text:
\section{Introduction}\label{DHY_2101_introduction}
The Gaussian measure space, denoted by $\l(\bR^d, |\cdot|, \gamma\r)$, is the Euclidean space $\bR^d$ endowed with the Euclidean distance $|\cdot|$ and the Gaussian probability measure $\gamma$, where
\[\ddd\gamma(x):=\pi^{-d/2} e^{-|x|^2} \ddd x.\]
The Gaussian harmonic analysis extends the classical results obtained in harmonic analysis of trigonometric expansions to orthogonal polynomial expansions~\cite{MS_1965}. On the other hand, Gaussian harmonic analysis is widely used in the second quantization~\cite{Nelson_1973}, Malliavin calculus~\cite{Nualart_2006}, hypercontractivity~\cite{BE_1985}, and geometric applications\cite{Bakry_1988}, etc.

On Gaussian measure spaces, due to the special locally doubling and reverse doubling property which will be shown in Section \ref{DHY_2101_preliminaries}, we concentrate on the \textit{local Hardy-Littlewood maximal operator $M^{a}$} defined by
\[M^{a}(f)(x):=\sup_{Q\in\mathscr{Q}_a(x)} \frac{1}{\gamma(Q)} \int_Q |f(y)|\ddd \gamma(y),\]
where
\[\mathscr{Q}_a(x)=\{Q\in\mathscr{Q}_a: Q\ni x\}, \quad \mathscr{Q}_a=\{Q\subset \bR^d: \ell(Q)\leq am(c_Q)\}, \quad  m(x)=\min\{1, {1}/{|x|}\}\]
with $c_Q$ denoting the center of the cube $Q$ and $\ell(Q)$ denoting the side length of the cube $Q$. Furthermore, the \textit{local fractional Hardy-Littlewood maximal operator $M_{\alpha}^{a}$} is defined by
\[M^{a}_{\alpha}(f)(x):=\sup_{Q\in\mathscr{Q}_a(x)}\frac{1}{[\gamma(Q)]^{1-\dot{\alpha}}}\int_Q |f(y)|\ddd \gamma(y)\]
with $\dot{\alpha}=\alpha/d$. Here all the sides of the cubes are parallel to the coordinate axes.

Recently the study of Gaussian harmonic analysis has aroused extensive attention. In 2007, Mauceri and Meda~\cite{MM_2007} introduced the local maximal operator and used it to develop the singular integral operator theory on the Gaussian measure spaces; in the same year, Aimar~et~al.~\cite{AFS_2007} obtained the weak type $(1,1)$ inequalities for the Gaussian Riesz transform and a general maximal operator which dominates the Ornstein-Uhlenbeck maximal operator; then in 2010, Liu and Yang\cite{LY_2010} obtained the boundedness of the \textit{local fractional integral operator} defined by
\[I_{\alpha}^{a}(f)(x):=\int_{B(x, am(x))} \frac{f(y)}{[\gamma(B(x,|x-y|))]^{1-\dot{\alpha}}}\ddd\gamma(y)\]
and the local fractional maximal operator $M^{a}_{\alpha}$ on the Gaussian Lebesgue spaces; in 2014, Liu~et~al.~\cite{LSY_2014} obtained the boundedness of the local fractional integral operator and the local maximal operator on the Gaussian Morrey-type spaces; later in 2016, Wang~et~al.~\cite{WZL_2016} characterized the one-weight boundedness of $M^a$ on Gaussian Lebesgue spaces by the \textit{local $A_{p}^a$ weights} defined as
\[[\omega]_{A_{p}^a}=\sup_{B\in\mathscr{B}_{a}}\l(\frac{1}{\gamma(B)}\int_B \omega(x) \ddd\gamma(x)\r)\l(\frac{1}{\gamma(B)}\int_B \omega(x)^{1-p'}\ddd\gamma(x)\r)^{p-1}<\infty,\]
and obtained the one-weight boundedness of $M_{\alpha}^a$ on the Gaussian Lebesgue spaces by the \textit{local $A_{p,q}^a$ weights} defined as
\begin{equation*}
[w]_{A_{p,q}^a}=\sup_{B\in\mathscr{B}_{a}} \l(\frac{1}{\gamma(B)}\int_B \omega(x)^q\ddd \gamma(x)\r)^{1/q}\l(\frac{1}{\gamma(B)}\int_B \omega(x)^{-p'}\ddd \gamma(x)\r)^{1/{p'}}<\infty;
\end{equation*}
more recently in 2020,  Lin and Mao~\cite{LM_2020} established the one-weight norm inequalities associated with the local $A_{p,q}^a$ weights for the fractional operators $I_{\alpha}^a$ and $M_{\alpha}^a$ on the Gaussian measure spaces.

Based on the results above, this article aims to establish the two-weight norm inequalities for the local fractional maximal and integral operators with respect to the local Muckenhoupt type weights and the local Sawyer type weights. To state our main results, we introduce the following local Muckenhoupt type weights.
\begin{definition}
Given $0<a<\infty$, $0\leq\dot{\alpha}<1$, $\alpha=\dot{\alpha}d$ and $1< p, q<\infty$, we say that a pair of weights $(u,v)\in A_{p,q,\alpha}^a$ if
\begin{align*}
[u,v]_{A_{p,q,\alpha}^a}:=\sup_{Q\in\mathscr{Q}_a} \gamma(Q)^{(\dot{\alpha}+1/q-1/p)}\l(\frac{1}{\gamma(Q)} \int_Q u(x) \ddd \gamma(x)\r)^{1/q}\l(\frac{1}{\gamma(Q)} \int_Q v(x)^{1-p'}\ddd\gamma(x)\r)^{1/{p'}}<\infty.
\end{align*}
For $p=1$, we say $(u,v)\in A_{1,q,\alpha}^a$ if
\[[u,v]_{A_{1,q,\alpha}^a}:=\sup_{Q\in\mathscr{Q}_a} \mathop{\mathrm{ess}\sup}\limits_{x\in Q}^{}\gamma(Q)^{(\dot{\alpha}+1/q-1)} \l(\frac{1}{\gamma(Q)} \int_Q  u(x) \ddd \gamma(x)\r)^{1/q} v(x)^{-1}<\infty,\]
where the essential supremum is associated with the measure $\gamma$.
\end{definition}

\begin{remark}\label{DHY_1}
Note that if $\alpha+d/q-d/p=0$ and
\[u(x)=\omega(x)^q,\quad v(x)=\omega(x)^p,\]
then the two-weight condition $A_{p,q,\alpha}^a$ goes back to the one-weight condition $A_{p,q}^a$ first introduced by Wang~et~al~\cite{WZL_2016}; if $\alpha+d/q-d/p<0$, by letting suitable $\gamma(Q)\to 0$, we conclude the fact $A_{p,q,\alpha}^a=\emptyset$; finally if $\alpha+d/q-d/p>0$, it is obvious that $u(x)=v(x)=1$ satisfy the $A_{p,q,\alpha}^a$ condition, i.e., $A_{p,q,\alpha}^a\neq\emptyset$.
\end{remark}
One of our main results in this paper is the following two-weight weak-type estimate for $M_{\alpha}^a$.

\begin{thm}\label{GM_2001_thm4.3}
Given $a\in(0,\infty)$, $1\leq p\leq q<\infty$, $0\leq\alpha<d$ and a pair of weights $(u,v)$, if
\[(u,v)\in \bigcup_{b'>a}A_{p,q,\alpha}^{b'},\]
then $M_{\alpha}^a$ is bounded from $L^p\l(\bR^d, v,\gamma\r)$ to $L^{q,\infty}\l(\bR^d, u,\gamma\r)$, that is, there exists a constant $C>0$ such that
\[\int_{\{x\in\bR^d: M_{\alpha}^a(f)(x)>\lambda\}}u(x)\ddd\gamma(x)\leq \frac{C}{\lambda^q}\l(\int_{\bR^d}|f(x)|^p v(x)\ddd\gamma(x)\r)^{q/p}\]
holds for all $\lambda>0$.
\end{thm}
In addition, we get the following two-weight weak-type estimate for the \textit{local fractional integral opeartor $\tilde{I}_{\alpha}^a$ with cubes on Gaussian measure spaces} defined by
\[\tilde{I}_{\alpha}^a(f)(x):=\int_{Q\l(x, am(x)\r)} \frac{f(y)}{\l[\gamma\l(Q(x,|x-y|)\r)\r]^{1-\dot{\alpha}}} \ddd\gamma(y),\]
where $Q\l(x, am(x)\r)$ denotes the cube with $c_Q=x$ and $\ell(Q)=am(x)$.
\begin{thm}\label{TWWTEFI}
Given $a\in(0,\infty)$, $1\leq p\leq q<\infty$, $0<\alpha<d$ and a pair of weights $(u,v)$, if
\[(u,v)\in\bigcup_{a<b'}\bigcup_{0<\beta'<\alpha} A_{p,q,\beta'}^{b'},\]
then $\tilde{I}_{\alpha}^a$ is bounded from $L^p(\bR^d, v, \gamma)$ to $L^{q,\infty}(\bR^d, u, \gamma)$, that is, there exists a constant $C>0$ such that
\[\int_{\{x\in\bR^d: \tilde{I}_{\alpha}^a(f)(x)>\lambda\}} u(x) \ddd\gamma(x)\leq \frac{C}{\lambda^q}\l(\int_{\bR^d} |f(x)|^p v(x)\ddd\gamma(x)\r)^{q/p}\]
holds for all $\lambda>0$.
\end{thm}

Based on the radialization method in \cite{DHY_2012} and inspired by \cite{GM_2001}, we prove Theorem~\ref{GM_2001_thm4.3} by introducing a radial version of the local fractional maximal operator $M_{\alpha}^a$ and the $A_{p,q,\alpha}^a$ weights. So that we can use the dyadic analysis on Gaussian measure spaces to prove the desired conclusion. Then Theorem~\ref{TWWTEFI} comes from an extended Welland type inequality on Gaussian measure spaces (see Lemma \ref{LM_2020_lemma3.4} below) and the boundedness of $M_{\alpha}^a$. Furthermore using a similar method, we also establish the two-weight strong-type estimates for $M_{\alpha}^a$ and $\tilde{I}_{\alpha}^a$ under the following local Sawyer type condition.

\begin{definition}
Let $a\in(0,\infty)$, $\alpha\in[0,d)$ and $1<p\leq q<\infty$. We say that a pair of weights $(u,v)$ satisfies the \textit{local-$a$ testing condition} if
\[[u,v]_{\mathscr{M}_{p,q,\alpha}^a}:=\sup_{Q\in\mathscr{Q}_a} \l[\int_Q \l(M_{\alpha}^a(v^{1-p'}\chi_Q)(x)\r)^q u(x) \ddd  \gamma(x)\r]^{1/q}\l(\int_Q v(x)^{1-p'}\ddd\gamma(x)\r)^{-1/p}<\infty.\]
In this situation, we write $(u,v)\in \mathscr{M}_{p,q,\alpha}^a$.
\end{definition}

\begin{remark}
We can rewrite the definition of $[u,v]_{\mathscr{M}_{p,q,\alpha}^a}$ as
\begin{align*}
\sup_{Q\in\mathscr{Q}_a}\l[\int_Q\l(\sup_{Q'\in\mathscr{Q}_a(x)}\frac{1}{\gamma(Q')^{1-\dot{\alpha}}}\int_{Q'\cap Q} v(y)^{1-p'}\ddd\gamma'(y)\r)^q u(x)\ddd\gamma(x)\r]^{\frac{1}{q}}\l(\int_Q v(x)^{1-p'}\ddd\gamma(x)\r)^{-\frac{1}{p}}.
\end{align*}
Thereby when $d/q-d/p+\alpha<0$, similar to Remark \ref{DHY_1}, we conclude $\mathscr{M}_{p,q,\alpha}^a =\emptyset$ by letting suitable $Q=Q'$ and $\gamma(Q')=\gamma(Q)\to 0$.
\end{remark}

Then we obtain the two-weight strong-type boundedness of $M_{\alpha}^a$ and $\tilde{I}_{\alpha}^a$ as follows.
\begin{thm}\label{GM_2001_thm3.1}
Given $a\in(0,\infty)$, $1< p\leq q<\infty$, $0\leq\alpha<d$ and a pair of weights $(u,v)$, if
\[(u,v)\in \mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2},\]
then $M_{\alpha}^a$ is bounded from $L^p(\bR^d, v, \gamma)$ to $L^{q}(\bR^d, u, \gamma)$, that is, there exists a constant $C>0$ such that
\[\l(\int_{\bR^d}\Big(M_{\alpha}^a(f)(x)\Big)^qu(x)\ddd\gamma(x)\r)^{1/q}\leq C \l(\int_{\bR^d}|f(x)|^p v(x)\ddd\gamma(x)\r)^{1/p}.\]
\end{thm}
\begin{thm}\label{GM_2001_thm6.5}
Given $a\in(0,\infty)$, $1< p\leq q<\infty$, $0<\alpha<d$ and a pair of weights $(u,v)$, if
\[(u,v)\in\bigcup_{6a+9\sqrt{d}a^2<b'}\bigcup_{0<\beta'<\alpha} \mathscr{M}_{p,q,\beta'}^{b'},\]
then $\tilde{I}_{\alpha}^a$ is bounded from $L^p(\bR^d, v, \gamma)$ to $L^{q}(\bR^d, u, \gamma)$, that is, there exists a constant $C>0$ such that
\[\l(\int_{\bR^d}\l(\tilde{I}_{\alpha}^a(f)(x)\r)^q u(x)\ddd\gamma(x)\r)^{1/q}\leq C\l(\int_{\bR^d} |f(x)|^p v(x)\ddd\gamma(x)\r)^{1/p}.\]
\end{thm}

The highlights of the paper are as follows. In Section \ref{DHY_2101_preliminaries}, we give some basic facts used in the proofs of the desired main results, as well as the radial versions of $M_{\alpha}^a$ and $A_{p,q,\alpha}^a$. In Section \ref{DHY_2101_TLMTW}, we investigate some properties for the local Muckenhoupt type weights and prove the two-weight weak-type boundedness of $M_{\alpha}^a$ and $\tilde{I}_{\alpha}^a$. Meanwhile a natural question (see Question \ref{DHY_2101_question1} below) arises here. In Section \ref{DHY_2101_TLSTW}, we study the similar properties for the local Sawyer type weights and prove the two-weight strong-type boundedness of $M_{\alpha}^a$ and $\tilde{I}_{\alpha}^a$. Furthermore, a similar question (see Question \ref{DHY_2101_question2} below) is stated at the end of Section \ref{DHY_2101_TLSTW}.

We end this section with some notions and notations. Hereafter, we will be working in $\bR^d$ and $d$ will always denote the dimension. We will denote by $C$ or its variants a positive constant independent of the main involved parameters, and use $f\lesssim g$ to denote $f\leq C g$; particularly, if $f\lesssim g\lesssim f$, then we will write $f\sim g$. If necessary, we will denote the dependence of the constants parenthetically, e.g., $C=C(a,d)$ or $C=C_{a,d}$. Similarly, $f\lesssim_{a,d} g$ will denote $f\leq C_{a,d} g$ and $f\sim_{a,d}g$ will denote $C_{a,d} f\leq g\leq C_{a,d} f$. By a weight we will always mean a locally integrable function and non-negative almost everywhere on $\bR^d$ (with respect to the associated measure $\gamma$). For a given weight $\omega$, the weighted Gaussian Lebesgue norms on $\bR^d$ will be denoted by
\[\|f\|_{L^p(\bR^d, \omega, \gamma)}:=\l(\int_{\bR^d} |f(x)|^p\omega(x)\ddd\gamma(x)\r)^{\frac{1}{p}},\]
 and
  \[\|f\|_{L^{p,\infty}(\bR^d, \omega, \gamma)}:=\sup_{\lambda>0}\frac{1}{\lambda}\l(\int_{\{x\in\bR^d: f(x)>\lambda\}} \omega(x)\ddd\gamma(x)\r)^{\frac{1}{p}}.\]

\section{Preliminaries}\label{DHY_2101_preliminaries}
It is easy to see that the probability measure $\gamma$ is highly concentrated around the origin with exponential decay at infinity. Thereby it is not a doubling measure, i.e., there is no constant $C>0$, independent of $x\in\bR^d$ and $r>0$, such that
\[\gamma\l(B(x,2r)\r) \leq C \gamma\l(B(x,x)\r)\]
holds for all $x\in\bR^d$ and $r>0$. Here $\gamma(B):=\int_B \ddd\gamma(x)$. See~\cite[Appendix 10.3]{GHA_2019} for more details. Hence we know that the Gaussian measure space is not a homogeneous type space in the sense of Coifman and Weiss~\cite{CW_1977}. However, as we have mentioned in Section \ref{DHY_2101_introduction}, if we define the family of admissible cubes $\mathscr{Q}_a$, then \cite[Proposition 2.1]{MM_2007} points out that
\begin{equation}\label{DHY_2101_equivalent}
e^{|c_Q|^2}\sim_{a}e^{|x|^2}
\end{equation}
holds for all $Q\in \mathscr{Q}_a$ and all $x\in Q$. From this estimate we conclude that
\begin{equation}\label{DHY_2101_equivalent_2}
\gamma(Q)=\int_Q \ddd\gamma(x)\sim_{a,d} \;e^{-|c_Q|^2}\int_Q \ddd x =e^{-|c_Q|^2}\ell(Q)^d
\end{equation}
holds for all $Q\in \mathscr{Q}_a$. Furthermore the Gaussian measure is doubling and reverse doubling if we restrict it to $\mathscr{Q}_a$. In other words, there exist constants $C_1=C_{a,d}\geq 1$ and $C_2=C'_{a,d}>1$ such that for all $Q\in \mathscr{Q}_a$ we have
\[\gamma(2Q)\leq C_1 \gamma(Q), \quad \gamma(2Q)\geq C_2\gamma(Q).\]
Hence we say $\gamma$ satisfies the locally doubling condition and locally reverse doubling condition on $\mathscr{Q}_a$.

