text:
\section{Introduction}

For a measurable function $F: \mathbb{R} ^{n+1}_+:= \mathbb{R} ^n\times (0,\infty) \to  \mathbb{C} $ and $0<r<\infty$, let \begin{equation}
\label{eq:A}
\mathcal{A}_r(F)(x):=\left(\int_0^{\infty}\int_{B(x,t)}|F(y,t)|^r\frac{dy\,dt}{t^{n+1}}\right)^{\frac{1}{r}}, \quad x\in  \mathbb{R} ^n.
\end{equation}

Consider the tent space $T_r^q$, $0<q,r<\infty$, defined as the space of all measurable functions $F$ such that $\mathcal{A}_r(F)\in L^q( \mathbb{R} ^n)$. We also define the weak tent space $wT^q_r$ as the space of all measurable functions $F$ such that $\mathcal{A}_r(F)\in L^{q,\infty}( \mathbb{R} ^n)$.

These spaces play an important role in harmonic analysis as evidenced in \cite{CoifmanMeyerStein}, starting from the use of Lusin area functional on harmonic functions.
They are heavily used in the recent theory of Hardy spaces associated with operators (\cite{HofmannMayboroda}, \cite{HofmannMayborodaMcIntosh}). They also appear if one wants to study maximal regularity operators arising from  some linear or nonlinear partial differential equations (\cite{KT}, \cite{AKMP}). In particular, one wants to  understand  how some (sub)linear operators act on them.  More precisely the following two types of operators appear. First,
$$\mathcal{T}(F)(x,t):= T_{t}(F(\cdot, t))(x),$$
where $T_{t}$ acts on functions on $ \mathbb{R} ^n$. Second,  $$\mathcal{T}(F)(x,t): = \int_{0}^\infty T_{t,s}(F(\cdot, s))(x) \frac{ds}{s}$$ where $T_{t,s}$ acts on functions on $ \mathbb{R} ^n$.  For the second type, we refer to  \cite{AMcR}, \cite{HNP}, \cite{AKMP}.  Positive results on $T^q_{2}$ all rely on the use of $L^2$ off-diagonal estimates (or improved $L^{\min(q,2)}-L^{\max(q,2)}$ off diagonal estimates) and change of angle in the tent space norms.

For the first type, there is a simple sufficient condition that also depends on the change of angle.
Let us assume that $T_{t}$ acts on $L^2$ functions with compact support and
\begin{equation}
\label{eq:off}
\int_{B(x,t)} |T_{t}(f)(y)|^2\, dy \lesssim 2^{-2j\gamma} \int_{C_{j}(B(x,t))} |f(y)|^2\, dy
\end{equation}
with some $\gamma\ge 0$, provided $f$ is supported in $C_{j}(B(x,t))$ where $C_{j}(B(x,t))=B(x,4t)$ if $j=1$ and
$C_{j}(B(x,t))=B(x,2^{j+1}t)\setminus B(x,2^jt)$ when $j\ge2$.
Then
$$
\mathcal{A}_2(\mathcal{T}(F))(x) \lesssim \sum_{j\ge1} 2^{-j\gamma}\mathcal{A}_2^{(2^{j+1})}(F)(x)
$$
where
$\mathcal{A}_2^{(\alpha)}$ is defined as $\mathcal{A}_2$ with $B(x,t)$ replaced by $B(x,\alpha t)$ in \eqref{eq:A}. Using the well known change of angle inequality
$$
\| \mathcal{A}_2^{(\alpha)}F\|_{L^q( \mathbb{R} ^n)}\lesssim \alpha^{n\max(1/2, 1/q)} \|\mathcal{A}_2F\|_{L^q( \mathbb{R} ^n)}
$$
we can conclude for the $T^q_{2}$ boundedness of $\mathcal{T}$ if $\gamma>n\max(1/2, 1/q)$.
Note in particular that if $\gamma\le n/2$, this argument gives no boundedness, even for $q=2$. Often, the operators  $T_{t}$ are assumed to be uniformly  bounded on $L^2( \mathbb{R} ^n)$, which gives $T^2_{2}$ boundedness of  $\mathcal{T}$, whatever $\gamma$. Still, a condition $\gamma>0$ does not seem to guarantee boundedness on $T^q_{2}$ for a range of $q$ about $2$ in general. Thus, there is no available general criterion when $\gamma\le n/2$.

If we let $T_{t}=T$ be independent of $t$ and be a Calder\'on-Zygmund operator, then one obtains
\eqref{eq:off} with $\gamma=n/2$. Similarly we get $\gamma=n/2$ if we let $T_{t}=\mathcal{M}$ be the centered maximal operator. As said, this argument does not apply.

On the other hand, it is well-known that if we replace $\mathcal{A}_2$ by the vertical norm
$\mathcal{V}_2$ where $$
\mathcal{V}_r(F)(x)=\left(\int_0^{\infty}|F(x,t)|^r\frac{dt}{t}\right)^{\frac{1}{r}} , \quad x\in  \mathbb{R} ^n,
$$
then for $T$ being the maximal operator $\mathcal{M}$,
\begin{equation}
\label{eq:FS}
\|\mathcal{V}_r(\mathcal{M}(F))\|_{L^q( \mathbb{R} ^n)} \lesssim \|\mathcal{V}_r(F)\|_{L^q( \mathbb{R} ^n)}
\end{equation}
is the vector-valued maximal inequality of Fefferman-Stein, valid when $1<q,r<\infty$ (\cite{FeffermanStein}). It is thus a natural question whether $\mathcal{V}_r$ can be replaced by $\mathcal{A}_r$, that is whether
the maximal operator, identified with its tensor product with the identity on functions of the $t$ variable, is bounded on $T^q_{r}$.

A modern simple proof of \eqref{eq:FS} is by invoking  extrapolation (see \cite{CruzMartellPerez}): it suffices to prove
$$
\|\mathcal{V}_r(\mathcal{M}(F))\|_{L^r(w)} \lesssim \|\mathcal{V}_r(F)\|_{L^r(w)}
$$
for any $w\in A_{r}$ to obtain \eqref{eq:FS}, and the latter follows from Muckenhoupt's theorem.
Thus we are tempted to follow the same route and indeed, we shall prove
$$
\|\mathcal{A}_r(\mathcal{M}(F))\|_{L^r(w)} \lesssim \|\mathcal{A}_r(F)\|_{L^r(w)}
$$
for any $w\in A_{r}$ using simple upper bounds and known results. We note that the functionals
$\mathcal{V}_r$ and $\mathcal{A}_r$ do not compare on $L^q$ when $q\ne r$,  as shown in   \cite{AuscherHofmannMartell}. Hence,  one cannot deduce such results directly.

For other operators, we shall also show how extrapolation allows us to conclude tent space boundedness:  we will consider Calder\'on-Zygmund operators,  Riesz potentials and fractional maximal functions, in which case, one looks for $T^p_{r}$ to $T^{q}_{r}$ boundedness for some $q>p$.
We will also consider singular non integral operators such as the Riesz transform of elliptic operators to test applicability of our methods. In this case, it is a representation of the operator in the form
$\int_{0}^\infty \theta_{s}\, \frac{ds}{s}
$
that is essential. We obtain tent space boundedness with limited range in $q$ and $r$ that is consistent with that of the $L^p$ theory.

For Calder\'on-Zygmund operators, we shall explore  what happens when $q\le 1$. At $q=1$, we prove a weak-type inequality. We can also take advantage of cancellations in using atomic decompositions at the level of tent spaces.  Then,  atoms need to satisfy the additional condition
$$
\int_{ \mathbb{R} ^n} A(x,t)\, dx=0, \quad \textrm{for a.e.}\, t>0
$$
and we get results for $q> \frac{n}{n+1}$. Imposing more vanishing moments against polynomials allows to get smaller values of $q$ as it is the case with Hardy spaces on $ \mathbb{R} ^n$.

As easy corollaries, we obtain  results in amalgam spaces  in Section  \ref{section:amalgam}.

\section{Main results}\label{sec:main}

As mentioned, if $(T_{t})_{t>0}$ is a family of operators on $ \mathbb{R} ^n$ acting on (some) measurable functions, we let $\mathcal{T}$ defined by
$$
\mathcal{T}(F)(x,t)=T_{t}(F(\cdot, t))(x),
$$
provided the formula makes sense, that is provided $F(\cdot, t)$ belongs to an appropriate domain of $T_{t}$.
If $T$ is a single operator and $T_{t}=T$ for each $t>0$ then $\mathcal{T}= T\otimes I$. In that case and from now on,  we use the same notation by a slight abuse.

We are ready to state our main results. Precise definitions will be given later.

\begin{theorem}\label{thm:maximal}
Let $\mathcal{M}$ be the centered Hardy-Littlewood maximal operator. For all $1<r<\infty$,
\begin{list}{$(\theenumi)$}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm\itemsep=0.2cm\topsep=.0cm \renewcommand{\theenumi}{\alph{enumi}}}

\item
$
\mathcal{M}:T^{q}_r\rightarrow T^q_r
$, for all $1<q<\infty;$

\item $\mathcal{M}:T^{1}_r\rightarrow wT^1_r$.

\end{list}

\end{theorem}
\begin{theorem}\label{thm:C-Zoperator}
Let $\mathcal{T}$ be a Calder\'on-Zygmund operator on $ \mathbb{R} ^n$ of order $\delta\in (0,1]$	.  For all $1<r<\infty,
$
\begin{list}{$(\theenumi)$}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm\itemsep=0.2cm\topsep=.0cm \renewcommand{\theenumi}{\alph{enumi}}}

\item $
\mathcal{T}:T^q_r\rightarrow T^q_r$, for all $1<q<\infty;$

\item $
\mathcal{T}:T^1_r\rightarrow wT^1_r;$

\item  $\mathcal{T}:\mathfrak{T}^q_r\rightarrow T_r^q$,\, for all $\frac{n}{n+\delta}<q\leq 1;$

\item  $\mathcal{T}:\mathfrak{T}^q_r\rightarrow \mathfrak{T}^q_r$,\, for all $\frac{n}{n+\delta}<q\leq 1$, if $\mathcal{T}^*(1)=0$.

