Pick $0<\varepsilon<\min\{s-\frac{1}{p},L-s\}$ and let $x_1'=(x_2,\cdots,x_n)\in\bbbr^{n-1}$ be fixed for now. When $l\leq k$, we use Lemma <ref> and (<ref>) and calculate as follows

≲   sup_k∈2^k(s-L)∑_l≤k2^lε·2^l(L-ε)
_1 f_l(·,x_1')(x_1)

≲   sup_l∈sup_k≥l2^k(s+ε-L)·2^l(L-ε)_1 f_l(·,x_1')(x_1)

≲   sup_l∈2^ls_1(|f_l(·,x_1')|^r)(x_1)^1/r,

where we also used the $1$-dimensional version of Lemma <ref> and $0<r<p$. Inserting (<ref>) into $\|\cdot\|_{L^p(\bbbr^n)}$ quasinorm and invoking the mapping property of the $1$-dimensional Hardy-Littlewood maximal function, we can estimate the first term in (<ref>) from above by the following

≲   (∫_^n-1∫__1(sup_l∈2^lsr|f_l(·,x_1')|^r)(x_1)^p/r
dx_1 dx_1')^1/p

≲   (∫_^n-1∫_sup_l∈2^lsp|f_l(x_1,x_1')|^p dx_1 dx_1')^1/p

When $l>k$ and $|t|\lesssim 2^{-k}$, we use (<ref>) and (<ref>) with temporarily fixed $x_1'\in\bbbr^{n-1}$ to obtain

\begin{equation}\label{eq567}
\sup_{t\in[2^{-k},2^{1-k})}|\varDelta^L_{t,1}f_l(x_1,x_1')|\lesssim 2^{\frac{l-k}{r}}\HLmax_1(|f_l(\cdot,x_1')|^r)(x_1)^{\frac{1}{r}}.
\end{equation}

Since $\frac{1}{p}<s<L$ by the assumption, we can pick $\varepsilon>0$ to be sufficiently close to $0$ and $r$ to be sufficiently close to $p$ so that $s-\frac{1}{r}-\varepsilon$ is a positive finite number, and then we can obtain

≲   sup_k∈2^k(s-1/r)∑_l>k2^-lε·2^l(ε+1/r)

≲   sup_l∈sup_k<l2^k(s-1/r-ε)·2^l(ε+1/r)

≲   sup_l∈2^ls_1(|f_l(·,x_1')|^r)(x_1)^1/r.

Inserting (<ref>) into $\|\cdot\|_{L^p(\bbbr^n)}$ quasinorm and proceeding as in (<ref>), we can estimate the second term in (<ref>) from above by $\|f\|_{\Fspinf}$. Therefore we have obtained the inequality

\begin{equation}\label{eq569}
\|\sup_{k\in\bbbz}\sup_{t\in[2^{-k},2^{1-k})} 2^{ks}\sum_{l\in\bbbz}|\varDelta^L_{t,1}f_l(\cdot)|\|_{L^p(\bbbr^n)}\lesssim\|f\|_{\Fspinf},
\end{equation}

when $0<p<\infty$, $q=\infty$ and $\frac{1}{p}<s<L$. The assumption $f\in\Fspinf$ and the above inequality also show $\sum_{l\in\bbbz}|\varDelta^L_{t,1}f_l(x)|<\infty$ for every $k\in\bbbz$ and almost every $t\in[2^{-k},2^{1-k}),x\in\bbbr^n$. In conjunction with (<ref>), we have reached the conclusion that

\begin{equation}\label{eq570}
\varDelta^L_{t,1}f=\sum_{l\in\bbbz}\varDelta^L_{t,1}f_l(x)
\end{equation}

in the sense of $\Sw'(\bbbr^n)/\mathscr{P}(\bbbr^n)$ for every $k\in\bbbz$ and almost every $t\in[2^{-k},2^{1-k}),x\in\bbbr^n$ when $0<p<\infty$, $q=\infty$ and $\frac{1}{p}<s<L$, furthermore we also obtain

\begin{equation}\label{eq571}
\|\sup_{t>0}\frac{|\varDelta^L_{t,1}f|}{t^s}\|_{L^p(\bbbr^n)}\lesssim
\|\sup_{k\in\bbbz}\sup_{t\in[2^{-k},2^{1-k})}2^{ks}\sum_{l\in\bbbz}|\varDelta^L_{t,1}f_l(\cdot)|\|_{L^p(\bbbr^n)}.
\end{equation}

Inequalities (<ref>) and (<ref>) conclude the proof of Theorem <ref> (iii).

Finally we come to the proof of Theorem <ref> (iv). We assume the right side of (<ref>) is finite, otherwise the inequality is trivial. We also use the sufficiently large positive integer $m_0$ given in (<ref>) and let $0<r<p$. We continue using the partition of unity $\{\rho_j\}_{j=1}^n$ associated with the set $A'=\{\xi\in\bbbr^n:\frac{1}{2}\leq|\xi|<2\}$ introduced at the beginning of the proof of Theorem <ref> (ii), and also continue assuming $\frac{1}{2}\leq|\xi|<2$ and $\delta\leq|\xi_j|<2$ for $\xi\in spt.\rho_j$ and $\delta$ being a sufficiently small positive number. Then by using Remark <ref>, Lemma <ref> and the mapping property of Hardy-Littlewood maximal function, we can obtain the estimate
≲   2^-m_0 s∑_j=1^nsup_k∈2^ks|_n(_nψ(2^m_0-kξ)ρ_j(2^m_0-kξ)
_n f)(·)|_L^p(^n)

≲   ∑_j=1^nsup_k∈2^ks_n(_n[ρ_j(2^m_0-kξ)_n f])(·)_L^p(^n)

≲   ∑_j=1^nsup_k∈2^ks
_n(|_n[ρ_j(2^m_0-kξ)_n f]|^r)(·)^1/r_L^p(^n)

≲   ∑_j=1^nsup_k∈2^ks|_n[ρ_j(2^m_0-kξ)_n f](·)|_L^p(^n).

We now estimate each term in the sum of (<ref>). It suffices to provide estimate for the term $\|\sup_{k\in\bbbz}2^{ks}|\iFT_n[\rho_1(2^{m_0-k}\xi)\FT_n f](\cdot)|\|_{L^p(\bbbr^n)}$ when $j=1$, and estimates for the terms when $j=2,\cdots,n$ can be obtained in the same way. When $1<p<\infty$, we have
|_n[ρ_1(2^m_0-kξ)_n f](x)|

=   sup_t∈[1,2]|_n[ρ_1(2^m_0-kξ)_n f](x)|

=   sup_t∈[1,2]|_n[ρ_1(2^m_0-kξ)/(e^2πi·2^-ktξ_1-1)^L
·(e^2πi·2^-ktξ_1-1)^L_n f](x)|

=   sup_t∈[1,2]|_n[ρ_1(2^m_0-kξ)/(e^2πi·2^-ktξ_1-1)^L]*^L_2^-kt,1f(x)|.

We use (<ref>) and the following estimate

\begin{equation}\label{eq573}
\mvint_{A_{k-l}}\sup_{t\in[1,2]}|\varDelta^L_{2^{-k}t,1}f(x-y)|dy\lesssim
\HLmax_n(\sup_{t\in[1,2]}|\varDelta^L_{2^{-k}t,1}f(\cdot)|)(x),
\end{equation}

and then we can estimate $|\iFT_n[\rho_1(2^{m_0-k}\xi)\FT_n f](x)|$ from above by the following

\begin{equation}\label{eq574}
(\sum_{l\leq 0}2^{ln}+\sum_{l>0}2^{l(n-N)})\HLmax_n(\sup_{t\in[1,2]}|\varDelta^L_{2^{-k}t,1}f|)(x)
\lesssim\HLmax_n(\sup_{t\in[1,2]}|\varDelta^L_{2^{-k}t,1}f|)(x)
\end{equation}

for every $x\in\bbbr^n$ if we pick $N>n$. Therefore when $1<p<\infty$, $q=\infty$ and $s\in\bbbr$, we invoke the mapping property of Hardy-Littlewood maximal function and obtain
sup_k∈2^ks|_n[ρ_1(2^m_0-kξ)_n f]|_L^p(^n)

≲   _n(sup_k∈sup_t∈[1,2]2^ks|^L_2^-kt,1f|)_L^p(^n)

Inequality (<ref>) is still true if we replace $\rho_1,\varDelta^L_{2^{-k}t,1},\varDelta^L_{t,1}$ by $\rho_j,\varDelta^L_{2^{-k}t,j},\varDelta^L_{t,j}$ respectively for $j=2,\cdots,n$. Inserting these inequalities back into (<ref>) proves the first part of Theorem <ref> (iv). To prove the second part of Theorem <ref> (iv), we use the function $\phi$ satisfying conditions (<ref>), (<ref>), (<ref>), and the same $m_0$ as in (<ref>), and the large positive integer $J>m_0$ whose value will be determined later. From (<ref>), (<ref>) and the inequality

\begin{equation}\label{eq576}
\sup_{t\in[1,2]}\mvint_{A_{k-l}}|\varDelta^L_{2^{-k}t,1}(\phi_{2^{m_0-J-k}}*f)(x-y)|^r dy\lesssim
\HLmax_n(\sup_{t\in[1,2]}|\varDelta^L_{2^{-k}t,1}(\phi_{2^{m_0-J-k}}*f)|^r)(x),
\end{equation}

we can obtain
|_n[ρ_1(2^m_0-kξ)_n f](x)|^r

=   sup_t∈[1,2]|_n[ρ_1(2^m_0-kξ)/(e^2πi·2^-ktξ_1-1)^L]*

≲   2^Jn(1-r)·(∑_l≤02^ln+∑_l>02^l(n-N'r))

≲   2^Jn(1-r)_n(sup_t∈[1,2]|^L_2^-kt,1(ϕ_2^m_0-J-k*f)|^r)(x),

if we pick $N'>\frac{n}{r}$. From this inequality and the mapping property of Hardy-Littlewood maximal function, we deduce the estimate
sup_k∈2^ks|_n[ρ_1(2^m_0-kξ)_n f](·)|_L^p(^n)

≲   2^Jn(1/r-1)

Assuming the validity of decomposition for now, then (<ref>) can be estimated from above by the sum of

\begin{equation}\label{eq579}
\|\sup_{k\in\bbbz}\sup_{t\in[1,2]}2^{ks}|\varDelta^L_{2^{-k}t,1}f(\cdot)|\|_{L^p(\bbbr^n)}\sim
\|\sup_{t>0}\frac{|\varDelta^L_{t,1}f(\cdot)|}{t^s}\|_{L^p(\bbbr^n)}<\infty,
\end{equation}

\begin{equation}\label{eq580}
\|\sup_{k\in\bbbz}\sup_{t\in[1,2]}2^{ks}|\varDelta^L_{2^{-k}t,1}(\sum_{l=J+1}^{\infty}f_{k+l-m_0})(\cdot)|\|_{L^p(\bbbr^n)}.
\end{equation}

To estimate (<ref>), we use (<ref>), (<ref>), Lebesgue's differentiation theorem and the fact that $k+l-m_0>k+J-m_0>k$ to obtain for $k\in\bbbz$, $l>J$ and $t\in[1,2]$,

\begin{equation}\label{eq581}
|\varDelta^L_{2^{-k}t,1}f_{k+l-m_0}(x)|\lesssim 2^{\frac{l}{r}}\HLmax_1(|f_{k+l-m_0}(\cdot,x_1')|^r)(x_1)^{\frac{1}{r}},
\end{equation}

where the constant is independent of $l,k,t$. Therefore we have

=   2^Jn(1/r-1)
sup_k∈sup_t∈[1,2]2^ks∑_l=J+1^∞2^-lε·2^lε |^L_2^-kt,1f_k+l-m_0(·)|_L^p(^n)

≲   2^Jn(1/r-1)-Jε

≲   2^Jn(1/r-1)-Jε

The assumption $\sigma_p+\frac{1}{p}<s<\infty$ indicates

\begin{equation}\label{eq583}
\varepsilon+\frac{1}{r}<n(\frac{1}{r}-1)+\frac{1}{r}<s
\end{equation}

if we pick $\varepsilon>0$ to be sufficiently small and $r$ to be sufficiently close to $p$ when $0<r<p\leq 1$. Thus by the mapping property of the $1$-dimensional Hardy-Littlewood maximal function, we can continue from (<ref>) and obtain the following estimate

≲   2^J[n(1/r-1)+1/r-s]
(∫_^n-1∫__1(sup_k∈2^ksr|f_k(·,x_1')|^r)(x_1)^p/rdx_1 dx_1')^1/p

≲   2^J[n(1/r-1)+1/r-s]f_.

The assumption $f\in\Fspinf$ implies $\sum_{l=J+1}^{\infty}|\varDelta^L_{2^{-k}t,1}f_{k+l-m_0}(x)|<\infty$ for every $k\in\bbbz$ and almost every $t\in[1,2],x\in\bbbr^n$. From this implication, (<ref>) and (<ref>), we deduce

\begin{equation}\label{eq585}
\varDelta^L_{2^{-k}t,1}(\phi_{2^{m_0-J-k}}*f)(x)=\varDelta^L_{2^{-k}t,1}f(x)-
\sum_{l=J+1}^{\infty}\varDelta^L_{2^{-k}t,1}f_{k+l-m_0}(x)
\end{equation}

in the sense of $\Sw'(\bbbr^n)$ for every $k\in\bbbz$ and almost every $t\in[1,2],x\in\bbbr^n$ when $0<p\leq 1$, $q=\infty$ and $\sigma_p+\frac{1}{p}<s<\infty$, furthermore estimating (<ref>) from above by the sum of (<ref>) and (<ref>) is justified, moreover (<ref>) can be estimated from above by (<ref>). We have obtained
sup_k∈2^ks|_n[ρ_1(2^m_0-kξ)_n f](·)|_L^p(^n)

≤   C_1'2^Jn(1/r-1)sup_t>0|^L_t,1f(·)|/t^s_L^p(^n)

Inequality (<ref>) is still true if we replace $\rho_1,\varDelta^L_{t,1},C_1',C_1''$ by $\rho_j,\varDelta^L_{t,j},C_j',C_j''$ respectively for $j=2,\cdots,n$. Inserting these inequalities back into (<ref>) yields

\begin{equation}\label{eq588}
\|f\|_{\Fspinf}\leq
\end{equation}

Due to (<ref>), we can pick $J$ to be sufficiently large so that the coefficient $C''2^{J[n(\frac{1}{r}-1)+\frac{1}{r}-s]}<\frac{1}{2}$ and shift the second term on the right side of (<ref>) to its left side to complete the proof of Theorem <ref> (iv). Now we conclude the proof of Theorem <ref>.

§ BESOV-LIPSCHITZ SPACE AND ITERATED DIFFERENCES ALONG COORDINATE AXES

PROOF OF THEOREM <REF>

To prove Theorem <ref> (i), it suffices to prove inequality (<ref>) for $j=1$, and the other cases for $j=2,\cdots,n$ can be proven in the same way. We begin with estimating the following two terms
(∑_k∈2^k(sq+1)∫_2^-k^2^1-k∑_l≤k|^L_t,1f_l|_L^p(^n)^q dt)^1/q

≲   (∑_k∈2^ksqsup_t∈[1,2]∑_l≤k|^L_2^-kt,1f_l|_L^p(^n)^q)^1/q,

(∑_k∈2^k(sq+1)∫_2^-k^2^1-k∑_l>k|^L_t,1f_l|_L^p(^n)^q dt)^1/q

≲   (∑_k∈2^ksqsup_t∈[1,2]∑_l>k|^L_2^-kt,1f_l|_L^p(^n)^q)^1/q.

We pick $0<\varepsilon<\min\{s,L-s\}$. If $1\leq p\leq\infty$, we use Minkowski's inequality for $\|\cdot\|_{L^p(\bbbr^n)}$-norm and obtain

≲   (∑_l≤k2^lε·2^-lε^L_2^-kt,1f_l_L^p(^n))^q

≲   (∑_l≤k2^lε)^qsup_j≤k2^-jqε^L_2^-kt,1f_j_L^p(^n)^q

≲   2^kqε∑_l≤k2^-lqε^L_2^-kt,1f_l_L^p(^n)^q,

≲   (∑_l>k2^-lε·2^lε^L_2^-kt,1f_l_L^p(^n))^q

≲   (∑_l>k2^-lε)^qsup_j>k2^jqε^L_2^-kt,1f_j_L^p(^n)^q

≲   2^-kqε∑_l>k2^lqε^L_2^-kt,1f_l_L^p(^n)^q.