On the other hand, the Gaussian measure is trivially a $d$-dimensional measure in $\bR^d$, i.e.,
\[\gamma(Q)\leq \ell(Q)^d\]
holds for all cubes $Q$ in $\bR^d$. Therefore some results on the $d$-dimensional measure, such as~\cite{Tolsa_2001} and~\cite{GM_2001}, may be useful in Gaussian harmonic analysis.

By using the estimates (\ref{DHY_2101_equivalent}) and (\ref{DHY_2101_equivalent_2}), we conclude that
\begin{align*}
\frac{1}{[\gamma(Q)]^{1-\dot{\alpha}}}\int_Q |f(y)|\ddd \gamma(y)
&\sim_{a,d} e^{-|c_Q|^2}\frac{1}{[\gamma(Q)]^{1-\dot{\alpha}}}\int_Q |f(y)|\ddd y \\
&\sim_{a,d} e^{-\dot{\alpha}|c_Q|^2}\frac{1}{\ell(Q)^{d-\alpha}}\int_Q |f(y)|\ddd y \\
&\sim_{a,d} \frac{1}{\ell(Q)^{d-\alpha}} \int_Q |f(y)| e^{-\dot{\alpha}|y|^2} \ddd y
\end{align*}
holds for all $Q\in\mathscr{Q}_a$. Now we can give the pointwise equivalent radial version of $M_{\alpha}^a$ as follows.
\begin{definition}
Let $a\in(0,\infty)$, $\dot{\alpha}\in [0,1)$, $\alpha=\dot{\alpha}d$ and $f\in L_{loc}^{1}(\gamma)$. We define the \textit{local fractional maximal operator $M^{a}_{\alpha}$ on Gaussian measure spaces} by setting
\begin{align*}
M^{a}_{\alpha}(f)(x)&:=\sup_{Q\in\mathscr{Q}_a(x)}\frac{1}{[\gamma(Q)]^{1-\dot{\alpha}}}\int_Q |f(y)|\ddd \gamma(y) \\
&:\sim_{a,d} \sup_{Q\in\mathscr{Q}_a(x)} \frac{1}{\ell(Q)^{d-\alpha}} \int_Q |f(y)| \ddd \gamma'(y),
\end{align*}
where $\ddd \gamma'(y)=e^{-\dot{\alpha}|y|^2}\ddd y=e^{-\alpha|y|^2/d}\ddd y$.
\end{definition}

To use the dyadic analysis on Gaussian measure spaces more conveniently and adapt the $A_{p,q,\alpha}^a$ condition to our radialization method, we introduce the following $\mathscr{A}_{p,q,\alpha}^a$ condition.
\begin{definition}
Given $0<a<\infty$, $0\leq\dot{\alpha}<1$, $\alpha=\dot{\alpha}d$ and $1< p\leq q<\infty$, we say that a pair of weights $(u,v)\in\mathscr{A}_{p,q,\alpha}^a$ if
\begin{align*}
[u,v]_{\mathscr{A}_{p,q,\alpha}^a}=\sup_{Q\in\mathscr{Q}_a} \frac{1}{\ell(Q)^{d-\alpha}}\l(\int_Q  u(x) \ddd  \gamma'(x)\r)^{1/q}\l(\int_Q v(x)^{1-p'}\ddd\gamma'(x)\r)^{1/{p'}}<\infty.
\end{align*}
In the case $p=1$, we say $(u,v)\in \mathscr{A}_{1,q,\alpha}^a$ if there exists a constant $C$ such that for every cube $Q\in\mathscr{Q}_a$ the inequality
\[\frac{1}{\ell(Q)^{d-\alpha}} \l(\int_Q  u(x) \ddd  \gamma'(x)\r)^{1/q} \leq Cv(x)\]
holds for $\gamma'$-a.e. $x\in Q$.
\end{definition}

We can show the following close relation between the $A_{p,q,\alpha}^a$ condition and the $\mathscr{A}_{p,q,\alpha}^a$ condition.
\begin{prop}\label{DHY_RBAA}
Let
\[u'(x):=u(x)e^{-(1-\dot{\alpha})|x|^2},\quad v'(x):=v(x)e^{-(1-\dot{\alpha})|x|^2}.\]
Then
\[(u,v)\in A_{p,q,\alpha}^a\Leftrightarrow (u',v')\in \mathscr{A}_{p,q,\alpha}^a.\]
\end{prop}

\begin{proof}
We focus on proving the case $p>1$ since the case $p=1$ is essentially the same. For every fixed $Q\in\mathscr{Q}_a$, we have known that
\begin{equation}\label{DHY_LDP}
e^{-|x|^2}\sim_a e^{|y|^2}
\end{equation}
holds for any $x\in Q$ and any $y\in Q$. Hence we conclude
\begin{align*}
&\gamma(Q)^{(\dot{\alpha}+1/q-1/p)}\l(\frac{1}{\gamma(Q)} \int_Q u(x) \ddd \gamma(x)\r)^{1/q}\l(\frac{1}{\gamma(Q)} \int_Q v(x)^{1-p'}\ddd\gamma(x)\r)^{1/{p'}} \\
&\sim_{a,d} \ell(Q)^{\alpha+d/q-d/p} e^{-|c_Q|^2(\dot{\alpha}+1/q-1/p)} \l(\frac{e^{-|c_Q|^2}}{\ell(Q)^d e^{-|c_Q|^2}} \int_Q u(x) \ddd x\r)^{1/q}\l(\frac{e^{-|c_Q|^2}}{\ell(Q)^d e^{-|c_Q|^2}} \int_Q v(x)^{1-p'}\ddd x\r)^{1/{p'}} \\
&=\ell(Q)^{\alpha+d/q-d/p} e^{-|c_Q|^2(\dot{\alpha}+1/q-1/p)} \l(\frac{1}{\ell(Q)^d}\int_Q u(x)\ddd x\r)^{1/q} \l(\frac{1}{\ell(Q)^d}\int_Q v(x)^{1-p'}\ddd x\r)^{1/{p'}}.
\end{align*}
On the other hand, using the estimate (\ref{DHY_LDP}) again we have
\begin{align*}
&\frac{1}{\ell(Q)^{d-\alpha}}\l(\int_Q u(x)e^{-(1-\dot{\alpha})|x|^2}\ddd\gamma'(x)\r)^{1/q}\l(\int_Q v(x)^{1-p'}e^{-(1-\dot{\alpha})|x|^2(1-p')}\ddd\gamma'(x)\r)^{1/{p'}} \\
&=\frac{1}{\ell(Q)^{d-\alpha}} \l(\int_Q u(x)\ddd x\r)^{1/q} e^{-|c_Q|^2/q}\l(\int_Q v(x)^{1-p'}\ddd x\r)^{1/{p'}} e^{-|c_Q|^2(\dot{\alpha}-1/p)} \\
&=\ell(Q)^{\alpha+d/q-d/p} e^{-|c_Q|^2(\dot{\alpha}+1/q-1/p)} \l(\frac{1}{\ell(Q)^d}\int_Q u(x)\ddd x\r)^{1/q} \l(\frac{1}{\ell(Q)^d}\int_Q v(x)^{1-p'}\ddd x\r)^{1/{p'}}.
\end{align*}
These two facts yield the desired result.
\end{proof}

\section{The local Muckenhoupt type weights}\label{DHY_2101_TLMTW}
Due to the Proposition \ref{DHY_RBAA} above, we investigate some properties of the $\mathscr{A}_{p,q,\alpha}^a$ condition. In the one-weight case , Wang~et~al.~\cite{WZL_2016} point out that the local Moukenhoupt weights $A_{p}^a$ on Gaussian measure spaces have the property $A_{p}^a=A_{p}^b$ and then $A_{p,q}^a=A_{p,q}^b$; but in the two-weight case, the similar result $\mathscr{A}_{p,q,\alpha}^a=\mathscr{A}_{p,q,\alpha}^b$ is not always true. To see this fact, we need the following example first.
\begin{example}\label{WZL_2016_prop3.2}
Let $0<a<b<\infty$, $n\in \bZ_{+}$, $u(x)$ and $v(x)$ be even functions on $\bR^1$. When $x\in\bR^{+}$,
\[u(x)=\l\{\begin{array}{ll}
1, & x\in (0,1), \\
n^q, & x\in \l(1+(n-1)a+(n-1)b, 1+(n-1)a+(n-1)b+\frac{b-a}{2}\r), \\
1/{n^q}, & x\in \l(1+(n-1)a+(n-1)b+\frac{b-a}{2}, 1+na+nb\r),
\end{array}\r.\]
with $n\geq1$ and
\[v(x)=\l\{\begin{array}{ll}
1, & x\in(0,1+b), \\
n^p, & x\in \l(1+(n-2)a+(n-1)b, 1+(n-1)a+nb-\frac{b-a}{2}\r),\\
1/{n^p}, & x\in \l(1+(n-1)a+nb-\frac{b-a}{2}, 1+(n-1)a+nb\r),
\end{array}\r.\]
with $n\geq2$. Define
\[[u,v]_{\mathfrak{A}_{p,q,\alpha}^a}=\sup_{Q\in \mathscr{Q}'_a}\frac{1}{\ell(Q)^{1-\alpha}}\l(\int_Q u(x)\ddd x\r)^{1/q} \l(\int_Q v(x)^{1-p'} \ddd x\r)^{1/{p'}},\]
where $\mathscr{Q}'_a=\{Q\subset \bR^d: \ell(Q)\leq a\}$.
If $1/q-1/p+\alpha\geq0$, then
\[[u,v]_{\mathfrak{A}_{p,q,\alpha}^a}<\infty, \quad [u,v]_{\mathfrak{A}_{p,q,\alpha}^b}=\infty.\]
\end{example}

\begin{proof}
We only need to consider all the cubes $Q\in\mathscr{Q}'_{a}$ on $\bR^{+}$ due to the symmetry. Setting $\sigma(x)=v(x)^{1-p'}$, we can draw the function graphs of $u(x)^{1/q}$ and $\sigma(x)^{1/p}$ on $\bR^{+}$.

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\draw[color=black] (7.480414345286288,-0.1957683491553985) node {$A_3$};
\draw [fill=black] (8.5,0) circle (1.5pt);
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\draw [fill=ffffff] (1,1) circle (1pt);
\draw [fill=ffffff] (1.5,1) circle (1pt);
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\draw [fill=ffffff] (4.5,2) circle (1pt);
\end{scriptsize}
\end{axis}
\end{tikzpicture}
\caption{The fuctions $u(x)^{1/q}$ and $\sigma(x)^{1/p}$ on $\bR^+$}
\label{figure_1}
\end{figure}

As shown in \autoref{figure_1}, the green full line segments are the function graph of $\sigma(x)^{1/p}$ and the purple dotted line segments are the function graph of $u(x)^{1/q}$. The lengths of these intervals satisfy the following
\[(B_i,B'_i)=b,\quad(A_i,A'_i)=a,\quad(B_i,A_i)=(A'_i,B'_i)=\frac{b-a}{2}, \quad (B'_i,B_{i+1})=a\]
for every $i\in\bZ_{+}$. Based on this \autoref{figure_1}, we shall prove the desired results more intuitively. Set
\[Q_{n_1}=(B_n,A_n)=\l(1+(n-1)a+(n-1)b, 1+(n-1)a+(n-1)b+\frac{b-a}{2}\r),\]
\[Q_{n_2}=(A'_n,B'_n)=\l(1+(n-1)a+nb-\frac{b-a}{2}, 1+(n-1)a+nb\r)\]
and
\[Q_n=(B_n,B'_n)=\Big(1+(n-1)a+(n-1)b, 1+(n-1)a+nb\Big).\]
For fixed $Q\in\mathscr{Q}'_a$, since $(A_i,A'_i)=(B'_i,B_{i+1})=a$ and $\ell(Q)\leq a$, the open interval $Q$ cannot intersect with more than one of these intervals $Q_{n_1}$ and $Q_{n_2}$. Therefore, when $Q\cap Q_{n_1}\neq\emptyset$ we obtain
\begin{align*}
\frac{1}{\ell(Q)^{1-\alpha}}\l(\int_Q u(x)\ddd x\r)^{1/q} \l(\int_Q v(x)^{1-p'} \ddd x\r)^{1/{p'}}
&\leq \frac{1}{\ell(Q)^{1-\alpha}}\l(\int_Q n^q\ddd x\r)^{1/q} \l(\int_Q \frac{1}{n^{p'}} \ddd x\r)^{1/{p'}} \\
&= \frac{1}{\ell(Q)^{1-\alpha}} \ell(Q)^{1/q+1/{p'}} \\
&= \ell(Q)^{1/q-1/p+\alpha} \leq a^{1/q-1/p+\alpha};
\end{align*}
and when $Q\cap Q_{n_2}\neq\emptyset$ we also obtain
\begin{align*}
\frac{1}{\ell(Q)^{1-\alpha}}\l(\int_Q u(x)\ddd x\r)^{1/q} \l(\int_Q v(x)^{1-p'} \ddd x\r)^{1/{p'}}
&\leq \frac{1}{\ell(Q)^{1-\alpha}}\l(\int_Q \frac{1}{n^q} \ddd x\r)^{1/q} \l(\int_Q n^{p'} \ddd x\r)^{1/{p'}} \\
&= \frac{1}{\ell(Q)^{1-\alpha}} \ell(Q)^{1/q+1/{p'}} \\
&= \ell(Q)^{1/q-1/p+\alpha} \leq a^{1/q-1/p+\alpha},
\end{align*}
where we have used the assumption that $1/q-1/p+\alpha\geq 0$. Hence we have proved the result
\[[u,v]_{\mathfrak{A}_{p,q,\alpha}^a}\leq a^{1/q-1/p+\alpha}<\infty.\]
On the other hand, by chosing $Q=Q_n\in \mathscr{Q}'_b$, we conclude that
\begin{align*}
[u,v]_{\mathfrak{A}_{p,q,\alpha}^b}&=\sup_{Q\in \mathscr{Q}'_b}\frac{1}{\ell(Q)^{d-\alpha}}\l(\int_Q u(x)\ddd x\r)^{1/q} \l(\int_Q v(x)^{1-p'} \ddd x\r)^{1/{p'}} \\
&\geq \frac{1}{\ell(Q_n)^{1-\alpha}}\l(\int_{Q_{n_1}} u(x)\ddd x\r)^{1/q} \l(\int_{Q_{n_2}} v(x)^{1-p'} \ddd x\r)^{1/{p'}} \\
&=\frac{1}{\ell(Q_n)^{1-\alpha}} \cdot n^2 \cdot \ell(Q_{n_1})^{d/q} \cdot \ell(Q_{n_2})^{d/{p'}} \\
&=n^2\frac{\l(\frac{b-a}{2}\r)^{1/q+1/{p'}}}{b^{d-\alpha}}.
\end{align*}
Letting $n\to\infty$, we get the desired result $[u,v]_{\mathfrak{A}_{p,q,\alpha}^b}=\infty$.  This completes the proof.
\end{proof}

