f(x)=\left(\int_{0}^{\infty}\left[r^{-1}\fint_{B_{r}}|\Delta_{y}^{2,\mathrm{sym}}f(x)|dy\right]^{2}\frac{dr}{r}\right)^{1/2}\lesssim_{G,d}\left(\int_{0}^{\infty}\left[M\beta_{f,1}(B_{r}(\cdot))(x)\right]^{2}\frac{dr}{r}\right)^{1/2}.$

Now we argue that

$\left(\int_{G}\left(\int_{0}^{\infty}\left[M\beta_{f,1}(B_{r}(\cdot))(x)\right]^{2}\frac{dr}{r}\right)^{p/2}dx\right)^{1/p}\lesssim_{G}\max\\{p,\frac{1}{p-1}\\}\left(\int_{G}\left(\int_{0}^{\infty}\left[\beta_{f,1}(B_{r}(\cdot))(x)\right]^{2}\frac{dr}{r}\right)^{p/2}dx\right)^{1/p}$

This is an instance of the Fefferman–Stein vector-valued maximal function
inequality

$\left(\int_{G}\left[\sum_{j\in\mathbb{Z}}M(g_{j})^{u}(x)\right]^{v/u}dx\right)^{1/v}\lesssim_{G}\max\\{1,\frac{1}{u-1}\\}\max\\{v,\frac{1}{v-1}\\}\left(\int_{G}\left[\sum_{j\in\mathbb{Z}}g_{j}(x)^{u}\right]^{v/u}dx\right)^{1/v}$
(61)

on $n_{h}$-regular metric measure spaces for measurable functions
$(g_{j}:G\to\mathbb{R})_{j\in\mathbb{Z}}$ and $u,v\in(1,\infty)$, due to
[GLY09, Theorem 1.2]. Indeed, we apply (61) to the functions

$g_{j}(x)=\beta_{f,1}(B_{2^{-j}}(x))$

and exponents $u=2>1$ and $v=p>1$, noting by Corollary 47 that

$M(\beta_{f,1}(B_{r}(\cdot)))\lesssim_{G}M(g_{j}),\quad
r\in[2^{-j-1},2^{-j}],$

and

$g_{j}\lesssim_{G}\beta_{f,1}(B_{r}(\cdot)),\quad r\in[2^{-j},2^{-j+1}],$

Thus, we have
$\|S_{1}f\|_{L^{p}(G)}\lesssim_{G,p}\|\mathfrak{G}_{1}f\|_{L^{p}(G)}$,
$1<p<\infty$, as desired.

####  7.3.2. The case $\alpha>1$

We use induction on $\alpha$.

The following lemma is an analogue of [Dor85, Lemma 1], and the proof proceeds
in the same manner.

######  Lemma 56 (Analogue of [Dor85, Lemma 1]).

Let $f\in L^{p}(G)$ such that $\mathfrak{G}_{\alpha}f\in L^{p}(G)$, and
$\gamma$ be a multi-index with $|\gamma|<\alpha$. With $f_{\gamma}$ as in
Proposition 54,
$\|\mathfrak{G}_{\alpha-|\gamma|}f_{\gamma}\|_{L^{p}(G)}\lesssim_{G,\alpha}\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}$.

###### Proof.

Fix $x\in G$ and $r>0$. For $y\in B_{r}(x)$, by Proposition 54 and (56),

$\displaystyle|f_{\gamma}(y)-\frac{1}{\gamma!}\operatorname{Sym}(X^{\gamma})A^{d}_{x,2r}f(y)|$
$\displaystyle\leq|f_{\gamma}(y)-f_{\gamma}(y,r)|+|\frac{1}{\gamma!}\operatorname{Sym}(X^{\gamma})(A^{d}_{y,r}-A^{d}_{x,2r}f)(y)|$
$\displaystyle\lesssim_{G,\alpha}\int_{0}^{r}\beta_{f,d}(B_{s}(y))s^{-|\gamma|-1}ds+r^{-|\gamma|}\beta_{f,d}(B_{2r}(x)).$