If $0<p<1$, then we have

≲   (∑_l≤k2^lpε·2^-lpε^L_2^-kt,1f_l_L^p(^n)^p)^q/p

≲   2^kqε∑_l≤k2^-lqε^L_2^-kt,1f_l_L^p(^n)^q,

≲   (∑_l>k2^-lpε·2^lpε^L_2^-kt,1f_l_L^p(^n)^p)^q/p

≲   2^-kqε∑_l>k2^lqε^L_2^-kt,1f_l_L^p(^n)^q.

When $t\in[1,2]$ and $l\leq k$, we use estimate (<ref>) with $0<r<p$, the $1$-dimensional version of Lemma <ref> and the mapping property of the $1$-dimensional Hardy-Littlewood maximal function to obtain

\begin{equation}\label{eq610}
\|\varDelta^L_{2^{-k}t,1}f_l\|_{L^p(\bbbr^n)}\lesssim 2^{(l-k)L}\|f_l\|_{L^p(\bbbr^n)}\text{ for }0<p\leq\infty,
\end{equation}

where the constant is independent of $t,l,k$. Inserting (<ref>), (<ref>) and (<ref>) into (<ref>) yields

\begin{equation}\label{eq611}
(\sum_{l\in\bbbz}\sum_{k\geq l}2^{kq(s+\varepsilon-L)}\cdot 2^{lq(L-\varepsilon)}\|f_l\|_{L^p(\bbbr^n)}^q)^{\frac{1}{q}}
\lesssim\|f\|_{\Bspq}.
\end{equation}

We can also use estimate (<ref>) and proper change of variable to obtain

\begin{equation}\label{eq612}
\|\varDelta^L_{2^{-k}t,1}f_l\|_{L^p(\bbbr^n)}\lesssim\sum_{m=0}^L\|f_l(\cdot+2^{-k}mte_1)\|_{L^p(\bbbr^n)}
\lesssim\|f_l\|_{L^p(\bbbr^n)},
\end{equation}

where constants are independent of $t,k,l$, and (<ref>) is true for all $0<p\leq\infty$. Inserting (<ref>), (<ref>) and (<ref>) into (<ref>) yields

\begin{equation}\label{eq613}
(\sum_{l\in\bbbz}\sum_{k<l}2^{kq(s-\varepsilon)}\cdot 2^{lq\varepsilon}\|f_l\|_{L^p(\bbbr^n)}^q)^{\frac{1}{q}}
\lesssim\|f\|_{\Bspq}.
\end{equation}

Combining (<ref>), (<ref>), (<ref>) and (<ref>), we have proven

≲   (∑_k∈2^ksq

≲   f_,

if $0<p\leq\infty$, $0<q<\infty$ and $0<s<L$. The assumption $f\in\Bspq$ implies $\sum_{l\in\bbbz}|\varDelta^L_{t,1}f_l(x)|<\infty$ for almost every $t\in(0,\infty)$ and $x\in\bbbr^n$. In conjunction with (<ref>), we have reached the conclusion that

\begin{equation}\label{eq615}
\varDelta^L_{t,1}f=\sum_{l\in\bbbz}\varDelta^L_{t,1}f_l(x)
\end{equation}

in the sense of $\Sw'(\bbbr^n)/\mathscr{P}(\bbbr^n)$ for almost every $t\in(0,\infty)$ and $x\in\bbbr^n$ when $0<p\leq\infty$, $0<q<\infty$ and $0<s<L$. Hence we also have the estimate

\begin{equation}\label{eq616}
\end{equation}

Inequalities (<ref>) and (<ref>) conclude the proof of Theorem <ref> (i).

Now we prove the first and the second parts of Theorem <ref> (ii). We can assume the right side of (<ref>) is finite and the left side of (<ref>) is positive, otherwise inequality (<ref>) will be trivial. In the definition of Peetre-Fefferman-Stein maximal function, we pick the number $r$ so that $0<r<p$. We also use the positive integer $m_0$ given in (<ref>) and $\{\rho_j\}_{j=1}^n$ is the partition of unity given in (<ref>) and (<ref>). Then we have
=   (∑_k∈2^(k-m_0)sqψ_2^m_0-k*f_L^p(^n)^q)^1/q

≲   ∑_j=1^n(∑_k∈2^ksq
_n[_nψ(2^m_0-kξ)ρ_j(2^m_0-kξ)_n f]_L^p(^n)^q)^1/q.

By Remark <ref> and Remark <ref>, we can obtain the following pointwise estimate
_n[_nψ(2^m_0-kξ)ρ_j(2^m_0-kξ)_n f](x)

≲   _n{_n[_nψ(2^m_0-kξ)ρ_j(2^m_0-kξ)_n f]}(x)

≲   _n{_n[ρ_j(2^m_0-kξ)_n f]}(x)

≲   _n(|_n[ρ_j(2^m_0-kξ)_n f]|^r)(x)^1/r,

for $j\in\{1,\cdots,n\}$ and $x\in\bbbr^n$. Inserting (<ref>) into (<ref>) and invoking the mapping property of Hardy-Littlewood maximal function yield

\begin{equation}\label{eq619}
\|f\|_{\Bspq}\lesssim\sum_{j=1}^n\|\{2^{ks}\iFT_n[\rho_j(2^{m_0-k}\xi)\FT_n f](\cdot)\}_{k\in\bbbz}\|_{l^q(L^p)}.
\end{equation}

We estimate the term with $j=1$ and estimates for other terms with $j=2,\cdots,n$ can be obtained in the same way. If $1<p\leq\infty$, $1\leq q<\infty$ and $s\in\bbbr$, we use (<ref>), (<ref>), (<ref>), the mapping property of Hardy-Littlewood maximal function, Minkowski's inequality for $\|\cdot\|_{L^p(\bbbr^n)}$-norm and Hölder's inequality for $1\leq q<\infty$ in a sequence and then we have
{2^ks_n[ρ_1(2^m_0-kξ)_n f](·)}_k∈_l^q(L^p)

≲   {2^ks_n(∫_[1,2]|^L_2^-kt,1f(·)|dt/t)(·)}_k∈_l^q(L^p)

≲   (∑_k∈2^ksq(∫_[1,2]^L_2^-kt,1f_L^p(^n)dt/t)^q)^1/q

≲   (∑_k∈2^ksq∫_[1,2]^L_2^-kt,1f_L^p(^n)^qdt/t)^1/q

≲   (∫_0^∞t^-sq^L_t,1f_L^p(^n)^qdt/t)^1/q.

Inequality (<ref>) is also true if we replace $\rho_1,\varDelta^L_{2^{-k}t,1},\varDelta^L_{t,1}$ by $\rho_j,\varDelta^L_{2^{-k}t,j},\varDelta^L_{t,j}$ respectively for $j=2,\cdots,n$. Inserting these inequalities into (<ref>) proves inequality (<ref>) when $1<p\leq\infty$, $1\leq q<\infty$ and $s\in\bbbr$. If $1<p\leq\infty$, $0<q<1$ and $0<s<L$, we can still obtain estimate (<ref>). We continue from there and estimate each term in the summation of (<ref>) as below

=   (∫_[1,2]2^ksq^L_2^-kt,1f_L^p(^n)^q·2^ks(1-q)^L_2^-kt,1f_L^p(^n)^1-qdt/t)^q

≲   (∫_[1,2]2^ksq^L_2^-kt,1f_L^p(^n)^qdt/t)^q·2^ks(1-q)qsup_t∈[1,2]^L_2^-kt,1f_L^p(^n)^(1-q)q,

and then by using Hölder's inequality with conjugates $\frac{1}{q}$ and $\frac{1}{1-q}$, we can estimate (<ref>) from above by the product of the following two terms,

\begin{equation}\label{eq623}
\sum_{k\in\bbbz}\int_{[1,2]}2^{ksq}\|\varDelta^L_{2^{-k}t,1}f\|_{L^p(\bbbr^n)}^q\frac{dt}{t}\lesssim
\int_0^{\infty}t^{-sq}\|\varDelta^L_{t,1}f\|_{L^p(\bbbr^n)}^q\frac{dt}{t}
\end{equation}

\begin{equation}\label{eq624}
\end{equation}

Furthermore (<ref>) can be estimated from above by $\|f\|_{\Bspq}^{1-q}$ if we apply (<ref>) and the argument for justifying the decomposition afterward. Combining these estimates together, we have shown

\begin{equation}\label{eq625}
\|\{2^{ks}\iFT_n[\rho_1(2^{m_0-k}\xi)\FT_n f](\cdot)\}_{k\in\bbbz}\|_{l^q(L^p)}\lesssim
\int_0^{\infty}t^{-sq}\|\varDelta^L_{t,1}f\|_{L^p(\bbbr^n)}^q\frac{dt}{t}\cdot\|f\|_{\Bspq}^{1-q}.
\end{equation}

Inequality (<ref>) is still true if we replace $\rho_1,\varDelta^L_{t,1}$ by $\rho_j,\varDelta^L_{t,j}$ respectively for $j=2,\cdots,n$. Inserting these inequalities into (<ref>) yields

\begin{equation}\label{eq626}
\|f\|_{\Bspq}\lesssim\sum_{j=1}^n
\int_0^{\infty}t^{-sq}\|\varDelta^L_{t,j}f\|_{L^p(\bbbr^n)}^q\frac{dt}{t}\cdot\|f\|_{\Bspq}^{1-q},
\end{equation}

when $1<p\leq\infty$, $0<q<1$ and $0<s<L$. Then inequality (<ref>) indicates the desired inequality (<ref>). Next we come to the proof of the third part of Theorem <ref> (ii). We prove (<ref>) when $0<p\leq 1$, $0<q<\infty$ and $\sigma_p<s<L$, we pick $0<r<p$ and estimate the term $\|\{2^{ks}\iFT_n[\rho_1(2^{m_0-k}\xi)\FT_n f](\cdot)\}_{k\in\bbbz}\|_{l^q(L^p)}$. We still use the sufficiently large positive integer $m_0$ given in (<ref>), the function $\phi$ satisfying conditions (<ref>), (<ref>), (<ref>), and $J>m_0$ is a large positive integer whose value will be determined later. From (<ref>), the mapping property of Hardy-Littlewood maximal function and Minkowski's inequality for $\|\cdot\|_{L^{p/r}(\bbbr^n)}$-norm, we can deduce the following
{2^ks_n[ρ_1(2^m_0-kξ)_n f](·)}_k∈_l^q(L^p)

≲   2^Jn(1/r-1)(∑_k∈2^ksq

≲   2^Jn(1/r-1)(∑_k∈2^ksq

If furthermore $q$ and $r$ satisfy $q\geq r$, then by using Hölder's inequality we have

\begin{equation}\label{eq628}
\int_1^2\|\varDelta^L_{2^{-k}t,1}(\phi_{2^{m_0-J-k}}*f)\|_{L^p(\bbbr^n)}^q\frac{dt}{t}.
\end{equation}

Hence we can estimate (<ref>) from above by

\begin{equation}\label{eq629}
\int_1^2\|\varDelta^L_{2^{-k}t,1}(\phi_{2^{m_0-J-k}}*f)\|_{L^p(\bbbr^n)}^q\frac{dt}{t})^{\frac{1}{q}}.
\end{equation}

From (<ref>) and (<ref>), we can infer

\begin{equation}\label{eq630}
\varDelta^L_{2^{-k}t,1}(\phi_{2^{m_0-J-k}}*f)(x)=\varDelta^L_{2^{-k}t,1}f(x)
\end{equation}

in the sense of $\Sw'(\bbbr^n)$. Assuming the validity of decomposition for now, we can estimate (<ref>) from above by the sum of the following two terms,

\begin{equation}\label{eq631}
\int_1^2\!\!\!\!\|\varDelta^L_{2^{-k}t,1}f\|_{L^p(\bbbr^n)}^q\frac{dt}{t})^{\frac{1}{q}}
\sim 2^{Jn(\frac{1}{r}-1)}(\int_0^{\infty}\!\!\!\!t^{-sq}\|\varDelta^L_{t,1}f\|_{L^p(\bbbr^n)}^q\frac{dt}{t})^{\frac{1}{q}}
\end{equation}

\begin{equation}\label{eq632}
\int_1^2\|\varDelta^L_{2^{-k}t,1}(\sum_{j=J+1}^{\infty}f_{k+j-m_0})\|_{L^p(\bbbr^n)}^q\frac{dt}{t})^{\frac{1}{q}}.
\end{equation}

To estimate (<ref>), we use (<ref>) and the condition $0<p\leq 1$ and begin with the following

≲   2^Jn(1/r-1)(∑_k∈2^ksq(∑_j=J+1^∞

In case $q\leq p$, since $0<\sigma_p<s$, we have
≲   2^Jn(1/r-1)(∑_k∈2^ksq∑_j=J+1^∞f_k+j-m_0_L^p(^n)^q)^1/q

≲   2^Jn(1/r-1)(∑_j=J+1^∞2^(m_0-j)sq

≲   2^J[n(1/r-1)-s]f_.

In case $p<q$, we use $0<\varepsilon<\min\{s,L-s\}$ and estimate as follows
=   2^Jn(1/r-1)(∑_k∈2^ksq(∑_j=J+1^∞2^-jpε·2^jpε

≲   2^Jn(1/r-1)(∑_k∈2^ksq(∑_j=J+1^∞2^-jpε)^q/p

≲   2^J[n(1/r-1)-ε]

≲   2^J[n(1/r-1)-s]f_.

Combining (<ref>), (<ref>), (<ref>), (<ref>) and the assumption $f\in\Bspq$ altogether, we find $\sum_{j=J+1}^{\infty}|\varDelta^L_{2^{-k}t,1}f_{k+j-m_0}(x)|<\infty$ for every $k\in\bbbz$ and almost every $t\in[1,2],x\in\bbbr^n$. In conjunction with (<ref>) and (<ref>), we have proven (<ref>) is true not only in the sense of $\Sw'(\bbbr^n)$ but also for every $k\in\bbbz$ and almost every $t\in[1,2],x\in\bbbr^n$ when $0<r<p\leq 1$, $r\leq q<\infty$ and $\sigma_p<s<L$. Furthermore estimating (<ref>) from above by the sum of (<ref>) and (<ref>) is justified, moreover (<ref>) can be estimated from above by (<ref>) and hence by the term $2^{J[n(\frac{1}{r}-1)-s]}\|f\|_{\Bspq}$. Recall (<ref>) and (<ref>), then we have obtained
{2^ks_n[ρ_1(2^m_0-kξ)_n f](·)}_k∈_l^q(L^p)

≤   C_1'2^Jn(1/r-1)(∫_0^∞t^-sq^L_t,1f_L^p(^n)^qdt/t)^1/q

Inequality (<ref>) is also true if we replace $\rho_1,\varDelta^L_{t,1},C_1',C_1''$ by $\rho_j,\varDelta^L_{t,j},C_j',C_j''$ respectively for $j=2,\cdots,n$. Inserting these inequalities into (<ref>) yields

\begin{equation}\label{eq638}
\|f\|_{\Bspq}\leq C'2^{Jn(\frac{1}{r}-1)}\sum_{j=1}^n
\end{equation}

The condition $\sigma_p<s<L$ implies $n(\frac{1}{r}-1)-s<0$ if $r$ is sufficiently close to $p$, and hence the coefficient $C''2^{J[n(\frac{1}{r}-1)-s]}$ is less than $\frac{1}{2}$ when $J$ is a sufficiently large positive integer. Shifting the second term on the right side of (<ref>) to its left side proves the desired inequality (<ref>) when $0<r<p\leq 1$, $r\leq q<\infty$ and $\sigma_p<s<L$. If $0<q<r<p\leq 1$ and $\sigma_p<s<L$, then we continue from (<ref>). Applying (<ref>) and assuming the validity of decomposition for now, then (<ref>) can be estimated from above by the sum of the following two terms,

\begin{equation}\label{eq639}
\end{equation}

\begin{equation}\label{eq640}
\end{equation}

To estimate (<ref>), we first rewrite each term in the summation of (<ref>) as follows

=   (∫_1^2 2^ksq^L_2^-kt,1f_L^p(^n)^q·2^ks(r-q)^L_2^-kt,1f_L^p(^n)^r-qdt/t)^q/r

≲   (∫_1^2 2^ksq^L_2^-kt,1f_L^p(^n)^qdt/t)^q/r·(2^kssup_t∈[1,2]^L_2^-kt,1f_L^p(^n))^(r-q)q/r,

and then we apply Hölder's inequality with conjugates $\frac{r}{q}$ and $\frac{r}{r-q}$ to obtain
≲   2^Jn(1/r-1)
(∑_k∈∫_1^2 2^ksq^L_2^-kt,1f_L^p(^n)^qdt/t)^1/r·(∑_k∈2^ksqsup_t∈[1,2]^L_2^-kt,1f_L^p(^n)^q)^1/q-1/r

≲   2^Jn(1/r-1)

where in (<ref>) we used estimate (<ref>) and the argument afterward to justify the decomposition. And (<ref>) requires $0<s<L$. To estimate (<ref>), we use (<ref>) and the condition $0<q<r<p\leq 1$ to get

\begin{equation}\label{eq643}
\|\sum_{j=J+1}^{\infty}|\varDelta^L_{2^{-k}t,1}f_{k+j-m_0}|\|_{L^p(\bbbr^n)}^r\lesssim
\sum_{j=J+1}^{\infty}\|f_{k+j-m_0}\|_{L^p(\bbbr^n)}^r,
\end{equation}

and then we can insert (<ref>) into the following term and obtain

≲   2^Jn(1/r-1)(∑_k∈2^ksq(∑_j=J+1^∞f_k+j-m_0_L^p(^n)^r

≲   2^Jn(1/r-1)(∑_j=J+1^∞2^(m_0-j)sq∑_k∈2^(k+j-m_0)sq

≲   2^J[n(1/r-1)-s]f_.