\begin{prop}\label{WZL_2016_prop3.2_2}
Let $0<a<b<\infty$, $0\leq\alpha<d$ and $1<p\leq q<\infty$. Then
\[\mathscr{A}_{p,q,\alpha}^b\subsetneqq\mathscr{A}_{p,q,\alpha}^a.\]
\end{prop}

\begin{proof}
It's trivial that $\mathscr{A}_{p,q,\alpha}^b\subset\mathscr{A}_{p,q,\alpha}^a$ and by Remark \ref{DHY_1} together with Propsition \ref{DHY_RBAA} we only need to concentrate on the situation $1/q-1/p+\alpha\geq0$. In other words, it is enough to find $(u,v)$ belongs to $\mathscr{A}_{p,q,\alpha}^a$ but not in $\mathscr{A}_{p,q,\alpha}^b$ under the assumption $1/q-1/p+\alpha\geq0$. Inspired by Example \ref{WZL_2016_prop3.2}, on $\bR^{+}$, we set
\[u(x)=\l\{\begin{array}{ll}
e^{\dot{\alpha}|x|^2}, & x\in (0,1), \\
n^{kq}e^{\dot{\alpha}|x|^2}, & x\in \l(1+(n-1)a+\sum_{i=1}^{n-1}b_i, 1+(n-1)a+\sum_{i=1}^{n-1}b_i+\frac{b_n-a_n}{2}\r), \\
n^{-kq}e^{\dot{\alpha}|x|^2}, & x\in \l(1+(n-1)a+\sum_{i=1}^{n-1}b_i+\frac{b_n-a_n}{2}, 1+na+\sum_{i=1}^{n}b_i\r),
\end{array}\r.\]
with $n\geq1$ and
\[v(x)=\l\{\begin{array}{ll}
e^{\frac{\dot{\alpha}|x|^2}{1-p'}}, & x\in(0,1), \\
n^{kp}e^{\frac{\dot{\alpha}|x|^2}{1-p'}}, & x\in \l(1+(n-2)a+\sum_{i=1}^{n-1}b_i, 1+(n-1)a+\sum_{i=1}^{n}b_i-\frac{b_n-a_n}{2}\r), \\
n^{-kp}e^{\frac{\dot{\alpha}|x|^2}{1-p'}}, & x\in \l(1+(n-1)a+\sum_{i=1}^{n}b_i-\frac{b_n-a_n}{2}, 1+(n-1)a+\sum_{i=1}^{n}b_i\r),
\end{array}\r.\]
with $n\geq2$ where $a_n, b_n$ satisfy the condition
\begin{equation}\label{condition_1}
\l\{\begin{array}{l}
x_1-\frac{b}{2x_1}=1 \\
b_n=\frac{b}{x_n}>0 \\
1+(n-1)a+\sum_{i=1}^{n-1}b_i=x_n-\frac{b}{2x_n} \\
a_n=\frac{a}{x_n}
\end{array}\r.,
\end{equation}
and the parameter $k$ will be chosen later. The exponent terms in $u(x)$, $v(x)$ are used to offset the measure $\ddd\gamma'(x)$ to Lebesgue measure $\ddd x$ and the condition (\ref{condition_1}) can be shown in \autoref{figure_2}.

\begin{figure}[htbp]\centering
\definecolor{ffqqqq}{rgb}{1,0,0}
\definecolor{xdxdff}{rgb}{0.49019607843137253,0.49019607843137253,1}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm]
\begin{axis}[
x=1cm,y=1cm,
axis lines=middle,
xmin=-0.3,
xmax=14.6,
ymin=-1,
ymax=1,
xtick={0,1},
ytick={0},]
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\draw[color=black] (7.52458373510695,0.22481729680176477) node {$A_2$};
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\draw[color=black] (9.1,0.22481729680176477) node {$A'_2$};
\draw [fill=ffqqqq] (8.2,0) circle (2.5pt);
\draw[color=ffqqqq] (8.318196698273647,0.26890690586658217) node {$x_2$};
\draw [fill=ffqqqq] (12.4,0) circle (1.5pt);
\draw[color=ffqqqq] (12.35,0.22481729680176477) node {$B_3$};
\draw [fill=ffqqqq] (14.2,0) circle (1.5pt);
\draw[color=ffqqqq] (14.36341832997056,0.22481729680176477) node {$B'_3$};
\draw [fill=black] (12.6,0) circle (1.5pt);
\draw[color=black] (12.721080392306146,0.22481729680176477) node {$A_3$};
\draw [fill=black] (14,0) circle (1.5pt);
\draw[color=black] (13.95,0.22481729680176477) node {$A'_3$};
\draw [fill=ffqqqq] (13.3,0) circle (2.5pt);
\draw[color=ffqqqq] (13.42651413734321,0.26890690586658217) node {$x_3$};
\end{scriptsize}
\end{axis}
\end{tikzpicture}
\caption{The admissible partition of $\bR^+$}
\label{figure_2}
\end{figure}

As you can see in \autoref{figure_2}, the red intervals $(B_n,B'_n)$ and the black intervals $(A_n,A'_n)$ have the same centers $x_n$. Furthermore, the lengths of these intervals satisfy the following
\[B'_n-B_n=b/{x_n}=b_n,\quad A'_n-A_n=a/{x_n}=a_n, \quad B_{n+1}-B'_n=a\]
so that $(B_n,B'_n)\in\mathscr{Q}_b$, $(A_n,A'_n)\in\mathscr{Q}_a$ and these intervals $(B_n,B'_n)$ are ``far enough to each other''.

By even extension, we get $u(x)$ and $v(x)$ on $\bR^1$. Similarly set
\[Q_{n_1}=(B_n,A_n)=\l(1+(n-1)a+\sum_{i=1}^{n-1}b_i, 1+(n-1)a+\sum_{i=1}^{n-1}b_i+\frac{b_n-a_n}{2}\r),\]
\[Q_{n_2}=(A'_n,B'_n)=\l(1+(n-1)a+\sum_{i=1}^{n}b_i-\frac{b_n-a_n}{2}, 1+(n-1)a+\sum_{i=1}^{n}b_i\r)\]
and
\[Q_n=(B_n,B'_n)=\l(1+(n-1)a+\sum_{i=1}^{n-1}b_i, 1+(n-1)a+\sum_{i=1}^{n}b_i\r).\]
For fixed $Q\in\mathscr{Q}_a$, we should avoid the situation that $Q\cap Q_{n_1}\neq \emptyset$ and $Q\cap Q_{n_2}\neq \emptyset$ hold simultaneously. Actually, this situation may appear if $n=1$. However, we claim that the open interval $Q\in\mathscr{Q}_a$ can not intersect with more than one of these intervals $Q_{n_1}$ and $Q_{n_2}$ if $n\geq2$.

Based on this claim and the assumption $1/q-1/p+\alpha\geq0$, we can follow the scheme of the proof in Example \ref{WZL_2016_prop3.2}. For every fixed $Q\in\mathscr{Q}_a$, if $Q\cap Q_{n}\neq\emptyset$ with $n\geq2$, we conclude that
\begin{align*}
\frac{1}{\ell(Q)^{1-\alpha}}\l(\int_Q  u(x) \ddd  \gamma'(x)\r)^{1/q}\l(\int_Q v(x)^{1-p'}\ddd\gamma'(x)\r)^{1/{p'}}
&\leq \frac{1}{\ell(Q)^{1-\alpha}}\cdot n^k\cdot n^{-k} \cdot |Q|^{1/q} \cdot |Q|^{1/{p'}} \\
&=\ell(Q)^{1/q-1/p+\alpha} \leq a^{1/q-1/p+\alpha},
\end{align*}
where $|Q|$ is the Lebesgue measure of $Q$; if $Q\cap Q_1\neq \emptyset$, we also conclude that
\begin{align*}
&\frac{1}{\ell(Q)^{1-\alpha}}\l(\int_Q  u(x) \ddd  \gamma'(x)\r)^{1/q}\l(\int_Q v(x)^{1-p'}\ddd\gamma'(x)\r)^{1/{p'}} \\
&\leq \frac{1}{\ell(Q)^{1-\alpha}}\cdot |Q|^{1/q} \cdot |Q|^{1/{p'}} \\
&=\ell(Q)^{1/q-1/p+\alpha} \leq a^{1/q-1/p+\alpha}.
\end{align*}
Hence we can obtain the result
\[[u,v]_{\mathscr{A}_{p,q,\alpha}^a}\leq a^{1/q-1/p+\alpha}<\infty.\]
However by chosing $Q=Q_n\in \mathscr{Q}'_b$, we can similarly get
\begin{align*}
[u,v]_{\mathscr{A}_{p,q,\alpha}^b}&\geq \frac{1}{\ell(Q_n)^{1-\alpha}}\l(\int_{Q_{n_1}}  u(x) \ddd  \gamma'(x)\r)^{1/q}\l(\int_{Q_{n_2}} v(x)^{1-p'}\ddd\gamma'(x)\r)^{1/{p'}} \\
&= n^{2k} \cdot \frac{1}{\ell(Q_n)^{1-\alpha}} \cdot |Q_{n_1}|^{1/q} \cdot |Q_{n_2}|^{1/{p'}} \\
&=n^{2k} \cdot \frac{1}{b_n^{1-\alpha}} \cdot \l(\frac{b_n-a_n}{2}\r)^{1/q+1/{p'}} \\
&=\frac{\l(\frac{b-a}{2}\r)^{1/q+1/{p'}}}{b^{1-\alpha}} n^{2k} \cdot |x_n|^{-\alpha-1/q+1/p} \\
&\geq \frac{\l(\frac{b-a}{2}\r)^{1/q+1/{p'}}}{b^{1-\alpha}} n^{2k} \cdot (1+na+nb)^{-\alpha-1/q+1/p},
\end{align*}
where we have used the facts that $\alpha+1/q-1/p\geq0$ and
\[|x_n|\leq 1+na+\sum_{i=1}^n b_i\leq 1+na+nb.\]
Finally by taking $k=\alpha+1/q-1/p+1>0$, we get the desired weights $u(x)$ and $v(x)$.

It remains to prove the claim above. It's easy to see that if $n\geq 2$, then $x_{n}>1+a>\sqrt{a/2}$. Notice that when $x>\sqrt{a/2}$, the function $f(x)=x+\frac{a}{2x}$ is increasing. Hence if $n\geq2$, then for all $0<c_Q\leq1$, we have
\[x_n+\frac{a}{2x_n}>a+1\geq c_Q+a;\]
and for all $1<c_Q<x_n$, we have
\[x_n+\frac{a}{2x_n}>c_Q+\frac{a}{2c_Q}.\]
On the other hand, it is obvious that if $c_Q\geq x_n$, then
\[c_Q-\frac{a}{2c_Q}\geq x_n-\frac{a}{2x_n}.\]
These comments deduce that there is no interval $Q\in\mathscr{Q}_a$ satisfies the following two conditions simultaneously
\begin{equation*}
\l\{
\begin{array}{l}
c_Q+\frac{am(c_Q)}{2}>A'_n=x_n+\frac{a}{2x_n} \\
c_Q-\frac{am(c_Q)}{2}<A_n=x_n-\frac{a}{2x_n}
\end{array}\r.
\end{equation*}
when $n\geq 2$. This finishes the proof of the claim.
\end{proof}

Now we turn to the proof of Theorem \ref{GM_2001_thm4.3}. Firstly, based on the radial versions of the local operators and weights constructed above, we can prove the following two lemmas by imitating the proofs in \cite{GM_2001} and using some basic facts on the Gaussian measure spaces.
\begin{lemma}\label{GM_2001_lemma4.2}
Let $0<a<\infty$ and $1\leq q\leq \frac{d}{d-\alpha}$. Then the pair of weights $(u,v)\in \mathscr{A}_{1,q,\alpha}^a$ if and only if
\begin{equation}\label{GM_2001_lemma4.2_1}
\l(M_{\beta}^au(x)\r)^{1/q}\leq C v(x)
\end{equation}
with $\beta=d-(d-\alpha)q$ for $\gamma'$-almost everywhere $x\in\bR^d$. Indeed, both conditions hold with the same constant.
\end{lemma}

\begin{proof}
Set
\[\widetilde{\mathscr{Q}}=\l\{Q\subset \bR^d: c_Q\in \bQ^d \;\;\text{and}\;\; \ell(Q)\in \bQ^{+}\r\}, \quad \widetilde{\mathscr{Q}}_a=\widetilde{\mathscr{Q}}\cap \mathscr{Q}_a.\]
By a continuity argument, it is enough to consider the cubes just in $\widetilde{\mathscr{Q}}_a$ in the definition of $M_{\alpha}^a$.

Suppose that $(u,v)\in \mathscr{A}_{1,q,\alpha}^a$ with constant $C_0$. For every $Q\in \widetilde{\mathscr{Q}}_a$, define
\[N(Q)=\l\{x\in Q: \frac{1}{\ell(Q)^{d-\alpha}}\l(\int_Q u(y)\ddd\gamma'(y)\r)^{1/q}>C_0 v(x)\r\}, \quad  N=\bigcup_{Q\in\widetilde{\mathscr{Q}}_a} N(Q).\]
Then by the definition of $\mathscr{A}_{1,q,\alpha}^a$ we have $\gamma'(N(Q))=0$ and, since $\widetilde{\mathscr{Q}}_a$ is countable, $\gamma'(N)=0$. Set
\[F=\l\{y\in\bR^d: \l(M_{\beta}^a(u)(y)\r)^{1/q}>C_0 v(y)\r\},\]
where $\frac{d-\beta}{q}=d-\alpha$. Hence for every $y\in F$, there exists a cube $Q\in\widetilde{\mathscr{Q}}_a(y)$ such that
\[\frac{1}{\ell(Q)^{d-\alpha}}\l(\int_Q u(x)\ddd\gamma'(x)\r)^{1/q}>C_0 v(y).\]
This yields that $F\subset N$ and $\gamma'(F)=0$. Equivalently, we have proved
\[\l(M_{\beta}^au(x)\r)^{1/q}\leq C_0 v(x) \quad \gamma'-a.e. \;\;x\in\bR^d.\]

On the other hand if (\ref{GM_2001_lemma4.2_1}) holds with constant $C_0$, then for any fixed $Q\in\widetilde{\mathscr{Q}}_a$ and $\gamma'$-a.e. $y\in Q$ we directly conclude
\begin{align*}
\frac{1}{\ell(Q)^{d-\alpha}}\l(\int_Q u(y)\ddd\gamma'(y)\r)^{1/q}&\leq \l(\sup_{Q\in\mathscr{Q}_a(y)} \frac{1}{\ell(Q)^{(d-\alpha)q}} \int_Q u(x)\ddd\gamma'(x)\r)^{1/q} \\
&=\l(M_{\beta}^a (u)(y)\r)^{1/q} \\
&\leq C_0 v(y).
\end{align*}
\end{proof}

\begin{lemma}\label{GM_2001_thm4.3.1}
Given $a\in(0,\infty)$, $1\leq p\leq q<\infty$, $0\leq\alpha<d$ and a pair of weights $(u,v)$, then the following two statements are equivalent:
\begin{itemize}
\item[(i)] $(u,v)\in \mathscr{A}_{p,q,\alpha}^{a}$.
\item[(ii)] For every $f\geq0$ and every cube $Q\in\mathscr{Q}_a$,
\[\l(\frac{1}{\ell(Q)^{d-\alpha}}\int_Q f(x)\ddd\gamma'(x)\r)^q \l(\int_Q u(x)\ddd\gamma'(x)\r)\leq C\l(\int_Q f(x)^p v(x)\ddd\gamma'(x)\r)^{q/p}.\]
\end{itemize}
\end{lemma}

\begin{proof}
We adapt some ideas from \cite{GM_2001}. Firstly, let us prove the conclusion (ii) with the assumption that $(u,v)\in \mathscr{A}_{p,q,\alpha}^{a}$. In the case $p=1$, for every $f\geq0$ and every $Q\in \mathscr{Q}_a$ we have
\begin{align*}
\l(\frac{1}{\ell(Q)^{d-\alpha}}\int_Q f(x)\ddd\gamma'(x)\r)^q \l(\int_Q u(x)\ddd\gamma'(x)\r)
&=\l[\int_Q f(x)\frac{1}{\ell(Q)^{d-\alpha}} \l(\int_Q u(x)\ddd\gamma'(x)\r)^{1/q}\ddd\gamma'(x)\r]^q \\
&\leq C \l(\int_Q f(x) v(x)\ddd\gamma'(x)\r),
\end{align*}
where the last inequality comes from $(u,v)\in \mathscr{A}_{1,q,\alpha}^a$. In the case $1<p<\infty$, by $\holder$'s inequality
\[\l(\frac{1}{\ell(Q)^{d-\alpha}}\int_Q f(x)\ddd\gamma'(x)\r)^q
\leq \frac{1}{\ell(Q)^{(d-\alpha)q}}\l(\int_Q f(x)^p v(x) \ddd\gamma'(x)\r)^{q/p} \l(\int_Q v(x)^{1-p'} \ddd \gamma'(x)\r)^{q/{p'}}.\]
Thus the condition $(u,v)\in\mathscr{A}_{p,q,\alpha}^a$ yields that
\begin{align*}
&\l(\frac{1}{\ell(Q)^{d-\alpha}}\int_Q f(x)\ddd\gamma'(x)\r)^q \l(\int_Q u(x)\ddd\gamma'(x)\r) \\
&\leq \frac{1}{\ell(Q)^{(d-\alpha)q}}\l(\int_Q f(x)^p v(x) \ddd\gamma'(x)\r)^{q/p} \l(\int_Q v(x)^{1-p'} \ddd \gamma'(x)\r)^{q/{p'}} \l(\int_Q u(x)\ddd\gamma'(x)\r) \\
&\leq [u,v]^q_{\mathscr{A}_{p,q,\alpha}^a} \l(\int_Q f(x)^p v(x) \ddd\gamma'(x)\r)^{q/p}.
\end{align*}