Thus, by Lemma 46,

$\displaystyle\beta_{f_{\gamma},d-|\gamma|}(B_{r}(x))$
$\displaystyle\lesssim_{G,\alpha}\fint_{B_{r}(x)}\left(\int_{0}^{r}\beta_{f,d}(B_{s}(y))s^{-|\gamma|-1}dsdy+r^{-|\gamma|}\beta_{f,d}(B_{2r}(x))\right)dy$
$\displaystyle\leq\int_{0}^{r}M(\beta_{f,d}(B_{s}(\cdot)))(x)s^{-|\gamma|-1}ds+r^{-|\gamma|}\beta_{f,d}(B_{2r}(x)).$

By Hardy’s inequality with $\nu=\alpha-|\gamma|+\frac{1}{2}$ and $p=2$,

$G_{\alpha-|\gamma|}f_{\gamma}(x)\lesssim_{G,\alpha}\mathfrak{G}_{\alpha}f(x)+\left(\int_{0}^{\infty}\left[r^{-\alpha}M\beta_{f,d}(B_{r}(\cdot))(x)\right]^{2}\frac{dr}{r}\right)^{1/2},$

and the lemma follows by the aforementioned Fefferman–Stein inequality (61) on
vector-valued maximal operators with $u=v=2>1$ and

$g_{j}(x)=2^{\alpha j}\beta_{f,d}(B_{2^{-j+1}}(x)).$

∎

The following lemma is an analogue of [Dor85, Lemma 2], and the proof proceeds
in the same manner.

######  Lemma 57 (Analogue of [Dor85, Lemma 2]).

Let $f\in L^{p}(G)$ with $\mathfrak{G}_{\alpha}f\in L^{p}(G)$, $\alpha>1$.
Then the weak partials $X_{i}f$ exist and coincide with $f_{(1,i)}$ a.e.,
where $(1,i)$ stands for the multi-index $\gamma$ with
$\gamma_{r,j}=\delta_{(r,j),(1,i)}$.

###### Proof.

Denote $e_{r,i}=\exp(x_{r,i})$ for $r=1,\cdots,s$ and $i=1,\cdots,k_{r}$. It
is enough to show that $(f(xe_{1,i}^{r})-f(x))/r$ tends to $f_{(1,i)}$ in
$L^{p}$ as $r\to 0$. First,

$\displaystyle\left|\frac{f(xe_{1,i}^{r})-f(x)}{r}-f_{(1,i)}(x)\right|$
$\displaystyle\leq\frac{|f(xe_{1,i}^{r})-f_{0}(xe_{1,i}^{r},2r)|}{r}+\frac{|f(x)-f_{0}(x,2r)|}{r}+\left|\frac{f_{0}(xe_{1,i}^{r},2r)-f_{0}(x,2r)}{r}-f_{(1,i)}(x)\right|$
$\displaystyle\stackrel{{\scriptstyle\mathrm{Proposition}~{}\ref{prop:measurableTaylor}(2)}}{{\lesssim_{G,\alpha}}}r^{\alpha-1}\left(\mathfrak{G}_{\alpha}f(xe_{1,i}^{r})+\mathfrak{G}_{\alpha}f(x)\right)+\left|\frac{f_{0}(xe_{1,i}^{r},2r)-f_{0}(x,2r)}{r}-f_{(1,i)}(x)\right|.$

Next, since $f_{0}(y,r)=A_{y,r}^{\lfloor\alpha\rfloor}f(y)$,

$\displaystyle f_{0}(xe_{1,i}^{r},2r)-f_{0}(x,2r)$
$\displaystyle=A^{\lfloor\alpha\rfloor}_{xe_{1,i}^{r},2r}f(xe_{1,i}^{r})-A^{\lfloor\alpha\rfloor}_{x,2r}f(xe_{1,i}^{r})+A^{\lfloor\alpha\rfloor}_{x,2r}f(xe_{1,i}^{r})-A^{\lfloor\alpha\rfloor}_{x,2r}f(x)$
$\displaystyle=A^{\lfloor\alpha\rfloor}_{xe_{1,i}^{r},2r}f(xe_{1,i}^{r})-A^{\lfloor\alpha\rfloor}_{x,2r}f(xe_{1,i}^{r})+\sum_{j=1}^{\lfloor\alpha\rfloor}f_{j(1,i)}(x,2r)r^{j}/j!.$