The assumption $f\in\Bspq$ tells us $\sum_{j=J+1}^{\infty}|\varDelta^L_{2^{-k}t,1}f_{k+j-m_0}(x)|<\infty$ for every $k\in\bbbz$ and almost every $t\in[1,2],x\in\bbbr^n$. In conjunction with (<ref>) and (<ref>), we have proven (<ref>) is true not only in the sense of $\Sw'(\bbbr^n)$ but also for every $k\in\bbbz$ and almost every $t\in[1,2],x\in\bbbr^n$ when $0<q<r<p\leq 1$ and $\sigma_p<s<L$. Furthermore estimating (<ref>) from above by the sum of (<ref>) and (<ref>) is justified, moreover (<ref>) can be estimated from above by (<ref>) and hence by (<ref>). Combining (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) altogether, we have obtained
{2^ks_n[ρ_1(2^m_0-kξ)_n f](·)}_k∈_l^q(L^p)

≤   C_1'2^Jn(1/r-1)(∫_0^∞t^-sq^L_t,1f_L^p(^n)^qdt/t)^1/r

Inequality (<ref>) is also true if we replace $\rho_1,\varDelta^L_{t,1},C_1',C_1''$ by $\rho_j,\varDelta^L_{t,j},C_j',C_j''$ respectively for $j=2,\cdots,n$. Inserting these inequalities into (<ref>) yields
≤   C'2^Jn(1/r-1)∑_j=1^n

The condition $\sigma_p<s<L$ implies $n(\frac{1}{r}-1)-s<0$ if $r$ is sufficiently close to $p$, and hence the coefficient $C''2^{J[n(\frac{1}{r}-1)-s]}$ is less than $\frac{1}{2}$ when $J$ is a sufficiently large positive integer. And then we can shift the second term on the right side of (<ref>) to its left side, divide both sides of the resulting inequality by $\|f\|_{\Bspq}^{1-\frac{q}{r}}$ and raise the power to $\frac{r}{q}$, finally we can reach the desired inequality (<ref>) when $0<q<r<p\leq 1$ and $\sigma_p<s<L$. We have finished the proof for the third part of Theorem <ref> (ii).

Now we prove Theorem <ref> (iii). We only need to prove inequality (<ref>) for $j=1$, and the other cases for $j=2,\cdots,n$ can be proved in the same way. We pick $0<\varepsilon<\min\{s,L-s\}$ and begin with estimating the following term

\begin{equation}\label{eq648}
\sup_{k\in\bbbz}\sup_{2^{-k}\leq t<2^{1-k}}\!\!\!\!2^{ks}\|\sum_{l\leq k}|\varDelta^L_{t,1}f_l|\|_{L^p(\bbbr^n)}+
\sup_{k\in\bbbz}\sup_{2^{-k}\leq t<2^{1-k}}\!\!\!\!2^{ks}\|\sum_{l>k}|\varDelta^L_{t,1}f_l|\|_{L^p(\bbbr^n)}.
\end{equation}

If $1\leq p\leq\infty$, by using Minkowski's inequality and (<ref>), we can estimate the first term of (<ref>) from above by

≲   sup_k∈2^k(s+ε-L)sup_l≤k2^l(L-ε)f_l_L^p(^n)

≲   sup_l∈sup_k≥l2^k(s+ε-L)·2^l(L-ε)f_l_L^p(^n)

and by using Minkowski's inequality and (<ref>), we can estimate the second term of (<ref>) from above by

≲   sup_k∈2^k(s-ε)sup_l>k2^lεf_l_L^p(^n)

≲   sup_l∈sup_k<l2^k(s-ε)·2^lεf_l_L^p(^n)

If $0<p<1$, by using (<ref>) for $l\leq k$, we can estimate the first term of (<ref>) from above by

\begin{equation}\label{eq651}
\sup_{k\in\bbbz}2^{ks}\sup_{2^{-k}\leq t<2^{1-k}}(\sum_{l\leq k}2^{lp\varepsilon}\cdot 2^{-lp\varepsilon}
\|\varDelta^L_{t,1}f_l\|_{L^p(\bbbr^n)}^p)^{\frac{1}{p}}\lesssim\|f\|_{\Bspinf},
\end{equation}

and by using (<ref>) for $l>k$, we can estimate the second term of (<ref>) from above by

\begin{equation}\label{eq652}
\sup_{k\in\bbbz}2^{ks}\sup_{2^{-k}\leq t<2^{1-k}}(\sum_{l>k}2^{-lp\varepsilon}\cdot 2^{lp\varepsilon}
\|\varDelta^L_{t,1}f_l\|_{L^p(\bbbr^n)}^p)^{\frac{1}{p}}\lesssim\|f\|_{\Bspinf}.
\end{equation}

From (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we have proven

\begin{equation}\label{eq653}
\sup_{t>0}t^{-s}\|\sum_{l\in\bbbz}|\varDelta^L_{t,1}f_l|\|_{L^p(\bbbr^n)}\lesssim(\ref{eq648})\lesssim\|f\|_{\Bspinf},
\end{equation}

when $0<p\leq\infty$, $q=\infty$ and $0<s<L$. The assumption $f\in\Bspinf$ implies $\sum_{l\in\bbbz}|\varDelta^L_{t,1}f_l(x)|<\infty$ for almost every $t>0,x\in\bbbr^n$. In conjunction with (<ref>), we have shown

\begin{equation}\label{eq654}
\varDelta^L_{t,1}f=\sum_{l\in\bbbz}\varDelta^L_{t,1}f_l(x)
\end{equation}

in the sense of $\Sw'(\bbbr^n)/\mathscr{P}(\bbbr^n)$ for almost every $t>0,x\in\bbbr^n$ when $0<p\leq\infty$, $q=\infty$ and $0<s<L$. Therefore we have obtained the inequality

\begin{equation}\label{eq655}
\sup_{t>0}t^{-s}\|\varDelta^L_{t,1}f\|_{L^p(\bbbr^n)}\lesssim
\sup_{t>0}t^{-s}\|\sum_{l\in\bbbz}|\varDelta^L_{t,1}f_l|\|_{L^p(\bbbr^n)}.
\end{equation}

Inequalities (<ref>) and (<ref>) complete the proof of Theorem <ref> (iii).

Finally we come to the proof of Theorem <ref> (iv). We assume the right side of (<ref>) is finite, otherwise inequality (<ref>) is trivial. In the definition of Peetre-Fefferman-Stein maximal function, we pick the number $r$ so that $0<r<p$. We also use the positive integer $m_0$ given in (<ref>) and $\{\rho_j\}_{j=1}^n$ is the partition of unity given in (<ref>) and (<ref>). Then we have
≲   ∑_j=1^nsup_k∈2^ks
_n[_nψ(2^m_0-kξ)ρ_j(2^m_0-kξ)_n f]_L^p(^n)

≲   ∑_j=1^nsup_k∈2^ks_n[ρ_j(2^m_0-kξ)_n f]_L^p(^n),

where (<ref>) is a consequence by Remark <ref>, Remark <ref> and the mapping property of Hardy-Littlewood maximal function. It suffices to estimate the term for $j=1$ in (<ref>), and estimates for the other terms in (<ref>) when $j=2,\cdots,n$ can be obtained in the same way. If $1<p\leq\infty$, $q=\infty$ and $s\in\bbbr$, we can infer from the calculation method displayed in (<ref>) that
_n[ρ_1(2^m_0-kξ)_n f]_L^p(^n)

=   sup_t∈[1,2]_n[ρ_1(2^m_0-kξ)_n f]_L^p(^n)

=   sup_t∈[1,2]_n[ρ_1(2^m_0-kξ)/(e^2πi·2^-ktξ_1-1)^L]*^L_2^-kt,1f

We use (<ref>), Minkowski's inequality for $\|\cdot\|_{L^p(\bbbr^n)}$-norm, the mapping property of Hardy-Littlewood maximal function and the following estimate

\begin{equation}\label{eq659}
\|\mvint_{A_{k-l}}|\varDelta^L_{2^{-k}t,1}f(\cdot-y)|dy\|_{L^p(\bbbr^n)}\lesssim
\|\HLmax_n(|\varDelta^L_{2^{-k}t,1}f|)(\cdot)\|_{L^p(\bbbr^n)}\lesssim
\|\varDelta^L_{2^{-k}t,1}f\|_{L^p(\bbbr^n)},
\end{equation}

where $A_{k-l}$ denotes the annulus $\{y\in\bbbr^n:2^{l-k}\leq|y|<2^{1+l-k}\}$, and then we can estimate (<ref>) from above by

\begin{equation}\label{eq660}
\sup_{t\in[1,2]}(\sum_{l\leq 0}2^{ln}+\sum_{l>0}2^{l(n-N)})\|\varDelta^L_{2^{-k}t,1}f\|_{L^p(\bbbr^n)}\lesssim
\sup_{t\in[1,2]}\|\varDelta^L_{2^{-k}t,1}f\|_{L^p(\bbbr^n)},
\end{equation}

if we pick $N>n$. Combining (<ref>) and (<ref>), we have proven

\begin{equation}\label{eq661}
\sup_{k\in\bbbz}2^{ks}\|\iFT_n[\rho_1(2^{m_0-k}\xi)\FT_n f]\|_{L^p(\bbbr^n)}\lesssim
\sup_{t>0}t^{-s}\|\varDelta^L_{t,1}f\|_{L^p(\bbbr^n)}
\end{equation}

for any $s\in\bbbr$. And inequality (<ref>) is true if we replace $\rho_1,\varDelta^L_{t,1}$ by $\rho_j,\varDelta^L_{t,j}$ respectively for $j=2,\cdots,n$. Inserting these inequalities into (<ref>) yields the desired inequality (<ref>). If $0<p\leq 1$, $q=\infty$ and $\sigma_p<s<\infty$, then we use the function $\phi$ satisfying conditions (<ref>), (<ref>), (<ref>), and $J>m_0$ is a large positive integer whose value will be determined later. Since $0<r<p\leq 1$, we use the following inequality

\begin{equation}\label{eq662}
\mvint_{A_{k-l}}|\varDelta^L_{2^{-k}t,1}(\phi_{2^{m_0-J-k}}*f)(x-y)|^rdy\lesssim
\HLmax_n(|\varDelta^L_{2^{-k}t,1}(\phi_{2^{m_0-J-k}}*f)|^r)(x),
\end{equation}

and deduce from (<ref>) and (<ref>) the estimate below

≲   2^Jn(1/r-1)_n(|^L_2^-kt,1(ϕ_2^m_0-J-k*f)|^r)(x)^1/r,

where we let $N'>\frac{n}{r}$ in (<ref>). From (<ref>) we know that when $\rho_1(2^{m_0-k}\xi)\neq 0$ and $t\in[1,2]$, $|(e^{2\pi i\cdot 2^{-k}t\xi_1}-1)^L|>0$ and we can infer from the calculation method displayed in (<ref>) that
sup_k∈2^ks_n[ρ_1(2^m_0-kξ)_n f]_L^p(^n)

=   sup_k∈2^kssup_t∈[1,2]_n[ρ_1(2^m_0-kξ)/(e^2πi·2^-ktξ_1-1)^L]*

≲   sup_k∈2^ks+Jn(1/r-1)sup_t∈[1,2]

≲   sup_k∈2^ks+Jn(1/r-1)sup_t∈[1,2]

Recall (<ref>) and assume the validity of decomposition for now, then we can estimate (<ref>) from above by the sum of the following two terms,

\begin{equation}\label{eq665}
\end{equation}

\begin{equation}\label{eq666}
\end{equation}

To estimate (<ref>), we pick $0<\varepsilon<s$ and use (<ref>) to obtain

≲   (∑_j=J+1^∞^L_2^-kt,1f_k+j-m_0_L^p(^n)^p)^1/p

≲   (∑_j=J+1^∞2^-jpε·2^jpεf_k+j-m_0_L^p(^n)^p)^1/p

≲   2^-Jεsup_j>J2^jεf_k+j-m_0_L^p(^n),

where the constants are independent of $t$. And then we can have the following estimate

≲   2^J[n(1/r-1)-ε]sup_j>J2^j(ε-s)+m_0 ssup_k∈2^(k+j-m_0)s

≲   2^J[n(1/r-1)-s]f_.

The assumption $f\in\Bspinf$ implies $\sum_{j=J+1}^{\infty}|\varDelta^L_{2^{-k}t,1}f_{k+j-m_0}(x)|<\infty$ for every $k\in\bbbz$ and almost every $t\in[1,2],x\in\bbbr^n$. In conjunction with (<ref>) and (<ref>), we have shown (<ref>) is true not only in the sense of $\Sw'(\bbbr^n)$ but also for every $k\in\bbbz$ and almost every $t\in[1,2],x\in\bbbr^n$ when $0<p\leq 1$, $q=\infty$ and $\sigma_p<s<\infty$. Furthermore estimating (<ref>) from above by the sum of (<ref>) and (<ref>) is justified, moreover (<ref>) can be estimated from above by (<ref>) and hence by (<ref>). We have obtained the inequality
sup_k∈2^ks_n[ρ_1(2^m_0-kξ)_n f]_L^p(^n)

≤   C_1'2^Jn(1/r-1)sup_t>0t^-s^L_t,1f_L^p(^n)+

Inequality (<ref>) is also true if we replace $\rho_1,\varDelta^L_{t,1},C_1',C_1''$ by $\rho_j,\varDelta^L_{t,j},C_j',C_j''$ respectively for $j=2,\cdots,n$. Inserting these inequalities into (<ref>) yields

\begin{equation}\label{eq671}
\|f\|_{\Bspinf}\leq C'2^{Jn(\frac{1}{r}-1)}\sum_{j=1}^n\sup_{t>0}t^{-s}\|\varDelta^L_{t,j}f\|_{L^p(\bbbr^n)}+
\end{equation}

The assumption $\sigma_p<s<\infty$ allows $n(\frac{1}{r}-1)-s<0$ when $r$ is sufficiently close to $p$. Thus when $J$ is a sufficiently large positive integer, the coefficient $C''2^{J[n(\frac{1}{r}-1)-s]}$ is less than $\frac{1}{2}$, and we can shift the second term on the right side of (<ref>) to its left side and then the desired inequality (<ref>) is proved. Now the proof of Theorem <ref> is complete.

§ BESOV-LIPSCHITZ SPACE AND ITERATED DIFFERENCES

PROOF OF THEOREM <REF>

We first prove Theorem <ref> (i). We continue using the notation $A_k=\{h\in\bbbr^n:2^{-k}\leq|h|<2^{1-k}\}$ for $k\in\bbbz$ and thus $\bbbr^n\setminus\{0\}=\bigcup_{k\in\bbbz}A_k$. We also pick the number $r$ in the definition of the Peetre-Fefferman-Stein maximal function so that $0<r<p$. We begin with estimating $(\int_{\bbbr^n}|h|^{-sq}\|
\sum_{j\in\bbbz}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}^q\frac{dh}{|h|^n})^{\frac{1}{q}}$ from above by the following

\begin{equation}\label{eq704}
(\sum_{k\in\bbbz}2^{k(sq+n)}\int_{A_k}\|\sum_{j\leq k}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}^q dh)^{\frac{1}{q}}+
(\sum_{k\in\bbbz}2^{k(sq+n)}\int_{A_k}\|\sum_{j>k}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}^q dh)^{\frac{1}{q}}.
\end{equation}

We can pick $0<\varepsilon<\min\{s,L-s\}$ and use the same calculation method as in (<ref>), (<ref>), (<ref>), (<ref>) to obtain
∑_j≤k|^L_h f_j|_L^p(^n)^q   ≲
2^kqε∑_j≤k2^-jqε^L_h f_j_L^p(^n)^q,

∑_j>k|^L_h f_j|_L^p(^n)^q   ≲
2^-kqε∑_j>k2^jqε^L_h f_j_L^p(^n)^q.