On the other hand, suppose the statement (ii) is right. Take $f\geq 0$. For any $S\subset Q\in\mathscr{Q}_a$, using (ii) with $f\chi_{S}(x)$ we get the following
\begin{equation}\label{GM_2001_thm4.3.1_1}
\l(\frac{1}{\ell(Q)^{d-\alpha}}\int_{S} f(x)\ddd \gamma'(x)\r)^q\l(\int_Q u(x)\ddd\gamma'(x)\r)\leq C\l(\int_S f(x)^p v(x)\ddd \gamma'(x)\r)^{q/p}.
\end{equation}
Then by taking $f\equiv1$ we obtain
\begin{equation}\label{GM_2001_thm4.3.1_2}
\l(\frac{\gamma'(S)}{\ell(Q)^{d-\alpha}}\r)^q\l[u(Q,\gamma')\r]\leq C \l[v(S,\gamma')\r]^{q/p},
\end{equation}
where
\[\gamma'(S)=\int_{S} \ddd \gamma'(x),\quad u(Q,\gamma')=\int_Q u(x)\ddd\gamma'(x), \quad v(S,\gamma')=\int_S v(x)\ddd \gamma'(x).\]
For any $Q\in\mathscr{Q}_a$, from this inequality (\ref{GM_2001_thm4.3.1_2}) we claim that
\[v(x)>0 \;\;\;\gamma'-a.e. \;x\in Q \quad\text{unless} \quad u(x)=0 \;\;\;\gamma'-a.e. \;x\in Q;\]
\[u(Q,\gamma')<\infty \quad \text{unless} \quad v(x)=\infty \;\;\;\gamma'- a.e. \;x\in Q.\]
Indeed, if $v(x)=0$ on some $S\subset Q$ with $\gamma'(S)>0$, from the inequality (\ref{GM_2001_thm4.3.1_2}) we have $u(Q,\gamma')=0$; if $u(Q,\gamma')=\infty$, using this estimate (\ref{GM_2001_thm4.3.1_2}) again, we conclude that $v(S,\gamma')=\infty$ for all $S\subset Q$ with $\gamma'(S)>0$. Hence we finish the proof of the claim.

Once we have done these observations shown above, we can prove this lemma by following the scheme of the proof in \cite[Theorem 4.3]{GM_2001}. For the case $1<p<\infty$, we can obtain the desired conclusion by setting
\[f(x)=v(x)^{1-p'}, \quad S_j=\l\{x\in Q: v(x)>\frac{1}{j}\r\}\]
where $Q\in\mathscr{Q}_a$ and using some locally integrable arguments together with the fact $v(x)>0$ for $\gamma'$-a.e. $x\in Q$; for the case $p=1$, we can get the desired result by rewriting the estimate (\ref{GM_2001_thm4.3.1_2}) as
\[\frac{1}{\ell(Q)^{d-\alpha}}\l(\int_Q u(x) \ddd \gamma'(x)\r)^{1/q}\leq C \frac{v(S,\gamma')}{\gamma'(S)} \quad \forall S\subset Q\in\mathscr{Q}_a \;\;\text{with}\;\; \gamma'(S)>0,\]
considering
\[a>\mathop{\mathrm{ess}\inf}\limits_{Q}:=\inf\l\{t>0: \gamma'\{x\in Q: v(x)<t\}>0\r\}, \quad S_a=\{x\in Q: v(x)<a\}\subset Q\in\mathscr{Q}_a\]
and using a continuity argument. This part is a little long but essentially the same as~\cite[Theorem 4.3]{GM_2001} since the restriction $Q\in\mathscr{Q}_a$ does not metter. Thereby we omit the detailed proof here.
\end{proof}

Next we can use the dyadic analysis and covering theorem together with some techniques on Gaussian measure spaces to establish the two-weight weak-type boundedness of $M_{\alpha}^a$ with respect to the $A_{p,q,\alpha}^a$ condition on Gaussian Lebesgue spaces.
\begin{proof}[\textbf{Proof of Theorem \ref{GM_2001_thm4.3}}]
By using Proposition \ref{DHY_RBAA}, we will prove that if $(u,v)\in\mathscr{A}_{p,q,\alpha}^b$ for some $b>a$, then
\begin{equation}\label{GM_2001_thm4.3_1}
\int_{\{x\in\bR^d: M_{\alpha}^a(f)(x)>\lambda\}} u(x)\ddd\gamma'(x)\lesssim \frac{1}{\lambda^q} \l(\int_{\bR^d}|f(x)|^p v(x)\ddd\gamma'(x)\r)^{q/p}
\end{equation}
holds for every $\lambda>0$.

It is sufficient to concentrate on the nonnegative functions $f\in L^p(\bR^d, v,\gamma')$. By Lemma \ref{GM_2001_thm4.3.1}, we know that the following estimate
\begin{equation}\label{GM_2001_thm4.3_2}
\l(\frac{1}{\ell(Q)^{d-\alpha}}\int_Q f(x)\ddd\gamma'(x)\r)^q \l(\int_Q u(x)\ddd\gamma'(x)\r)\lesssim\l(\int_Q f(x)^p v(x)\ddd\gamma'(x)\r)^{q/p}
\end{equation}
holds for every $f\geq0$ and every $Q\in\mathscr{Q}_b$. When $f\in L^p(\bR^d, v,\gamma')$ and $Q\in \mathscr{Q}_b$ with $u(Q,\gamma')>0$, the inequality (\ref{GM_2001_thm4.3_2}) yields that
\[\int_Q f(x)\ddd\gamma'(x)<\infty.\]
It follows that if $\int_{Q_0} f(x)\ddd\gamma'(x)=\infty$ for some cube $Q_0$, then $u(Q_0,\gamma')=0$ and there is a improper point $x_0$ of the function $f(x)$ contained in the closure of the cube $Q_0$. For this $x_0$, using the estimate (\ref{GM_2001_thm4.3_2}) again we conclude that $u(Q',\gamma')=0$ for all $Q'\in \mathscr{Q}_b(x_0)$. On the other hand, by the definition of $M_{\alpha}^a$ we konw that $M_{\alpha}^a(f)(x)=\infty$ for all $x\in Q'\in \mathscr{Q}_{a}(x_0)$.
Thus we observe that
\[u(x)\equiv0, \quad M_{\alpha}^a(f)(x)\equiv\infty\]
for $\gamma'$-a.e. $x\in\bigcup_{Q'\in \mathscr{Q}_a(x_0)} Q'$. Notice that the desired result (\ref{GM_2001_thm4.3_1}) also holds in this case. Thereby we only need to focus on the situation $x\notin\bigcup_{Q'\in \mathscr{Q}_b(x_0)} Q'$ where $M_{\alpha}^a(f)(x)<\infty$. In this way, without loss of generality, we can assume that
\[\int_Q f(x)\ddd\gamma'(x)<\infty\]
for all $Q\in \mathscr{Q}_a$. By the Heine-Borel theorem, we can assume $f\in L^1_{\mathrm{loc}}(\bR^d, \gamma')$. Moreover, by defining $f_k=f\chi_{Q(0,k)}$, if the estimate (\ref{GM_2001_thm4.3_1}) holds for each $f_k\in L^1(\bR^d, \gamma')$ with the constant independent of $k$, then using Fatou's lemma we can obtain (\ref{GM_2001_thm4.3_1}) for all $f\in L^1_{\mathrm{loc}}(\bR^d, \gamma')$. Hence we can assume $f\in L^1(\bR^d, \gamma')$. Take all these remarks into account, we only need to prove the desired result for
\[f\geq0, \quad f\in L^p(\bR^d, v, \gamma')\cap L^1(\bR^d, \gamma').\]
Define
\[E_{\lambda}:=\{x\in\bR^d: M_{\alpha}^a(f)(x)>\lambda\}.\] Then for every $x\in E_{\lambda}$, there exists a cube $Q_x\in\mathscr{Q}_a(x)$ such that
\[\frac{1}{\ell(Q_x)^{d-\alpha}}\int_{Q_x} |f(y)|\ddd \gamma' (y)>\lambda.\]
Note that $\ell(Q_x)\leq a$. By \cite[Theorem 1.5]{Sawano_2005}, there exists a constant $N=N(d,k)$ depending only on the dimension $d$ and the ratio $k=b/a$ such that
\[E_{\lambda}\subset \bigcup_{x\in E_{\lambda}}Q_x \subset \bigcup_{j=1}^{N} \bigcup_{x_{\rho}\in E_{\lambda, j}} k Q_{x_{\rho}},\]
where the cubes in the subfamilies $\{Q_{x_{\rho}}\}_{x_{\rho}\in E_{\lambda, j}}$ are disjiont for fixed $j$.
Recall that (\ref{GM_2001_thm4.3_2}) leads to (\ref{GM_2001_thm4.3.1_1}). By taking $S=Q_{x_\rho}$ and $Q=kQ_{x_\rho}\in\mathscr{Q}_b$ in this estimate (\ref{GM_2001_thm4.3.1_1}) we can obtain
\begin{align*}
\int_{E_{\lambda}} u(x)\ddd\gamma'(x)&\leq \sum_{j=1}^N \sum_{x_{\rho} \in E_{\lambda, j}} \int_{kQ_{x_\rho}} u(x)\ddd\gamma'(x) \\
&\lesssim \sum_{j=1}^N \sum_{x_{\rho} \in E_{\lambda, j}} \l(\frac{1}{\ell(Q_{x_\rho})^{d-\alpha}} \int_{Q_{x_\rho}} f(x)\ddd\gamma'(x)\r)^{-q} \l(\int_{Q_{x_\rho}} f(x)^p v(x) \ddd\gamma'(x)\r)^{q/p} \\
&\leq \sum_{j=1}^N \sum_{x_{\rho} \in E_{\lambda, j}} \frac{1}{\lambda^q} \l(\int_{Q_{x_\rho}} f(x)^p v(x) \ddd\gamma'(x)\r)^{q/p} \\
&\leq \sum_{j=1}^N \frac{1}{\lambda^q} \l(\sum_{x_{\rho} \in E_{\lambda, j}} \int_{Q_{x_\rho}} f(x)^p v(x) \ddd\gamma'(x)\r)^{q/p} \\
&\leq \frac{N}{\lambda^q}\l(\int_{\bR^d} f(x)^p v(x)\ddd\gamma'(x)\r)^{q/p},
\end{align*}
where we have used the fact $p\leq q$ and the disjointness of $\{Q_{x_{\rho}}\}_{E_{\lambda, j}}$ for fixed $j$.
\end{proof}

Now we turn to the proof of the two-weight weak-type estimate for the local fractional integral operator $\tilde{I}_{\alpha}^a$ with cubes on Gaussian measure spaces. We need to introduce the following lemma first.
\begin{lemma}\label{LM_2020_lemma3.4}
Let $a\in (0,\infty)$, $0\leq \alpha_1<\alpha<\alpha_2\leq d$ and $f\in L_{\mathrm{loc}}^1(\gamma)$. If $\tilde{b}>a$, then there exists a constant $C$ independent of the function $f$ such that
\[|\tilde{I}_{\alpha}^{a}(f)(x)| \leq C\l(M_{\alpha_1}^a (f)(x)\r)^{\frac{\alpha_2-\alpha}{\alpha_2-\alpha_1}} \l(M_{\alpha_2}^{\tilde{b}} (f)(x)\r)^{\frac{\alpha-\alpha_1}{\alpha_2-\alpha_1}}\]
holds for all $x\in \bR^d$.
\end{lemma}
\begin{proof}
We adapt some ideas from \cite[Lemma 3.4]{LM_2020} and \cite[Lemma 4.1]{LX_2017}. Set $\dot{\alpha}_1=\alpha_1/d$ and $\dot{\alpha}_2=\alpha_2/d$. If $x=(x_1, \ldots, x_d) \in \bR^d$, we define the norm $\|\cdot\|$ on $\bR^d$ by
\[\|x\|=\max\{|x_1|,\ldots, |x_d|\}.\]
Then for any $x\in\bR^d$, we divide the proof into two cases.

\textbf{Case 1:} $\gamma\l[Q(x, am(x))\r]\leq \l(\frac{M_{\alpha_2}^{\tilde{b}} (f)(x)}{M_{\alpha_1}^a (f)(x)}\r)^{\frac{1}{\dot{\alpha}_2-\dot{\alpha}_1}}$.

By using the locally reverse doubling property of $\gamma$ on the admissiable cubes $\mathscr{Q}_a$, we obtain a constant $R=R_{d,a}\in (1,\infty)$ such that
\begin{align*}
|\tilde{I}_{\alpha}^a (f)(x)|&\leq \sum_{j=0}^{\infty} \int_{2^{-j-1} am(x)\leq 2\|x-y\|<2^{-j}am(x)} \frac{|f(y)|}{[\gamma(Q(x, |x-y|))]^{1-\dot{\alpha}}} \ddd\gamma(y) \\
&\lesssim \sum_{j=0}^{\infty} \frac{[\gamma(Q(x,2^{-j}am(x)))]^{\dot{\alpha}-\dot{\alpha}_1}}{[\gamma(Q(x,2^{-j}am(x)))]^{1-\dot{\alpha}_1}} \int_{Q(x, 2^{-j}am(x))} |f(y)| \ddd\gamma(y) \\
&\leq [\gamma(Q(x, am(x)))]^{\dot{\alpha}-\dot{\alpha}_1} \sum_{j=0}^{\infty} R^{-j(\dot{\alpha}-\dot{\alpha}_1)} M_{\alpha_1}^a (f)(x) \\
&\lesssim [\gamma(Q(x, am(x)))]^{\dot{\alpha}-\dot{\alpha}_1} M_{\alpha_1}^a (f)(x).
\end{align*}
Combining this estimate and the assumption in this case, we get the desired conclusion that
\begin{align*}
|\tilde{I}_{\alpha}^{a}(f)(x)|&\lesssim \l(M_{\alpha_1}^a (f)(x)\r)^{\frac{\dot{\alpha}_2-\dot{\alpha}}{\dot{\alpha}_2-\dot{\alpha}_1}} \l(M_{\alpha_2}^{\tilde{b}} (f)(x)\r)^{\frac{\dot{\alpha}-\dot{\alpha}_1}{\dot{\alpha}_2-\dot{\alpha}_1}} \\
&=\l(M_{\alpha_1}^a (f)(x)\r)^{\frac{\alpha_2-\alpha}{\alpha_2-\alpha_1}} \l(M_{\alpha_2}^{\tilde{b}} (f)(x)\r)^{\frac{\alpha-\alpha_1}{\alpha_2-\alpha_1}}.
\end{align*}

\textbf{Case 2:} $\gamma\l[Q(x, am(x))\r]> \l(\frac{M_{\alpha_2}^{\tilde{b}} (f)(x)}{M_{\alpha_1}^a (f)(x)}\r)^{\frac{1}{\dot{\alpha}_2-\dot{\alpha}_1}}$.