\end{list}
\end{theorem}

\begin{theorem}\label{thm:rieszpotential}
For $0<\alpha<n$, $\frac{n}{n-\alpha}<r<\infty$, and $1<p<q<\infty$ such that $\frac{1}{p}-\frac{1}{q}=\frac{\alpha}{n}$,
$$
\mathcal{I}_{\alpha}, \mathcal{M}_{\alpha}:T^p_{r}\rightarrow T^q_{r}.
$$
\end{theorem}%
\begin{theorem}\label{thm:riesztransform} Let $L=-\mathop{\rm div}(A\nabla)$ be an elliptic operator with complex-valued coefficients.
For $q_-(L)<q,r<q_+(L)$ we have
$$
\nabla L^{-\frac{1}{2}}:T^q_r\rightarrow T^q_r.
$$
\end{theorem}

Here is an interesting corollary.

\begin{corollary}\label{co:maximal}
Assume $(T_{t})_{t>0}$ is a family of operators with $\sup_{t>0}|T_{t}(f)|\le (\mathcal{M}|f|^\rho)^{1/\rho}$ for some $\rho\ge 1$. For all $\rho<q,r<\infty$,
\begin{align}\label{maximal:2}
\mathcal{T}:T^{q}_r\rightarrow T^q_r.
\end{align}
\end{corollary}

This applies to the heat semigroup $e^{t^2\Delta}$ or the Poisson semigroup $e^{-t\sqrt {-\Delta}}$.
Note that, in both cases, there is enough decay. Often, the sup norm is too strong an hypothesis. Here is a weaker one, applying for example to semigroups $e^{-t^2L}$ associated to elliptic operators such as the ones in Section \ref{sec:RT}.

\begin{corollary}\label{co:maximal2}
Assume $(T_{t})_{t>0}$ is a family of operators with a kind of reverse H\"older estimate
$$\left(\Xint-_{B(x,t)}|T_{t}(f)(y)|^s dy\right)^{\frac{1}{s}}\le\left(\Xint-_{B(x,\alpha t)}|\mathcal{M} (|f|^\rho)(y)|\ dy\right)^{\frac{1}{\rho}} $$ for some $\alpha>1$ and  some $1 \le \rho < s$, uniformly for all $(x,t)\in  \mathbb{R} ^{n+1}_{+}$. Then, for all $(r,q)$ with $\rho<r \le s$ and $\rho<q<\infty,$
\begin{align}\label{maximal:3}
\mathcal{T}:T^{q}_r\rightarrow T^q_r.
\end{align}
\end{corollary}

This follows from the pointwise inequality
$$
\mathcal{A}_r(\mathcal{T}(F))(x) \le \left(\mathcal{A}^{(\alpha)}_{\frac{r}{\rho}}(\mathcal{M}(|F|^\rho))(x)\right)^{\frac{1}{\rho}}
$$
 with $\mathcal{T}(F)(x,t)= T_{t}(F(\cdot, t))(x)$, and Theorem \ref{thm:maximal}.

\section{Weights}

Since we are going to use some weight theory, let us recall  definitions and some properties. We say that a  function, $w$, is a  weight if  $w\in L^1_{loc}$ and $w(x)>0$ for a.e. $x\in  \mathbb{R} ^n$. For $1<p<\infty$ if $B$ represents a ball in $ \mathbb{R} ^n$ we say that $w\in A_p$ if
$$
\left(\Xint-_Bw(x)dx\right)\left(\Xint-_Bw(x)^{1-p'}dx\right)^{p-1}\leq C, \,\textrm{for all } B\subset  \mathbb{R} ^n.
$$
For $p=1$, $w\in A_1$ if
$$
\Xint-_Bw(y)dy\leq C w(x), \,\textrm{for }a.e.\,x\in B\,\textrm{and for all }B\subset \mathbb{R} ^n.
$$
We introduce also the reverse H\"older classes. For $1<q<\infty$ we say that $w\in RH_{q}$ if
$$
\left(\Xint-_Bw(x)^{q}dx\right)^{\frac{1}{q}}\leq C\Xint-_Bw(x)dx, \,\textrm{for all } B\subset  \mathbb{R} ^n.
$$
And for $q=\infty$, $w\in RH_{\infty}$ if
$$
w(x)\leq C \Xint-_Bw(y)dy, \,\textrm{for }a.e.\,x\in B\,\textrm{and for all }B\subset \mathbb{R} ^n.
$$

We sum up some of the properties of these classes in the following result, see for instance \cite{GCRF85},
\cite{Duo}, or \cite{Grafakos}.

\begin{proposition}\label{prop:weights}\
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\labelenumi}{$(\theenumi)$}
\addtolength{\itemsep}{0.2cm}

\item $A_1\subset A_p\subset A_q$ for $1\le p\le q<\infty$.

\item $RH_{\infty}\subset RH_q\subset RH_p$ for $1<p\le q\le \infty$.

\item If $w\in A_p$, $1<p<\infty$, then there exists $1<q<p$ such
that $w\in A_q$.

\item If $w\in RH_s$, $1<s<\infty$, then there exists $s<r<\infty$ such
that $w\in RH_r$.

\item $\displaystyle A_\infty=\bigcup_{1\le p<\infty} A_p=\bigcup_{1<s\le
\infty} RH_s$.

\item If $1<p<\infty$, $w\in A_p$ if and only if $w^{1-p'}\in
A_{p'}$.

\end{enumerate}
\end{proposition}
\section{Hardy Littlewood maximal operator}
The centered Hardy-Littlewood maximal operator is defined for locally integrable $f$ by
\begin{align*}
\mathcal{M}(f)(x)=\sup_{\tau>0}\Xint-_{B(x,\tau)}|f(y)|\, dy.
\end{align*}
For this operator we use the following pointwise inequality.
\begin{lemma}\label{lemma:H-Lmaximal}
For all $x\in  \mathbb{R} ^n$, $t>0$, and $1<r<\infty$, and all $f$ locally $r$ integrable,  we have that
\begin{align}\label{pointwisemaximal}
\left(\Xint-_{B(x,t)}|\mathcal{M}(f)(y)|^r\, dy\right)^{\frac{1}{r}}
\lesssim
\left(\Xint-_{B(x,2t)}|f(y)|^r\, dy\right)^{\frac{1}{r}}
+
\mathcal{M}_u\left(\Xint-_{B(\cdot,t)}|f(z)|dz\right)(x),
\end{align}
where $\mathcal{M}_u$ is the uncentered maximal operator.
\end{lemma}
\begin{proof}
Fix $x\in  \mathbb{R} ^n$ and $t>0$, and split the  supremum into $0<\tau\leq t$ and $t<\tau$. Then,
\begin{multline*}
\left(\Xint-_{B(x,t)}|\mathcal{M}(f)(y)|^r\, dy\right)^{\frac{1}{r}}
\leq
\left(\Xint-_{B(x,t)}\left(\sup_{0<\tau\leq t}\Xint-_{B(y,\tau)}|f(z)|dz\right)^r\, dy\right)^{\frac{1}{r}}
\\
+
\left(\Xint-_{B(x,t)}\left(\sup_{\tau> t}\Xint-_{B(y,\tau)}|f(z)|dz\right)^r\, dy\right)^{\frac{1}{r}}=:I+II.
\end{multline*}
Now, since, for $0<\tau\leq t$ and $y\in B(x,t)$ it happens that $B(y,\tau)\subset B(x,2t)$,
\begin{multline*}
I
\leq
\left(\Xint-_{B(x,t)}\left(\sup_{0<\tau\leq t}\Xint-_{B(y,\tau)}|f(z)\chi_{B(x,2t)}(z)|dz\right)^r\, dy\right)^{\frac{1}{r}}
\\
\leq
\left(\Xint-_{B(x,t)}|\mathcal{M}(f\chi_{B(x,2t)})(y)|^r\, dy\right)^{\frac{1}{r}}
\lesssim
\left(\Xint-_{B(x,2t)}|f(y)|^r\, dy\right)^{\frac{1}{r}},
\end{multline*}
where in the last inequality we have used that $\mathcal{M}:L^r( \mathbb{R} ^n)\rightarrow L^r( \mathbb{R} ^n)$ (\cite[Theorem 2.5]{Duo}).

As for $II$, note that, for $\xi, z\in  \mathbb{R} ^n$,  $\xi\in B(z,t)\Leftrightarrow z\in B(\xi,t)$, and also that if $z\in B(y,\tau)$, $\xi\in B(z,t)$, and $\tau>t$, then $\xi\in B(y,2\tau)$. Besides, observe that the fact that $x\in B(y,t)$ and $\tau>t$ implies that $x\in B(y,2\tau)$. Hence, applying Fubini's theorem,
\begin{multline*}
II
=
\left(\Xint-_{B(x,t)}\left(\sup_{\tau> t}\Xint-_{B(y,\tau)}|f(y)|\Xint-_{B(z,t)}d\xi\,dz\right)^r\, dy\right)^{\frac{1}{r}}
\\
\leq
\left(\Xint-_{B(x,t)}\left(\sup_{\tau> t}\Xint-_{B(y,2\tau)}\Xint-_{B(\xi,t)}|f(z)|dz\,d\xi\right)^r\, dy\right)^{\frac{1}{r}}
\\
\lesssim
\mathcal{M}_u\left(\Xint-_{B(\cdot,t)}|f(z)|dz\right)(x).
\end{multline*}
Gathering the estimates obtained for $I$ and $II$, we conclude
\eqref{pointwisemaximal}.
\end{proof}

\subsection{Proof of Theorem \ref{thm:maximal}, part $(a)$}
Let $w$ be a Muckenhoupt weight.
We shall prove for all $w\in A_r$ that  for all $F\in T^r_{r}$ (hence $F(\cdot, t)$ is locally $r$ integrable for almost every $t>0$)
\begin{align}\label{maximal:tentspace2}
\int_{ \mathbb{R} ^n}|\mathcal{A}_r(\mathcal{M}(F))(x)|^rw(x)dx
\leq
C \int_{ \mathbb{R} ^n}|\mathcal{A}_r(F)(x)|^rw(x)dx.
\end{align}
From this,  by \cite[Theorem 3.9]{CruzMartellPerez}, we have that, for all $1<q<\infty$, $F\in T^r_{r}$ and $w_0\in A_q$,
\begin{align*}
\int_{ \mathbb{R} ^n}|\mathcal{A}_r(\mathcal{M}(F))(x)|^qw_0(x)dx
\leq
C \int_{ \mathbb{R} ^n}|\mathcal{A}_r(F)(x)|^qw_0(x)dx.
\end{align*}
In particular, for $w_0\equiv 1$, we have that $w_0\in A_q$ for all $1<q<\infty$, then, for all $F\in T^r_{r}$,
\begin{align}\label{maximaldensity}
\|\mathcal{M}F\|_{T^q_r}=\left(\int_{ \mathbb{R} ^n}|\mathcal{A}_r(\mathcal{M}(F))(x)|^qdx\right)^{\frac{1}{q}}
\leq
C \left(\int_{ \mathbb{R} ^n}|\mathcal{A}_r(F)(x)|^qdx\right)^{\frac{1}{q}}=
C\|F\|_{T^q_r}.
\end{align}
Approximating $ T^q_{r}$ functions by compactly supported $T^r_{r}$ functions,  we conclude \eqref{maximaldensity} holds for functions $F\in T^q_r$ by the monotone convergence theorem.