And (<ref>), (<ref>) are true for $0<p\leq\infty$, $0<q<\infty$. To estimate the first term in (<ref>), we use (<ref>), Remark <ref> and the mapping property of Hardy-Littlewood maximal function for $\|\cdot\|_{L^{\frac{p}{r}}(\bbbr^n)}$-norm to obtain

\begin{equation}\label{eq707}
\|\varDelta^L_h f_j\|_{L^p(\bbbr^n)}\lesssim 2^{(j-k)L}(1+2^{j-k})^{\frac{n}{r}}\|\PFSmax_n f_j\|_{L^p(\bbbr^n)}
\lesssim 2^{(j-k)L}\|f_j\|_{L^p(\bbbr^n)},
\end{equation}

for $j\leq k$, $|h|\lesssim 2^{-k}$ and constants are independent of $h\in\bbbr^n$ and $j,k\in\bbbz$. We put (<ref>) and (<ref>) into the first term of (<ref>) then we have
(∑_k∈2^k(sq+n)∫_A_k∑_j≤k|^L_h f_j|_L^p(^n)^q dh)^1/q

≲   (∑_k∈∑_j≤k2^kq(s+ε-L)·2^jq(L-ε)f_j_L^p(^n)^q)^1/q

≲   (∑_j∈∑_k≥j2^kq(s+ε-L)·2^jq(L-ε)f_j_L^p(^n)^q)^1/q≲f_.

To estimate the second term of (<ref>), we can use (<ref>) and proper change of variable to obtain

\begin{equation}\label{eq708}
\|\varDelta^L_h f_j\|_{L^p(\bbbr^n)}\lesssim\|f_j\|_{L^p(\bbbr^n)}\qquad\text{ for all }j\in\bbbz,
\end{equation}

and the constant is independent of $j\in\bbbz$ and $h\in\bbbr^n$. We put (<ref>) and (<ref>) into the second term of (<ref>) then we have
(∑_k∈2^k(sq+n)∫_A_k∑_j>k|^L_h f_j|_L^p(^n)^q dh)^1/q

≲   (∑_k∈∑_j>k2^kq(s-ε)·2^jqεf_j_L^p(^n)^q)^1/q

≲   (∑_j∈∑_k<j2^kq(s-ε)·2^jqεf_j_L^p(^n)^q)^1/q

Combining (<ref>), (<ref>) and (<ref>), we have proven

\begin{equation}\label{eq711}
(\int_{\bbbr^n}|h|^{-sq}\|\sum_{j\in\bbbz}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}^q\frac{dh}{|h|^n})^{\frac{1}{q}}
\lesssim\|f\|_{\Bspq}.
\end{equation}

The assumption $f\in\Bspq$ implies $\sum_{j\in\bbbz}|\varDelta^L_h f_j(x)|<\infty$ for almost every $h\in\bbbr^n$ and $x\in\bbbr^n$. In conjunction with (<ref>), we have shown

\begin{equation}\label{eq712}
\varDelta^L_h f=\sum_{j\in\bbbz}\varDelta^L_h f_j(x)
\end{equation}

in the sense of $\Sw'(\bbbr^n)/\mathscr{P}(\bbbr^n)$ for almost every $h\in\bbbr^n$ and $x\in\bbbr^n$ when $0<p\leq\infty$, $0<q<\infty$ and $0<s<L$. Therefore we obtain

\begin{equation}\label{eq713}
(\int_{\bbbr^n}|h|^{-sq}\|\varDelta^L_h f\|_{L^p(\bbbr^n)}^q\frac{dh}{|h|^n})^{\frac{1}{q}}\lesssim
(\int_{\bbbr^n}|h|^{-sq}\|\sum_{j\in\bbbz}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}^q\frac{dh}{|h|^n})^{\frac{1}{q}}.
\end{equation}

Inequalities (<ref>) and (<ref>) conclude the proof of Theorem <ref> (i).

To prove Theorem <ref> (ii), we assume the right side of inequality (<ref>) is finite otherwise the inequality is trivial. We recall that $spt.\FT_n\psi\subseteq A'=\{\xi\in\bbbr^n:\frac{1}{2}\leq|\xi|<2\}$ and use the positive integer $m_0$ satisfying (<ref>). We also continue using the spherical caps $\{C_l\}_{l=1}^M$ constructed right after (<ref>), the corresponding sets $\{P_l\}_{l=1}^M$ given in (<ref>), (<ref>), (<ref>), and the associated smooth partition of unity $\{\rho_l\}_{l=1}^M$ satisfying (<ref>). We have the apparent estimate

\begin{equation}\label{eq714}
\|f\|_{\Bspq}=\|\{2^{k-m_0}f_{k-m_0}\}_{k\in\bbbz}\|_{l^q(L^p)}\lesssim
\end{equation}

When $1<p\leq\infty$, we can obtain from (<ref>) the following

\begin{equation}\label{eq715}
\psi_{2^{m_0-k}}*f(x)=\sum_{l=1}^M\iFT_n[\FT_n\psi(2^{m_0-k}\xi)\rho_l(2^{m_0-k}\xi)\FT_n f](x).
\end{equation}

For each $l\in\{1,\cdots,M\}$, $\theta\in C_l$ and $1\leq\tau\leq 2$, we can infer from (<ref>), (<ref>) and (<ref>) the following
_n[_nψ(2^m_0-kξ)ρ_l(2^m_0-kξ)_n f](x)

=   _n[_nψ(2^m_0-kξ)ρ_l(2^m_0-kξ)/(e^2πi·2^-kτθ·ξ-1)^L·(e^2πi·2^-kτθ·ξ-1)^L_n f](x)

=   _nλ_l,τθ(2^-kξ)(·)*(^L_2^-kτθf)(x).

Due to Lemma <ref> and hence (<ref>), we have

≲   ∫_2^k|y|<12^kn|_nλ_l,τθ(2^ky)|·|(^L_2^-kτθf)(x-y)|dy

≲   _2^k|y|<1|(^L_2^-kτθf)(x-y)|dy+∑_l=0^∞2^l(n-N)

≲   (1+∑_l=0^∞2^l(n-N))_n(|^L_2^-kτθf|)(x)≲_n(|^L_2^-kτθf|)(x),

if in the last step above we pick $N>n$. Since $1<p\leq\infty$, we invoke the mapping property of Hardy-Littlewood maximal function and obtain for $0<q<\infty$,
≲   ∑_l=1^M_[1,2]_C_l_n[_nψ(2^m_0-kξ)ρ_l(2^m_0-kξ)_n f]_L^p(^n)^q

≲   ∑_l=1^M∫_1^2∫_C_l_nλ_l,τθ(2^-kξ)(·)*(^L_2^-kτθf)
_L^p(^n)^q d^n-1(θ)dτ/τ

≲   ∫_1^2∫_^L_2^-kτθf_L^p(^n)^q d^n-1(θ)dτ/τ

≲   ∫_A_k^L_h f_L^p(^n)^qdh/|h|^n,

for every $k\in\bbbz$. Inserting (<ref>) into (<ref>) proves (<ref>) when $1<p\leq\infty$, $0<q<\infty$ and $s\in\bbbr$. When $0<p<1$, $0<q<\infty$ and $\sigma_p<s<\infty$, we use the function $\phi$ satisfying conditions (<ref>), (<ref>), (<ref>), and $J>m_0$ is a large positive integer whose value will be determined later. Then we have

\begin{equation}\label{eq719}
\psi_{2^{m_0-k}}*f(x)=\sum_{l=1}^M\iFT_n[\FT_n\psi(2^{m_0-k}\xi)\rho_l(2^{m_0-k}\xi)
\FT_n\phi(2^{m_0-J-k}\xi)\FT_n f](x).
\end{equation}

Furthermore for each $l\in\{1,\cdots,M\}$, $\theta\in C_l$ and $1\leq\tau\leq 2$, we can infer from (<ref>), (<ref>) and (<ref>) the following
_n[_nψ(2^m_0-kξ)ρ_l(2^m_0-kξ)_nϕ(2^m_0-J-kξ)_n f](x)

=   _n[_nψ(2^m_0-kξ)ρ_l(2^m_0-kξ)/(e^2πi·2^-kτθ·ξ-1)^L·(e^2πi·2^-kτθ·ξ-1)^L_nϕ(2^m_0-J-kξ)_n f](x)

=   _nλ_l,τθ(2^-kξ)(·)*(^L_2^-kτθ(ϕ_2^m_0-J-k*f))(x).

The Fourier transform of the Schwartz function

$$y\mapsto 2^{kn}\iFT_n\lambda_{l,\tau\theta}(2^{k}y)\cdot\varDelta^L_{2^{-k}\tau\theta}(\phi_{2^{m_0-J-k}}*f)(x-y)$$

is compactly supported in a ball centered at origin of radius about $2^{k+J-m_0}$ in $\bbbr^n$, therefore by invoking Remark <ref> or the more general Lemma <ref> of Plancherel-Polya-Nikol'skij inequality, we can estimate (<ref>) from above by

≲   2^(k+J-m_0)n(1/r-1)2^kn_nλ_l,τθ(2^k·)·^L_2^-kτθ(ϕ_2^m_0-J-k*f)(x-·)_L^r(^n),

for $0<r<p<1$. We insert (<ref>) with $n-Nr<0$ into (<ref>) and use the following inequality

\begin{equation}\label{eq722}
\mvint_{A_{k-m}}|\varDelta^L_{2^{-k}\tau\theta}(\phi_{2^{m_0-k-J}}*f)(x-y)|^r dy\lesssim
\HLmax_n(|\varDelta^L_{2^{-k}\tau\theta}(\phi_{2^{m_0-k-J}}*f)|^r)(x),
\end{equation}

and then combine the result with (<ref>) to obtain
|_n[_nψ(2^m_0-kξ)ρ_l(2^m_0-kξ)_nϕ(2^m_0-J-kξ)_n f](x)|

≲   2^Jn(1/r-1)_n(|^L_2^-kτθ(ϕ_2^m_0-k-J*f)|^r)(x)^1/r,

for every $l\in\{1,\cdots,M\}$, $\theta\in C_l$ and $\tau\in[1,2]$. Then we use (<ref>), (<ref>), the calculation method displayed in (<ref>) and the mapping property of Hardy-Littlewood maximal function for $\|\cdot\|_{L^{\frac{p}{r}}(\bbbr^n)}$-norm to obtain
≲   2^Jnq(1/r-1)∑_l=1^M∫_1^2∫_C_l

≲   2^Jnq(1/r-1)∫_1^2∫_

≲   2^Jnq(1/r-1)∫_A_k^L_h(ϕ_2^m_0-k-J*f)_L^p(^n)^qdh/|h|^n,

and (<ref>) is true for $0<q<\infty$. We insert (<ref>) into (<ref>), recall (<ref>), (<ref>) and (<ref>), and we also assume the validity of decomposition for now, then $\|f\|_{\Bspq}$ can be estimated from above by the sum of the following two terms,

\begin{equation}\label{eq725}
2^{Jn(\frac{1}{r}-1)}(\sum_{k\in\bbbz}2^{ksq}\!\!\int_{A_k}\!\!\!\!\|\varDelta^L_h f\|_{L^p(\bbbr^n)}^q\frac{dh}{|h|^n})^{\frac{1}{q}}\sim
2^{Jn(\frac{1}{r}-1)}(\int_{\bbbr^n}\!\!\!\!|h|^{-sq}\|\varDelta^L_h f\|_{L^p(\bbbr^n)}^q\frac{dh}{|h|^n})^{\frac{1}{q}}
\end{equation}

\begin{equation}\label{eq726}
\|_{L^p(\bbbr^n)}^q\frac{dh}{|h|^n})^{\frac{1}{q}}.
\end{equation}

To estimate (<ref>), we begin with
2^Jn(1/r-1)(∑_k∈2^ksq∫_A_k∑_j=J+1^∞|^L_h f_k+j-m_0|

≲   2^Jn(1/r-1)(∑_k∈2^k(sq+n)∫_A_k
(∑_j=J+1^∞^L_h f_k+j-m_0_L^p(^n)^p)^q/pdh)^1/q,

since $0<p<1$. When $0<q\leq p$, we use (<ref>) and the following inequality

\begin{equation}\label{eq728}
(\sum_{j=J+1}^{\infty}\|\varDelta^L_h f_{k+j-m_0}\|_{L^p(\bbbr^n)}^p)^{\frac{q}{p}}\lesssim
\sum_{j=J+1}^{\infty}\|\varDelta^L_h f_{k+j-m_0}\|_{L^p(\bbbr^n)}^q\lesssim
\sum_{j=J+1}^{\infty}\|f_{k+j-m_0}\|_{L^p(\bbbr^n)}^q,
\end{equation}

where the constants are independent of $j,k\in\bbbz$ and $h\in\bbbr^n$. When $p<q<\infty$, we pick $0<\varepsilon<s$, use (<ref>) and the following inequality
(∑_j=J+1^∞2^-jpε·2^jpε^L_h f_k+j-m_0_L^p(^n)^p)^q/p

≲   (∑_l=J+1^∞2^-lpε)^q/psup_j>J
2^jqε^L_h f_k+j-m_0_L^p(^n)^q

≲   2^-Jqε∑_j=J+1^∞2^jqεf_k+j-m_0_L^p(^n)^q,

where the constants are independent of $j,k\in\bbbz$ and $h\in\bbbr^n$. Insert (<ref>) and (<ref>) into (<ref>) and exchange the order of summation, and then we can estimate (<ref>) from above by $2^{J[n(\frac{1}{r}-1)-s]}\|f\|_{\Bspq}$. The assumption $f\in\Bspq$ implies $\sum_{j=J+1}^{\infty}|\varDelta^L_h f_{k+j-m_0}(x)|<\infty$ for every $k\in\bbbz$ and almost every $h\in A_k,x\in\bbbr^n$. In conjunction with (<ref>), (<ref>) and (<ref>), we have shown

\begin{equation}\label{eq730}
\varDelta^L_{h}(\phi_{2^{m_0-k-J}}*f)(x)=\varDelta^L_{h}f(x)-\sum_{j=J+1}^{\infty}\varDelta^L_{h}f_{k+j-m_0}(x)
\end{equation}

is true not only in the sense of $\Sw'(\bbbr^n)$ but also for every $k\in\bbbz$ and almost every $h\in A_k,x\in\bbbr^n$ when $0<p<1$, $0<q<\infty$ and $\sigma_p<s<\infty$. Furthermore estimating $\|f\|_{\Bspq}$ from above by the sum of (<ref>) and (<ref>) is justified, moreover (<ref>) can be estimated from above by (<ref>) and hence by $2^{J[n(\frac{1}{r}-1)-s]}\|f\|_{\Bspq}$. We have obtained the inequality

\begin{equation}\label{eq731}
\|f\|_{\Bspq}\leq
C'2^{Jn(\frac{1}{r}-1)}(\int_{\bbbr^n}\!\!\!\!|h|^{-sq}\|\varDelta^L_h f\|_{L^p(\bbbr^n)}^q\frac{dh}{|h|^n})^{\frac{1}{q}}+
\end{equation}

The condition $\sigma_p<s<\infty$ implies $n(\frac{1}{r}-1)-s<0$ when $r$ is sufficiently close to $p$. Hence when the positive integer $J$ is sufficiently large, the coefficient $C''2^{J[n(\frac{1}{r}-1)-s]}$ is less than $\frac{1}{2}$ and then we can shift the second term on the right side of (<ref>) to its left side and prove the inequality (<ref>) when $0<p<1$, $0<q<\infty$ and $\sigma_p<s<\infty$. The proof for Theorem <ref> (ii) is now concluded.