From \cite[Lemma 2.11]{LSY_2014} we know there exists an $r\in (0,a)$ such that
\begin{equation}\label{LM_2020_lemma3.4_1}
\frac{1}{2}\l(\frac{M_{\alpha_2}^{\tilde{b}} (f)(x)}{M_{\alpha_1}^a (f)(x)}\r)^{\frac{1}{\dot{\alpha}_2-\dot{\alpha}_1}}< \gamma\l[Q(x, rm(x))\r]< \l(\frac{M_{\alpha_2}^{\tilde{b}} (f)(x)}{M_{\alpha_1}^a (f)(x)}\r)^{\frac{1}{\dot{\alpha}_2-\dot{\alpha}_1}}.
\end{equation}
Then we can write the following
\[|\tilde{I}_{\alpha}^a(f)(x)|\leq \int_{2\|x-y\|<rm(x)} \frac{|f(y)|}{[\gamma(Q(x, |x-y|))]^{1-\dot{\alpha}}} \ddd\gamma(y) + \int_{rm(x)\leq 2\|x-y\|<am(x)} \cdots \ddd\gamma(y)=: I_1+I_2.\]
For the term $I_1$, using the similar method in the first case, except for the $a$ replaced by $r$, we conclude that
\[I_1\lesssim [\gamma(Q(x, rm(x)))]^{\dot{\alpha}-\dot{\alpha}_1} M_{\alpha_1}^a (f)(x).\]
For the term $I_2$, setting $N_r:=\lfloor\log_{\frac{\tilde{b}}{a}} \l(\frac{a}{r}\r)\rfloor$ be the maximal integer no more than $\log_{\frac{\tilde{b}}{a}} \l(\frac{a}{r}\r)$ we can get
\begin{align*}
I_2&\leq \sum_{j=0}^{N_r} \int_{\l(\frac{\tilde{b}}{a}\r)^j rm(x)\leq 2\|x-y\|<\l(\frac{\tilde{b}}{a}\r)^{j+1} rm(x)} \frac{|f(y)|}{[\gamma(Q(x, |x-y|))]^{1-\dot{\alpha}}} \ddd\gamma(y) \\
&\leq \sum_{j=0}^{N_r} \frac{1}{\l[\gamma\l(Q\l(x,\l(\frac{\tilde{b}}{a}\r)^{j}rm(x)\r)\r)\r]^{1-\dot{\alpha}}} \int_{Q\l(x,\l(\frac{\tilde{b}}{a}\r)^{j+1}rm(x)\r)} |f(y)| \ddd\gamma(y).
\end{align*}
Note that $Q\l(x,\l(\frac{\tilde{b}}{a}\r)^{j+1} rm(x)\r) \in \mathscr{Q}_{\tilde{b}}$ for all $j=1,\ldots, N_r$. Hence using the similar approach in the first case, by the locally reverse doubling property of $\gamma$ on $\mathscr{Q}_{\tilde{b}}$ we know there exists a constant $\tilde{R}=R_{d,\tilde{b}}\in (1,\infty)$ such that
\begin{align*}
I_2&\lesssim \sum_{j=0}^{N_r} \frac{\l[\gamma\l(Q\l(x, \l(\frac{\tilde{b}}{a}\r)^j rm(x)\r)\r)\r]^{\dot{\alpha}-\dot{\alpha}_2}}{\l[\gamma\l(Q\l(x, \l(\frac{\tilde{b}}{a}\r)^{j+1} rm(x)\r)\r)\r]^{1-\dot{\alpha}_2}} \int_{Q\l(x,\l(\frac{\tilde{b}}{a}\r)^{j+1} rm(x)\r)} |f(y)| \ddd\gamma(y) \\
&\leq \l[\gamma\l(Q\l(x, rm(x)\r)\r)\r]^{\dot{\alpha}-\dot{\alpha}_2} \sum_{j=0}^{N_r} \tilde{R}^{j(\dot{\alpha}-\dot{\alpha}_2)} M_{\alpha_2}^{\tilde{b}} (f)(x) \\
&\leq \l[\gamma\l(Q\l(x, rm(x)\r)\r)\r]^{\dot{\alpha}-\dot{\alpha}_2} \sum_{j=0}^{\infty} \tilde{R}^{j(\dot{\alpha}-\dot{\alpha}_2)} M_{\alpha_2}^{\tilde{b}} (f)(x) \\
&\lesssim  \l[\gamma\l(Q\l(x, rm(x)\r)\r)\r]^{\dot{\alpha}-\dot{\alpha}_2} M_{\alpha_2}^{\tilde{b}} (f)(x).
\end{align*}
Combining these two estimates of $I_1$ and $I_2$, then taking the condition (\ref{LM_2020_lemma3.4_1}) into account we conclude the following desired result
\begin{align*}
|\tilde{I}_{\alpha}^a(f)(x)|&\lesssim [\gamma(Q(x, rm(x)))]^{\dot{\alpha}-\dot{\alpha}_1} M_{\alpha_1}^a (f)(x)+ \l[\gamma\l(Q\l(x, rm(x)\r)\r)\r]^{\dot{\alpha}-\dot{\alpha}_2} M_{\alpha_2}^{\tilde{b}} (f)(x) \\
&\lesssim \l(M_{\alpha_1}^a (f)(x)\r)^{\frac{\alpha_2-\alpha}{\alpha_2-\alpha_1}} \l(M_{\alpha_2}^{\tilde{b}} (f)(x)\r)^{\frac{\alpha-\alpha_1}{\alpha_2-\alpha_1}}.
\end{align*}
Thereby we complete the proof of this lemma.
\end{proof}

\begin{proof}[\textbf{Proof of Theorem \ref{TWWTEFI}}]
Assume that $(u,v)\in A_{p,q,\alpha-\varepsilon}^b$ for some $b>a$ and $\varepsilon>0$. First, the fact $\gamma(Q)<1$ yields that
\[A_{p,q,\alpha_1}^{b}\subset A_{p,q,\alpha_2}^{b}\]
if $0<\alpha_1<\alpha_2<d$. Hence it is sufficient to consider the case $0<\varepsilon<\min\{\alpha, d-\alpha\}$. Then using Lemma \ref{LM_2020_lemma3.4} with
\[\tilde{b}=\frac{a+b}{2}, \quad \alpha_2-\alpha=\alpha-\alpha_1=\varepsilon\]
and using the $\holder$'s inequality for weak spaces (see \cite[Exercise 1.1.15]{GTM_249}) with $q_1=q_2=2q$ we conclude that
\begin{align*}
\l\|\tilde{I}_{\alpha}^a(f)\r\|_{L^{q,\infty}(\bR^d, u, \gamma)}
&\lesssim \l\|\l(M_{\alpha+\varepsilon}^{\frac{a+b}{2}}(f)(\cdot)\r)^{1/2}\l(M_{\alpha-\varepsilon}^a(f)(\cdot)\r)^{1/2}\r\|_{L^{q,\infty}(\bR^d, u, \gamma)} \\
&\leq \l\|\l(M_{\alpha+\varepsilon}^{\frac{a+b}{2}}(f)(\cdot)\r)^{1/2}\r\|_{L^{q_1,\infty}(\bR^d, u, \gamma)} \l\|\l(M_{\alpha-\varepsilon}^a(f)(\cdot)\r)^{1/2}\r\|_{L^{q_2,\infty}(\bR^d, u, \gamma)} \\
&=\l\|M_{\alpha+\varepsilon}^{\frac{a+b}{2}}(f)\r\|^{1/2}_{L^{q,\infty}(\bR^d, u, \gamma)} \l\|M_{\alpha-\varepsilon}^a(f)\r\|^{1/2}_{L^{q,\infty}(\bR^d, u, \gamma)} \\
&\lesssim \|f\|_{L^p(\bR^d, u, \gamma)},
\end{align*}
where the last inequality comes from Theorem \ref{GM_2001_thm4.3}.
\end{proof}

Based on the Theorem \ref{GM_2001_thm4.3} proved above and the one-weight results in~\cite{WZL_2016}, it is natural to ask the following question.
\begin{question}\label{DHY_2101_question1}
Is $(u,v)\in A_{p,q,\alpha}^a$ rather than $(u,v)\in\bigcup_{b'>a} A_{p,q,\alpha}^{b'}$ sufficient for the two-weight weak-type boundedness of $M_{\alpha}^a$?
\end{question}
It is not hard to see that $(u,v)\in A_{p,q,\alpha}^a$ is necessary for the two-weight weak-type boundedness of $M_{\alpha}^a$. This problem (the equivalence) may be solved by means of using other covering theorem instead of \cite[Theorem 1.5]{Sawano_2005} or other more delicate approach, but we have no progress yet. On the other hand, we have the following example to yield that the relation
\[\mathscr{A}_{p,q,\alpha}^a=\bigcup_{b'>a}\mathscr{A}_{p,q,\alpha}^{b'}\]
is not always true. Here we only give the example for the case $a=1$, the method is also valid for general $a>0$.

\begin{example}
Let $u(x)$ and $v(x)$ be even functions. When $x\in\bR^+$,
\[v(x)=\l\{\begin{array}{ll}
x^{\frac{1}{p'-1}}e^{\frac{\dot{\alpha}x^2}{1-p'}}, & 0<x<1, \\
e^{\frac{\dot{\alpha}x^2}{1-p'}}, & otherwise,
\end{array}\r.\]
and
\[u(x)=\l\{\begin{array}{ll}
0, & 0<x<1, \\
e^{\dot{\alpha}x^2}, &otherwise.
\end{array}\r.\]
Then for any $b'>1$ we have
\[[u,v]_{\mathscr{A}_{p,q,\alpha}^1}<\infty, \quad [u,v]_{\mathscr{A}_{p,q,\alpha}^{b'}}=\infty.\]
\end{example}
\begin{proof}
Let's first consider the following
\begin{align*}
[u,v]_{\mathscr{A}_{p,q,\alpha}^{b'}}&=\sup_{(c,c_1)\in\mathscr{Q}_{b'}}\frac{1}{|c_1-c|^{1-\alpha}} \l(\int_{c}^{c_1} u(x)\ddd\gamma'(x)\r)^{1/q}\l(\int_{c}^{c_1} v(x)^{1-p'}\ddd\gamma'(x)\r)^{1/{p'}} \\
&=\sup_{(c,c_1)\in\mathscr{Q}_{b'}}\frac{1}{|c_1-c|^{1-\alpha}} \l(\int_{c}^{c_1} u'(x)\ddd x\r)^{1/q}\l(\int_{c}^{c_1} \sigma'(x) \ddd x\r)^{1/{p'}}
\end{align*}
where $b'>1$ and on $\bR^{+}$
\[\sigma'(x)=\l\{\begin{array}{ll}
1/x, & 0<x<1, \\
1, & otherwise,
\end{array}\r.\quad
u'(x)=\l\{\begin{array}{ll}
0, & 0<x<1, \\
1, &otherwise.
\end{array}\r.\]
We take the interval $(c,c_1):=(0,c')$ where $1<c'<\min\{2,b'\}$. It's easy to see that $(0,c')\in\mathscr{Q}_{b'}$. Hence we conclude
\begin{align*}
[u,v]_{\mathscr{A}_{p,q,\alpha}^{b'}}&\geq \frac{1}{{c'}^{1-\alpha}}\l(\int_1^{c'} \ddd x\r)^{1/q}\l(\int_0^{c'} \frac{1}{x} \ddd x\r)^{1/{p'}} \\
&=\frac{1}{{c'}^{1-\alpha}} \l(c'-1\r)^{1/q} \l(\int_0^{c'} \frac{1}{x} \ddd x\r)^{1/{p'}} \\
&=\infty.
\end{align*}
On the other hand, since
\begin{align*}
[u,v]_{\mathscr{A}_{p,q,\alpha}^{1}}&=\sup_{(c,c_1)\in\mathscr{Q}_1}\frac{1}{|c_1-c|^{1-\alpha}} \l(\int_{c}^{c_1} u(x)\ddd\gamma'(x)\r)^{1/q}\l(\int_{c}^{c_1} v(x)^{1-p'}\ddd\gamma'(x)\r)^{1/{p'}} \\
&=\sup_{(c,c_1)\in\mathscr{Q}_1}\frac{1}{|c_1-c|^{1-\alpha}} \l(\int_{c}^{c_1} u'(x)\ddd x\r)^{1/q}\l(\int_{c}^{c_1} \sigma'(x) \ddd x\r)^{1/{p'}},
\end{align*}
we divide the proof into three cases as follows.

\textbf{Case 1:} If the interval $[c,c_1]\subset [-1,1]$, we have $[u,v]_{\mathscr{A}_{p,q,\alpha}^{1}}=0$ by the definition of $u(x)$;

\textbf{Case 2:} If $c>1$ or $c_1<-1$, we deduce that
\[[u,v]_{\mathscr{A}_{p,q,\alpha}^{1}}\leq\frac{(c_1-c)^{1/q+1/{p'}}}{|c_1-c|^{1-\alpha}}\leq (c_1-c)^{\alpha+1/q-1/p}\leq1;\]

\textbf{Case 3:} If $1\in(c,c_1)$ or $-1\in(c,c_1)$, we only need to prove the result under the situation $1\in(c,c_1)$ by the symmetry. Moreover, by a continuity argument it is enough to consider the behavior of $c_1\to 1$ with $c_1-c=1$. In this case, set $y=c_1-1$. Then it follows that
\begin{align*}
&\frac{1}{|c_1-c|^{1-\alpha}} \l(\int_{c}^{c_1} u(x)\ddd\gamma'(x)\r)^{1/q}\l(\int_{c}^{c_1} v(x)^{1-p'}\ddd\gamma'(x)\r)^{1/{p'}} \\
&=\frac{1}{|c_1-c|^{1-\alpha}} \l(\int_{1}^{1+y} \ddd x\r)^{1/q}\l(\int_{y}^{1} \frac{1}{x}\ddd x+\int_{1}^{1+y} \ddd x\r)^{1/{p'}} \\
&=y^{1/q}\l(-\ln{y}+y\r)^{1/{p'}} \to 0
\end{align*}
as $y\to 0$. Combining all these three cases, we get the desired result $[u,v]_{\mathscr{A}_{p,q,\alpha}^{1}}<\infty$.
\end{proof}