Therefore, to finish the proof it just remains to show \eqref{maximal:tentspace2}. This follows by \eqref{pointwisemaximal}  applied to $f=F(\cdot, t)$
and the fact that $\mathcal{M}_u:L^r(w)\rightarrow L^r(w)$, for all $w\in A_r$ (\cite{Muckenhoupt}). Using these two facts and \cite[Proposition 3.2]{MartellPrisuelos}, for all $w\in A_r$,
\begin{multline*}
\int_{ \mathbb{R} ^n}|\mathcal{A}_r(\mathcal{M}(F))(x)|^rw(x)dx
=
\int_{ \mathbb{R} ^n}\int_0^{\infty}\Xint-_{B(x,t)}|\mathcal{M}(F(\cdot,t))(y)|^r\frac{dy\,dt}{t}\,w(x)dx
\\
\lesssim
\int_{ \mathbb{R} ^n}\int_0^{\infty}\Xint-_{B(x,2t)}|F(y,t)|^r\frac{dy\,dt}{t}\,w(x)dx
+
\int_{ \mathbb{R} ^n}\int_0^{\infty}\left|\mathcal{M}_{u}\left(\Xint-_{B(\cdot,t)}|F(y,t)|dy\right)(x)\right|^r\frac{dt}{t}\,w(x)dx
\\
\lesssim
\int_{ \mathbb{R} ^n}|\mathcal{A}_r(F)(x)|^r\,w(x)dx
+
\int_0^{\infty}\int_{ \mathbb{R} ^n}\left|\mathcal{M}_{u}\left(\Xint-_{B(\cdot,t)}|F(y,t)|dy\right)(x)\right|^rw(x)dx\frac{dt}{t}\,
\\
\lesssim
\int_{ \mathbb{R} ^n}|\mathcal{A}_r(F)(x)|^r\,w(x)dx
+
\int_0^{\infty}\int_{ \mathbb{R} ^n}\left(\Xint-_{B(x,t)}|F(y,t)|dy\right)^rw(x)dx\frac{dt}{t}\,
\\
\lesssim
\int_{ \mathbb{R} ^n}|\mathcal{A}_r(F)(x)|^r\,w(x)dx.
\end{multline*}
\qed
\subsection{Proof of Theorem \ref{thm:maximal}, part $(b)$}
By \eqref{pointwisemaximal} and the change of angle in tent spaces, for all $\lambda>0$, we have that
\begin{align*}
\lambda|\{x\in  \mathbb{R} ^n:\mathcal{A}_r(\mathcal{M}(F))(x)>\lambda\}|\lesssim \|F\|_{T^1_r}+\lambda\left|\left\{x\in  \mathbb{R} ^n:\mathcal{V}_r(\mathcal{M}_u(\widetilde{F}))(x)>\frac{\lambda}{2}\right\}\right|,
\end{align*}
where $\widetilde{F}(x,t):=\Xint-_{B(x,t)}|F(z,t)|\,dz$. Then, applying  the Fefferman-Stein vector-valued weak type $(1,1)$ inequality \cite{FeffermanStein}, we control the second term in the above sum  by
$$
C\int_{ \mathbb{R} ^n}\left(\int_{0}^{\infty}|\widetilde{F}(x,t)|^r\frac{dt}{t}\right)^{\frac{1}{r}}dx\lesssim \|F\|_{T^1_r},
$$
for some constant $C>0$. Therefore, taking the supremum over all $\lambda>0$, conclude that
$$
\|\mathcal{M}F\|_{wT^1_r}\lesssim \|F\|_{T^1_r}.
$$
\qed
\section{Calder\'on-Zygmund operators}\label{section:C-Z}
Recall that $\mathcal{T}$ is a Calder\'on-Zygmund operator of order $\delta \in (0,1]$ if
$\mathcal{T}$ is bounded on $L^2( \mathbb{R} ^n)$ and has a kernel representation
\begin{align*}
\mathcal{T}(f)(x)=\int_{ \mathbb{R} ^n}K(x,y)f(y)\,dy,
\end{align*}
for almost every $x$ not in the support of $f\in L^2( \mathbb{R} ^n)$,
with the kernel, $K$, satisfying the standard conditions: for some $\delta>0$,
\begin{align}\label{kernel1}
|K(x,y)|\leq \frac{C}{|x-y|^n}, \textrm{ for } x\ne y;
\end{align}
\begin{align}\label{kernel2}
|K(x,y)-K(x,z)|\leq C\frac{|y-z|^{\delta}}{|x-y|^{n+\delta}},
 \textrm{ for } |x-y|>2|y-z|;
\end{align}
\begin{align}\label{kernel3}
|K(x,y)-K(w,y)|\leq C \frac{|x-w|^{\delta}}{|x-y|^{n+\delta}}, \textrm{ for } |x-y|>2|x-w|.
\end{align}
Classically, $\mathcal{T}$ extends to a bounded operator on $L^r( \mathbb{R} ^n)$ for $1<r<\infty$  (see for instance \cite[Theorem 5.10]{Duo}) and the kernel representation holds also when $f\in L^r( \mathbb{R} ^n)$.
The following lemma gives us a useful pointwise inequality for Calder\'on-Zygmund operators.
\begin{lemma}\label{lemma:C_Z}
Let $\mathcal{T}$ be a Calder\'on-Zygmund operator and   $f\in L^r( \mathbb{R} ^n)$. We have, for $1<r<\infty$, and for each $x\in  \mathbb{R} ^n$ and all $t>0$,
\begin{align}\label{C-Z:pointwise}
\left(\Xint-_{B(x,t)}|\mathcal{T}(f)(y)|^rdy\right)^{\frac{1}{r}}
\lesssim
\left(\Xint-_{B(x,2t)}|f(y)|^rdy\right)^{\frac{1}{r}}
+
\mathcal{T}_{*}(f)(x)
+\mathcal{M}(f)(x),
\end{align}
where $\mathcal{T}_{*}(f)(x):=\sup_{\varepsilon>0}\left|\int_{|x-y|>\varepsilon}K(x,y)f(y)dy\right|.$
\end{lemma}
\begin{proof}
Fix $x\in  \mathbb{R} ^n$ and $t>0$, consider the ball $B(x,2t)$ and write  $f=f\chi_{B(x,2t)}+f\chi_{ \mathbb{R} ^n\setminus B(x,2t)}=:f_{loc}+f_{glob}$. Then
\begin{equation*}
 \left(\Xint-_{B(x,t)}|\mathcal{T}(f)(y)|^rdy\right)^{\frac{1}{r}}
 \leq
  \left(\Xint-_{B(x,t)}|\mathcal{T}(f_{loc})(y)|^rdy\right)^{\frac{1}{r}}
 +
 \left(\Xint-_{B(x,t)}|\mathcal{T}(f_{glob})(y)|^rdy\right)^{\frac{1}{r}}
 =:I+II.
\end{equation*}
Since  $\mathcal{T}:L^r( \mathbb{R} ^n)\rightarrow L^r( \mathbb{R} ^n)$,
$$
I\lesssim \left(\Xint-_{B(x,2t)}|f(y)|^rdy\right)^{\frac{1}{r}}.
$$
As for $II$,  apply the fact that, for $y\in B(x,t)$, $\{z:|x-z|>2t\}\subset\{z:
|x-z|>2|x-y|\}$ and  \eqref{kernel3}. Then,
\begin{align*}
II&
=
\left(\Xint-_{B(x,t)}\left|\int_{ \mathbb{R} ^n}K(y,z)f_{glob}(z)\,dz\right|^r\,dy\right)^{\frac{1}{r}}
=
\left(\Xint-_{B(x,t)}\left|\int_{|x-z|>2t}K(y,z)f(z)\,dz\right|^r\,dy\right)^{\frac{1}{r}}
\\
&
=
\left(\Xint-_{B(x,t)}\left|\int_{|x-z|>2t}(K(y,z)-K(x,z)+K(x,z))f(z)\,dz\right|^rdy\right)^{\frac{1}{r}}
\\
&\leq
 \left(\Xint-_{B(x,t)}\left(\int_{|x-z|>2t}|K(y,z)-K(x,z)|\, |f(z)|\,dz\right)^rdy\right)^{\frac{1}{r}}
\\
& \quad
 +
 \left(\Xint-_{B(x,t)}\left|\int_{|x-z|>2t}K(x,z)f(z)\,dz\right|^rdy\right)^{\frac{1}{r}}
 \\
&\lesssim
 \left(\Xint-_{B(x,t)}\left(\int_{|x-z|>2t}\frac{|x-y|^{\delta}}{|x-z|^{n+\delta}}\, |f(z)|\,dz\right)^rdy\right)^{\frac{1}{r}}
 +
\left|\int_{|x-z|>2t}K(x,z)f(z)\,dz\right|
 \\
&\lesssim
 \left(\Xint-_{B(x,t)}\left(\sum_{k=0}^{\infty}\int_{2^{k+1}t<|x-z|\leq 2^{k+2}t}\frac{|x-y|^{\delta}}{|x-z|^{n+\delta}}\, |f(z)|\,dz\right)^rdy\right)^{\frac{1}{r}}
+\left|\int_{|x-z|>2t}K(x,z)f(z)\,dz\right|
\\
&\lesssim \sum_{k=0}^{\infty}\frac{1}{2^{k\delta}}
 \left(\Xint-_{B(x,t)}\left(\Xint-_{B(x,2^{k+2}t)}|f(z)|\,dz\right)^rdy\right)^{\frac{1}{r}}
+\left|\int_{|x-z|>2t}K(x,z)f(z)\,dz\right|
\\
&\lesssim \sum_{k=0}^{\infty}\frac{1}{2^{k\delta}}
 \Xint-_{B(x,2^{k+2}t)}|f(z)|\,dz
+\left|\int_{|x-z|>2t}K(x,z)f(z)\,dz\right|
\\
&\lesssim \mathcal{M}(f)(x)
+\mathcal{T}_*(f)(x).
\end{align*}
\end{proof}

\subsection{Proof of Theorem \ref{thm:C-Zoperator}, part $(a)$}
As we said above we first use \eqref{C-Z:pointwise} to prove a weighted version of the case $q=r$ for $\mathcal{T}$. We recall that we use the same notation $\mathcal{T}$ for its extension to tent spaces.