Thus

$\displaystyle\left|\frac{f_{0}(xe_{1,i}^{r},2r)-f_{0}(x,2r)}{r}-f_{(1,i)}(x)\right|$
$\displaystyle\leq\left|\frac{f_{0}(xe_{1,i}^{r},2r)-f_{0}(x,2r)}{r}-f_{(1,i)}(x,2r)\right|+|f_{(1,i)}(x,2r)-f_{(1,i)}(x)|$
$\displaystyle\stackrel{{\scriptstyle\mathrm{Proposition}~{}\ref{prop:measurableTaylor}(2)}}{{\lesssim_{G,\alpha}}}\frac{|A^{\lfloor\alpha\rfloor}_{xe_{1,i}^{r},2r}f(xe_{1,i}^{r})-A^{\lfloor\alpha\rfloor}_{x,2r}f(xe_{1,i}^{r})|}{r}+\sum_{j=2}^{\lfloor\alpha\rfloor}|f_{j(1,i)}(x,2r)|r^{j-1}+r^{\alpha-1}\mathfrak{G}_{\alpha}f(x)$
$\displaystyle\lesssim_{G,\alpha}r^{-1}\fint_{B_{r}(xe_{1,i}^{r})}|A^{\lfloor\alpha\rfloor}_{xe_{1,i}^{r},2r}f(y)-A^{\lfloor\alpha\rfloor}_{x,2r}f(y)|dy+\sum_{j=2}^{\lfloor\alpha\rfloor}|f_{j(1,i)}(x,2r)|r^{j-1}+r^{\alpha-1}\mathfrak{G}_{\alpha}f(x)$
$\displaystyle\leq
r^{-1}\fint_{B_{r}(xe_{1,i}^{r})}|f-A^{\lfloor\alpha\rfloor}_{xe_{1,i}^{r},2r}f(y)|dy+r^{-1}\fint_{B_{r}(xe_{1,i}^{r})}|f-A^{\lfloor\alpha\rfloor}_{x,2r}f(y)|dy+\sum_{j=2}^{\lfloor\alpha\rfloor}|f_{j(1,i)}(x,2r)|r^{j-1}+r^{\alpha-1}\mathfrak{G}_{\alpha}f(x)$
$\displaystyle=r^{-1}\beta_{f,\lfloor\alpha\rfloor}(B_{2r}(xe_{1,i}^{r}))+r^{-1}\beta_{f,\lfloor\alpha\rfloor}(B_{2r}(x))+\sum_{j=2}^{\lfloor\alpha\rfloor}|f_{j(1,i)}(x,2r)|r^{j-1}+r^{\alpha-1}\mathfrak{G}_{\alpha}f(x),$

where in the penultimate inequality we used the fact that the $L^{1}(B_{1})$
and $L^{\infty}(B_{1})$ norms on the space of polynomials of weighted degree
$\leq\lfloor\alpha\rfloor$ are equivalent. Taking $L^{p}$ norms,

$\left\|\frac{f(\cdot
e_{1,i}^{r})-f(\cdot)}{r}-f_{(1,i)}(\cdot)\right\|_{L^{p}(G)}\lesssim_{G,\alpha}r^{-1}\|\beta_{f,\lfloor\alpha\rfloor}(B_{2r}(\cdot))\|_{L^{p}(G)}+\sum_{j=2}^{\lfloor\alpha\rfloor}r^{j-1}\|f_{j(1,i)}(\cdot,2r)\|_{L^{p}(G)}+r^{\alpha-1}\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}.$

By Corollary 47,
$\|\beta_{f,\lfloor\alpha\rfloor}(B_{2r}(\cdot))\|_{L^{p}(G)}\lesssim_{G,\alpha}r^{\alpha}\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}$.
By Proposition 54, if $j<\lfloor\alpha\rfloor$, then

$\|f_{j(1,i)}(\cdot,2r)\|_{L^{p}(G)}\lesssim_{G,\alpha}\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}+\|f\|_{L^{p}(G)},$

and if $j=\lfloor\alpha\rfloor=\alpha$, then

$\|f_{j(1,i)}(\cdot,2r)\|_{L^{p}(G)}\lesssim_{G,\alpha}(\log
r^{-1})(\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}+\|f\|_{L^{p}(G)}).$

Thus $\left\|\frac{f(\cdot
e_{1,i}^{r})-f(\cdot)}{r}-f_{(1,i)}(\cdot)\right\|_{L^{p}(G)}\to 0$ as $r\to
0$. ∎

The following lemma is an analogue of [Dor85, Lemma 3], and the proof proceeds
in the same manner.