\section{The local Sawyer type weights}\label{DHY_2101_TLSTW}
In this section, we begin with a result about the strict inclusion relation on the local-$a$ testing condition. The similar result on the $\mathscr{A}_{p,q,\alpha}^a$ condition has been proved in Proposition \ref{WZL_2016_prop3.2_2}.
\begin{prop}\label{WZL_2016_prop3.2_1}
Let $0<a<b<\infty$, $0\leq\alpha<d$ and $1<p\leq q<\infty$. Then
\[\mathscr{M}_{p,q,\alpha}^b\subsetneqq\mathscr{M}_{p,q,\alpha}^a.\]
\end{prop}

\begin{proof}
It's trivial that $\mathscr{M}_{p,q,\alpha}^b\subset\mathscr{M}_{p,q,\alpha}^a$ and it is enough to focus on the situation $1/q-1/p+\alpha\geq0$ due to the fact $\mathscr{M}_{p,q,\alpha}^a =\emptyset$ when $1/q-1/p+\alpha<0$. The desired weights are analogous to that in Proposition \ref{WZL_2016_prop3.2_2}. On $\bR^{+}$, we set
\[u(x)=\l\{\begin{array}{ll}
e^{(\dot{\alpha}-1)q|x|^2+|x|^2}, & x\in (0,1), \\
n^{-kq(p-1)}e^{(\dot{\alpha}-1)q|x|^2+|x|^2}, & x\in \l(1+(n-1)a+\sum_{i=1}^{n-1}b_i, 1+(n-1)a+\sum_{i=1}^{n-1}b_i+\frac{b_n-a_n}{2}\r),\\
n^{-kq(p+1)}e^{(\dot{\alpha}-1)q|x|^2+|x|^2}, & x\in \l(1+(n-1)a+\sum_{i=1}^{n-1}b_i+\frac{b_n-a_n}{2}, 1+na+\sum_{i=1}^{n}b_i\r),
\end{array}\r.\]
with $n\geq1$ and then for $n\geq 2$ we set
\[v(x)=\l\{\begin{array}{ll}
e^{\frac{|x|^2}{1-p'}}, & x\in(0,1+b_1), \\
n^{-kp(p-1)}e^{\frac{|x|^2}{1-p'}}, & x\in \l(1+(n-2)a+\sum_{i=1}^{n-1}b_i, 1+(n-1)a+\sum_{i=1}^{n}b_i-\frac{b_n-a_n}{2}\r),\\
n^{kp(p-1)}e^{\frac{|x|^2}{1-p'}}, & x\in \l(1+(n-1)a+\sum_{i=1}^{n}b_i-\frac{b_n-a_n}{2}, 1+(n-1)a+\sum_{i=1}^{n}b_i\r),
\end{array}\r.\]
where $a_n, b_n$ satisfy the same condition (\ref{condition_1}). Then by even extension, we can get $u(x)$ and $v(x)$ on $\bR^1$. For similar reasons we consider the following expression
\begin{equation}\label{WZL_2016_prop3.2_1.1}
\sup_{Q\in\mathscr{Q}_a}\l[\int_Q\l(\sup_{Q'\in\mathscr{Q}_a(x)}\frac{1}{\ell(Q')^{1-\alpha}}\int_{Q'\cap Q} v(y)^{1-p'}\ddd\gamma'(y)\r)^q u(x)\ddd\gamma(x)\r]^{\frac{1}{q}}\l(\int_Q v(x)^{1-p'}\ddd\gamma(x)\r)^{-\frac{1}{p}}
\end{equation}
only on $\bR^{+}$. Using the same notations as in Proposition \ref{WZL_2016_prop3.2_2}, we have
\begin{equation}\label{DHY_2101_equ_1}
u(x)\leq e^{(\dot{\alpha}-1)q|x|^2+|x|^2},\quad v(x)^{1-p'}\leq e^{|x|^2}
\end{equation}
on $(0,1+b_1+a)$ and on $\bR^1\setminus \{\cup Q_{n_1}\cup Q_{n_2}\}=:S$. Hence if $Q\subset S$ or if $Q\cap Q_1\neq\emptyset$ which means $Q\subset (0,1+b_1+a)$, we conclude that
\begin{align*}
&\l[\int_Q\l(\sup_{Q'\in\mathscr{Q}_a(x)}\frac{1}{\ell(Q')^{1-\alpha}}\int_{Q'\cap Q} v(y)^{1-p'}\ddd\gamma'(y)\r)^q u(x)\ddd\gamma(x)\r]^{\frac{1}{q}}\l(\int_Q v(x)^{1-p'}\ddd\gamma(x)\r)^{-\frac{1}{p}} \\
&\lesssim \l[\l(\frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}}\r)^q \ell(Q)\r]^{1/q} \ell(Q)^{-1/p} \\
&\leq \frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}} \ell(Q)^{1/q-1/p} \\
&\leq \ell(Q'\cap Q)^{\alpha}\ell(Q)^{1/q-1/p} \leq \ell(Q)^{\alpha+1/q-1/p} \leq a^{\alpha+1/q-1/p}
\end{align*}
by the estimates (\ref{DHY_2101_equ_1}). Thereby for the expression (\ref{WZL_2016_prop3.2_1.1}), we only need to focus on the case $Q\cap Q_n\neq\emptyset$ with $n\geq 2$ and furthermore the situations $Q\cap Q_{n_1}\neq \emptyset$ or $Q\cap Q_{n_2}\neq\emptyset$ where $n\geq2$.

If $Q\cap Q_{n_1}\neq \emptyset$ with $n\geq2$, then $Q'\in \mathscr{Q}_a$ yields that $Q'\cap Q_{n_2}=\emptyset$ and
\begin{equation}\label{WZL_2016_prop3.2_1.2}
\l(\frac{1}{\ell(Q')^{1-\alpha}}\int_{Q'\cap Q} v(y)^{1-p'}\ddd\gamma'(y)\r)^q \sim_{a} \l(n^{kp}e^{(1-\dot{\alpha})|x|^2} \frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}}\r)^q.
\end{equation}
If $Q\cap Q_{n_2}\neq\emptyset$ with $n\geq2$, from $n^{-kp(p-1)}\leq n^{kp(p-1)}$ and $1-p'\leq 0$ we similarly have
\begin{equation}\label{WZL_2016_prop3.2_1.3}
\l(\frac{1}{\ell(Q')^{1-\alpha}}\int_{Q'\cap Q} v(y)^{1-p'}\ddd\gamma'(y)\r)^q \lesssim_{a} \l(n^{kp}e^{(1-\dot{\alpha})|x|^2}\frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}}\r)^q.
\end{equation}
Based on these two estimates above, when $n\geq2$, for every $Q\in \mathscr{Q}_a$ we divide the proof into two cases. Firstly, if $Q\cap Q_{n_1}\neq \emptyset$, by the estimate (\ref{WZL_2016_prop3.2_1.2}) we have
\begin{align*}
&\l[\int_Q\l(\sup_{Q'\in\mathscr{Q}_a(x)}\frac{1}{\ell(Q')^{1-\alpha}}\int_{Q'\cap Q} v(y)^{1-p'}\ddd\gamma'(y)\r)^q u(x)\ddd\gamma(x)\r]^{1/q} \\
&\lesssim_{a} \l[\int_{Q\cap Q_{n_1}}\l(n^{kp}e^{(1-\dot{\alpha})|x|^2}\frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}}\r)^q e^{(\dot{\alpha}-1)q |x|^2+|x|^2} n^{-kq(p-1)}e^{-|x|^2}\ddd x\r]^{1/q}  \\
&= n^k \frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}} |Q\cap Q_{n_1}|^{1/q} \leq n^k \frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}}\ell(Q)^{1/q}.
\end{align*}
Then from the assumption $1/q-1/p+\alpha\geq0$ we directly get
\begin{align*}
[u,v]_{\mathscr{M}_{p,q,\alpha}^a}&\lesssim_{a} n^k \frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}}\ell(Q)^{1/q} \cdot \l(\int_Q v(x)^{1-p'}\ddd\gamma(x)\r)^{-1/p} \\
&\sim n^k \frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}}\ell(Q)^{1/q} \cdot n^{-k} |Q|^{-1/p} \\
&= \frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}}\ell(Q)^{1/q-1/p} \leq a^{\alpha+1/q-1/p}.
\end{align*}
Secondly, if $Q\cap Q_{n_2}\neq \emptyset$, then the inequality (\ref{WZL_2016_prop3.2_1.3}) holds. Noticing that $u(x)=n^{-kq(p+1)}e^{(\dot{\alpha}-1)q|x|^2+|x|^2}$ on $Q_{n_2}$, we can use an analogous method as in the first case to obtain
\begin{align*}
[u,v]_{\mathscr{M}_{p,q,\alpha}^a}&\lesssim_{a} n^{-k} \frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}} \ell(Q)^{1/q} \cdot \l(\int_Q v(x)^{1-p'}\ddd\gamma(x)\r)^{-1/p} \\
&\lesssim_{a} n^{-k} \frac{\ell(Q'\cap Q)}{\ell(Q')^{1-\alpha}} \ell(Q)^{1/q} \cdot n^{k} |Q|^{-1/p} \\
&\leq a^{\alpha+1/q-1/p}.
\end{align*}
Combining these two cases together with the former cases $Q\cap Q_1\neq\emptyset$ and $Q\subset S$, we have proved the result
\[[u,v]_{\mathscr{M}_{p,q,\alpha}^a}\lesssim a^{\alpha+1/q-1/p}<\infty.\]
Next we consider the term $[u,v]_{\mathscr{M}_{p,q,\alpha}^b}$. By taking $Q=Q_n\in \mathscr{Q}_b$ and $Q'=Q_{n_1}\in \mathscr{Q}_b(y)$ for $y\in Q_{n_1}$, similar to the proof in Proposition \ref{WZL_2016_prop3.2_2}, we deduce the following
\begin{align*}
[u,v]_{\mathscr{m}_{p,q,\alpha}^b}&\geq\l[\int_{Q_{n_1}}\l(\frac{1}{\ell(Q_{n_1})^{d-\alpha}}\int_{Q_{n_1}} v(y)^{1-p'}\ddd \gamma'(y)\r)^q u(x)\ddd \gamma(x)\r]^{1/q}\cdot \l(\int_{Q_{n_2}}v(x)^{1-p'}\ddd\gamma(x)\r)^{-1/p} \\
&\sim_{b} \l(\int_{Q_{n_1}}n^{kpq}e^{(1-\dot{\alpha})q|x|^2}\ell(Q_{n_1})^{\alpha q}\cdot n^{-kq(p-1)}e^{(\dot{\alpha}-1)q|x|^2+|x|^2}e^{-|x|^2}\ddd x\r)^{1/q}\cdot n^k|Q_{n_2}|^{-1/p} \\
&=n^{2k}\ell(Q_{n_1})^{\alpha}\cdot |Q_{n_1}|^{1/q} |Q_{n_2}|^{-1/p} \\
&\geq \l(\frac{b-a}{2}\r)^{\alpha+1/q-1/p} n^{2k} (1+na+nb)^{-\alpha-1/q+1/p},
\end{align*}
where we have used the facts that $\alpha+1/q-1/p\geq0$ and
\[|x_n|\leq 1+na+\sum_{i=1}^n b_i\leq 1+na+nb.\]
Finally taking $k=\alpha+1/q-1/p+1>0$, we get the desired weights.
\end{proof}

Then we consider the two-weight strong-type estimate for $M_{\alpha}^a$ with respect to the local-$a$ testing condition on Gaussian Lebesgue spaces. To use the dyadic analysis in this situation, we need the following lemma first.
\begin{lemma}\label{GM_2001_lemma3.2}
Consider $a\in(0,\infty)$, $\alpha\in[0,d)$, $f\in L_{\mathrm{loc}}^1(\bR^d, \gamma')$ and $f\geq0$. If for some cube $Q\in\mathscr{Q}_a$ and for some $t>0$ we have
\[\frac{1}{\ell(Q)^{d-\alpha}}\int_Q f(y) \ddd\gamma'(y)>t,\]
then there exists a dyadic cube $P\in \mathscr{Q}_{2a+3\sqrt{d}a^2}$ such that $Q\subset 3P$ and
\[\frac{1}{\ell(P)^{d-\alpha}}\int_{P} f(y)\ddd\gamma'(y)>2^{\alpha-2d}t.\]
\end{lemma}

\begin{proof}
Take $k\in\bZ$ such that $2^{k-1}\leq \ell(Q)<2^k$. Then there exist at most $2^d$ dyadic cubes $\{P_i\}_{i=1}^N$ which have the following two properties:
\[\ell(P_i)= 2^k \quad \text{and} \quad Q\cap P_i\neq \emptyset.\]
From $\ell(Q)<\ell(P_i)\leq 2\ell(Q)$ and $Q\cap P_i\neq \emptyset$ we conclude that $Q\subset 3P_i$. Moreover, it is easy to see that at least one of these dyadic cubes, say $P$, satisfies the condition
\[\frac{\int_P f(y) \ddd\gamma'(y)}{\ell(Q)^{d-\alpha}}> \frac{t}{2^d}.\]
As a consequence, for this dyadic cube $P$ we have
\[\frac{1}{\ell(P)^{d-\alpha}}\int_P f(y) \ddd\gamma'(x)>\frac{\ell(Q)^{d-\alpha}t}{\ell(P)^{d-\alpha}2^d}\geq2^{\alpha-2d} t.\]
Then it remains to prove that $P\in \mathscr{Q}_{2a+3\sqrt{d}a^2}$. We divide the proof into four cases as follows.

\textbf{Case 1:} $|c_{Q}|\leq 1$ and $|c_P|\leq 1$, then $\ell(Q)\leq a$ and $\ell(P)\leq2\ell(Q)\leq2a$;

\textbf{Case 2:} $|c_{Q}|\leq 1$ and $|c_P|> 1$, then $\ell(Q)\leq a$ and
\[\ell(P)\cdot|c_P|\leq 2\ell(Q)\cdot\l(|c_Q|+\frac{3\sqrt{d}\ell(Q)}{2}\r)=2\ell(Q)|c_Q|+2\ell(Q)\frac{3\sqrt{d}\ell(Q)}{2}\leq 2a+3\sqrt{d} a^2;\]

\textbf{Case 3:} $|c_{Q}|> 1$ and $|c_P|\leq 1$, then $\ell(Q)\leq a/|c_Q|\leq a$ and $\ell(P)\leq2\ell(Q)\leq2a$;

\textbf{Case 4:} $|c_{Q}|> 1$ and $|c_P|> 1$, then $\ell(Q)\leq a/|c_Q|\leq a$ and
\[\ell(P)\cdot|c_P|\leq 2\ell(Q)\cdot\l(|c_Q|+\frac{3\sqrt{d}\ell(Q)}{2}\r)=2\ell(Q)|c_Q|+2\ell(Q)\frac{3\sqrt{d}\ell(Q)}{2}\leq 2a+3\sqrt{d} a^2.\]
Combining all these four cases, we obtain the desired conclusion $P\in \mathscr{Q}_{2a+3\sqrt{d}a^2}$.
\end{proof}