We consider $F\in T^r_{r}$   so that for almost every $t>0$, $F(\cdot, t)\in L^r( \mathbb{R} ^n)$ and all calculations make sense. For a weight $w\in A_r\cap RH_{\infty}$, by \eqref{C-Z:pointwise}, Fubini, the fact that $\mathcal{T}^*, \mathcal{M}:L^r(w)\rightarrow L^r(w)$ (see for instance \cite{CoifmanFefferman}, \cite[Theorem 7.13]{Duo}), and applying \cite[Proof of Proposition 2.3]{AuscherHofmannMartell}, and \cite[Proposition 3.2]{MartellPrisuelos},
\begin{align*}
\Bigg(\int_{ \mathbb{R} ^n}\int_{0}^{\infty}\int_{B(x,t)}&|\mathcal{T}(F(\cdot,t))(y)|^r\frac{dy\,dt}{t^{n+1}}w(x)\,dx\Bigg)^{\frac{1}{r}}
\\
&
\lesssim
\Bigg(\int_{ \mathbb{R} ^n}\int_{0}^{\infty}\int_{B(x,2t)}|F(y,t)|^r\frac{dy\,dt}{t^{n+1}}w(x)\,dx\Bigg)^{\frac{1}{r}}
\\
&\quad+
\left(\int_{ \mathbb{R} ^n}\int_{0}^{\infty}|\mathcal{T}^*(F(\cdot,t))(x)|^r\frac{dt}{t}w(x)\,dx\right)^{\frac{1}{r}}
+
\left(\int_{ \mathbb{R} ^n}\int_{0}^{\infty}|\mathcal{M}(F(\cdot,t))(x)|^r\frac{dt}{t}w(x)\,dx\right)^{\frac{1}{r}}
\\
&\lesssim
\left(\int_{ \mathbb{R} ^n}\int_{0}^{\infty}\int_{B(x,t)}|F(y,t)|^r\frac{dy\,dt}{t^{n+1}}w(x)\,dx\right)^{\frac{1}{r}}
+
\left(\int_{ \mathbb{R} ^n}\int_{0}^{\infty}|F(x,t)|^r\frac{dt}{t}w(x)\,dx\right)^{\frac{1}{2}}
\\
&\lesssim
\left(\int_{ \mathbb{R} ^n}\int_{0}^{\infty}\int_{B(x,t)}|F(y,t)|^r\frac{dy\,dt}{t^{n+1}}w(x)\,dx\right)^{\frac{1}{r}}.
\end{align*}
Therefore, for all $w\in A_r\cap RH_{\infty}$ and $F\in T^r_{r}$,
\begin{align}\label{extra:C-Z}
\int_{ \mathbb{R} ^n}\int_{0}^{\infty}\int_{B(x,t)}|\mathcal{T}(F(\cdot,t))(y)|^r\frac{dy\,dt}{t^{n+1}}w(x)\,dx
\lesssim
\int_{ \mathbb{R} ^n}\int_{0}^{\infty}\int_{B(x,t)}|F(y,t)|^r\frac{dy\,dt}{t^{n+1}}w(x)\,dx.
\end{align}
In particular for $w\equiv 1$ and $F$ as above,
$$
\|\mathcal{T}(F)\|_{T_r^r}\lesssim \|F\|_{T^r_r},
$$
where the estimate does not depend on $F$. This proves the case $q=r$.
Note now that in view of \eqref{extra:C-Z}, we can apply \cite[Theorem 3.31]{CruzMartellPerez}, for $p_-=1$ and $p_+=r$. Then, we obtain that, for all
$1<q<r$ and $w_0\in A_q\cap RH_{\left(\frac{r}{q}\right)'}$, and all $F\in T^r_{r}$,
\begin{align*}
\int_{ \mathbb{R} ^n}\left(\int_{0}^{\infty}\int_{B(x,t)}|\mathcal{T}(F(\cdot,t))(y)|^r\frac{dy\,dt}{t^{n+1}}\right)^{\frac{q}{r}}w_0(x)\,dx
\lesssim
\int_{ \mathbb{R} ^n}\left(\int_{0}^{\infty}\int_{B(x,t)}|F(y,t)|^r\frac{dy\,dt}{t^{n+1}}\right)^{\frac{q}{r}}w_0(x)\,dx.
\end{align*}
Hence, taking $w_0\equiv 1$, we have in particular that $w_0\in A_q\cap RH_{\left(\frac{r}{q}\right)'}$. Then, for $1<q<r$ and all $F\in T^r_{r}$,
$$
\|\mathcal{T}(F)\|_{T_r^q}\lesssim \|F\|_{T^q_r}.
$$
We conclude by density of $T^r_{r}\cap T^q_{r}$ into $T^q_{r}$.

 In order to prove the boundedness for $1<r<q<\infty$, we use a duality argument. Take $F\in T_r^q\cap T^r_{r}$ and $G\in T^{q'}_{r'}\cap T^{r'}_{r'}$.
By the previous argument and dualization we obtain,
$$
\int_{ \mathbb{R} ^n}\int_0^{\infty}|F(y,t)\widetilde{\mathcal{T}}(G(\cdot,t))(y)|\frac{dt\,dy}{t}
\lesssim \|F\|_{T^q_r}\|G\|_{T^{q'}_{r'}},
$$
where $\widetilde{\mathcal{T}}$ is the adjoint of $\mathcal{T}$.
Also
$$
\int_{ \mathbb{R} ^n}\int_0^{\infty}\left|\mathcal{T}(F(\cdot,t))(x)G(x,t)\right|\frac{dt\,dx}{t} \lesssim  \|F\|_{T^r_r}\|G\|_{T^{r'}_{r'}} <\infty$$
 Thus,   Fubini's theorem
and
$$
\int_{ \mathbb{R} ^n}\mathcal{T}(F(\cdot,t))(x)G(x,t)\, dx= \int_{ \mathbb{R} ^n}F(y,t)\widetilde{\mathcal{T}}(G(\cdot,t))(y)\,dy
$$
yield
$$
\left|\int_{ \mathbb{R} ^n}\int_0^{\infty}\mathcal{T}(F(\cdot,t))(x)G(x,t)\frac{dt\,dx}{t}\right|
=
\left|\int_{ \mathbb{R} ^n}\int_0^{\infty}F(y,t)\widetilde{\mathcal{T}}(G(\cdot,t))(y)\frac{dt\,dy}{t}\right|
\lesssim \|F\|_{T^q_r}\|G\|_{T^{q'}_{r'}}.
$$
Finally, taking the supremum over all $G$ as above, such that $\|G\|_{T^{q'}_{r'}}\leq 1$,  we conclude that, for all $F \in T^q_{r}\cap T^r_{r}$,
$
\|\mathcal{T}(F)\|_{T^q_r}\lesssim \|F\|_{T^q_r}.
$
By density, this allows to extend the action of $\mathcal{T}$ to all $F\in T^q_{r}$.
\qed

Remark that
$$
\int_{ \mathbb{R} ^n}\int_0^{\infty}\left|\mathcal{T}(F(\cdot,t))(x)G(x,t)\right|\frac{dt\,dx}{t}<\infty
$$
 for all $F\in T^q_{r}$ and all $G\in T^{q'}_{{r'}}$ when $q=r$. But when $q\ne r$, the argument does not allow to conclude for the convergence of this integral for arbitrary  $F\in T^q_{r}$ and $G\in T^{q'}_{{r'}}$. Of course, this inequality holds for the extension of $\mathcal{T}$ on $T^q_{r}$.
\subsection{Proof of Theorem \ref{thm:C-Zoperator}, part $(b)$} Let $F\in T^r_{r} \cap T^1_{r}$, which is dense in $T^1_{r}$.  It follows from \eqref{C-Z:pointwise} that
$$
\mathcal{A}_r(\mathcal{T}(F)) \lesssim \mathcal{A}_r^{(2)}(F) + \mathcal{V}_r(\mathcal{M}(F)) + \mathcal{V}_r(\mathcal{T}_{*}(F)).
$$
We need to estimate the $L^{1,\infty}( \mathbb{R} ^n)$ norm of each term.

The first  term  has $L^1( \mathbb{R} ^n)$ norm  controlled  by $c\|F\|_{T^1_r}$ for some constant $c>0$ by change of angle in tent spaces.

For the second one, one applies Fefferman-Stein  vector-valued weak type $(1,1)$ inequality and then, the fact that the norm in $L^1( \mathbb{R} ^n)$ of the vertical function $\mathcal{V}_{r}(F)$ is controlled by the norm in $L^1( \mathbb{R} ^n)$ of the conical function $\mathcal{A}_r(F)$ (see \cite{AuscherHofmannMartell}).

For the third term, the needed weak type estimate is
$$
\left|\left\{x\in  \mathbb{R} ^n:\mathcal{V}_r(\mathcal{T}_{*}(F))(x)>{\lambda}\right\}\right|\lesssim \frac{\|\mathcal{V}_r(F)\|_{L^1( \mathbb{R} ^n)}}{\lambda}.
$$
It should be known but as we have not been able to locate a proof, we provide one for the reader's comfort. Once this is proved, we use again the  result in \cite{AuscherHofmannMartell} mentioned above.