Next we prove Theorem <ref> (iii). We pick $0<\varepsilon<\min\{s,L-s\}$ and begin with estimating $\sup_{h\in\bbbr^n}|h|^{-s}\|\sum_{j\in\bbbz}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}$ from above by

\begin{equation}\label{eq732}
\sup_{k\in\bbbz}\sup_{h\in A_k}2^{ks}\|\sum_{j\leq k}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}+
\sup_{k\in\bbbz}\sup_{h\in A_k}2^{ks}\|\sum_{j>k}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}.
\end{equation}

If $1\leq p\leq\infty$, by using Minkowski's inequality, (<ref>) and the calculation method displayed in (<ref>), we can estimate the first term in (<ref>) from above by $\|f\|_{\Bspinf}$. And by using Minkowski's inequality, (<ref>) and the calculation method displayed in (<ref>), we can estimate the second term in (<ref>) from above by $\|f\|_{\Bspinf}$. If $0<p<1$, by applying (<ref>) and the calculation method given in (<ref>) to the first term of (<ref>), and by applying (<ref>) and the calculation method given in (<ref>) to the second term of (<ref>), we can still estimate (<ref>) from above by $\|f\|_{\Bspinf}$. Thus we can obtain the inequality

\begin{equation}\label{eq733}
\sup_{h\in\bbbr^n}|h|^{-s}\|\sum_{j\in\bbbz}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}\lesssim\|f\|_{\Bspinf}.
\end{equation}

The assumption $f\in\Bspinf$ shows $\sum_{j\in\bbbz}|\varDelta^L_h f_j(x)|<\infty$ for almost every $h\in\bbbr^n,x\in\bbbr^n$. In conjunction with (<ref>), we have shown (<ref>) is true not only in the sense of $\Sw'(\bbbr^n)/\mathscr{P}(\bbbr^n)$ but also for almost every $h\in\bbbr^n,x\in\bbbr^n$ when $0<p\leq\infty$, $q=\infty$ and $0<s<L$. Therefore we have

\begin{equation}\label{eq734}
\sup_{h\in\bbbr^n}|h|^{-s}\|\varDelta^L_h f\|_{L^p(\bbbr^n)}\lesssim
\sup_{h\in\bbbr^n}|h|^{-s}\|\sum_{j\in\bbbz}|\varDelta^L_h f_j|\|_{L^p(\bbbr^n)}.
\end{equation}

Inequalities (<ref>) and (<ref>) conclude the proof of Theorem <ref> (iii).

Now we prove Theorem <ref> (iv). We assume the right side of (<ref>) is finite, otherwise the inequality is trivial. We still use the positive integer $m_0$ satisfying (<ref>), the spherical caps $\{C_l\}_{l=1}^M$ constructed right after (<ref>), the corresponding sets $\{P_l\}_{l=1}^M$ given in (<ref>), (<ref>), (<ref>), and the associated smooth partition of unity $\{\rho_l\}_{l=1}^M$ satisfying (<ref>). When $1<p\leq\infty$, we use (<ref>), (<ref>) and (<ref>) to obtain
≲   ∑_l=1^Msup_τ∈[1,2]sup_θ∈C_l
_n[_nψ(2^m_0-kξ)ρ_l(2^m_0-kξ)_n f]_L^p(^n)

≲   ∑_l=1^Msup_τ∈[1,2]sup_θ∈C_l

≲   ∑_l=1^Msup_τ∈[1,2]sup_θ∈C_l

≲   sup_τ∈[1,2]sup_θ∈^L_2^-kτθf_L^p(^n)
=sup_h∈A_k^L_h f_L^p(^n).

Therefore we have the inequality

\begin{equation}\label{eq736}
\|f\|_{\Bspinf}\lesssim\sup_{k\in\bbbz}2^{ks}\|\psi_{2^{m_0-k}}*f\|_{L^p(\bbbr^n)}\lesssim
\sup_{h\in\bbbr^n}|h|^{-s}\|\varDelta^L_h f\|_{L^p(\bbbr^n)},
\end{equation}

and inequality (<ref>) is true for $1<p\leq\infty$, $q=\infty$ and $s\in\bbbr$. When $0<p<1$, we also use the function $\phi$ satisfying conditions (<ref>), (<ref>), (<ref>), and $J>m_0$ is a large positive integer whose value will be determined later. Then we can use (<ref>) and (<ref>) with $0<r<p<1$ to obtain
≲   ∑_l=1^Msup_τ∈[1,2]sup_θ∈C_l
_n[_nψ(2^m_0-kξ)ρ_l(2^m_0-kξ)_nϕ(2^m_0-J-kξ)_n f]_L^p(^n)

≲   2^Jn(1/r-1)∑_l=1^Msup_τ∈[1,2]sup_θ∈C_l

≲   2^Jn(1/r-1)sup_τ∈[1,2]sup_θ∈

=   2^Jn(1/r-1)sup_h∈A_k^L_h(ϕ_2^m_0-k-J*f)_L^p(^n).

Insert (<ref>) into $\|f\|_{\Bspinf}$, recall (<ref>) and assume the validity of decomposition for now, then we can estimate $\|f\|_{\Bspinf}$ from above by the sum of the following two terms,

\begin{equation}\label{eq738}
2^{Jn(\frac{1}{r}-1)}\sup_{h\in\bbbr^n}|h|^{-s}\|\varDelta^L_h f\|_{L^p(\bbbr^n)}<\infty
\end{equation}

\begin{equation}\label{eq739}
2^{Jn(\frac{1}{r}-1)}\sup_{k\in\bbbz}\sup_{h\in A_k}2^{ks}
\|\varDelta^L_h(\sum_{j=J+1}^{\infty}f_{k+j-m_0})\|_{L^p(\bbbr^n)}.
\end{equation}

To estimate (<ref>), we use (<ref>) and begin with the following
∑_j=J+1^∞|^L_h f_k+j-m_0|_L^p(^n)

≲   2^Jn(1/r-1)sup_k∈sup_h∈A_k2^ks
(∑_j=J+1^∞^L_h f_k+j-m_0_L^p(^n)^p)^1/p

≲   2^Jn(1/r-1)sup_k∈

≲   2^J[n(1/r-1)-s]f_.

The assumption $f\in\Bspinf$ implies $\sum_{j=J+1}^{\infty}|\varDelta^L_h f_{k+j-m_0}(x)|<\infty$ for every $k\in\bbbz$ and almost every $h\in A_k,x\in\bbbr^n$. In conjunction with (<ref>), (<ref>) and (<ref>), we have shown (<ref>) is true not only in the sense of $\Sw'(\bbbr^n)$ but also for every $k\in\bbbz$ and almost every $h\in A_k,x\in\bbbr^n$ when $0<p<1$, $q=\infty$ and $\sigma_p<s<\infty$. Furthermore estimating $\|f\|_{\Bspinf}$ from above by the sum of (<ref>) and (<ref>) is justified, moreover (<ref>) can be estimated from above by (<ref>) and hence by (<ref>). We have obtained the inequality

\begin{equation}\label{eq742}
\|f\|_{\Bspinf}\leq C'2^{Jn(\frac{1}{r}-1)}\sup_{h\in\bbbr^n}|h|^{-s}\|\varDelta^L_h f\|_{L^p(\bbbr^n)}+
\end{equation}

The condition $\sigma_p<s<\infty$ indicates $n(\frac{1}{r}-1)-s<0$ when $r$ is sufficiently close to $p$. Thus the coefficient $C''2^{J[n(\frac{1}{r}-1)-s]}$ is less than $\frac{1}{2}$ when the positive integer $J$ is sufficiently large, and then we can shift the second term on the right side of (<ref>) to its left side and prove the desired inequality (<ref>). The proof of Theorem <ref> (iv) is now complete.

Finally we come to the proof of Theorem <ref> (v). By using a different method, some optimal conditions can be obtained in the case $p=1$. We assume the right sides of (<ref>) and (<ref>) are finite. If $1\leq q\leq\infty$ and $-n<s<\infty$, we use the radial Schwartz function $g$ satisfying (<ref>), (<ref>), (<ref>), and the kernel $G_k(\cdot)=\sum^L_{j=1}d_j g_{2^{-k}j}(\cdot)$ satisfying (<ref>), (<ref>), (<ref>) with $0<r<p=1$. Then from (<ref>), (<ref>) and (<ref>), we deduce
≲   ∑_l≥02^-l(n+s)·2^(k+l)s∫_A_0^L_2^-k-lhf_L^1(^n)dh

When $1\leq q<\infty$, we use Minkowski's inequality for $\|\cdot\|_{l^q}$-norm, Hölder's inequality and compute as in (<ref>) and (<ref>) with $N'>n+s>0$ then we can obtain

≲   (∑_k∈2^ksq∫_A_0^L_2^-khf_L^1(^n)^q dh)^1/q

≲   (∫_^n|h|^-sq^L_hf_L^1(^n)^qdh/|h|^n)^1/q.

When $q=\infty$, we use Minkowski's inequality for $\|\cdot\|_{l^{\infty}}$-norm and the inequality (<ref>) with $N'>n+s>0$ to obtain

≲   sup_k∈2^ks∫_A_0^L_2^-khf_L^1(^n)dh

≲   sup_k∈2^kssup_h∈A_0^L_2^-khf_L^1(^n)

≲   sup_h∈^n|h|^-s^L_hf_L^1(^n).

Indicated by (<ref>), (<ref>), (<ref>) and (<ref>), we know that the integral on the left end of (<ref>) is absolutely convergent and hence well-defined for every $k\in\bbbz$ and almost every $x\in\bbbr^n$. By (<ref>), (<ref>), Remark <ref> and the mapping property of Hardy-Littlewood maximal function, we have $\|f\|_{\dot{B}^s_{1,q}(\bbbr^n)}\lesssim\|\{2^{ks}G_k*f\}_{k\in\bbbz}\|_{l^q(L^1)}$ for $1\leq q\leq\infty$ and any $s\in\bbbr$. In conjunction with (<ref>) and (<ref>), we have proven the inequality (<ref>) and the inequality (<ref>) when $p=1$, $1\leq q<\infty$ and $-n<s<\infty$. To prove (<ref>) is true for $p=1$, $0<q<1$ and $0<s<\infty$, we notice that by picking $0<r<p=1$, the method given for the proof of the second part of Theorem <ref> (ii) still applies and $\sigma_1=0$. The proof of Theorem <ref> is now complete.

[1]
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci.
Hitchhiker's guide to the fractional sobolev spaces.
Bulletin des sciences mathématiques, 136(5):521–573, 2012.

[2]
Loukas Grafakos.
Classical Fourier analysis, volume 249 of Graduate Texts
in Mathematics.
Springer, New York, third edition, 2014.

[3]
Loukas Grafakos.
Modern Fourier analysis, volume 250 of Graduate Texts in
Springer, New York, third edition, 2014.

[4]
Martí Prats.
Measuring triebel-lizorkin fractional smoothness on domains in terms
of first-order differences.
Journal of the London Mathematical Society, 100(2):692–716,

[5]
Andreas Seeger.
A note on triebel-lizorkin spaces.
Banach Center Publications, 22(1):391–400, 1989.

[6]
E. M. Stein.
The characterization of functions arising as potentials.
Bull. Amer. Math. Soc., 67(1):102–104, 01 1961.

[7]
H. Triebel.
Theory of Function Spaces.
Monographs in Mathematics. Springer Basel, 1983.

[8]
H. Triebel.
Theory of Function Spaces II.
Monographs in Mathematics. Springer Basel, 1992.


# Quantifier elimination in II1 factors

Ilijas Farah  Department of Mathematics and Statistics, York University, 4700
Keele Street, Toronto, Ontario, Canada, M3J 1P3  Matematički Institut SANU,
Kneza Mihaila 36, 11 000 Beograd, p.p. 367, Serbia  email<EMAIL_ADDRESS>http://www.math.yorku.ca/$\sim$ifarah To the memory of Eberhard Kirchberg.

###### Abstract.

No type II1 tracial von Neumann algebra has theory that admits quantifier
elimination.

Partially supported by NSERC

ORCID iD https://orcid.org/0000-0001-7703-6931

Model theory is largely the study of definable sets, more precisely sets
definable by first order (continuous or discrete) formulas. If the theory of a
structure admits elimination of quantifiers, then its definable subsets are
definable by quantifier-free formulas and therefore easier to grasp. This is
the case with atomless Boolean algebras, dense linear orderings without
endpoints, real closed fields, divisible abelian groups…See for example [37].
Quantifier elimination is equivalent to an assertion about embeddings between
finitely-generated substructures of an ultrapower that resembles the well-
known fact that all embeddings of the hyperfinite II1 factor into its
ultrapower are unitarily equivalent (Proposition 1.2, Proposition 3.2). Our
motivation for studying quantifier elimination in tracial von Neumann stems
from Jekel’s work on 1-bounded entropy in $\mathrm{W}^{*}$-algebras defined on
types ([31], [30]). Our main result implies that 1-bounded entropy is a
genuine generalization of Hayes’s 1-bounded entropy (see [27], [28]). This is
because the latter is defined for quantifier-free types, while the former is
defined for full types.

In [24, Theorem 2.1] it was proven that the theory of the hyperfinite II1
factor does not admit quantifier elimination (see §1) and that if $N$ is a II1
McDuff factor such that every separable II1 factor is
$N^{\mathcal{U}}$-embeddable111All ultraproducts in this paper are associated
to nonprincipal ultrafilters on ${\mathbb{N}}$ and almost all of them are
tracial. then the theory of $N$ does not admit elimination of quantifiers
([24, Theorem 2.2]).

In spite of the slow start, the question of quantifier elimination in
$\mathrm{C}^{*}$-algebras has been answered completely. In [17, Theorem 1.1]
it was proven that the only $\mathrm{C}^{*}$-algebras whose theories in the
language of unital $\mathrm{C}^{*}$-algebras admit quantifier elimination are
(with $K$ denoting the Cantor space) ${\mathbb{C}}$, ${\mathbb{C}}^{2}$,
$M_{2}({\mathbb{C}})$ and $C(K)$, and that the only $\mathrm{C}^{*}$-algebras
whose theories in the language without a symbol for a unit admit quantifier
elimination are ${\mathbb{C}}$ and $C_{0}(K\setminus\\{0\\})$.

The key component of [17] is an observation due to Eberhard Kirchberg,222This
is the paragraph with norm ultraproducts. that there are two very different
embeddings of $\mathrm{C}^{*}_{r}(F_{2})$ into the ultrapower of the Cuntz
algebra $\mathcal{O}_{2}$. First, by [36], every exact
$\mathrm{C}^{*}$-algebra embeds into $\mathcal{O}_{2}$ and diagonally into its
ultrapower. Second, by [25], $\mathrm{C}^{*}_{r}(F_{2})$ embeds into
$\prod_{\mathcal{U}}M_{n}({\mathbb{C}})$, giving a nontrivial embedding into
the ultrapower of $\mathcal{O}_{2}$.

Later on, in [23], paragraph preceding Lemma 5.2, it was pointed out without a
proof that the argument from [24] implies that no McDuff II1 factor admits
elimination of quantifiers. Also, in [22, Proposition 4.17] it was proven that
if $N$ is a II1 factor such that $N$ and $M_{2}(N)$ have the same universal
theory and $N$ is existentially closed for its univeral theory, then $N$ is
McDuff. This implies that if $N$ is not McDuff and $M_{2}(N)$ embeds unitally
into $N^{\mathcal{U}}$, then $N$ does not admit elimination of quantifiers.

During a reading seminar on model theory and free probability based on [31] at
York University in the fall semester of 2022, the author (unaware of the
recent developments described in the last paragraph) rediscovered an easy
argument that the theory of a non-McDuff factor $N$ such that $M_{2}(N)$
embeds unitally into $N^{\mathcal{U}}$, does not admit quantifier elimination
and asserted that closer introspection of the proof of [24, Theorem 2.1] ought
to yield the same result for all II1 factors. This is indeed the case; the
following answers the [31, Question 2.18] in case of tracial von Neumann
algebras of type II1.

######  Theorem 1.

If $N$ is a tracial von Neumann algebra with a direct summand of type II1,
then the theory of $N$ does not admit elimination of quantifiers.

The key lemma is proven in §2 and the proof of Theorem 1 can be found at the
end of this section. The well-known criterion for quantifier elimination is
proven in §3 (in spite of it being well-known, a self-contained proof of this
fact was not available in the literature until it appeared in [26, Proposition
9.4]). In §4 we state conjectures on when the theory of a tracial von Neumann
algebra is model-completene and when it admits quantifier elimination.

Our terminology is standard. For model theory see [9], [26], for general
operator algebras [12], for II1 factors [1], and for model theory of tracial
von Neumann algebras [31] and [23].

For simplicity of notation, every tracial state is denoted $\tau$, except
those on $M_{n}({\mathbb{C}})$, denoted ${\rm tr}_{n}$.

A tracial von Neumann algebra is _$R^{\mathcal{U}}$ -embeddable_ if it embeds
into some (equivalently, any) ultrapower of the hyperfinite II1 factor $R$
associated with a nonprincipal ultrafilter on ${\mathbb{N}}$. By [32], not all
tracial von Neumann algebras with separable dual are
$R^{\mathcal{U}}$-embeddable.