######  Lemma 58 (Analogue of [Dor85, Lemma 3]).

Let $f\in L^{p}(G)$ with $\mathfrak{G}_{\alpha}f\in L^{p}(G)$, $\alpha>1$.
Then $\mathfrak{G}_{\alpha-1}f\in L^{p}(G)$ with

$\|\mathfrak{G}_{\alpha-1}f\|_{L^{p}(G)}\lesssim_{G,\alpha}\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}+\|f\|_{L^{p}(G)}.$

###### Proof.

Write

$A^{\lfloor\alpha\rfloor}_{x,r}f(xy)=\underbrace{\sum_{|\gamma|<\lfloor\alpha\rfloor}f_{\gamma}(x,r)y^{\gamma}}_{\eqqcolon
A^{<\lfloor\alpha\rfloor}_{x,r}f(xy)}+\underbrace{\sum_{|\gamma|=\lfloor\alpha\rfloor}f_{\gamma}(x,r)y^{\gamma}}_{\eqqcolon
A^{=\lfloor\alpha\rfloor}_{x,r}f(xy)}.$

Then by Lemma 46,

$\beta_{f,\lfloor\alpha\rfloor-1}(B_{r}(x))\lesssim_{G,\alpha}\fint_{B_{r}(x)}|f-A^{<\lfloor\alpha\rfloor}_{x,r}f|\leq\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))+\|A^{=\lfloor\alpha\rfloor}_{x,r}f\|_{L^{\infty}(B_{r}(x))}.$
(62)

By (56), for $i\in\mathbb{Z}_{\geq 0}$ and $|\gamma|=\lfloor\alpha\rfloor$,

$|f_{\gamma}(x,2^{i}r)|\lesssim_{G,\alpha}(2^{i}r)^{-\lfloor\alpha\rfloor}\fint_{B_{2^{i}r}(x)}|f|\lesssim_{G}(2^{i}r)^{-\lfloor\alpha\rfloor-
n_{h}/p}\|f\|_{L^{p}(G)}\to 0\quad\mathrm{as~{}}i\to\infty$

and thus

$\displaystyle|f_{\gamma}(x,r)|$
$\displaystyle\leq\sum_{i=0}^{\infty}|f_{\gamma}(x,2^{i}r)-f_{\gamma}(x,2^{i+1}r)|\stackrel{{\scriptstyle\eqref{polydiff}}}{{\lesssim_{G,\alpha}}}\sum_{i=0}^{\infty}(2^{i}r)^{-\lfloor\alpha\rfloor}\|A^{\lfloor\alpha\rfloor}_{x,2^{i}r}f-A^{\lfloor\alpha\rfloor}_{x,2^{i+1}r}f\|_{L^{\infty}(B_{2^{i}r}(x))}$
$\displaystyle\stackrel{{\scriptstyle\eqref{claim}}}{{\lesssim_{G,\alpha}}}\sum_{i=0}^{\infty}(2^{i}r)^{-\lfloor\alpha\rfloor}\fint_{B_{2^{i}r}(x)}|f-A^{\lfloor\alpha\rfloor}_{x,2^{i+1}r}f|\lesssim_{G,\alpha}\sum_{i=0}^{\infty}(2^{i}r)^{-\lfloor\alpha\rfloor}\beta_{f,\lfloor\alpha\rfloor}(B_{2^{i+1}r}(x)).$
$\displaystyle\stackrel{{\scriptstyle\mathrm{Corollary~{}}\ref{monotoneBeta}}}{{\lesssim_{G,\alpha}}}\int_{r}^{\infty}\beta_{f,\lfloor\alpha\rfloor}(B_{\rho}(x))\rho^{-\lfloor\alpha\rfloor-1}d\rho.$

Plugging these results into (62), we have

$\beta_{f,\lfloor\alpha\rfloor-1}(B_{r}(x))\lesssim_{G,\alpha}\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))+r^{\lfloor\alpha\rfloor}\int_{r}^{\infty}\beta_{f,\lfloor\alpha\rfloor}(B_{\rho}(x))\rho^{-\lfloor\alpha\rfloor-1}d\rho,$