Fix $\lambda>0$ and consider the set
$$
\Omega_{\lambda}:=\{x\in  \mathbb{R} ^n:\mathcal{M}_u(\mathcal{V}_r(F))(x)>\lambda\},
$$
where we  recall that $\mathcal{M}_u$ represents the uncentered Hardy-Littlewood maximal operator. We have that $\Omega_\lambda$ is open and, since $\|\mathcal{V}_r(F)\|_{L^1( \mathbb{R} ^n)}<\infty$, we conclude that $|\Omega_{\lambda}|<\infty$. Therefore, we can take a Whitney decomposition $\Omega_{\lambda}=\bigcup_{i\in  \mathbb{N} }Q_i$, where $Q_i$ are dyadic and disjoint cubes such that
$$
\sqrt{n}\ \ell(Q_i)\le \textrm{dist}(Q_i, \mathbb{R} ^n\setminus \Omega_{\lambda})<4
\sqrt{n}\ \ell(Q_i).
$$
Hence,
$$
\mathcal{V}_r(F)(x)\leq \lambda, \, \textrm{ for a.e.}\, x\in  \mathbb{R} ^{n}\setminus \Omega_{\lambda} , \quad \Xint-_{Q_i}|\mathcal{V}_r(F)(x)|\,dx\leq 8^n\lambda, \quad \textrm{and}\quad |\Omega_{\lambda}|\leq\frac{1}{\lambda}\int_{\Omega_\lambda}|\mathcal{V}_r(F)(x)|\,dx.
$$
Then if we set
$$
\mathcal{G}=\mathcal{V}_r(F)\chi_{ \mathbb{R} ^n\setminus \Omega_{\lambda}}+\sum_{i\in  \mathbb{N} }\left(\Xint-_{Q_i}\mathcal{V}_r(F)\right)\chi_{Q_i}, \,\textrm{ and }\,
\mathcal{B}=\sum_{i\in  \mathbb{N} }\left(\mathcal{V}_r(F)-\Xint-_{Q_i}\mathcal{V}_r(F)\right)\chi_{Q_i}
$$
we have that $\mathcal{V}_r(F)=\mathcal{G}+\mathcal{B}$ is a Calder\'on-Zygmund decomposition of $\mathcal{V}_r(F)$ at heigh $\lambda$ satisfiying:
$$
|\mathcal{G}(x)|\leq 10^n\lambda,\,\textrm{for a.e. }x\in  \mathbb{R} ^n,\quad \|\mathcal{G}\|^r_{L^{r}( \mathbb{R} ^n)}\leq (10^n\lambda)^{r-1}\|\mathcal{V}_r(F)\|_{L^1( \mathbb{R} ^n)},
$$
$$
\int_{Q_i}\mathcal{B}(x)dx=0,\,\Xint-_{Q_i}|\mathcal{B}(x)|dx\leq 2
\Xint-_{Q_i}|\mathcal{V}_r(F)(x)|dx,\,\textrm{ and }\, \|\mathcal{B}\|_{L^1( \mathbb{R} ^n)}\leq 2\|\mathcal{V}_r(F)\|_{L^1( \mathbb{R} ^n)}.
$$
Now set $F=G+H,$
where
$$
G(x,t)=F(x,t)\chi_{ \mathbb{R} ^{n}\setminus\Omega_{\lambda}}(x)+\sum_{i\in  \mathbb{N} }\chi_{Q_i}(x)\Xint-_{Q_i}F(y,t)\, dy,$$
and
$$
H(x,t)=\sum_{i\in  \mathbb{N} }\chi_{Q_i}(x)\left(F(x,t)-\Xint-_{Q_i}F(y,t)\, dy\right)=:\sum_{i\in  \mathbb{N} }H_i(x,t).
$$
Then,
\begin{multline*}
\left|\left\{x\in  \mathbb{R} ^n:\mathcal{V}_r(\mathcal{T}_{*}(F))(x)>{\lambda}\right\}\right|
\\
\leq
\left|\left\{x\in  \mathbb{R} ^n:\mathcal{V}_r(\mathcal{T}_{*}(G))(x)>\frac{\lambda}{2}\right\}\right|
+
\left|\left\{x\in  \mathbb{R} ^n:\mathcal{V}_r(\mathcal{T}_{*}(H))(x)>\frac{\lambda}{2}\right\}\right|=:I+II.
\end{multline*}
Applying Chebychev's inequality and the $ L^r( \mathbb{R} ^n)$ boundedness of $\mathcal{T}_{*}$, we obtain
\begin{multline*}
I
\lesssim \frac{1}{\lambda^r}\int_{ \mathbb{R} ^n}\mathcal{V}_r(\mathcal{T}_{*}(G))(x)^r\,dx
\\
=
\frac{1}{\lambda^r}\int_0^{\infty}\int_{ \mathbb{R} ^n}|\mathcal{T}_{*}(G(\cdot, t))(x)|^r\,dx\frac{dt}{t}
\lesssim
\frac{1}{\lambda^r}\int_0^{\infty}\int_{ \mathbb{R} ^n}|G(x,t)|^r\,dx\frac{dt}{t}
\\
=\frac{1}{\lambda^r}\int_{ \mathbb{R} ^n}\int_0^{\infty}\left|F(x,t)\chi_{ \mathbb{R} ^n\setminus \Omega_{\lambda}}(x)+\sum_{i\in  \mathbb{N} }\chi_{Q_i}(x)\Xint-_{Q_i}|F(y,t)|\,dy
\right|^r\frac{dt}{t}\,dx
\\
\lesssim
\frac{1}{\lambda^r}\int_{ \mathbb{R} ^n}\left|\chi_{ \mathbb{R} ^n\setminus \Omega_{\lambda}}(x)\mathcal{V}_r(F)(x)+\sum_{i\in  \mathbb{N} }\chi_{Q_i}(x)\Xint-_{Q_i}\mathcal{V}_r(F)(y)\,dy
\right|^r\,dx
\\
= \frac{1}{\lambda^r}
\|\mathcal{G}\|^r_{L^r( \mathbb{R} ^n)}
\lesssim
\frac{1}{\lambda}\|\mathcal{V}_r(F)\|_{L^1( \mathbb{R} ^n)}.
\end{multline*}
As for the estimate of $II$, note that
\begin{equation*}
II
\lesssim
\left|\bigcup_{i\in  \mathbb{N} }2\sqrt n\, Q_i\right|+
\left|\left\{x\in  \mathbb{R} ^n\setminus\left(\bigcup_{i\in  \mathbb{N} }2\sqrt n\, Q_i\right) :\mathcal{V}_r(\mathcal{T}_{*}(H))(x)>\frac{\lambda}{2}\right\}\right|.
\end{equation*}
Then, since
$$
\left|\bigcup_{i\in  \mathbb{N} }2\sqrt n\, Q_i\right|
\lesssim
|\Omega_{\lambda}|
\lesssim
\frac{1}{\lambda}\|\mathcal{V}_r(F)\|_{L^1( \mathbb{R} ^n)},
$$
we just need to consider the second term in the previous sum.  For  $t>0$, and $x\in  \mathbb{R} ^n\setminus\left(\bigcup_{j\in  \mathbb{N} }2\sqrt n\, Q_j\right)$, let us study the $\mathcal{T}_{*}(H)(x,t)$. Pick $\varepsilon>0$ and consider
$$
\left|\int_{|x-y|>\varepsilon}
K(x,y)H(y,t)\,dy\right|= \left| \sum_{i\in  \mathbb{N} }\int_{|x-y|>\varepsilon}
K(x,y)H_{i}(y,t)\,dy\right|.
$$
We distinguish three possible cases in the series. Case $1$: $Q_i\subset B(x,\varepsilon)$. Then,
$Q_i\cap( \mathbb{R} ^n\setminus B(x,\varepsilon))=\emptyset$, and consequently
$$
\left|\int_{|x-y|>\varepsilon}
K(x,y)H_i(y,t)\,dy\right|=0.
$$
Case $2$: $Q_i\subset  \mathbb{R} ^n\setminus B(x,\varepsilon)$. Call $y_{i}$ the centre of $Q_{i}$ and $\ell(Q_{i})$ its length. Since $x\in  \mathbb{R} ^n\setminus 2\sqrt n\, Q_i$,  we have   $|x-y_i|>2|y-y_i|$ for any $y\in Q_{i}$. As   $\supp(H_i)\subset Q_i\subset  \mathbb{R} ^n\setminus B(x,\varepsilon)$, we can use the mean value $\int_{ \mathbb{R} ^n}H_i(y,t)\, dy=0$ to obtain
 \begin{align*}
\left|\int_{|x-y|>\varepsilon}
K(x,y)H_i(y,t)\,dy\right|
& \leq
\int_{Q_{i}}
|K(x,y)-K(x,y_i)||H_i(y,t)|\,dy
\\ &
\lesssim
\frac{\ell(Q_i)^{\delta}}{|x-y_i|^{n+\delta}}\int_{Q_i}
|H_i(y,t)|\,dy \lesssim \frac{\ell(Q_i)^{\delta}}{|x-y_i|^{n+\delta}}\int_{Q_i}
|F(y,t)|\,dy.
\end{align*}
Case $3$: $B(x,\varepsilon)\cap Q_i\neq \emptyset$ but $Q_i\nsubset B(x,\varepsilon)$. Note that  $\varepsilon> \sqrt{n}\, \ell(Q_i)/2$. Indeed, if
$x_0\in B(x,\varepsilon)\cap Q_i$, $\sqrt{n}\, \ell(Q_i)\le |x-y_i|\leq |x_0-x|+|x_0-y_i|< \varepsilon+\frac{\sqrt{n}\, \ell(Q_i)}{2}$ hence $ \sqrt{n}\, \ell(Q_i)/2<\varepsilon$. It follows that $Q_i\subset B(x,3\varepsilon)$. Hence,
\begin{multline*}
\left|\int_{|x-y|>\varepsilon}
K(x,y)H_i(y,t)\,dy\right|
\leq
\int_{\varepsilon<|x-y|<3\varepsilon}\frac{1}{|x-y|^{n}}
|H_i(y,t)|\,dy
\\
\lesssim
\Xint-_{B(x,3\varepsilon)}
|F(y,t)|\chi_{Q_i}(y)\,dy+
\Xint-_{B(x,3\varepsilon)}
\Xint-_{Q_i}|F(z,t)|\,dz\chi_{Q_i}(y)\,dy
\lesssim
\Xint-_{B(x,3\varepsilon)}
|F(y,t)|\chi_{Q_i}(y)\,dy.
\end{multline*}
It follows that
$$
\left|\int_{|x-y|>\varepsilon}
K(x,y)H(y,t)\,dy\right| \lesssim   \sum_{i\in  \mathbb{N} } \frac{\ell(Q_i)^{\delta}}{|x-y_i|^{n+\delta}}\int_{Q_i}
|F(y,t)|\,dy + \Xint-_{B(x,3\varepsilon)}
|F(y,t)|\chi_{\Omega_{\lambda}}(y)\,dy.
$$
Taking the supremum over all $\varepsilon>0$, we obtain,
$$
\mathcal{T}_{*}(H)(x,t) \le \sum_{i\in  \mathbb{N} } \frac{\ell(Q_i)^{\delta}}{|x-y_i|^{n+\delta}}\int_{Q_i}
|F(y,t)|\,dy + \mathcal{M}(F)(x,t).
$$
Therefore, by Minkowski inequality
$$
\mathcal{V}_r(\mathcal{T}_{*}(H))(x) \le \sum_{i\in  \mathbb{N} }
\frac{\ell(Q_i)^{\delta}}{|x-y_i|^{n+\delta}}\int_{Q_i}
|\mathcal{V}_r(F)(y)|\,dy + \mathcal{V}_r(\mathcal{M}(F))(x).
$$

Consequently, applying Fefferman-Stein  weak type $(1,1)$ inequality  and Chebychev's inequality
\begin{multline*}
\left|\left\{x\in  \mathbb{R} ^n\setminus\left(\bigcup_{i\in  \mathbb{N} }2\sqrt n\, Q_i\right) :\mathcal{V}_r(\mathcal{T}_{*}(H))(x)>\frac{\lambda}{2}\right\}\right|
\\
\lesssim \left|\left\{x\in  \mathbb{R} ^n:\mathcal{V}_r(\mathcal{M}(F))(x)>\frac{\lambda}{4}\right\}\right|
+
\frac{1}{\lambda}\sum_{i\in  \mathbb{N} }
\int_{ \mathbb{R} ^{n}\setminus 2\sqrt n\, Q_i}
\frac{\ell(Q_i)^{\delta}}{|x-y_i|^{n+\delta}}\,dx\int_{Q_i}
|\mathcal{V}_r(F)(y)|\,dy
\\
\lesssim
\frac{1}{\lambda}\left(\|\mathcal{V}_r(F)\|_{L^1( \mathbb{R} ^n)}
+
\int_{\Omega_\lambda}
|\mathcal{V}_r(F)(y)|\,dy\right)\lesssim
\frac{1}{\lambda}\|\mathcal{V}_rF\|_{L^1( \mathbb{R} ^n)}.
\end{multline*}
\qed
\subsection{Proof of Theorem \ref{thm:C-Zoperator}, part $(c)$}
 For $q\le 1$, the space $T^q_{r}$  has an atomic decomposition.  An atom in $T^q_r$ is a measurable function $A(x,t)$ such that there exists a ball $B\subset  \mathbb{R} ^n$  with $\supp(A)\subset\widehat{B}:=\{(x,t)\in  \mathbb{R} ^{n+1}_+: d(x, \mathbb{R} ^n\setminus B)\geq t\}$, and
\begin{equation}
\label{eq:qatom}
\left(\int\!\!\!\int_{\widehat{B}}|A(x,t)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}\leq |B|^{\frac{1}{r}-\frac{1}{q}}.
\end{equation}
In order to keep things clear we write this as a proposition, whose proof can be found in \cite[Section 8, Proposition 5]{CoifmanMeyerStein}.
\begin{proposition}\label{prop:atoms}
Let $F\in T^q_r$, $0<q\leq 1$ and $1<r<\infty$. Then $F=\sum_{i=1}^{\infty}\lambda_iA_i$, where $A_i$ are $T^q_r$ atoms, $\lambda_i\in  \mathbb{C} $, and  $\left(\sum_{i=1}^{\infty}|\lambda_i|^q\right)^{\frac{1}{q}}\lesssim \|F\|_{T^q_r}$. Conversely, any such sum converges in $T^q_{r}$ and
$ \|\sum_{i=1}^{\infty}\lambda_iA_i\|_{T^q_{r}} \lesssim \left(\sum_{i=1}^{\infty}|\lambda_i|^q\right)^{\frac{1}{q}}.$
\end{proposition}

Let us now introduce, for $0<q\leq 1$ and $1<r<\infty$, a subspace of $T^q_r$ that we denote by $\mathfrak{T}_r^q$. We say that $A$ is a  $\mathfrak{T}^q_r$ atom if it is a $T^q_r$ atom and   satisfies $\int_{ \mathbb{R} ^n}A(x,t)dx=0$ for a.e. $t>0$. This integral makes sense as \begin{equation*}
\left(\int_{0}^{\infty}\left(\int_{ \mathbb{R} ^n}|A(x,t)|\,dx\right)^r\frac{dt}{t}\right)^{\frac{1}{r}}
\le
\left(\int\!\!\!\int_{\widehat{B}}|A(x,t)|^r\,dx\frac{dt}{t}\right)^{\frac{1}{r}}|B|^{1-\frac{1}{r}}
\le  |B|^{1-\frac{1}{q}}<\infty.
\end{equation*}
We define $\mathfrak{T}^q_r$ as the subspace of
$F\in T^q_r$ such that $F$ has    an atomic decomposition with $A_{i}$ being $\mathfrak{T}^q_r$ atom and  $\left(\sum_{i=1}^{\infty}|\lambda_i|^q\right)^{\frac{1}{q}}<\infty$.