### A personal note

Two personal notes are in order. First, at a dinner in Oberwolfach I pointed
out, politely and enthusiastically, that much of [35] can be construed as
model theory. Let’s just say that Eberhard made it clear (politely) that he
did not share my enthusiasm. A couple of years later we collaborated on a
model theory paper ([17]). The present note does for type II1 factors what
[17] did for $\mathrm{C}^{*}$-algebras. Second, I wish that I recorded all
conversations that I had with Eberhard. It took me weeks to process some of
the enlightening raw information that he dumped on me, parts of which may be
lost for posterity. He will be missed.

### Acknowledgments

I am indebted to the participants of the seminar on Model Theory and Free
Probability held at York University in the fall semester 2022, and in
particular to Saeed Ghasemi, Pavlos Motakis, and Paul Skoufranis for some
insightful remarks. I would also like to thank Srivatsav Kunnawalkam
Elayavalli for informing me that he proved Lemma 2.1 independently in February
2023 and for useful remarks on an early draft of this paper, and to Isaac
Goldbring for remarks on the first version of this paper and to the referee
for a delightfully opinionated (and very useful) report.

##  1\. Quantifier elimination in II1 factors

We specialize the definitions from [21, §2.6] to the case of tracial von
Neumann algebras. The ongoing discussion applies to any other countable
continuous language ${\mathcal{L}}$ and, with obvious modifications, to any
continuous language ${\mathcal{L}}$. For simplicity, we consider only formulas
in which all free variables range over the (operator) unit ball. This is not a
loss of generality, as an easy rescaling argument shows.

For $n\geq 0$ and an $n$-tuple of variables $\bar{x}=\langle
x_{0},\dots,x_{n-1}\rangle$ (if $n=0$ this is the empty sequence) let
$\operatorname{\mathfrak{F}^{\bar{x}}}$ be the $\mathbb{R}$-algebra of
formulas in the language of tracial von Neumann algebras with free variables
included in $\bar{x}$. For a fixed tracial von Neumann algebra $N$ define a
seminorm $\|\cdot\|_{N}$ on $\operatorname{\mathfrak{F}^{\bar{x}}}$ by
($\varphi^{M}(\bar{a})$ denotes the evaluation of $\varphi(\bar{x})$ at
$\bar{a}$ in $M$)

$\|\varphi(\bar{x})\|_{N}=\sup\varphi^{N^{\mathcal{U}}}(\bar{a})$

where $\bar{a}$ ranges over all $n$-tuples in the unit ball of
$N^{\mathcal{U}}$. (The standard definition takes supremum over all structures
$M$ elementarily equivalent to $N$ and all $n$-tuples in $M$ of the
appropriate sort, but by the universality of ultrapowers and the Downward
Löwenheim–Skolem theorem, the seminorms coincide.)

Let $\operatorname{\mathfrak{F}_{QF}^{\bar{x}}}$ denote the
$\mathbb{R}$-algebra of all quantifier-free formulas in
$\operatorname{\mathfrak{F}^{\bar{x}}}$. This is clearly a subalgebra of
$\operatorname{\mathfrak{F}^{\bar{x}}}$.

If $M$ is a subalgebra of $N$ and $\bar{a}$ in $M$, it is said that $M$ is an
_elementary submodel_ of $N$ if $\varphi^{M}(\bar{a})=\varphi^{N}(\bar{a})$
for all $\varphi\in\operatorname{\mathfrak{F}^{\bar{x}}}$. An isometric
embedding is _elementary_ if its range is an elementary submodel. For
quantifier-free formulas this equality is automatic, but this is a strong
assumption in general. For example, any two elementary embeddings of a
separable structure into an ultrapower333Ultrapower of a separable structure
associated with a nonprincipal ultrafilter on ${\mathbb{N}}$. are conjugate by
an automorphism of the latter if the Continuum Hypothesis holds444Continuum
Hypothesis has little bearing on the relations between separable structures by
Shoenfield’s Absoluteness Theorem; see e.g., [34]. ([18, Corollary 16.6.5]).

######  Definition 1.1.

The theory of a tracial von Neumann algebra $N$ _admits elimination of
quantifiers_ if $\operatorname{\mathfrak{F}_{QF}^{\bar{x}}}$ is
$\|\cdot\|_{N}$-dense in $\operatorname{\mathfrak{F}^{\bar{x}}}$ for every
$\bar{x}$.

Proposition 1.2 below is essentially a special case [9, Proposition 13.6]
stated with a reference to [29, pp. 84–91] in lieu of a proof; see also [31,
Lemma 2.14]. Until recently, a self-contained proof of this fact could not be
found in the literature. This has finally been remedied in [26, Proposition
9.4]. For reader’s convenience I include a proof using ultrapowers instead of
elementary extensions (see Proposition 3.2, the equivalence of (1), (2), and
(3)).

######  Proposition 1.2.

For every tracial von Neumann algebra555Needless to say, the analogous
statement holds for $\mathrm{C}^{*}$-algebras and for any other axiomatizable
category. $N$ the following are equivalent.

  1. (1)

The theory of $N$ admits elimination of quantifiers.

  2. (2)

For every finitely generated $\mathrm{W}^{*}$-subalgebra $G$ of
$N^{{\mathcal{U}}}$, every trace-preserving embedding $\Phi\colon F\to
N^{\mathcal{U}}$ of a finitely generated $\mathrm{W}^{*}$-subalgebra $F$ of
$G$ extends to a trace-preserving embedding $\Psi\colon G\to N^{\mathcal{U}}$.

$G$$N^{\mathcal{U}}$$F$$N^{\mathcal{U}}$$\subseteq$$\subseteq$$\Phi$$\Psi$

  3. (3)

Same as (2), but for arbitrary separable substructures.

Clause (2) resembles the well-known property of the hyperfinite II1 factor
$R$, that any two copies of $R$ in $R^{\mathcal{U}}$ are unitarily conjugate
([33], [14]), the analogous (well-known) fact about strongly self-absorbing
$\mathrm{C}^{*}$-algebras, as well as the defining property of ‘Generalized
Jung factors’ (in this case, automorphisms are not required to be inner, see
[4], also see [3]) but it is strictly stronger since neither $R$ nor any of
the strongly self-absorbing $\mathrm{C}^{*}$-algebras admit quantifier
elimination. The point is that $F$ and $G$ in (2) range over arbitrary
finitely-generated substructures of the ultrapower.

##  2\. Topological dynamical systems associated to II1 factors

Lemma 2.1 below is proven by unravelling of the proof of [14, Corollary 6.11].
Modulo the standard results (Proposition 1.2, Proposition 3.2) it implies
Theorem 1.

######  Lemma 2.1.

There are a II1 factor $M_{1}$ with separable predual, a subfactor $M_{0}$ of
$M_{1}$, and an automorphism $\alpha$ of $M_{1}$ such that

  1. (1)

$\alpha\restriction M_{0}=\operatorname{id}_{M_{0}}$, but
$\alpha\neq\operatorname{id}_{M_{1}}$,

and for every type II1 tracial von Neumann algebra $N$ and every ultrapower
$N^{{\mathcal{U}}}$ the following two conditions hold.

  1. (2)

There is a trace-preserving embedding $\Phi\colon M_{1}\to N^{{\mathcal{U}}}$
such that

$N^{{\mathcal{U}}}\cap\Phi[M_{0}]^{\prime}=N^{{\mathcal{U}}}\cap\Phi[M_{1}]^{\prime}.$

  2. (3)

There is a trace-preserving embedding $\Phi_{1}$ of the crossed product
$M_{1}\rtimes_{\alpha}{\mathbb{Z}}$ into $N^{\mathcal{U}}$, and
$N^{{\mathcal{U}}}\cap\Phi_{1}[M_{0}]^{\prime}\neq
N^{{\mathcal{U}}}\cap\Phi_{1}[M_{1}]^{\prime}$.

Moreover, one can choose $M_{0}=L(\operatorname{SL}(2k+1,{\mathbb{Z}}))$ for
any $k\geq 1$ and (regardless on the choice of $M_{0}$) $M_{1}=M_{0}*P$, for
any $R^{\mathcal{U}}$-embeddable tracial von Neumann algebra $P$ with
separable predual $P$.

By for example taking $P=L^{\infty}([0,1])$, we can take
$M_{0}=L(\operatorname{SL}(3,{\mathbb{Z}})$ and
$M_{1}=L(\operatorname{SL}_{3}({\mathbb{Z}})*{\mathbb{Z}})$.

###### Proof.

Let $\Gamma$ be a property (T) group with infinitely many inequivalent
irreducible representations on finite-dimensional Hilbert spaces. For the
‘moreover’ part, fix $k\geq 1$ and an $R^{\mathcal{U}}$-embeddable tracial von
Neumann algebra $P$ with separable predual, let
$\Gamma=\operatorname{SL}(2k+1,{\mathbb{Z}})$ for any $k\geq 1$, let
$M_{0}=L(\Gamma)$, $M_{1}=L(\Gamma)*P$, and
$\alpha=\operatorname{id}_{M_{0}}*\alpha_{0}$, for some nontrivial
automorphism $\alpha_{0}$ of $L({\mathbb{Z}})$. Thus (1) holds. Since $M_{1}$
is $R^{{\mathcal{U}}}$-embeddable, (3) follows by [2, Proposition 3.4(2)].

It remains to prove (2). For $n\geq 2$ let
$\rho_{n}\colon\Gamma\curvearrowright\ell_{2}(n)$ be an irreducible action of
$\Gamma$ on $\ell_{2}(n)$, if such action exists, and trivial action
otherwise. (The choice of $\rho_{n}$ in the latter case will be completely
irrelevant.) Then $\rho_{n}$ defines a unital *-homomorphism of the group
algebra ${\mathbb{C}}\Gamma$ into $M_{n}({\mathbb{C}})$, also denoted
$\rho_{n}$. Let
$\rho=\bigoplus_{n}\rho_{n}\colon{\mathbb{C}}\Gamma\to\prod_{n}M_{n}({\mathbb{C}})$.

Fix a nonprincipal ultrafilter ${\mathcal{U}}$ on ${\mathbb{N}}$ that
concentrates on the set $\\{n\mid\rho_{n}$ is irreducible$\\}$. If
$q\colon\prod_{n}M_{n}({\mathbb{C}})\to\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$
is the quotient map, then
$q\circ\rho\colon{\mathbb{C}}\Gamma\to\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$
is a unital *-homomorphism. Let $M_{0}$ be the ultraweak closure of
$q\circ\rho[{\mathbb{C}}\Gamma]$. If
$\Gamma=\operatorname{SL}(2k+1,{\mathbb{Z}})$, then [6, Theorem 1] implies
that $M_{0}$ isomorphic to the group factor $L(\Gamma)$; this nice fact
however does not affect the remaining part of our proof.

For $g\in\Gamma$ we have a representation $(u(g)_{n})/{\mathcal{U}}$, where
$u(g)_{n}=\rho_{n}(g)$ is a unitary in $M_{n}({\mathbb{C}})$ for every $n$.
Since $P$ is $R^{\mathcal{U}}$-embeddable, it is also embeddable into
$\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$. Let
$K\subseteq\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$ be a countable set that
that generates an isomorphic copy of $P$.

Our copy of $M_{0}$ in $\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$ and the copy
of $P$ generated by $K$ need not generate an isomorphic copy of
$M_{1}=M_{0}*P$. In order to ‘correct’ this, we invoke some standard results.

At this point we need terminology from free probability. A unitary $u$ in a
tracial von Neumann algebra $(M,\tau)$ is a _Haar unitary_ if $\tau(u^{m})=0$
whenever $m\neq 0$. Such Haar unitary is _free from_ some $X\subseteq M$ if
for every $m\geq 1$, if $a_{j}$ is such that $\tau(a_{j})=0$ and in the linear
span of $X$ and if $k(j)\neq 0$ for $0\leq j<m$, then (note that
$\tau(u^{k(j)})=0$ since $u$ is a Haar unitary)

$\tau(a_{0}u^{k(0)}a_{1}u^{k(1)}\dots a_{m-1}u^{k(m-1)})=0.$

More generally, if $X$ and $Y$ are subsets of $M$ then $X$ is _free from_ $Y$
if for every $m\geq 1$, if $a_{j}$ is such that $\tau(a_{j})=0$ and in the
linear span of $X$ and $b_{j}$ is such that $\tau(b_{j})=0$ and in the linear
span of $Y$ for $j<m$, then

$\tau(a_{0}b_{0}a_{1}b_{1}\dots a_{m-1}b_{m-1})=0.$

Since $\Gamma$ is a Kazhdan group ([7]), we can fix a Kazhdan pair
$F,\varepsilon$ for $\Gamma$. By [39, Theorem 2.2], there is a Haar unitary
$w\in\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$ free from
$X=\\{(u(g)_{n})/{\mathcal{U}}\mid g\in F\\}\cup\\{K\\}$.

A routine calculation shows that $K^{\prime}=\\{waw^{*}\mid a\in K\\}$ is free
from $X$ and that $K^{\prime}$ generates an isomorphic copy of $P$. Therefore
the $\mathrm{W}^{*}$-subalgebra of $\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$
generated by $\\{(u(g)_{n})/{\mathcal{U}}\mid g\in F\\}\cup\\{K^{\prime}\\}$
is isomorphic to $M_{1}\cong M_{0}*P$. This defines an embedding
$\Phi_{0}\colon M_{1}\to\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$.

Fix a type II1 tracial von Neumann algebra $N$. In order to find an embedding
of $M_{1}$ into $N^{\mathcal{U}}$ as required in (2), for every $n\geq 1$
write $N=M_{n}({\mathbb{C}})\bar{\otimes}N^{1/n}$ (where $N^{1/n}$ is the
corner of $N$ associated to a projection whose center-valued trace is $1/n$).
Define an embedding of $\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$ into
$N^{\mathcal{U}}$ by (the far right side is $N^{\mathcal{U}}$ in disguise)

$\textstyle\prod^{{\mathcal{U}}}M_{n}({\mathbb{C}})\ni(a_{n})/{\mathcal{U}}\mapsto(a_{n}\otimes
1_{N^{1/n}})/{\mathcal{U}}\in\prod^{{\mathcal{U}}}(M_{n}({\mathbb{C}})\bar{\otimes}N^{1/n}).$

Let $\Phi\colon M_{1}\to N^{\mathcal{U}}$ be the composition of the embedding
$\Phi_{0}$ of $M_{1}$ into $\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$ with this
embedding.

It remains to prove that
$N^{\mathcal{U}}\cap\Phi[M_{0}]^{\prime}=N^{\mathcal{U}}\cap\Phi[M_{1}]^{\prime}$.
Since $M_{0}\subseteq M_{1}$, only the forward inclusion requires a proof.
Suppose that $b\in N^{\mathcal{U}}\cap\Phi[M_{0}]^{\prime}$ and fix a
representing sequence $(b_{n})$ for $b$. Fix for a moment $\delta>0$. Then the
set ($F,\varepsilon$ is a Kazhdan pair for $\Gamma$)

$Z_{\delta}=\\{n\mid\max_{g\in
F}\|u^{g}_{n}b_{n}-b_{n}u^{g}_{n}\|_{2}<\varepsilon\delta\\}$

belongs to ${\mathcal{U}}$.666The set $Z_{\delta}$ would have belonged to
${\mathcal{U}}$ regardless of whether the maximum had been taken over $F$ or
some other finite subset of $\Gamma$. The maximum has been taken over $F$
because we need to bound the norm of the commutator of $b$ with all elements
of $F$, and not elements of some other finite subset of $\Gamma$.

For each $n$ we define an action $\sigma_{n}\colon\Gamma\curvearrowright
L^{2}(N,\tau)$ as follows. For $f\in\Gamma$ let
$\tilde{\rho}_{n}(f)=\rho_{n}(f)\otimes 1_{N^{1/n}}$. For $c\in N$, let

$f.c=\tilde{\rho}_{n}(f)c\tilde{\rho}_{n}(f)^{*}.$

This gives an action by isometries of $\Gamma$ on $(N,\|\cdot\|_{2,\tau})$,
which continuously extends to action $\sigma_{n}$ of $\Gamma$ on
$L^{2}(N,\tau)$.

For $g\in F$ and $n\in Z_{\delta}$ we have
$\|g.b_{n}-b_{n}\|_{2}=\|u^{g}_{n}b_{n}-b_{n}u^{g}_{n}\|_{2}<\varepsilon\delta$.
Since $\rho_{n}$ is an irreducible representation, the space of
$\sigma_{n}$-invariant vectors in $L^{2}(N,\tau)$ is equal to
$1_{M_{n}({\mathbb{C}})}\otimes L^{2}(N^{1/n},\tau)$. Let $P_{n}$ be the
projection onto this space. Since $F,\varepsilon$ is a Kazhdan pair for
$\Gamma$, by [13, Proposition 12.1.6] (with $\Gamma=\Lambda$) or [7,
Proposition 1.1.9] (the case when $F$ is compact, and modulo rescaling) we
have $\|b_{n}-P_{n}(b_{n})\|_{2}<\varepsilon$.