and

$\displaystyle\mathfrak{G}_{\alpha-1}f(x)$
$\displaystyle=\left(\int_{0}^{\infty}\left[\frac{\beta_{f,\lfloor\alpha\rfloor-1}(B_{r}(x))}{r^{\alpha-1}}\right]^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\lesssim_{G,\alpha}\left(\int_{0}^{\infty}\left[\frac{\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))}{r^{\alpha-1}}\right]^{2}\frac{dr}{r}\right)^{1/2}+\left(\int_{0}^{\infty}\left[r^{\lfloor\alpha\rfloor-\alpha+1/2}\int_{r}^{\infty}\beta_{f,\lfloor\alpha\rfloor}(B_{\rho}(x))\rho^{-\lfloor\alpha\rfloor-1}d\rho\right]^{2}dr\right)^{1/2}$
$\displaystyle\lesssim\left(\int_{0}^{\infty}\left[\frac{\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))}{r^{\alpha-1}}\right]^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\leq\left(\int_{0}^{1}\left[\frac{\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))}{r^{\alpha-1}}\right]^{2}\frac{dr}{r}\right)^{1/2}+\left(\int_{1}^{\infty}\left[\frac{\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))}{r^{\alpha-1}}\right]^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{claim}}}{{\lesssim_{G,\alpha}}}\left(\int_{0}^{1}\left[\frac{\beta_{f,\lfloor\alpha\rfloor}(B_{r}(x))}{r^{\alpha}}\right]^{2}\frac{dr}{r}\right)^{1/2}+\left(\int_{1}^{\infty}\left[\frac{Mf(x)}{r^{\alpha-1}}\right]^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\lesssim_{\alpha}\mathfrak{G}_{\alpha}f(x)+Mf(x),$

where the second inequality uses the second form of Hardy’s inequality (25)
with $p=2$ and $\nu=\lfloor\alpha\rfloor-\alpha+1/2$. This completes the
proof. ∎

To complete the induction, suppose the $\gtrsim$ direction of Theorem 8 holds
for $\alpha-1$. If $f\in L^{p}(G)$ with $\mathfrak{G}_{\alpha}f\in L^{p}(G)$,
by Lemma 58, $\mathfrak{G}_{\alpha-1}f\in L^{p}(G)$, so that $f\in
S^{p}_{\alpha-1}$ by induction. On the other hand, by Proposition 54 (2) and
Lemmas 56 and 57, for $i=1,\cdots,k$, $X_{i}f\in L^{p}(G)$ with
$\mathfrak{G}_{\alpha-1}(X_{i}f)\in L^{p}(G)$, so again by induction
$X_{i}f\in S^{p}_{\alpha-1}(G)$. Now, by Proposition 50, $f\in
S^{p}_{\alpha}(G)$. Tracking the norms involved in this argument, we have that

$\displaystyle\|\mathfrak{G}_{\alpha}f\|_{L^{p}(G)}\stackrel{{\scriptstyle\mathrm{Lemma~{}}\ref{lem:inductOnG}}}{{\gtrsim_{G,\alpha}}}\sum_{i=1}^{k}\|\mathfrak{G}_{\alpha-1}f_{(1,i)}\|_{L^{p}(G)}\stackrel{{\scriptstyle\mathrm{Lemma~{}}\ref{lem:weakFirstDeriv}}}{{=}}\sum_{i=1}^{k}\|\mathfrak{G}_{\alpha-1}(X_{i}f)\|_{L^{p}(G)}$
$\displaystyle\stackrel{{\scriptstyle\mathrm{induction}}}{{\gtrsim_{G.\alpha,p}}}\sum_{i=1}^{k}\|(-\Delta_{p})^{(\alpha-1)/2}X_{i}f\|_{L^{p}(G)}\stackrel{{\scriptstyle\mathrm{Proposition~{}}\ref{prop:inductOnDelta}}}{{\asymp_{G,\alpha,p}}}\|(-\Delta_{p})^{\alpha/2}f\|_{L^{p}(G)}.$

This completes the $\gtrsim$ direction of Theorem 8.

###  7.4. $L^{q}$ $\beta$-numbers, $q>1$

The extension of Theorem 22 to $\beta$ numbers defined in the $L^{q}$ sense
are discussed in [Dor85, Section 5] and [FO20a, Section 6]. We will follow
their presentation.

By the discussion so far, it is enough to show that the left-hand side of (8)
is maximized up to constants when $q=1$.