The reason to introduce those spaces is because, for $0<q\leq 1$,
we can not obtain boundedness of singular integrals (and in general of Calder\'on-Zygmund operators)
from the tent space $T^q_r$ to itself. If we want to arrive into $T^q_r$, an option is to take functions in $\mathfrak{T}^q_r$. Note that $T^q_r$ atoms, hence $\mathfrak{T}^q_r$ atoms,  belong to $T^r_{r}$.

\begin{lemma}\label{lem:extension} Suppose that $\mathcal{U}: {T}^r_r\rightarrow T^r_r$ is a linear and bounded operator and that there exists $C<\infty$ such that for all
 $\mathfrak{T}_{r}^{q}$ atom $A$,   $\|\mathcal{U}(A)\|_{T^q_r}\le C$.
 Then, $\mathcal{U}$ has a bounded  extension from $\mathfrak{T}^q_r$ to $ T^q_r$.
 \end{lemma}

 \begin{proof} Let   $A$ be a $\mathfrak{T}^q_r$ atom such that $\supp(A)\subset \widehat{B}$, for some ball $B\subset  \mathbb{R} ^n$.
Defining, for $0<\eta<\rho$, where $\rho$ is the radius of $B$,
$$
A_{\eta}(y,t):=\begin{cases}
A(y,t)\, \textrm{if}\, t> \eta,
\\
0\, \textrm{if}\, t\leq \eta,
\end{cases}\,
$$
we have that $A-A_{\eta}$  are $\mathfrak{T}^q_r$ atoms, uniformly in $\eta$, thus
$$
\|A - A_{\eta}\|_{\mathfrak{T}^q_{r}} \le |B|^{\frac{1}{q}-\frac{1}{r}} \left(\int\!\!\!\int_{\widehat{B}}|A(x,t)- A_{\eta}(x,t)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}} \to 0
$$
by the dominated convergence theorem. This and the fact that finite linear combinations of $\mathfrak{T}^q_r$ atoms are dense in $\mathfrak{T}^q_r$ by definition,  imply that  the set $E_r$ of compactly supported functions $\varphi$ in $ \mathbb{R} ^{n+1}_{+}$ that are  $r$ integrable and $ \int_{ \mathbb{R} ^n}\varphi(x,t)\,dx=0$ {for a.e.} $t>0$ is dense in  $\mathfrak{T}^q_r$. Then, let $F\in E_{r}$ and take a decomposition $F=\sum_{i=0}^{\infty}\lambda_iA_i$, where $A_i$ are $\mathfrak{T}_{r}^{q}$ atoms and  $\left(\sum_{i=1}^{\infty}|\lambda_i|^q\right)^{\frac{1}{q}} \le 2 \|F\|_{\mathfrak{T}^q_{r}}$. Since  the $t$ support of $F$ is contained in some interval $[a,b]$, we may eliminate the atoms associated to balls with radii less than $a$. Following the proof of Theorem 4.9 in \cite{AMcR}, we obtain that the decomposition converges in $T^r_{r}$. Thus we may write
$$
\mathcal{U}(F)= \sum_{i=0}^\infty \lambda_i\,  \mathcal{U}(A_i)
$$
and use the hypothesis to conclude that $\|\mathcal{U}(F)\|_{T^q_{r}}\le 2 C\|F\|_{\mathfrak{T}^q_{r}}$. By density, we conclude the argument.
\end{proof}

We say that function $M$ is a $T^q_r$ molecule if there exists a ball $B\subset  \mathbb{R} ^n$ such that, for some $\varepsilon>0$,
\begin{align*}
\left(\int\!\!\!\int_{\widehat{4B}}|M(x,t)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}\leq |4B|^{\frac{1}{r}-\frac{1}{q}}
\end{align*}
and, for all $j\geq 2$,
\begin{align*}
\left(\int\!\!\!\int_{\widehat{C_{j}}}|M(x,t)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}\leq 2^{-({j+1})\varepsilon}|2^{j+1}B|^{\frac{1}{r}-\frac{1}{q}},
\end{align*}
where we define $\widehat{C_{j}}:=\widehat{2^{j+1}B}\setminus
\widehat{2^{j}B}$ and $\widehat{C_{1}}=\widehat{4B}$ .
 By writing $M= \sum_{j\ge 1} {\bf 1}_{ \widehat{C_{j}}} M$ and observing that the functions ${\bf 1}_{ \widehat{C_{j}}} M$ are $T^q_{r}$ atoms up to factor $2^{-(j+1)\varepsilon}$, we obtain $\|M\|_{T^{q}_{r}} \le \left(\sum_{j\geq 1}|2^{-(j+1)\varepsilon }|^q\right)^{\frac{1}{q}}$.

Let us finally prove part $(c)$ of Theorem \ref{thm:C-Zoperator}. We  follow the same scheme as in \cite{CoifmanWeiss} and show that Calder\'on-Zygmund operators of order $\delta \in (0,1]$ apply $\mathfrak{T}^q_r$ atoms  to $T^q_r$ molecules, provided that $q>\frac{n}{n+\delta}$, up to a constant that depends uniquely on $\delta, n, r, q$ and the properties of the operator. From the previous lemma, this suffices.

Let $A$ is a $\mathfrak{T}^q_r$ atom. Let $B$ be a ball such that
$\supp A\subset \widehat{B}$, and
\begin{align*}
\left(\int\!\!\!\int_{\widehat{B}}|A(x,t)|^r\right)^{\frac{1}{r}}\leq |B|^{\frac{1}{r}-\frac{1}{q}}.
\end{align*}
We shall show that, for $\varepsilon=n+\delta-\frac{n}{q}$ which is positive since $q>\frac{n}{n+\delta}$,
\begin{list}{$(\theenumi)$}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm\itemsep=0.2cm\topsep=.0cm \renewcommand{\theenumi}{\arabic{enumi}}}
\item $\left(\int\!\!\!\int_{\widehat{4B}}|\mathcal{T}(A(\cdot,t))(x)|^r\right)^{\frac{1}{r}}\lesssim |4B|^{\frac{1}{r}-\frac{1}{q}}$;

\item for $j\geq 2$, $\left(\int\!\!\!\int_{\widehat{C_j}}|\mathcal{T}(A(\cdot,t))(x)|^r\right)^{\frac{1}{r}}\lesssim 2^{-({j+1})\varepsilon}|B_{j+1}|^{\frac{1}{r}-\frac{1}{q}}$.

\end{list}
For each $j\geq 2$, denote by $r_j:=2^{j}r_B$ and $B_j:=B(x_B,r_j)$. Besides, recall that $\widehat{C_1}:=\widehat{B_{2}}$ and $\widehat{C_j}:=\widehat{B_{j+1}}\setminus \widehat{B_j}$, for all $j\geq 2$.