Therefore $(P_{n}(b_{n}))/{\mathcal{U}}=(b_{n})/{\mathcal{U}}$ belongs to
$\prod^{\mathcal{U}}(1_{M_{n}({\mathbb{C}})}\bar{\otimes}N^{1/n})$, which is
equal to
$N^{\mathcal{U}}\cap(\prod^{{\mathcal{U}}}M_{n}({\mathbb{C}}))^{\prime}$.
Recall that $\Phi[M_{1}]\subseteq\prod^{\mathcal{U}}M_{n}({\mathbb{C}})$.
Since $b$ in $N^{\mathcal{U}}\cap\Phi[M_{0}]^{\prime}$ was arbitrary,
$N^{\mathcal{U}}\cap\Phi[M_{0}]’$ is included in
$N^{\mathcal{U}}\cap\Phi[M_{1}]^{\prime}$ (and is therefore equal to it), as
required. ∎

Lemma 2.1 shows, in terminology of [14, Corollary 6.11] (also [15]), that for
all type II1 tracial von Neumann algebras $N$, all $k\geq 1$, and all
$R^{\mathcal{U}}$-embeddable tracial von Neumann algebras with separable
predual $M$, for the natural action of $\operatorname{Aut}M$ on ${\rm
Hom}(L(\operatorname{SL}(2k+1,{\mathbb{Z}})*M,N^{\mathcal{U}})$ there is an
extreme point with trivial stabilizer.

###### Proof of Theorem 1.

Suppose for a moment that $N$ is of type II1. Let $M_{0}$ and $M_{1}$ be as in
Lemma 2.1. The embeddings of $M_{1}$ and $M_{1}\rtimes_{\alpha}{\mathbb{Z}}$
into $N^{\mathcal{U}}$ provided by (2) and (3) of this lemma satisfy (2) of
Proposition 1.2, and therefore the theory of $N$ does not admit elimination of
quantifiers.

In the general case, when $N$ has a type II1 summand, we can write it as
$N=N_{0}\oplus N_{1}$, where $N_{0}$ is type I and $N_{1}$ is type II1 (e.g.,
[12, §III.1.4.7]). Then $N^{\mathcal{U}}=N_{0}^{\mathcal{U}}\oplus
N_{1}^{\mathcal{U}}$. Let $r=\tau(1_{N_{1}})$, $P_{0}={\mathbb{C}}\oplus
M_{1}$, and $P_{1}={\mathbb{C}}\oplus M_{1}\rtimes_{\alpha}{\mathbb{Z}}$, with
the tracial states $\tau_{0}$ and $\tau_{1}$ such that
$\tau_{0}(1_{M_{1}})=\tau_{1}(1_{M_{1}\rtimes_{\alpha}{\mathbb{Z}}})=r$. Since
type II1 algebra cannot be embedded into one of type I, this choice of tracial
states forces that every embedding of $M_{1}$ into $N^{\mathcal{U}}$ sends
$1_{M_{1}}$ to the image of $1_{N_{1}}$ under the diagonal map. An analogous
fact holds for $M_{1}\rtimes_{\alpha}{\mathbb{Z}}$. Therefore, as in the
factorial case, Proposition 1.2 implies that the theory of $N$ does not admit
elimination of quantifiers. ∎

##  3\. Proof of Proposition 1.2

As promised, here is a long overdue self-contained proof of [9, Proposition
13.6] (Proposition 3.2 below). We provide a proof in case when the language
${\mathcal{L}}$ is single-sorted and countable. The former is a matter of
convenience777Although the language of tracial von Neumann algebras has a sort
for every operator $n$-ball, by homogeneity one can assume that all free
variables range over the unit ball anyway. and the latter requires a minor
change of the statement, considering sufficiently saturated models instead of
ultrapowers (see [9, Proposition 13.6]).

Let $\bar{a}$ be an $n$-tuple of elements in a metric structure $M$ and let
$\bar{x}$ be an $n$-tuple of variables. Each expression of the form
($\varphi(\bar{x},y)$ is a formula and $r\in\mathbb{R}$)

$\varphi(\bar{a},y)=r,\qquad\varphi(\bar{a},y)\geq
r,\qquad\varphi(\bar{a},y)\leq r$

is a _condition_ (in some contexts called _closed condition_) in $y$ over
$\bar{a}$. A set of conditions over $\bar{a}$ is a _type_ over $\bar{a}$. If
all formulas occurring in conditions of a type are quantifier-free, then the
type is said to be quantifier-free. Some $c\in N^{\mathcal{U}}$ satisfies
condition $\varphi(\bar{a},y)=r$ if $\varphi^{N^{\mathcal{U}}}(\bar{a},c)=r$,
and it _realizes the type $\mathsf{t}(y)$_ if it satisfies all of its
conditions. A type $\mathsf{t}(y)$ over $\bar{a}$ in some metric structure $M$
is _consistent_ if for every finite set of conditions in $\mathsf{t}(y)$ and
every $\varepsilon>0$ some element of $M$ approximately realizes each of these
conditions, up to $\varepsilon$.

The salient property of ultrapowers (associated with nonprincipal ultrafilters
on ${\mathbb{N}}$) is that they are _countably saturated_ : every consistent
type over a separable set is realized (e.g., [9], [21], [18]).

The _quantifier-free type of an $n$-tuple $\bar{a}$_ in $M$ is the set of all
conditions of the form $\varphi^{M}(\bar{x})=\varphi^{M}(\bar{a})$, when
$\varphi$ ranges over $\operatorname{\mathfrak{F}_{QF}^{\bar{x}}}$. The _full
type of an $n$-tuple $\bar{a}$_ in $M$ is the set of all conditions of the
form $\varphi^{M}(\bar{x})=\varphi^{M}(\bar{a})$, when $\varphi$ ranges over
$\operatorname{\mathfrak{F}^{\bar{x}}}$. Quantifier-free and full types are
naturally identified with a homomorphism from
$\operatorname{\mathfrak{F}_{QF}^{\bar{x}}}$
($\operatorname{\mathfrak{F}^{\bar{x}}}$, respectively) into $\mathbb{R}$ (see
[21]). If $X$ is a subset of a metric structure $M$, we may expand the
language of $M$ by constant symbols for the element of $X$. The type of
$\bar{a}$ in $M$ in the expanded language is called the _type of $\bar{a}$
over $X$_.

The following well-known and straightforward lemma is somewhat illuminating.

######  Lemma 3.1.

Suppose that $\bar{a}$ and $\bar{a}^{\prime}$ are $n$-tuples in metric
structures in the same language. Then $a_{j}\mapsto a^{\prime}_{j}$ for $j<n$
defines an isometric isomorphism between the structures generated by $\bar{a}$
and $\bar{a}^{\prime}$ if and only if $\bar{a}$ and $\bar{a}^{\prime}$ have
the same quantifier-free type. ∎

While the quantifier-free type of a tuple codes only the isomorphism type of
an algebra generated by it, the full type codes first-order properties such as
the existence of square roots of unitaries and Murray–von Neumann equivalence
of projections. These existential properties are coded by K-theory, at least
in unital $\mathrm{C}^{*}$-algebras for which Groethendieck maps are
injective. In this case, a unitary $u$ has the $n$-th root if and only if the
$K_{1}$-class $[u]_{1}$ is divisible by $n$, and projections $p$ and $q$ are
Murray–von Neumann equivalent if and only if $[p]_{0}=[q]_{0}$ (see [38] or
[11]). However, the information coded by theory and types is different from
that coded by K-theory. On the one hand, there are separable AF-algebras with
nonisomorphic $K_{0}$ groups ([16]) and on the other hand there are separable,
nuclear $\mathrm{C}^{*}$-algebras with the same Elliott invariant but
different theories. All known counterexamples to the Elliott program fall into
this category; see ‘The theory as an invariant’ in [21, p. 4–5].

In the following $\equiv$ denotes the relation of elementary equivalence
(i.e., sharing the same theory).

######  Proposition 3.2.

If ${\mathcal{L}}$ is a countable metric language, then for every
${\mathcal{L}}$-structure $N$ the following are equivalent.

  1. (1)

The theory of $N$ admits elimination of quantifiers.

  2. (2)

For every finitely generated ${\mathcal{L}}$-substructure $G$ of an
ultrapower888Again, this ultrapower is associated to a nonprincipal
ultrafilter on ${\mathbb{N}}$. $N^{{\mathcal{U}}}$, every isometric embedding
of a finitely generated ${\mathcal{L}}$-substructure $F$ of $G$ into
$N^{\mathcal{U}}$ extends to an isometric embedding of $G$.

  3. (3)

Same as (2), but for arbitrary separable substructures.

  4. (4)

For every finitely generated ${\mathcal{L}}$-substructure $G$ of some
$M_{1}\equiv N$, for every isometric embedding of a finitely generated
${\mathcal{L}}$-substructure $F$ of $G$ into $M_{2}\equiv N$ there is an
elementary extension $M_{3}$ of $M_{2}$ such that the embedding of $F$ into
$M_{3}$ extends to an embedding of $G$ into $M_{3}$.

  5. (5)

If $\varphi(\bar{x},y)$ is quantifier free ${\mathcal{L}}$-formula, then the
formula $\inf_{y}\varphi(\bar{x},y)$ is a $\|\cdot\|_{N}$-limit of quantifier
free formulas.

###### Proof.

The equivalence of (2) and (4) is a well-known consequence of saturation of
ultraproducts (see [18, §16]), and (4) will not be used in this paper (it is
included only for completeness).

For simplicity and without loss of generality we assume that the language
${\mathcal{L}}$ is single-sorted. Assume (1) and fix finitely generated
${\mathcal{L}}$-substructures $F\leq G\leq N^{\mathcal{U}}$ and an isometric
embedding $\Phi\colon F\to N^{\mathcal{U}}$. By Lemma 3.1, $\bar{a}$ and
$\Phi(\bar{a})$ have the same quantifier-free type (over the empty set). It
suffices to prove that $\Phi$ extends to an isometric embedding of $G$ in case
when $G$ is generated by $b$ and $F$ for a single element $b$.

Let $r_{\varphi}=\varphi(\bar{a},b)$ for every
$\varphi(\bar{x},y)\in\operatorname{\mathfrak{F}_{QF}^{\bar{x},y}}$. Consider
the following quantifier-free type over $\Phi(\bar{a})$,999Since the formulas
are quantifier-free, we do not need to specify the algebra in which they are
being evaluated.

$\displaystyle\mathsf{t}(y)$
$\displaystyle=\\{\varphi(\Phi(\bar{a}),y)=r_{\varphi}\mid\varphi(\bar{x},y)\in\operatorname{\mathfrak{F}_{QF}^{\bar{x},y}})\\}.$

This type is uncountable, but separable in $\|\cdot\|_{N}$ since the space
$\operatorname{\mathfrak{F}^{\bar{x}}}$ is separable in $\|\cdot\|_{N}$.

In order to prove that $\mathsf{t}(y)$ is satisfiable, fix a finite set of
conditions in $\mathsf{t}(x)$, say $\varphi_{j}(\bar{a},y)=r_{j}$ for $j<m$,
$m\geq 1$. Consider the formula

$\psi(\bar{x})=\inf_{y}\max_{j<m}|\varphi_{j}(\bar{x},y)-r_{j}|.$

Then $\psi^{N^{\mathcal{U}}}(\bar{a})=0$, as witnessed by $b$. Since the
theory of $N$ admits elimination of quantifiers, there are quantifier-free
formulas $\psi_{k}(\bar{x})$, for $k\geq 1$, such that
$\|\psi(\bar{x})-\psi_{k}(\bar{x})\|_{N}<1/k$ for all $n$. Therefore, since
$\Phi$ is an isometry between the structures generated by $\bar{a}$ and
$\Phi(\bar{a})$, we have

$\psi^{N^{\mathcal{U}}}(\Phi(\bar{a}))=\lim_{n}\psi_{n}(\Phi(\bar{a}))=\lim_{n}\psi_{n}(\bar{a})=\psi^{N^{\mathcal{U}}}(\bar{a})=0.$

Thus $\mathsf{t}(y)$ is consistent, and by countable saturation ([18,
§16])some $c\in N^{\mathcal{U}}$ realizes it. Thus $\bar{a},b$ and
$\Phi(\bar{a}),c$ have the same quantifier-free type. By mapping $b$ to $c$
and Lemma 3.1, one finds an isometric extension of $\Phi$ to $G$ as required.

To prove that (2) implies (5), assume that (5) fails. Fix a quantifier-free
formula $\varphi(\bar{x},y)$ in $n+1$ variables and $\varepsilon>0$ such that
every $\psi\in\operatorname{\mathfrak{F}_{QF}^{\bar{x}}}$ satisfies

$\|\inf_{y}\varphi(\bar{x},y)-\psi(\bar{x})\|_{N}\geq\varepsilon.$

Let $\mathsf{t}(\bar{x}(0),\bar{x}(1))$ be the type in $2n$ variables with the
following conditions.

$\displaystyle\psi(\bar{x}(0))-\psi(\bar{x}(1))$ $\displaystyle=0,\text{ for
all $\psi\in\operatorname{\mathfrak{F}_{QF}^{\bar{x}}}$, and }$
$\displaystyle\min(\varepsilon,\inf_{y}\varphi(\bar{x}(1),y)-\inf_{y}\varphi(\bar{x}(0),y))$
$\displaystyle\geq\varepsilon.$

This type is consistent by our assumptions, and by countable saturation it is
realized in $N^{\mathcal{U}}$ by some $\bar{a}(0),\bar{a}(1)$. With $F$
denoting the ${\mathcal{L}}$-structure generated by $\bar{a}(0)$, we have an
isometric embedding $\Phi$ of $F$ into $N^{\mathcal{U}}$ that sends
$\bar{a}(0)$ to $\bar{a}(1)$.

By using countable saturation again, we can find $b\in N^{\mathcal{U}}$ such
that $\inf_{y}\varphi(\bar{a}(1),y)-\varphi(\bar{a}(0),b)\geq\varepsilon$. Let
$G$ be the ${\mathcal{L}}$-structure generated by $\bar{a}(0)$ and $b$. Then
$\Phi$ cannot be extended to an isometric embedding of $G$ into
$N^{\mathcal{U}}$, and (2) fails.

Since (5) implies that for every quantifier-free formula $\varphi(\bar{x},y)$,
the formula $\sup_{y}\varphi(\bar{x},y)$ is a uniform $\|\cdot\|_{N}$-limit of
quantifier-free formulas (by replacing $\varphi$ with $-\varphi$), the proof
that (5) implies (1) follows by induction on complexity of formulas (see [9]).
∎

##  4\. Concluding remarks

The theory of $L^{\infty}[0,1]$ (with respect to the Lebesgue trace) admits
elimination of quantifiers ([31, Theorem 2.13], see also [8, Fact 2.10] and
[10, Example 4.3]). By [31, Lemma 2.17], every matrix algebra
$M_{n}({\mathbb{C}})$ admits elimination of quantifiers. However, if $m\neq n$
then the algebra $M_{m}({\mathbb{C}})\oplus M_{n}({\mathbb{C}})$, with respect
to the tracial state $\tau=\frac{1}{2}({\rm tr}_{m}+{\rm tr}_{n})$ (where
${\rm tr}_{k}$ is the normalized trace on $M_{k}({\mathbb{C}})$), does not
admit elimination of quantifiers. This is because the units of the two
summands have the same quantifier-free type (the quantifier-free type of a
projection is determined by its trace by Lemma 3.1), but if $m>n$ then there
is no isometric embedding from $M_{m}({\mathbb{C}})$ into
$M_{m}({\mathbb{C}})\oplus M_{n}({\mathbb{C}})$ that sends $1_{m}$ to $1_{n}$.
On the other hand, we have the following.

######  Proposition 4.1.

Suppose that $M_{m}({\mathbb{C}})\oplus M_{n}({\mathbb{C}})$ is equipped with
a tracial state $\tau$ which has the property that two projections $p$ and $q$
in $M_{m}({\mathbb{N}})\oplus M_{n}({\mathbb{C}})$ are Murray–von Neumann
equivalent if and only if $\tau(p)=\tau(q)$.

Then the theory of this algebra admits elimination of quantifiers.