Denote $d=\lfloor\alpha\rfloor$. We compute that for all $x\in G$, $r>0$,
$y\in B_{r}(x)$ and $n\in\mathbb{N}$ so that $B_{2^{-n}r}(y)\subset B_{r}(x)$,
we have

$\displaystyle\fint_{B_{2^{-n}r}(y)}|f(z)-A^{d}_{x,r}f(z)|dz$
$\displaystyle\leq\fint_{B_{2^{-n}r}(y)}|f(z)-A^{d}_{y,2^{-n}r}f(z)|dz+\sum_{k=1}^{n+1}\fint_{B_{2^{-n}r}(y)}|A^{d}_{y,2^{k-1-n}r}f(z)-A^{d}_{y,2^{k-n}r}f(z)|dz$
$\displaystyle\quad+\fint_{B_{2^{-n}r}(y)}|A^{d}_{y,2r}f(z)-A^{d}_{x,r}f(z)|dz$
$\displaystyle=\beta_{f,d}(B_{2^{-n}r}(y))+\sum_{k=1}^{n+1}\fint_{B_{2^{-n}r}(y)}|A^{d}_{y,2^{k-1-n}r}[f-A^{d}_{y,2^{k-n}r}f]|+\fint_{B_{2^{-n}r}(y)}|A^{d}_{x,r}[A^{d}_{y,2r}f-f]|$
$\displaystyle\leq\beta_{f,d}(B_{2^{-n}r}(y))+\sum_{k=1}^{n+1}\left\|A^{d}_{y,2^{k-1-n}r}[f-A^{d}_{y,2^{k-n}r}f]\right\|_{L^{\infty}(B_{2^{k-1-n}r}(y))}+\left\|A^{d}_{x,r}[A^{d}_{y,2r}f-f]\right\|_{L^{\infty}(B_{r}(x))}$
$\displaystyle\stackrel{{\scriptstyle\eqref{claim}}}{{\lesssim_{G,d}}}\beta_{f,d}(B_{2^{-n}r}(y))+\sum_{k=1}^{n+1}\fint_{B_{2^{k-1-n}r}(y)}|f-A^{d}_{y,2^{k-n}r}f|+\fint_{B_{r}(x)}|A^{d}_{y,2r}f-f|$
$\displaystyle\lesssim_{G,d}\beta_{f,d}(B_{2^{-n}r}(y))+\sum_{k=1}^{n+1}\beta_{f,d}(B_{2^{k-n}r}(y))+\beta_{f,d}(B_{2r}(y))$
$\displaystyle\stackrel{{\scriptstyle\mathrm{Corollary}~{}\ref{monotoneBeta}}}{{\lesssim_{G,d}}}\int_{2^{-n}r}^{2^{-n+1}r}\beta_{f,d}(B_{s}(y))\frac{ds}{s}+\sum_{k=1}^{n+1}\int_{2^{k-n}r}^{2^{k+1-n}r}\beta_{f,d}(B_{s}(y))\frac{ds}{s}+\int_{2r}^{4r}\beta_{f,d}(B_{s}(y))\frac{ds}{s}$
$\displaystyle\lesssim\int_{2^{-n}r}^{4r}\beta_{f,d}(B_{s}(y))\frac{ds}{s}.$

Thus by Lebesgue’s differentiation theorem for doubling metric measure spaces
and Corollary 47, we infer that for $x\in G$ and $r>0$,

$|f(y)-A^{d}_{x,r}f(y)|\lesssim_{G,d}\int_{0}^{4r}\beta_{f,d}(B_{s}(y))\frac{ds}{s}\lesssim_{G,d}\int_{0}^{4r}\fint_{B_{s}(y)}\beta_{f,d}(B_{2s}(z))dz\frac{ds}{s},\quad\mbox{a.e.
}y\in B_{r}(x).$ (63)

Now fix

$p>1\quad\text{and}\quad 1\leq
q<\frac{\min\\{p,2\\}n_{h}}{n_{h}-\min\\{p,2\\}}.$

Then, choose some $1<w<\min\\{p,2\\}$ and $0<\eta<\min\\{n_{h}/w,\alpha\\}$
such that

$q=\frac{w{n_{h}}}{{n_{h}}-\eta w}.$ (64)

Given this choice of $q,w$, and $\eta$, [HLNT13, Theorem 4.1] tells us that in
${n_{h}}$-regular metric measure spaces, the following fractional
Hardy–Littlewood maximal function is a bounded operator $L^{w}(G)\to
L^{q}(G)$:

$M_{\eta}g(y):=\sup_{s>0}\left\\{s^{\eta}\fint_{B_{s}(y)}|g(z)|dz\right\\}$

It follows that

$\displaystyle r^{n_{h}/q}\beta_{f,d,q}(B_{r}(x))$
$\displaystyle\stackrel{{\scriptstyle\eqref{form9}}}{{\lesssim_{G,\alpha}}}\left(\int_{B_{r}(x)}\left[\int_{0}^{4r}\fint_{B_{s}(y)}\beta_{f,d}(B_{2s}(z))dz\frac{ds}{s}\right]^{q}dy\right)^{1/q}$
$\displaystyle\leq\int_{0}^{4r}\left(\int_{B_{r}(x)}\left[\fint_{B_{s}(y)}\beta_{f,d}(B_{2s}(z))dz\right]^{q}dy\right)^{1/q}\frac{ds}{s}$
$\displaystyle\leq\int_{0}^{4r}s^{-\eta}\left(\int_{B_{r}(x)}M_{\eta}(\beta_{f,d}(B_{2s}(\cdot))\chi_{B_{5r}(x)}(\cdot))(y)^{q}dy\right)^{1/q}\frac{ds}{s}$
$\displaystyle\lesssim_{p,q}\int_{0}^{4r}s^{-\eta}\left(\int_{B_{5r}(x)}\beta_{f,d}(B_{2s}(z))^{w}dz\right)^{1/w}\frac{ds}{s}$
$\displaystyle\lesssim_{G}\int_{0}^{4r}s^{-\eta}r^{n_{h}/w}\left[M(\beta_{f,d}(B_{2s}(\cdot))^{w})(x)\right]^{1/w}\frac{ds}{s},$

where in the second inequality we used Minkowski’s inequality and in the
fourth inequality we used the boundedness of $M_{\eta}:L^{w}(G)\to L^{q}(G)$.
By (64), we have

$\beta_{f,d,q}(B_{r}(x))\lesssim_{G,\alpha,p,q}r^{\eta}\int_{0}^{4r}s^{-\eta-1}\left[M(\beta_{f,d}(B_{2s}(\cdot))^{w})(x)\right]^{1/w}ds.$

Next, noting that $\alpha>\eta$ and using Hardy’s inequality (23) with
$\nu=\alpha-\eta+1/2$, we obtain

$\displaystyle\left(\int_{0}^{\infty}\left[\frac{1}{r^{\alpha}}\beta_{f,d,q}(B_{r}(x))\right]^{2}\frac{dr}{r}\right)^{1/2}$
$\displaystyle\lesssim_{G,\alpha,p,q}\left(\int_{0}^{\infty}r^{2\eta-2\alpha-1}\left[\int_{0}^{4r}s^{-\eta-1}\left[M(\beta_{f,d}(B_{2s}(\cdot))^{w})(x)\right]^{1/w}ds\right]^{2}dr\right)^{1/2}$
$\displaystyle\lesssim_{\alpha,q}\left(\int_{0}^{\infty}\left[M(r^{-w\alpha}\beta_{f,d}(B_{2r}(\cdot))^{w})(x)\right]^{2/w}\frac{dr}{r}\right)^{1/2}.$

We conclude that

$\displaystyle\left(\int_{G}\left(\int_{0}^{\infty}\left[\frac{1}{r^{\alpha}}\beta_{f,d,q}(B_{r}(x))\right]^{2}\frac{dr}{r}\right)^{p/2}dx\right)^{1/p}$
$\displaystyle\lesssim_{G,\alpha,p,q}\left(\int_{G}\left(\int_{0}^{\infty}\left[M(r^{-w\alpha}\beta_{f,d}(B_{2r}(\cdot))^{w})(x)\right]^{2/w}\frac{dr}{r}\right)^{p/2}dx\right)^{1/p}$
$\displaystyle\lesssim_{G,\alpha,p,q}\left(\int_{G}\left(\int_{0}^{\infty}\left[\frac{1}{r^{\alpha}}\beta_{f,d}(B_{r}(x))\right]^{2}\frac{dr}{r}\right)^{p/2}dx\right)^{1/p},$

where in the second inequality we applied the Fefferman–Stein vector-valued
maximal function inequality (61) with functions

$g_{j}(x)=2^{\alpha wj}\beta_{f,d}(B_{2^{-j+1}}(x))$

and exponents $u=2/w>1$ and $v=p/w>1$, along with Corollary 47.

This completes the proof of Theorem 22.

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