We start by proving $(1)$. Since $\mathcal{T}$ is bounded in $T_{r}^r$, we have that
\begin{align*}
\left(\int\!\!\!\int_{\widehat{4B}}|\mathcal{T}(A(\cdot,t))(x)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\lesssim
\left(\int\!\!\!\int_{\widehat{B}}|A(x,t)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\lesssim
|4B|^{\frac{1}{r}-\frac{1}{q}}.
\end{align*}
On the other hand, for $j\geq 2$, because $A(x,t)=0$ for $t>r_{B}$, the radius of $B$,
\begin{equation*}
\left(\int\!\!\!\int_{\widehat{C_j}}|\mathcal{T}(A(\cdot,t))(x)|^{r}\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\le
\left(\int_{0}^{r_{B}}\int_{B_{j+1}\setminus B_{j-1}}|\mathcal{T}(A(\cdot,t))(x)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}.
\end{equation*}
Now, applying the fact that $\int_{ \mathbb{R} ^n}A(x,t)\,dx=0$ for a.e. $t>0$, and the property \eqref{kernel2} of the kernel $K$, we obtain that
\begin{multline*}
I
\leq
\left(\int_{0}^{r_{B}}\int_{r_{j-1}\le |x-x_B|< r_{j+1}}\left|\int_{ \mathbb{R} ^n}K(x,y)A(y,t)\,dy\right|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\\
=
\left(\int_{0}^{r_{B}}\int_{r_{j-1}\le |x-x_B|< r_{j+1}}\left|\int_{ \mathbb{R} ^n}(K(x,y)-K(x,x_B))A(y,t)\,dy\right|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\\
\lesssim
\left(\int_{0}^{r_{B}}\int_{r_{j-1}\le |x-x_B|< r_{j+1}}\left(\int_{ \mathbb{R} ^n}\frac{|x_B-y|^{\delta}}{|x-x_B|^{n+\delta}}|A(y,t)|\,dy\right)^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\\
\lesssim
\left(\int_{0}^{r_{B}}\int_{r_{j-1}\le |x-x_B|< r_{j+1}}\Xint-_{B}|A(y,t)|^r\,dy\frac{dx\,dt}{t}\right)^{\frac{1}{r}} 2^{-(j+1)(n+\delta)}
\\
\lesssim
\left(\int_{0}^{r_{B}}\int_{B}|A(y,t)|^r\frac{dy\,dt}{t}\right)^{\frac{1}{r}} 2^{-(j+1)\left(n\left(1-\frac{1}{r}\right)+\delta\right)}
\\
\lesssim
 2^{-(j+1)\left(n\left(1-\frac{1}{r}\right)+\delta\right)}|B|^{\frac{1}{r}-\frac{1}{q}}
= 2^{-(j+1)\left(n+\delta-\frac{n}{q}\right)}|2^{j+1}B|^{\frac{1}{r}-\frac{1}{q}}.
\end{multline*}
This shows $(2)$.
\qed
\subsection{Proof of Theorem \ref{thm:C-Zoperator}, part $(d)$}
Remark that if $M$ is a $T^q_r$ molecule, then
\begin{multline*}
\left(\int_{0}^{\infty}\left(\int_{ \mathbb{R} ^n}|M(x,t))|\,dx\right)^r\frac{dt}{t}\right)^{\frac{1}{r}}
\lesssim
\sum_{j\geq 1}
\left(\int_{0}^{\infty}\left(\int_{ \mathbb{R} ^n}\chi_{\widehat{C_j}}(x,t)|M(x,t)|\,dx\right)^r\frac{dt}{t}\right)^{\frac{1}{r}}
\\
\lesssim
\sum_{j\geq 1}
\left(\int\!\!\!\int_{\widehat{C_j}}|M(x,t)|^r\,dx\frac{dt}{t}\right)^{\frac{1}{r}}|B_{j+1}|^{1-\frac{1}{r}}
\lesssim
\sum_{j\geq 1} 2^{-(j+1)\varepsilon}
|B_{j+1}|^{1-\frac{1}{q}}
\lesssim |B|^{1-\frac{1}{q}}<\infty
\end{multline*}
as $1-\frac{1}{q}\le 0$.
Therefore, if, in addition,  $\int_{ \mathbb{R} ^n}M(x,t)\,dx=0$, for a.e. $t>0$, we say that $M$ is a $\mathfrak{T}^q_{r}$ molecule.  A $\mathfrak{T}^q_r$ molecule can be written as a series of $\mathfrak{T}^q_r$ atoms. We see that in the next result.
\begin{proposition}\label{prop:moleculesumatoms} There exists a constant $C<\infty$ such that
given  a $\mathfrak{T}^q_r$ molecule $M$,
we have that $M\in \mathfrak{T}^q_r$, with
$\|M\|_{\mathfrak{T}^q_r}\leq C.$
\end{proposition}
\begin{proof}
Let $M$ be  a $\mathfrak{T}^q_r$ molecule with associated ball $B=B(x_B,r_B)$.
Following the notation in the previous proof, write
$$
M(x,t)=\sum_{j=1}^{\infty}\left(M(x,t)\chi_{\widehat{C_j}}(x,t)-
\int_{ \mathbb{R} ^n}\chi_{\widehat{C_j}}(y,t)M(y,t)\,dy\frac{\chi_{{B_{j+1}}}(x)}{|B_{j+1}|}\right)
+
\sum_{j=1}^{\infty}
\int_{ \mathbb{R} ^n}\chi_{\widehat{C_j}}(y,t)M(y,t)\,dy\frac{\chi_{{B_{j+1}}}(x)}{|B_{j+1}|}.
$$
For all $j\geq 1$, define
$$
\alpha_j(x,t):=
M(x,t)\chi_{\widehat{C_j}}(x,t)-
\int_{ \mathbb{R} ^n}\chi_{\widehat{C_j}}(y,t)M(y,t)\,dy\frac{\chi_{{B_{j+1}}}(x)}{|B_{j+1}|},
$$
and observe that $\supp \alpha_j\subset B_{j+1}\times (0,r_{j+1}] \subset \widehat{B_{j+2}}$  and
$$
\int_{ \mathbb{R} ^n}\alpha_j(x,t)\, dx=\int_{ \mathbb{R} ^n}\chi_{\widehat{C_j}}(y,t)M(y,t)\,dy\left(1-
\int_{ \mathbb{R} ^n}\frac{\chi_{B_{j+1}}(x)}{|B_{j+1}|}\,dx\right)=0.
$$
Besides,
\begin{multline*}
\left(\int\!\!\!\int_{\widehat{B_{j+2}}}|\alpha_j(x,t)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\leq
\left(\int\!\!\!\int_{\widehat{C_{j}}}|M(x,t)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\\
+
\left(\int\!\!\!\int_{\widehat{B_{j+2}}}\chi_{B_{j+1}}(x)\left(\frac{1}{|B_{j+1}|}\int_{B_{j+1}}\chi_{\widehat{C_j}}(y,t)|M(y,t)|\,dy\right)^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\leq 2 \left(\int\!\!\!\int_{\widehat{C_{j}}}|M(x,t)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\\
\leq
2^{-(j+1)\varepsilon+1}|B_{j+1}|^{\frac{1}{r}-\frac{1}{q}}
=c2^{-{j}\varepsilon}|B_{j+2}|^{\frac{1}{r}-\frac{1}{q}},
\end{multline*}
where $c$  depends on $\varepsilon,r,q$ only.
Therefore,
$A_j:=\frac{2^{j\varepsilon}}{c}\alpha_j$ is a $\mathfrak{T}^q_r$ atom, for all $j\geq 1$.

On the other hand,  note that
\begin{multline*}
\sum_{j=1}^{\infty}
\int_{ \mathbb{R} ^n}\chi_{\widehat{C_j}}(y,t)M(y,t)\,dy\frac{\chi_{{B_{j+1}}}(x)}{|B_{j+1}|}
=
\int_{ \mathbb{R} ^n}\chi_{\widehat{B_{2}}}(y,t)M(y,t)\,dy
\frac{\chi_{B_2}(x)}{|B_2|}
\\
+
\sum_{j=2}^{\infty}\left(
\int_{ \mathbb{R} ^n}\chi_{\widehat{B_{j+1}}}(y,t)M(y,t)\,dy-
\int_{ \mathbb{R} ^n}\chi_{\widehat{B_{j}}}(y,t)M(y,t)\,dy\right)
\frac{\chi_{{B_{j+1}}}(x)}{|B_{j+1}|}
\\
=
\sum_{j=1}^{\infty}
\int_{ \mathbb{R} ^n}\chi_{ \widehat{B_{j+1}}}(y,t)
M(y,t)\,dy\left(\frac{\chi_{{B_{j+1}}}(x)}{|B_{j+1}|}-\frac{\chi_{{B_{j+2}}}(x)}{|B_{j+2}|}\right).
\end{multline*}
Then, considering
$$
\beta_j(x,t):=
\int_{ \mathbb{R} ^n}\chi_{\widehat{B_{j+1}}}(y,t)
M(y,t)\,dy\left(\frac{\chi_{{B_{j+1}}}(x)}{|B_{j+1}|}-\frac{\chi_{{B_{j+2}}}(x)}{|B_{j+2}|}\right),
$$
we have that $\supp \beta_j\subset \widehat{B_{j+3}}$, and that
$$
\int_{ \mathbb{R} ^n}\beta_j(x,t)dx=0.
$$
Besides, since, for a. e. $t>0$,
$$
\int_{ \mathbb{R} ^n}M(y,t)\,dy=0,
$$
then, for each $j\geq 1$,
$$
\int_{ \mathbb{R} ^n}\chi_{ \mathbb{R} ^{n+1}_+\setminus \widehat{B_{j+1}}}(y,t)M(y,t)\,dy=-
\int_{ \mathbb{R} ^n}\chi_{\widehat{B_{j+1}}}(y,t)M(y,t)\,dy, \quad \textrm{for a.e. }t>0.
$$
This, together with the fact that
$$
\int_{ \mathbb{R} ^n}\chi_{\widehat{B_{j+1}}}(y,t)M(y,t)\,dy=
\chi_{(0,r_{j+1})}(t)\int_{ \mathbb{R} ^n}\chi_{\widehat{B_{j+1}}}(y,t)M(y,t)\,dy,
$$
 gives, for a.e. $t>0$,
$$
\int_{ \mathbb{R} ^n}\chi_{\widehat{B_{j+1}}}(y,t)M(y,t)\,dy=-
\chi_{(0,r_{j+1})}(t)\int_{ \mathbb{R} ^n}\chi_{ \mathbb{R} ^{n+1}_+\setminus \widehat{B_{j+1}}}(y,t)M(y,t)\,dy.
$$
Hence, for all $j\geq 1$,
$$
\beta_j(x,t)=
\chi_{(0,r_{j+1})}(t)\int_{ \mathbb{R} ^n}\chi_{ \mathbb{R} ^{n+1}_+\setminus \widehat{B_{j+1}}}(y,t)M(y,t)\,dy\left(\frac{\chi_{{B_{j+2}}}(x)}{|B_{j+2}|}-\frac{\chi_{{B_{j+1}}}(x)}{|B_{j+1}|}\right),\quad \textrm{for a.e. }t>0.
$$
Therefore,
\begin{multline*}
\left(\int\!\!\!\int_{\widehat{B_{j+3}}}|\beta_j(x,t)|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\lesssim
\sum_{i\geq j+1}\frac{|B_{i+1}|}{|B_{j+1}|}
\left(\int\!\!\!\int_{\widehat{B_{j+3}}}\chi_{B_{j+2}}(x)\left|\frac{1}{|B_{i+1}|}\int_{ \mathbb{R} ^n}\chi_{\widehat{C_i}}(y,t)
M(y,t)\,dy\right|^r\frac{dx\,dt}{t}\right)^{\frac{1}{r}}
\\
\lesssim \sum_{i\geq j+1}
 \left(\frac{|B_{i+1}|}{|B_{j+1}|}\right)^{1-\frac{1}{r}}
\left(\int\!\!\!\int_{\widehat{C_{i}}}
|M(y,t)|^r\frac{dy\,dt}{t}\right)^{\frac{1}{r}}
\lesssim
\sum_{i\geq j+1}2^{-(i+1)\varepsilon}
 |B_{j+1}|^{\frac{1}{r}-1}
 |B_{i+1}|^{1-\frac{1}{q}}
 \\
 \lesssim
  |B_{j+3}|^{\frac{1}{r}-\frac{1}{q}}\sum_{i\geq j+1}2^{-(i+1)\varepsilon}
 \leq c' 2^{-j\varepsilon}
 |B_{j+3}|^{\frac{1}{r}-\frac{1}{q}},
\end{multline*}
where $c'$ depends on $\varepsilon,r,q$ only.
Hence,
$A'_j(x,t):=\frac{2^{j\varepsilon}}{c'}\beta_j$ is
a $\mathfrak{T}^q_r$ atom.

Therefore, we have shown that $M= \sum_{j\ge 1} c2^{-j\varepsilon} A_{j}+ \sum_{j\ge 1} c'2^{-j\varepsilon} A'_{j}$, which evidently shows that $M\in \mathfrak{T}^q_r$ with norm bounded by  $(c+c')\left(\sum_{j\geq 1}|2^{-j\varepsilon q}|^q\right)^{\frac{1}{q}}.$
\end{proof}

Let us finally show that if $\mathcal{T}$ is a Calder\'on-Zygmund operator, then  $\mathcal{T}$ applies $\mathfrak{T}^q_r$ atoms to $\mathfrak{T}^q_r$ molecules, up to a uniform constant. Note that, from the above proposition,  and an adaptation of Lemma \ref{lem:extension},  this is enough to conclude the proof.