If it is not obvious that tracial states with the property required in
Proposition 4.1 exist, it may be easier to prove that there are only finitely
many tracial states that do not have this property. To wit, there are only
finitely many Murray-von Neumann equivalence classes of projections in
$M_{m}({\mathbb{C}})\oplus M_{n}({\mathbb{C}})$. For any two such distinct
classes $[p]$ and $[q]$, there is at most one tracial state $\tau$ such that
$\tau(p)=\tau(q)$. (Consider the system of two linear equations in two
variables $x$ and $y$, corresponding to the values of $\tau$ at rank-1
projections of $M_{m}({\mathbb{C}})$ and $M_{n}({\mathbb{C}})$. It has
infinitely many solutions if and only if $[p]=[q]$.)

###### Proof of Proposition 4.1.

Suppose that $\tau$ satisfies the assumption, $F$ is a
$\mathrm{C}^{*}$-subalgebra of $M_{m}({\mathbb{C}})\oplus
M_{n}({\mathbb{C}})$, and $\Phi\colon F\to M_{m}({\mathbb{C}})\oplus
M_{n}({\mathbb{C}})$ is a trace-preserving embedding. Then $F$ is a direct sum
of matrix algebras. Let $p_{j}$, for $j<k$, be the units of these matrix
algebras. Then $\tau(p_{j})=\tau(\Phi(p_{j}))$ for all $j$. By the assumption
on $\tau$ there is a partial isometry $v_{j}$ such that $v_{j}^{*}v_{j}=p_{j}$
and $v_{j}v_{j}^{*}=\Phi(p_{j})$. Therefore $u=\sum_{j<k}v_{j}$ is a unitary
such that $up_{j}u^{*}=\Phi(p_{j})$ for $j<k$. Since every automorphism of a
matrix algebra is implemented by a unitary, for every $j<k$ there is $w_{j}$
such that $w_{j}^{*}w_{j}=w_{j}w_{j}^{*}=p_{j}$ and for $a\in p_{j}F$ we have
$w_{j}v_{j}av_{j}^{*}w_{j}^{*}=\Phi(a)$. Therefore $\Phi$ coincides with
conjugation by the unitary $u^{\prime}=\sum_{j<k}w_{j}v_{j}$. This implies
that $\Phi$ automatically extends to an embedding of any $G$ such that
$F\subseteq G\subseteq M_{m}({\mathbb{C}})\oplus M_{n}({\mathbb{C}})$ into
$M_{m}({\mathbb{C}})\oplus M_{n}({\mathbb{C}})$. ∎

The fact that the theory of a fixed structure may or may not admit elimination
of quantifiers, depending on the choice of a language, is hardly surprising.
After all, this is exactly what happens with the full matrix algebras, as we
pass from the language of tracial von Neumann algebras to the language of
$\mathrm{C}^{*}$-algebras.

The idea of the proof of Proposition 4.1 should provide the first step towards
a confirmation of the following conjecture and a complete answer to [31,
Question 2.18].

######  Conjecture 4.2.

If $T$ is the theory of a tracial von Neumann algebra, then the following are
equivalent.

  1. (1)

$T$ admits elimination of quantifiers.

  2. (2)

Every model $N$ of $T$ is of type I, and if $N$ has separable predual then two
projections $p$ and $q$ in $N$ are conjugate by a trace preserving
automorphism if and only if $\tau(p)=\tau(q)$.

Model completeness is a useful weakening of quantifier elimination. A theory
is _model complete_ if every embedding between its models is elementary. As
the referee pointed out, the following can be extracted from a well-known
semantic characterization of elementary embeddings (see the paragraph between
Fact 2.1.2 and Fact 2.1.3 in [5]), but there is some merit in stating it
explicitly.

######  Proposition 4.3.

Assume the Continuum Hypothesis. For a continuous theory $T$ in a complete
language the following are equivalent.

  1. (1)

The theory $T$ is model-complete.

  2. (2)

If $M$ and $N$ are separable models of $T$ and $\Phi\colon M\to N$ is an
isometric embedding, then there is an isomorphism $\Psi\colon
M^{\mathcal{U}}\to N^{\mathcal{U}}$ such that the following diagram commutes
(the horizontal arrows are diagonal embeddings)

$M$$M^{\mathcal{U}}$$N$$N^{\mathcal{U}}$$\Phi$$\Psi$

The use of Continuum Hypothesis in (2) is innocuous, as it has no effect on
projective statements. This is a forcing argument well-known to set theorists
but poorly documented in the literature; see for example [17, Lemma 5.20]. It
is also a red herring, since in its absence (2) can be replaced with a
considerably more complex, but (once mastered) equally useful, assertion about
the existence of a $\sigma$-complete back-and-forth system of partial
isomorphisms between separable subalgebras of $M^{\mathcal{U}}$ and
$N^{\mathcal{U}}$ (see [18, §8.2, Theorem 16.6.4])

###### Proof.

Assume (1). Then $\Phi[M]$ is a separable elementary submodel of
$N^{\mathcal{U}}$, and the Continuum Hypothesis implies that
$M^{\mathcal{U}}\cong N^{\mathcal{U}}$, via an isomorphism $\Psi$ that sends
the diagonal copy of $M$ onto $\Phi[M]$. The $\mathrm{C}^{*}$-algebra case is
proven in [18, Theorem 16.7.5], and the proof of the general case is virtually
identical.

Assume (2) and let $\Phi\colon M\to N$ be an embedding between models of $T$.
We need to prove that $\Phi$ is elementary. By replacing $M$ and $N$ with
separable elementary submodels large enough to detect a given failure of
elementarity, we may assume they are separable. With $\Psi$ as guaranteed by
(2), the embedding of $M$ into $N^{\mathcal{U}}$ is, being the composition of
an elementary embedding and an isomorphism, elementary. Since the diagonal
image of $N$ in $N^{\mathcal{U}}$ is elementary, it follows that $\Phi$ is
elementary. ∎

######  Conjecture 4.4.

The theory of a tracial von Neumann algebra $N$ is model complete if and only
if $N$ is of type I.

A fact relevant to both conjectures is that by [20, Proposition 3.1],
elementarily equivalent type I tracial von Neumann algebras with separable
preduals are isometrically isomorphic. Following a referee’s advice, we state
the following one-time side remark in the theorem environment, with a proof.

######  Proposition 4.5.

The theory of every type I tracial von Neumann algebra is model complete.

###### Proof.

This is an immediate consequence of the results of [20]. Fix a type I tracial
von Neumann algebra $M$.

If $M$ is abelian, then $M\cong L^{\infty}(\mu)$ for a probability measure
$\mu$. Then [20, Lemma 3.2] implies that theory of $M$ determines the measures
of the atoms of $\mu$ (these are the values of $\rho(1,n)$ for $n\geq 1$,
using the notation from this lemma and taken in the decreasing order;
$\rho(1,0)$ is the measure of the diffuse part).

Therefore if $N\equiv M$ then $N\cong L^{\infty}(\nu)$, and the measures of
the atoms of $\nu$ (if any) are exactly the same as the measures of the atoms
of $\mu$. Every trace-preserving *-homomorphism $\Phi$ from $M$ into $N$ has
to send the diffuse part to the diffuse part. Since these parts are of the
same measure, it also sends atomic part to the atomic part. Finally, since the
atoms in $M$ and $N$ have the same measures (with multiplicities), and since
we are dealing with a probability measure, the restriction of $\Phi$ to the
atomic part is an isomorphism. The theory of $L^{\infty}$ space of a diffuse
measure admits elimination of quantifiers, hence the restriction of $\Phi$ to
the diffuse part is elementary. Thus $\Phi$ can be decomposed as a direct sum
of two elementary embeddings, and is therefore elementary by the second part
of [20, Corollary 2.6].

This proves Proposition in case when $M$ is abelian.

In general, when $M$ is not necessarily abelian, it is isomorphic to
$\bigoplus_{n\in X}A^{M}_{n}$, where $X\subseteq[1,\infty)$, and for every
$n\in X$ we have $A^{M}_{n}\cong M_{n}(B^{M}_{n})$, for a commutative von
Neumann algebra $B^{M}_{n}$. Suppose that $M\equiv N$. Then [20, Lemma 3.2]
implies that $N\cong\bigoplus_{n\in X}A^{N}_{n}$ (with the same set $X$) and
that $\tau(1_{A^{M}_{n}})=\tau(1_{A^{N}_{n}})$ for all $n\in X$. (Using the
notation from [20, Lemma 3.2], these quantities are both equal to
$\sum_{k}\rho_{M}(n,k)$, for every $n\in X$.)

Now assume that $\Phi\colon M\to N$ is a trace-preserving *-homomorphism. For
$n^{\prime}<n$ in $X$ we have that $\Phi(1_{A^{M}_{n}})$ and
$1_{A^{N}_{n^{\prime}}}$ are orthogonal. By induction on $n\in X$, this
implies that $\Phi(1_{A^{M}_{n}})=1_{A^{N}_{n}}$ for all $n$.

By the abelian case, the restriction of $\Phi$ to $A^{M}_{n}$ is elementary
for every $n$. By the second part of [20, Corollary 2.6], $\Phi$ is
elementary. ∎

The original version of this paper contained an appendix by Srivatsav
Kunnawalkam Elayavalli, removed following a suggestion of the referee. The
appendix was concerned with the two notions that seem to be sensitive to the
choice of additional axioms of ZFC, and the author’s strong opinion on such
definitions is expressed in a long footnote on p. 12 of [19].

## References

  * [1] C. Anantharaman and S. Popa.  An introduction to II1 factors.  available at https://idpoisson.fr/anantharaman/publications/IIun.pdf, 2017\.
  * [2] C. Anantharaman-Delaroche.  Amenable correspondences and approximation properties for von neumann algebras.  Pacific Journal of Mathematics, 171(2):309–341, 1995.
  * [3] S. Atkinson and S. K. Elayavalli.  On ultraproduct embeddings and amenability for tracial von Neumann algebras.  International Mathematics Research Notices, 2021(4):2882–2918, 2021\.
  * [4] S. Atkinson, I. Goldbring, and S. K. Elayavalli.  Factorial relative commutants and the generalized Jung property for II1 factors.  Advances in Mathematics, 396:108107, 2022.
  * [5] Scott Atkinson, Isaac Goldbring, and Srivatsav Kunnawalkam Elayavalli.  Factorial relative commutants and the generalized jung property for $II_{1}$ factors.  Advances in Mathematics, 396:108107, 2022.
  * [6] B. Bekka.  Operator-algebraic superridigity for ${\textrm{S}L}_{n}({\mathbb{Z}})$, $n\geq 3$.  Inventiones mathematicae, 2(169):401–425, 2007.
  * [7] B. Bekka, P. de La Harpe, and A. Valette.  Kazhdan’s property (T).  Cambridge university press, 2008.
  * [8] I. Ben Yaacov.  On theories of random variables.  Israel Journal of Mathematics, 194:957–1012, 2013.
  * [9] I. Ben Yaacov, A. Berenstein, C.W. Henson, and A. Usvyatsov.  Model theory for metric structures.  In Z. Chatzidakis et al., editors, Model Theory with Applications to Algebra and Analysis, Vol. II, number 350 in London Math. Soc. Lecture Notes Series, pages 315–427. London Math. Soc., 2008.
  * [10] I. Ben Yaacov and A. Usvyatsov.  Continuous first order logic and local stability.  Transactions of the American Mathematical Society, 362(10):5213–5259, 2010.
  * [11] B. Blackadar.  K-theory for operator algebras, volume 5 of MSRI Publications.  Cambridge University Press, 1998.
  * [12] B. Blackadar.  Operator algebras, volume 122 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2006.  Theory of $\mathrm{C}^{*}$-algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III.
  * [13] N. Brown and N. Ozawa.  $\mathrm{C}^{*}$-algebras and finite-dimensional approximations, volume 88 of Graduate Studies in Mathematics.  American Mathematical Society, 2008.
  * [14] N.P. Brown.  Topological dynamical systems associated to $II_{1}$ factors.  Adv. Math., 227(4):1665–1699, 2011.  With an appendix by Narutaka Ozawa.
  * [15] N.P. Brown and V. Capraro.  Groups associated to II1-factors.  Journal of Functional Analysis, 264(2):493–507, 2013.
  * [16] B. De Bondt, A. Vaccaro, B. Veličković, and A. Vignati.  Games on AF-Algebras.  International Mathematics Research Notices, 11 2022.  rnac313.
  * [17] C. J. Eagle, I. Farah, E. Kirchberg, and A. Vignati.  Quantifier elimination in $\mathrm{C}^{*}$-algebras.  International Mathematics Research Notices, 2017(24):7580–7606, 2017.
  * [18] I. Farah.  Combinatorial Set Theory and $\mathrm{C}^{*}$-algebras.  Springer Monographs in Mathematics. Springer, 2019.
  * [19] I. Farah.  Quantifier elimination in II1 factors.  version 2, with appendix by Srivatsav Kunnawalkam Elayavalli, arXiv preprint arXiv:2304.11371, https://arxiv.org/abs/2304.11371v2, 2023.
  * [20] I. Farah and S. Ghasemi.  Preservation of elementarity by tensor products of tracial von Neumann algebras.  arXiv preprint arXiv:2304.08437, 2023.
  * [21] I. Farah, B. Hart, M. Lupini, L. Robert, A. Tikuisis, A. Vignati, and W. Winter.  Model theory of $\mathrm{C}^{*}$-algebras.  Mem. Amer. Math. Soc., 271(1324):viii+127, 2021.
  * [22] I. Goldbring and B. Hart.  On the theories of Mcduff’s II1 factors.  International Mathematics Research Notices, 2017(18):5609–5628, 2017.
  * [23] I. Goldbring and B. Hart.  A survey on the model theory of tracial von Neumann algebras.  arXiv preprint arXiv:2210.15508, 2022.
  * [24] I. Goldbring, B. Hart, and T. Sinclair.  The theory of tracial von Neumann algebras does not have a model companion.  J. Symbolic Logic, 78(3):1000–1004, 2013.
  * [25] U. Haagerup and S. Thorbjørnsen.  A new application of random matrices: ${\mathrm{Ext}}(C^{*}_{\mathrm{red}}(F_{2}))$ is not a group.  Ann. of Math. (2), pages 711–775, 2005.
  * [26] B. Hart.  An introduction to continuous model theory.  arXiv preprint arXiv:2303.03969, 2023.
  * [27] B. Hayes.  1-bounded entropy and regularity problems in von neumann algebras.  International Mathematics Research Notices, 2018(1):57–137, 11 2016\.
  * [28] B. Hayes, D. Jekel, B. Nelson, and T. Sinclair.  A random matrix approach to absorption in free products.  International Mathematics Research Notices, 2021(3):1919–1979, 2021\.
  * [29] C.W. Henson and J. Iovino.  Ultraproducts in analysis.  In Catherine Finet and Christian Michaux, editors, Analysis and Logic, number 262 in London Mathematical Society Lecture Notes Series.
  * [30] D. Jekel.  Covering entropy for types in tracial $\mathrm{W}^{*}$-algebras.  arXiv preprint arXiv:2204.02582, 2022.
  * [31] D. Jekel.  Free probability and model theory of tracial $\mathrm{W}^{*}$-algebras.  arXiv preprint arXiv:2208.13867, 2022.
  * [32] Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen.  MIP*= RE.  arXiv preprint arXiv:2001.04383, 2020.
  * [33] K. Jung.  Amenability, tubularity, and embeddings into $R^{\omega}$.  Mathematische Annalen, 338(1):241–248, 2007.
  * [34] A. Kanamori.  The higher infinite: large cardinals in set theory from their beginnings.  Perspectives in Mathematical Logic. Springer, Berlin–Heidelberg–New York, 1995.
  * [35] E. Kirchberg.  Central sequences in $\mathrm{C}^{*}$-algebras and strongly purely infinite algebras.  In Operator Algebras: The Abel Symposium 2004, volume 1 of Abel Symp., pages 175–231. Springer, Berlin, 2006.
  * [36] E. Kirchberg and N.C. Phillips.  Embedding of exact $\mathrm{C}^{*}$-algebras in the Cuntz algebra ${\mathcal{O}}_{2}$.  J. Reine Angew. Math., 525:17–53, 2000.
  * [37] D. Marker.  Model theory, volume 217 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 2002.
  * [38] M. Rørdam, F. Larsen, and N.J. Laustsen.  An Introduction to K-theory for C∗ algebras.  Number 49 in London Mathematical Society Student Texts. Cambridge University Press, 2000.
  * [39] D. Voiculescu.  A strengthened asymptotic freeness result for random matrices with applications to free entropy.  International Mathematics Research Notices, 1998(1):41–63, 1998\.