On the basis of this Lemma \ref{GM_2001_lemma3.2}, by following the scheme of the proof in \cite[Theorem 3.1]{GM_2001} and using the radialization method mentioned before, we prove the two-weighted strong-type boundedness of the local fractional maximal operator $M_{\alpha}^a$ on Gaussian Lebesgue spaces.
\begin{proof}[\textbf{Proof of Theorem \ref{GM_2001_thm3.1}}]
Without loss of generality, we can assume that $f\in L^p(\bR^d, v,\gamma)$ is a nonnegative bounded function with compact support. Thereby we conclude that $M_{\alpha}^a(f)(x)$ is finite $\gamma$-almost everywhere. Decompose $\bR^d$ by
\[\bR^d=\bigcup_{k\in\bZ} \Omega_k, \quad \Omega_k=\{x\in\bR^d: 2^k<M_{\alpha}^a(f)(x)\leq 2^{k+1}\}.\]
Then for every $k\in\bZ$ and every $x\in\Omega_k$, there exists a cube $Q_{x}^k\in\mathscr{Q}_a$ satisfying the condition
\[\frac{1}{\ell(Q_{x}^k)^{d-\alpha}}\int_{Q_{x}^k} f(y)\ddd\gamma'(y)>2^k.\]
By Lemma \ref{GM_2001_lemma3.2} we get a dyadic cube $P_{x}^k\in\mathscr{Q}_{2a+3\sqrt{d}a^2}$ such that $Q_{x}^k\subset 3P_{x}^k$ and
\begin{equation}\label{GM_2001_thm3.1_1}
\frac{1}{\ell(P_{x}^k)^{d-\alpha}}\int_{P_{x}^k} f(y)\ddd\gamma'(y)>2^{\alpha-2d}2^k.
\end{equation}
Notice that $\ell(P_{x}^k)\leq 2a+3\sqrt{d}a^2$. Hence for every fixed $k$, there is a subcollection of maximal disjoint dyadic cubes $\{P_j^k\}_j$ in the sense that for any $Q_{x}^k$ there exists a $P_j^k$ satisfying $Q_{x}^k\subset 3P_{j}^k$. This construction yields that $\Omega_k\subset \bigcup_{j} 3P_{j}^k$. Thus we can decompose $\Omega_k$ by defining
\[E_{1}^k=3P_{1}^k\bigcap \Omega_k, E_2^k=\l(3P_2^k\setminus 3P_1^k\r)\bigcap \Omega_k,\ldots, E_j^k=\l(3P_j^k\setminus \bigcup_{r=1}^{j-1} 3P_r^k\r)\bigcap \Omega_k, \ldots.\]
In conclusion we obtain
\[\bR^d=\bigcup_{k\in\bZ} \Omega_k=\bigcup_{j,k} E_j^k\]
where the sets $E_j^k$ are disjoint and $\Omega_k$ are disjoint. For a fixed integer $K>0$, define
\[\Lambda_K=\{(j,k)\in \bN\times\bZ: |k|\leq K\}.\]
By using the fact $E_j^k\subset\Omega_k$ and the condition (\ref{GM_2001_thm3.1_1}) we deduce
\begin{align*}
I_K&:=\int_{\bigcup_{k=-K}^{K} \Omega_k} \l(M_{\alpha}^a(f)(x)\r)^q u(x)\ddd\gamma(x)=\sum_{(j,k)\in\Lambda_K} \int_{E_j^k} \l(M_{\alpha}^a(f)(x)\r)^q u(x)\ddd\gamma(x) \\
&\leq \sum_{(j,k)\in\Lambda_K} \int_{E_j^k} u(x)\ddd\gamma(x) 2^{(k+1)q} \\
&\leq 2^{(2d-\alpha+1)q} \sum_{(j,k)\in\Lambda_K} \int_{E_j^k} u(x)\ddd\gamma(x)\l(\frac{1}{\ell(P_{j}^k)^{d-\alpha}}\int_{P_{j}^k} f(y)\ddd\gamma'(y)\r)^q \\
&= 2^{(2d-\alpha+1)q}3^{(d-\alpha)q} \sum_{(j,k)\in\Lambda_K} \int_{E_j^k} u(x)\ddd\gamma(x) \l(\frac{1}{\ell(3P_{j}^k)^{d-\alpha}}\int_{3P_{j}^k} \sigma(y)\ddd\gamma'(y)\r)^q \l(\frac{\int_{P_{j}^k} (f\sigma^{-1})(y)\sigma(y)\ddd\gamma'(y)}{\int_{3P_{j}^k} \sigma(y)\ddd\gamma'(y)}\r)^q \\
&=2^{(2d-\alpha+1)q}3^{(d-\alpha)q} \int_{\mathscr{Y}} T_K(f\sigma^{-1})^q \ddd\nu,
\end{align*}
where $\mathscr{Y}=\bN\times\bZ$, $\sigma(x)=v(x)^{1-p'}$. Furthermore, the measure $\nu$ in $\mathscr{Y}$ is given by
\[\nu(j,k):=\int_{E_j^k} u(x)\ddd\gamma(x) \l(\frac{1}{\ell(3P_{j}^k)^{d-\alpha}}\int_{3P_{j}^k} \sigma(y)\ddd\gamma'(y)\r)^q\]
and the operator $T_K$ for measurable function $h$ is defined by the following three equivalent forms
\begin{align*}
T_K(h)(j,k)&:=\frac{\int_{P_{j}^k} h(y)\sigma(y)\ddd\gamma'(y)}{\int_{3P_{j}^k} \sigma(y)\ddd\gamma'(y)}\chi_{\Lambda_K}(j,k) \\
&:\sim_{a,d} \frac{\int_{P_{j}^k} h(y)\sigma(y)\ddd y}{\int_{3P_{j}^k} \sigma(y)\ddd y}\chi_{\Lambda_K}(j,k) \\
&:\sim_{a,d} \frac{\int_{P_{j}^k} h(y)\sigma(y)\ddd\gamma(y)}{\int_{3P_{j}^k} \sigma(y)\ddd\gamma(y)}\chi_{\Lambda_K}(j,k)
\end{align*}
due to the fact that $P_{j}^k\in\mathscr{Q}_{2a+3\sqrt{d}a^2}$. In this way, if we can prove that the operator $T_K$ is bounded from $L^p(\bR^d,\sigma, \gamma)$ to $L^q(\mathscr{Y},\nu)$ independently of $K$, we can obtain
\[I_K\lesssim \int_{\mathscr{Y}} T_K(f\sigma^{-1})^q \ddd\nu \lesssim \l(\int_{\bR^d} (f\sigma^{-1})^p \sigma\ddd\gamma\r)^{q/p}= \l(\int_{\bR^d} f(x)^p v(x)\ddd\gamma(x)\r)^{q/p}.\]
Based on the uniformity of $K$ and the monotone convergence theorem, we shall prove the desired result by letting $K\to\infty$.

Consequently, it remains to prove the uniform boundedness of $T_K$. It is easy to see that
\[T_K: L^{\infty}(\bR^d,\sigma,\gamma)\to L^{\infty}(\mathscr{Y},\nu)\]
with norm equal or less than $1$. Marcinkiewicz interpolation theorem yields that it is sufficient to show
\[T_K: L^{1}(\bR^d,\sigma,\gamma)\to L^{{q}/{p},\infty}(\mathscr{Y},\nu)\]
with norm independent of $K$. In other words, it is sufficient to prove that
\[\nu\l\{(j,k)\in\mathscr{Y}: T_K(h)(j,k)>\lambda\r\}\lesssim \l(\frac{1}{\lambda}\int_{\bR^d} |h(x)|\sigma\ddd\gamma(x)\r)^{q/p}\]
holds for all $\lambda>0$. To establish this estimate, for the fixed nonnegative bounded function $h(x)$ with compact support, set
\[F_{\lambda}=\l\{(j,k)\in\mathscr{Y}: T_K(h)(j,k)>\lambda\r\}=\l\{(j,k)\in\Lambda_K: T_K(h)(j,k)>\lambda\r\}.\]
Noticing the fact $E_j^k\subset 3P_j^k \in\mathscr{Q}_{6a+9\sqrt{d}a^2}$, we conclude
\begin{align*}
\nu(F_{\lambda})&=\sum_{(j,k)\in F_{\lambda}} \int_{E_j^k} u(x)\ddd\gamma(x) \l(\frac{1}{\ell(3P_{j}^k)^{d-\alpha}}\int_{3P_{j}^k} \sigma(y)\ddd\gamma'(y)\r)^q \\
&=\sum_{(j,k)\in F_{\lambda}} \int_{E_j^k} \l(\frac{1}{\ell(3P_{j}^k)^{d-\alpha}}\int_{3P_{j}^k} \sigma(y)\ddd\gamma'(y)\r)^q u(x) \ddd\gamma(x) \\
&\leq \sum_{(j,k)\in F_{\lambda}} \int_{E_j^k} \l(M_{\alpha}^{6a+9\sqrt{d}a^2}\l(\sigma \chi_{3P_j^k}\r)(x)\r)^q u(x) \ddd\gamma(x).
\end{align*}
Taking $\ell(P_j^k)\leq 2a+3\sqrt{d}a^2$ into account, we can extract a maximal disjoint subcollection $\{P_i\}_i$ from the collection $\{P_j^k: (j,k)\in F_{\lambda}\}$ in the sense that for every $(j,k)\in F_{\lambda}$ there exists an $i$ such that $P_{j}^k\subset P_i$. By the disjointness of $E_j^k$, the construction of $P_i$ and the fact $(u,v)\in \mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2}$ we can obtain
\begin{align*}
\nu(F_{\lambda})&\leq \sum_i\sum_{P_j^k\subset P_i} \int_{E_j^k} \l(M_{\alpha}^{6a+9\sqrt{d}a^2}\l(\sigma \chi_{3P_j^k}\r)(x)\r)^q u(x) \ddd\gamma(x) \\
&\leq \sum_i \int_{3P_i} \l(M_{\alpha}^{6a+9\sqrt{d}a^2}\l(\sigma \chi_{3P_i}\r)(x)\r)^q u(x) \ddd\gamma(x) \\
&\lesssim \sum_i\l(\int_{3P_i}\sigma(x) \ddd\gamma(x)\r)^{q/p}.
\end{align*}
Recall that these cubes $P_i$ were extracted from $\l\{P_j^k: (j,k)\in F_{\lambda}\r\}$ where $T_K(h)(j,k)>\lambda$. This yields
\[\int_{3P_i} \sigma(x)\ddd\gamma(x)<\frac{1}{\lambda}\int_{P_i}h(x)\sigma(x)\ddd\gamma(x).\]
Then by using $p\leq q$ and the disjointness of $P_i$ we shall conclude
\begin{align*}
\nu(F_{\lambda})&\lesssim \sum_i \l(\frac{1}{\lambda}\int_{P_i}h(x)\sigma(x)\ddd\gamma(x)\r)^{q/p} \\
&\leq \l(\sum_i \frac{1}{\lambda}\int_{P_i}h(x)\sigma(x)\ddd\gamma(x)\r)^{q/p} \\
&\leq \l(\frac{1}{\lambda}\int_{\bR^d}h(x)\sigma(x)\ddd\gamma(x)\r)^{q/p}.
\end{align*}
All these estimates above are independent of $K$, in other words, we have completed the proof.
\end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{GM_2001_thm6.5}}]
Assume that $(u,v)\in \mathscr{M}_{p,q,\alpha-\varepsilon}^{b}$ for some $b>6a+9\sqrt{d}a^2$ and $\varepsilon>0$. First, the fact $\gamma(Q)<1$ yields that
\[\mathscr{M}_{p,q,\alpha_1}^{b}\subset \mathscr{M}_{p,q,\alpha_2}^{b}\]
if $0<\alpha_1<\alpha_2<d$. Hence it is sufficient to consider the case $0<\varepsilon<\min\{\alpha, d-\alpha\}$. Since $b>6a+9\sqrt{d}a^2$, we can choose $\tilde{a}=\tilde{a}(a,b)>a$ such that $6\tilde{a}+9\sqrt{d}\tilde{a}^2<b$. For example, we can take
\[\tilde{a}=\frac{\frac{ab}{6a+9\sqrt{d}a^2}+a}{2}.\]
Then using Lemma \ref{LM_2020_lemma3.4} with
\[\tilde{b}=\tilde{a}, \quad \alpha_2-\alpha=\alpha-\alpha_1=\varepsilon\]
and using the $\holder$'s inequality with $q_1=q_2=2q$ we conclude that
\begin{align*}
\l\|\tilde{I}_{\alpha}^a(f)\r\|_{L^{q}(\bR^d, u, \gamma)}
&\lesssim \l\|\l(M_{\alpha+\varepsilon}^{\tilde{a}}(f)(\cdot)\r)^{1/2}\l(M_{\alpha-\varepsilon}^a(f)(\cdot)\r)^{1/2}\r\|_{L^{q}(\bR^d, u, \gamma)} \\
&\leq \l\|\l(M_{\alpha+\varepsilon}^{\tilde{a}}(f)(\cdot)\r)^{1/2}\r\|_{L^{q_1}(\bR^d, u, \gamma)} \l\|\l(M_{\alpha-\varepsilon}^a(f)(\cdot)\r)^{1/2}\r\|_{L^{q_2}(\bR^d, u, \gamma)} \\
&=\l\|M_{\alpha+\varepsilon}^{\tilde{a}}(f)\r\|^{1/2}_{L^{q}(\bR^d, u, \gamma)} \l\|M_{\alpha-\varepsilon}^a(f)\r\|^{1/2}_{L^{q}(\bR^d, u, \gamma)} \\
&\lesssim \|f\|_{L^p(\bR^d, u, \gamma)},
\end{align*}
where the last inequality comes from Theorem \ref{GM_2001_thm3.1}.
\end{proof}
As we have discussed in Question \ref{DHY_2101_question1}, there is also a similar problem for the local Sawyer type weights. We state the question as follows.
\begin{question}\label{DHY_2101_question2}
Is $(u,v)\in\mathscr{M}_{p,q,\alpha}^a$ rather than $(u,v)\in\mathscr{M}_{p,q,\alpha}^{6a+9\sqrt{d}a^2}$ sufficient for the two-weight strong-type boundedness of $M_{\alpha}^a$?
\end{question}
It is easy to see that $(u,v)\in\mathscr{M}_{p,q,\alpha}^a$ is necessary for the two-weight strong-type boundedness of $M_{\alpha}^a$. The parameter $6a+9\sqrt{d}a^2$ in our theorem may be smaller by finer estimates. We expect it can be $a$ but have no idea for this yet.

\bigskip\bigskip

\subsection*{Acknowledgements} This work was supported by National Natural Science Foundation of China (Grant Nos. 11871452 and 12071473), Beijing Information Science and Technology University Foundation (Grant Nos. 2025031).

\bigskip\bigskip

\vspace{-0.4cm}{\footnotesize

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

The generative modeling of audio signals is a fundamental problem at the intersection of signal processing and machine learning; recent learning-based algorithms have enabled advances in speech recognition~\citep{hinton2012deep}, audio synthesis \citep{DBLP:journals/corr/OordDZSVGKSK16, mehri2016samplernn}, music recommendation systems~\citep{coviello2012multivariate, wang2014improving, liang2015content}, and in many other areas~\citep{acevedo2009automated}. Audio processing also raises basic research questions pertaining to time series and generative modeling~\citep{haykin2005cocktail,bilmes2004graphical}.

One of the most significant recent advances in machine learning-based audio processing has been the ability to directly model {\em raw signals} in the time domain using neural networks \citep{DBLP:journals/corr/OordDZSVGKSK16, mehri2016samplernn}. Although this affords us the maximum modeling flexibility, it is also computationally expensive, requiring us to handle $>10,000$ audio samples at every second.

In this paper, we explore new lightweight modeling algorithms for audio.
In particular, we focus on a specific audio generation problem called {\em bandwidth extension}, in which the task is to reconstruct high-quality audio from a low-quality, down-sampled input containing only a small fraction (15-50\%) of the original samples. We introduce a new neural network-based technique for this problem that is inspired image super-resolution algorithms~\citep{Dong:2016:ISU:2914182.2914303}, which use machine learning techniques to interpolate a low-resolution image into a higher-resolution one. Learning-based methods often perform better in this context than general-purpose interpolation schemes such as splines because they leverage sophisticated domain-specific models of the appearance of natural signals.

As in image super-resolution, our model is trained on pairs of low and high-quality samples; at test-time, it predicts the missing samples of a low-resolution input signal. Unlike recent neural networks for generating raw audio, our model is fully feedforward and can be run in real-time. In addition to having multiple practical applications, our method also suggests new ways to improve existing generative models of audio.

\subsection{Contributions}

From a practical perspective, our technique has applications in telephony, compression, text-to-speech generation, forensic analysis, and in other domains. It outperforms baselines at $2\times$, $4\times$, and $6\times$ upscaling ratios, while also being significantly simpler than previous methods. Whereas most existing audio enhancement methods make substantial use of signal processing theory, our approach is conceptually very simple and requires no specialized knowledge to implement. Our neural networks are simply trained to map one audio time series into another. Our approach is also among the first to use convolutional architectures for bandwidth extension; as a result, it scales better with dataset size and computational resources relative to current alternatives.

From a generative modeling perspective, our work demonstrates that purely feedforward architectures operating in a non-discretized output space can achieve good performance on an important audio generation task.
This hints at the possibility of designing improved generative models for audio that combine both feedforward and recurrent components.

\section{Setup and background}

\paragraph{Audio processing.}
We represent an audio signal as a function $s(t) : [0,T] \to \mathbb{R}$, where $T$ is the duration of the signal (in seconds) and $s(t)$ is the amplitude at $t$. Taking a digital measurement of $s$ requires us to discretize the continuous function $s(t)$ into a vector $x(t) :  \{\frac{1}{R}, \frac{2}{R},..., \frac{RT}{R}\} \to \mathbb{R}$. We refer to $R$ as the {\em sampling rate} of $x$ (in Hz). Sampling rates may range from 4 KHz (low-quality telephone speech) to 44 Khz (high-fidelity music).

In this work, we interpret $R$ as the resolution of $x$; our goal is to increase the resolution of audio samples by predicting $x$ from a fraction of its samples taken at $\{\frac{1}{R}, \frac{2}{R},..., \frac{RT}{R}\}$. Note that by basic signal processing theory, this is equivalent to predicting the higher frequencies of $x$.

\paragraph{Bandwidth extension.}

Audio upsampling has been studied in the audio processing community under the name {\em bandwidth extension} \citep{ekstrand2002bandwidth,larsen2005audio}. Several learning-based approaches have been proposed, including Gaussian mixture models \citep{cheng1994statistical, park2000narrowband} and neural networks \citep{li2015dnn}. These methods typically involve hand-crafted features and use relatively simple models (e.g., neural networks with at most 2-3 densely connected layers) that are often part of a larger, more complex systems.
In comparison, our method is conceptually simple (operating directly on the raw audio signal), scalable (our neural networks are fully convolutional and fully feed-forward), more accurate, and is also among the few to have been tested on non-speech audio.

\section{Method}

\begin{figure}[t]
\vspace{-7mm}
\begin{center}
\includegraphics[width=12cm]{figures/generator2-compact.png}
\end{center}
\caption{Deep residual network used for audio super-resolution. We extract features via $B$ residual blocks; upscaling is done via stacked SubPixel layers.}
\end{figure}

\subsection{Setup}

Given a low resolution signal $x = \{x_{1/R_1}, ... x_{R_1 T_1 / R_1}\}$ sampled at a rate $R_1$, our goal is to reconstruct a high-resolution version $y = \{y_{1 / R_2}, ... y_{R_2 T_2 / R_2}\}$ of $x$ that has a sampling rate $R_2 > R_1$. For example, $x$ may be a voice signal transmitted via a standard telephone connection at 4 KHz; $y$ may be a high-resolution 16 KHz reconstruction of the orignal.
We use $r = R_2/R_1$ to denote the {\em upsampling ratio} of the two signals, which in our work equals $r=2,4,6$. We thus expect that $y_{r t / R_2} \approx x_{t / R_1}$ for $t = 1,2,...,T_1 R_1$.

To recover the under-defined signal, we learn a model $p(y \vert x)$ of the higher-resolution $y$, conditioned on its low-resolution instantiation $x$.
We assume that the relationship between the time series $x, y$ follows the equation
$y = f_\theta(x) + \epsilon,$
where $\epsilon \sim \mathcal{N}(0,1)$ is Gaussian noise and $f_\theta$ is a model parametrized by $\theta$. Our framework also extends to more complex noise models which the user may provide as a prior or that may be themselves parametrized by the model (similarly to how one parametrizes the normal distribution in a variational autoencoder).

The above formulation naturally leads to a mean squared error (MSE) objective
\begin{equation}
\ell(\mathcal{D}) = \frac{1}{n} \sqrt{ \sum_{i=1}^n ||y_i - f_\theta(x_i)||_2^2 }
\end{equation}
for determining the parameters $\theta$ based on a dataset $\mathcal{D} = \{x_i, y_i\}_{i=1}^n$ of source/target time series pairs. Since our model is fully convolutional, we may take the $x_i, y_i$ to be small patches sampled from the full time series.

\begin{figure}[t]
\begin{center}
\includegraphics[width=14cm]{figures/spectrogram2.pdf}
\end{center}
\vspace{-5mm}
\caption{Audio super-resolution visualized using spectrograms. A high-quality speech signal (leftmost) is subsampled at $r=4$, resulting in the loss of high frequencies (2nd from left). We recover the missing signal using a trained neural network (rightmost), greatly outperforming the cubic baseline (second from right). }
\end{figure}

\subsection{Model Architecture}

We parametrize the function $f$ with a deep convolutional neural network with residual connections; our neural network architecture is based on ideas from \citet{ShiCHTABRW16}, \citet{Dong:2016:ISU:2914182.2914303}, and \citet{pix2pix2016}, and is shown in Figure 1.
We highlight its main features below.

\paragraph{Bottleneck architecture.}

Our model contains $B$ successive downsampling and upsampling {\em blocks}: each performs a convolution, batch normalization, and applies a ReLU non-linearity. Downsampling block $b=1,2,...,B$ contains $\max(2^{6+b}, 512)$ convolutional filters of length $\min(2^{7-b}+1, 9)$ and a stride of $2$. Upsampling block $b$ has $\max(2^{7+(B-b+1)}, 512)$ filters of length $\min(2^{7-(B-b+1)}+1, 9)$.

Thus, at a downsampling step, we halve the spatial dimension and double the filter size; during upsampling, this is reversed. This bottleneck architecture is inspired by auto-encoders, and is known to encourage the model to learn a hierarchy of features. For example, on an audio task, bottom layers may extract wavelet-style features, while higher ones may correspond to phonemes \cite{DBLP:conf/nips/AytarVT16}. Note that the model is fully convolutional, and may run on input sequences of arbitrary length.

\paragraph{Skip connections.}

When the source series $x$ is similar to the target $y$, downsampling features will be also be useful for upsampling \citep{pix2pix2016}. We thus add additional skip connections which stack the tensor of $b$-th downsampling features with the $(B-b+1)$-th tensor of upsampling features. We also add an additive residual connection from the input to the final output: the model thus only needs to learn $y-x$, which in practice speeds up training.

\paragraph{Subpixel shuffling layer.}

In order to increase the time dimension during upscaling, we have implemented a one-dimensional version of the Subpixel layer of \citet{ShiCHTABRW16}, which has been shown to be less prone to produce artifacts \citep{odena2016deconvolution}.

An upscaling block's convolution maps an input tensor of dimension $F \times d$ into one of size $F/2 \times d$.
The subpixel layer reshuffles this $F/2 \times d$ tensor into another one of size $F/4 \times 2d$ (while preserving the tensor entries intact); these are concatenated with $F/4$ features from the downsampling stage, for a final output of size $F/2 \times 2d$. Thus, we have halved the number of filters and doubled the spatial dimension.

\section{Experiments}

\paragraph{Datasets.}

We use the VCTK dataset \citep{yamagishienglish} --- which contains 44 hours of data from 108 different speakers --- and the Piano dataset of \citet{mehri2016samplernn} (10 hours of  Beethoven sonatas).
We generate low-resolution audio signal from the 16 KHz originals by applying an order 8 Chebyshev type I low-pass filter before subsampling the signal by the desired scaling ratio.

We evaluate our method in three regimes.
The {\sc SingleSpeaker} task trains the model on the first 223 recordings of VCTK Speaker 1 (about 30 mins) and tests on the last 8 recordings.
The {\sc MultiSpeaker} task assesses our ability to generalize to new speakers. We train on the first 99 VCTK speakers and test on the 8 remaining ones; our recordings feature different voices and accents (Scottish, Indian, etc.)
Lastly, the {\sc Piano} task extends audio-super resolution to non-vocal data; we use the standard 88\%-6\%-6\% data split.

\paragraph{Methods.}

We compare our method relative to two baselines: a cubic B-spline --- which corresponds to the bicubic upsampling baseline used in image super-resolution --- and the recent neural network-based technique of \citet{li2015dnn},

The latter approach takes as input the short-time Fourier transform (STFT) of the input and predicts directly the phase and the magnitudes of the high frequency components using a dense neural network with three hidden layers of size 2048 and ReLU nonlinearities.
\citet{li2015dnn} have shown that this method is preferred over Gaussian Mixture Models in 84\% of cases in a user study.
This model requires that the scaling ratio be a power of $2$, hence it is not applicable when $r=6$.

We instantiate our model with $B=4$ blocks and train it for 400 epochs on patches of length 6000 (in the high-resolution space) using the ADAM optimizer with a learning rate of $10^{-4}$. To ensure source/target series are of the same length, the source input is pre-processed with cubic upscaling.
We do not compare against previously-proposed matrix factorization techniques \citep{Bansal05bandwidthexpansion,DBLP:conf/ismir/LiangHE13}, as they are typically trained on $<$ 10 input examples \citep{DBLP:conf/mlsp/SunM13} (due to the cost of jointly factorizing a large number of matrices), and do not scale to the size of our datasets.

\paragraph{Metrics}

Given a reference signal $y$ and an approximation $x$, the Signal to Noise Ratio (SNR) is defined as
\begin{equation}
\text{SNR}(x,y) = 10 \log \frac{|| y ||_2^2}{|| x-y ||_2^2}.
\end{equation}
The SNR is a standard metric used in the signal processing literature.
The Log-spectral distance (LSD) \citep{gray1976distance} measures the reconstruction quality of individual frequencies as follows:
\begin{equation}
\text{LSD}(x,y) = \frac{1}{L} \sum_{\ell=1}^L \sqrt{\frac{1}{K} \sum_{k=1}^K \left( X(\ell, k) - \hat X(\ell, k) \right)^2 },
\end{equation}
where $X$ and $\hat X$ are the log-spectral power magnitudes of $y$ and $x$, respectively. These are defined as $ X = \log |S|^2 $, where $S$ is the short-time Fourier transform (STFT) of the signal. We use $\ell$ and $k$ index frames and frequencies, respectively; in our experiments, we used frames of length 2048.

\paragraph{Evaluation}

\begin{wraptable}{r}{0.5\textwidth}
\vspace{-3mm}
\begin{tabular}{rcccc|c}
& \multicolumn{4}{c}{MultiSpeaker Sample} & \\
\hline
 & 1 & 2 & 3 & 4 & Average \\
\hline
Ours & 69 & 75 & 64 & 37 & 61.3 \\
DNN & 51 & 55 & 66 &  53 & 56.3 \\
Spline & 31 & 25 & 38 & 47 & 35.3 \\
\hline
\end{tabular}
\caption{MUSHRA user study scores. We show scores for each sample, averaged individual users. Average across all samples is also displayed}
\label{mos}
\end{wraptable}

The results of our experiments are summarized in Table 2. Our objective metrics show an improvement of 1-5 dB over the baselines, with the strongest improvements at higher upscaling factors.
Although, the spline baseline achieves a high SNR, its signal often lacks higher frequencies; the LSD metric is better at identifying this problem.
Our technique also improves over the DNN baseline; our convolutional architecture appears to use our  modeling capacity more efficiently than a dense neural network, and we expect such architectures will soon be more widely used in audio generation tasks.

\begin{table}[t]
\label{multispeaker}
\begin{center}
\begin{tabular}{lr|rrr|rrr|rrr}
& & \multicolumn{3}{c}{SingleSpeaker}  & \multicolumn{3}{c}{MultiSpeaker} & \multicolumn{3}{c}{Piano}                  \\
\hline
Ratio & Obj. & Spline & DNN & Ours  & Spline & DNN & Ours  & Spline & DNN & Ours \\
\hline
$r=2$    & SNR & 20.3 & 20.1 & 21.1 & 19.7 & 19.9 & 20.7 & 29.4 & 29.3 & 30.1 \\
    & LSD & 4.5 & 3,7 & 3.2 & 4.4 & 3.6 & 3.1 & 3.5 & 3.4 & 3.4 \\
$r=4$    & SNR & 14.8 & 15.9 & 17.1  & 13.0 & 14.9 & 16.1  & 22.2 & 23.0 & 23.5 \\
    & LSD & 8.2 & 4.9 & 3.6  & 8.0 & 5.8 & 3.5 & 5.8 & 5.2 & 3.6 \\
$r=6$    & SNR & 10.4 & n/a & 14.4  & 9.1 & n/a & 10.0 & 15.4 & n/a & 16.1 \\
    & LSD & 10.3 & n/a & 3.4  & 10.1 & n/a & 3.7 & 7.3 & n/a & 4.4 \\
\end{tabular}
\end{center}
\caption{Accuracy evaluation of audio-super resolution methods (in dB) on each of the three super-resolution tasks at upscaling ratios $r=2,4,6$.}
\end{table}

Next, we confirmed our objective experiments with a study in which human raters were asked to assess the quality of super-resolution using a MUSHRA (MUltiple Stimuli with Hidden Reference and Anchor) test. For each trial an audio sample was upscaled using different techniques\footnote{We have posted a our set of samples to: {\texttt https://kuleshov.github.io/audio-super-res/}.}. We collected four VCTK speaker recordings audio samples from the {\sc MultiSpeaker} testing set. For each recording, we collected the original utterance, a downsampled version at $r=4$, as well as signals super-resolved using Splines, DNNs, and our model (six versions in total). We recruited 10 subjects and used an online survey to ask each of them to rate each sample on a scale of 0 (extremely bad) to 100 (excellent) reconstruction.
The results from the experiment are summarized in Table \ref{mos}.
Our method ranked as being the best out of the three upscaling techniques.

\paragraph{Domain adaptation.}

\begin{wraptable}{r}{0.45\textwidth}
\vspace{-3mm}
\begin{small}
\begin{tabular}{rrrrr}
\hline
 & \multicolumn{2}{c}{LPF {\scriptsize (Test)}} & \multicolumn{2}{c}{No LPF {\scriptsize (Test)}} \\
  & SNR & LSD & SNR & LSD  \\
\hline
LPF {\scriptsize (Train)}  & 30.1 & 3.4 & 0.42 & 4.5   \\
No LPF {\scriptsize (Train)}  & 0.43 & 4.4 & 33.2 & 3.3  \\
\hline
\end{tabular}
\caption{Sensitivity of the model to whether low-resolution audio was subject to a low-pass filter (LPF) in dB.
}\label{scaling}
\vspace{-1mm}
\end{small}
\end{wraptable}

We tested the sensitivity of our method to out-of-distribution input via an audio super-resolution experiment in which the training set did not use a low-pass filter, while the test set did, and vice-versa.
We focused on the {\sc Piano} task and $r=2$. The output from the model was noisier than expected, indicating that generalization is an important practical concern.
We suspect this behavior may be common in super-resolution algorithms, but has not been widely documented. A potential solution would be to train on data that has been generated using multiple techniques.

In addition, we examined the ability of our model to generalize from speech to music and vice versa.
We found that switching domains produced noisy output, again highlighting the specialization of the model.

\paragraph{Architectural analysis.}

We examined the importance of our various architectural design choices via an ablation analysis on the {\sc MultiSpeaker} audio super-resolution task using an upscaling ratio of $r=4$. The adjacent figure displays the result: the green-ish line display the validation set $\ell_2$ loss of the original model over time; the yellow curve removes the additive residual connection; the green curve further removes the additive skip connection (while preserving the same total number of filters). This shows that symmetric skip connections are crucial for attaining good performance; additive connections add an additional small, but perceptible, improvement.

\begin{wrapfigure}{r}{0.53\textwidth}
\label{fig:curves}
\vspace{-5mm}
\includegraphics[width=8cm]{figures/ablation.pdf}
\vspace{-7mm}
\begin{small}
\caption{Model ablation analysis on the MultiSpeaker audio super-resolution task with $r=4$.
}\end{small}
\end{wrapfigure}

\paragraph{Computational performance.}

Our model is computationally efficient and can be run in real time.
On the {\sc Piano} task (where all input signals are 12s in length), our method processed a single second of audio in 0.11s on average on a Titan X GPU. Training our models, however, required about 2 days for the {\sc MultiSpeaker} task.
Unlike sequence-to-sequence architectures our model does not require the complete input sequence in order to begin generating an output sequence.

\subsection{Limitations}

Finally, to explore the limits of our approach, we evaluated our method on the MagnaTagATune dataset, which consists of about 200 hours of music from 188 different genres. This dataset is larger and much more diverse that the ones we considered so far. We found that our model underfit the dataset, with very little reduction in the training error, and no improvement over the spline baseline. Other learning-based baselines fared similarly. However, we expect improved results with a larger model and more computational resources.

\section{Previous Work and Discussion}

\paragraph{Time series modeling.}

In the machine learning literature, time series signals have most often been modeled with auto-regressive models, of which variants of recurrent networks are a special case \citep{gers2001applying,45168,mehri2016samplernn}.
Our approach instead generalizes conditional modeling ideas used in computer vision for tasks such as
image super-resolution \citep{Dong:2016:ISU:2914182.2914303,DBLP:journals/corr/LedigTHCATTWS16} or colorization \citep{zhang2016colorful}.

We identify a broad class of conditional time series modeling problems that arise in signal processing, biomedicine, and other fields and that are characterized by a natural alignment among source/target series pairs and differences that are well-represented by local transformations. We propose a general architecture for such problems and show that it works well in different domains.

\paragraph{Bandwidth extension.}

Existing learning-based approaches include Gaussian mixture models \citep{cheng1994statistical,park2000narrowband,pulakka2011speech}, linear predictive coding \citep{bradbury2000linear}, and neural networks \citep{li2015dnn}.
Our work proposes the first convolutional architecture, which we find to scale better with dataset size and outperform recent, specialized methods.
Moreover, while existing techniques involve many hand-crafted features (see e.g., \citet{pulakka2011speech}); our approach is fully domain-agnostic.

\paragraph{Audio applications.} In telephony, commercial efforts are underway to transmit voice at higher rates (typically 16 Khz) in specific handsets; audio-super resolution is a step towards recreating this experience in software. Similar applications could be found in compression, text-to-speech generation, and forensic analysis.
More generally, our work demonstrates the effectiveness of feedforward convolutional architectures on an audio generation task.

\section{Conclusion}

Machine learning techniques based on deep neural networks have been successful at solving under-defined problems in signal processing such as image super-resolution, colorization, in-painting, and many others.
Learning-based methods often perform better in this context than general-purpose algorithms because they leverage sophisticated domain-specific models of the appearance of natural signals.

In this work, we proposed new techniques that use this insight to upsample audio signals. Our technique extends previous work on image super-resolution to the audio domain; it outperforms previous bandwidth extension approaches on both speech and non-vocal music. Our approach is fast and simple to implement, and has applications in telephony, compression, and text-to-speech generation. It also demonstrates the effectiveness of feedforward architectures on an important audio generation task, suggesting new directions for generative audio modeling.