From the part (c) of the proof, we already know that $\mathcal{T}$ applies $\mathfrak{T}^q_r$ atoms to ${T}^q_r$ molecules, up to a uniform constant.  It remains to show $\int_{ \mathbb{R} ^n}\mathcal{T}(A(\cdot,t))(x)\,dx=0$.  Note for almost every $t>0$, $A(\cdot, t)$ is a multiple of an atom in the Hardy space $H^1( \mathbb{R} ^n)$. Indeed, its support is contained in $B$, it is in $L^r(B)$ with $r>1$ and has mean value 0. We knew that  $\mathcal{T}(A(\cdot,t))\in L^1( \mathbb{R} ^n)$ since $\mathcal{T}(A)$ has been shown to be a $T^q_{r}$ molecule. Thus, $\int_{ \mathbb{R} ^n}\mathcal{T}(A(\cdot,t))(x)\,dx=0$ as  $\mathcal{T}^*(1)=0$.
\section{Riesz potentials and fractional maximal functions}
For $0<\alpha <n$, consider the Riesz potential
\begin{align*}
\mathcal{I}_{\alpha}(f)(x):=\frac{1}{\gamma(\alpha)}\int_{ \mathbb{R} ^n}\frac{1}{|x-z|^{n-\alpha}}f(z)\,dz,
\end{align*}
where $\gamma(\alpha)=\pi^{\frac{n}{2}}2^{\alpha}\Gamma(\alpha/2)/\Gamma\left(\frac{n-\alpha}{2}\right)$,  and the fractional maximal function
\begin{align*}
\mathcal{M}_{\alpha}(f)(x)=\sup_{\tau>0}\tau^{\frac{\alpha}{n}}\Xint-_{B(x,\tau)}|f(y)|dy.
\end{align*}
Note that
\begin{align}\label{comparisonmaxriesz}
\mathcal{M}_{\alpha}(f)(x)\leq V_n^{-1}\mathcal{I}_{\alpha}(|f|)(x), \textrm{ for all }x\in  \mathbb{R} ^{n},
\end{align}
where $V_n$ is the volume of the unit ball in $ \mathbb{R} ^n$.

Consequently, it is enough to prove Theorem \ref{thm:rieszpotential} for Riesz potentials. Let us start by proving the following pointwise inequality.
\begin{lemma}\label{lemma:rieszpotential}
Let $0<\alpha<n$, $1< \vartheta<r<\infty$, and $\frac{\alpha}{n}=\frac{1}{\vartheta}-\frac{1}{r}$.  Then, for any $x\in  \mathbb{R} ^n$, $t>0$, if $f$ is locally  $\vartheta$ integrable,
\begin{align*}
\left(\Xint-_{B(x,t)}|\mathcal{I}_{\alpha}(f)(y)|^{r}dy\right)^{\frac{1}{r}}
\lesssim
t^{n\left(\frac{1}{\vartheta}-\frac{1}{r}\right)}\left(\Xint-_{B(x,5t)}|f(y)|^{\vartheta}dy\right)^{\frac{1}{\vartheta}}
+
\mathcal{I}_{\alpha}\left(\Xint-_{B(\cdot,t)}|f(z)|dz\right)(x).
\end{align*}
\end{lemma}
\begin{proof}
For each $x\in  \mathbb{R} ^n$ and $t>0$, split the support of $f$ into $B(x,5t)$ and $ \mathbb{R} ^n\setminus B(x,5t)$. Then,
\begin{multline*}
\left(\Xint-_{B(x,t)}|\mathcal{I}_{\alpha}(f)(y)|^{r}dy\right)^{\frac{1}{r}}
\leq
\left(\Xint-_{B(x,t)}|(\mathcal{I}_{\alpha}(\chi_{B(x,5t)}f))(y)|^{r}dy\right)^{\frac{1}{r}}
\\
+
\left(\Xint-_{B(x,t)}\left|\int_{|x-z|>5t}\frac{1}{|y-z|^{n-\alpha}}f(z)\,dz\right|^{r}dy\right)^{\frac{1}{r}}
=:I+II.
\end{multline*}
On the one hand, using that $\mathcal{I}_{\alpha}:L^{\vartheta}( \mathbb{R} ^n)\rightarrow L^{r}( \mathbb{R} ^n)$ (see \cite[Theorem 1, p.119]{Stein}), obtain that
\begin{align*}
I
\lesssim
t^{n\left(\frac{1}{\vartheta}-\frac{1}{r}\right)}\left(\Xint-_{B(x,5t)}|f(y)|^{\vartheta}dy\right)^{\frac{1}{\vartheta}}.
\end{align*}
On the other hand,
\begin{align*}
II
&
\lesssim
\left(\Xint-_{B(x,t)}\left(\int_{|x-z|>5t}\frac{1}{|y-z|^{n-\alpha}}|f(z)|\Xint-_{B(z,t)}\,d\xi\,dz\right)^{r}dy\right)^{\frac{1}{r}}
\\
&
\lesssim
\left(\Xint-_{B(x,t)}\left(\int_{|x-\xi|>4t}\Xint-_{B(\xi,t)}\frac{1}{|y-z|^{n-\alpha}}|f(z)|\,dz\,d\xi\right)^{r}dy\right)^{\frac{1}{r}}
\\
&
\lesssim
\left(\Xint-_{B(x,t)}\left(\int_{|x-\xi|>4t}\frac{1}{|x-\xi|^{n-\alpha}}\Xint-_{B(\xi,t)}\left(\frac{|x-\xi|}{|y-z|}\right)^{n-\alpha}|f(z)|\,dz\,d\xi\right)^{r}dy\right)^{\frac{1}{r}}
\\
&
\lesssim
\left(\Xint-_{B(x,t)}\left(\int_{|x-\xi|>4t}\frac{1}{|x-\xi|^{n-\alpha}}\Xint-_{B(\xi,t)}|f(z)|\,dz\,d\xi\right)^{r}dy\right)^{\frac{1}{r}}
\\
&
=
\int_{|x-\xi|> 4t}\frac{1}{|x-\xi|^{n-\alpha}}\Xint-_{B(\xi,t)}|f(z)|\,dz\,d\xi
\leq
\mathcal{I}_{\alpha}\left(\Xint-_{B(\cdot,t)}|f(z)|dz\right)(x).
\end{align*}
\end{proof}
\subsection{Proof of Theorem \ref{thm:rieszpotential}}

Let $F\in T^p_{r}$. Taking $\vartheta=\frac{nr}{\alpha r+n}$ in Lemma \ref{lemma:rieszpotential},
we obtain that
\begin{multline*}
\|\mathcal{I}_{\alpha}(F)\|_{T^q_{r}}
\lesssim
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}t^{n\left(\frac{r}{\vartheta}-1\right)}\left(\Xint-_{B(x,5t)}
|F(y,t)|^{\vartheta} dy\right)^{\frac{r}{\vartheta}}\frac{dt}{t}\right)^{\frac{q}{r}}dx\right)^{\frac{1}{q}}
\\
+
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\left(\mathcal{I}_{\alpha}
\left(\Xint-_{B(\cdot,t)}|F(y,t)|dy\right)(x)\right)^{r}\frac{dt}{t}\right)^{\frac{q}{r}}dx\right)^{\frac{1}{q}}=:I+II.
\end{multline*}
Since $r>\vartheta$, applying successively Jensen's inequality,  \cite[Theorem 2.19]{Amenta} for $s_1=\frac{1}{r}-\frac{1}{\vartheta}$, $s_0=0$, $p_0=p$, $p_{1}=q$, and $q=r$, and \cite[Section 3, Proposition 4]{CoifmanMeyerStein} (we use this proposition for $r$ instead of $2$, but the proof is the same),
\begin{multline*}
I
\lesssim
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\int_{B(x,5t)}
\left|t^{n\left(\frac{1}{\vartheta}-\frac{1}{r}\right)}F(y,t)\right|^{r}\frac{dy\,dt}{t^{n+1}}\right)^{\frac{q}{r}}dx\right)^{\frac{1}{q}}
\\
\lesssim
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\int_{B(x,5t)}
|F(y,t)|^{r}\frac{dy\,dt}{t^{n+1}}\right)^{\frac{p}{r}}dx\right)^{\frac{1}{p}}
\lesssim
\|F\|_{T^p_{r}}.
\end{multline*}
Finally, to estimate $II$, we shall proceed by extrapolation. We first recall some definitions. We say that a weight $w$ is a $A_{\tau,s}$ weight, for $1<\tau\leq s<\infty$, if it satisfies for every $B\subset  \mathbb{R} ^n$ that
$$
\left(\Xint-_{B}w(x)^s\,dx\right)^{\frac{1}{s}}
\left(\Xint-_{B}w(x)^{-\tau'}\,dx\right)^{\frac{1}{\tau'}}\leq C.
$$
Now, since $0<\alpha<n$ and $1<\vartheta<\frac{n}{\alpha}$ with  $\frac{1}{\vartheta}-\frac{1}{r}=\frac{\alpha}{n}$, by \cite[Theorem 4]{MuckenhouptWheede} for all $w\in A_{\vartheta,r}$
we have that $\mathcal{I}_{\alpha}:L^{\vartheta}(w^\vartheta)\rightarrow L^{r}(w^{r})$. This and Minkowski's integral inequality imply
\begin{multline*}
\left(\int_{ \mathbb{R} ^n}\int_0^{\infty}
\left|\mathcal{I}_{\alpha}\left(\Xint-_{B(\cdot,t)}|F(y,t)|dy\right)(x)\right|^{r}\frac{dt}{t}w(x)^{r}dx\right)^{\frac{1}{r}}
\\
\lesssim
\left(\int_0^{\infty}\int_{ \mathbb{R} ^n}\left|\mathcal{I}_{\alpha}
\left(\Xint-_{B(\cdot,t)}|F(y,t)|dy\right)(x)\right|^{r}w(x)^{r}dx\frac{dt}{t}\right)^{\frac{1}{r}}
\\
\lesssim
\left(\int_0^{\infty}\left(\int_{ \mathbb{R} ^n}\left(\Xint-_{B(x,t)}|F(y,t)|dy\right)^{\vartheta}w(x)^{\vartheta}dx\right)^{\frac{r}{\vartheta}}\frac{dt}{t}\right)^{\frac{1}{r}}
\\
\lesssim
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\left(\Xint-_{B(x,t)}|F(y,t)|dy\right)^{r}\frac{dt}{t}\right)^{\frac{\vartheta}{r}}w(x)^{\vartheta}dx
\right)^{\frac{1}{\vartheta}}
\\
\lesssim
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\int_{B(x,t)}|F(y,t)|^{r}\frac{dy\,dt}{t^{n+1}}\right)^{\frac{\vartheta}{r}}w(x)^{\vartheta}dx
\right)^{\frac{1}{\vartheta}}.
\end{multline*}
Then, since $1<\vartheta<r<\infty$ and $1<p<q<\infty$ with $\frac{1}{p}-\frac{1}{q}=\frac{1}{\vartheta}-\frac{1}{r}$, applying  \cite[Theorem 3.23]{CruzMartellPerez}, we have that, for all $w_0\in A_{p,q}$, and $F\in T^p_{r}$,
\begin{multline*}
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\left(
\mathcal{I}_{\alpha}\left(\Xint-_{B(\cdot,t)}|F(y,t)|dy\right)(x)\right)^{r}\frac{dt}{t}\right)^{\frac{q}{r}}w_0(x)^{q}dx\right)^{\frac{1}{q}}
\\
\lesssim
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\int_{B(x,t)}|F(y,t)|^{r}\frac{dy\,dt}{t^{n+1}}\right)^{\frac{p}{r}}w_0(x)^{p}dx
\right)^{\frac{1}{p}}.
\end{multline*}
In particular for $w_0\equiv 1$, we have that $w_0\in A_{p,q}$. Hence,
\begin{align*}
II
\lesssim
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\int_{B(x,t)}|F(y,t)|^{r}\frac{dy\,dt}{t^{n+1}}\right)^{\frac{p}{r}}dx
\right)^{\frac{1}{p}}=\|F\|_{T^p_{r}}.
\end{align*}
\qed
\section{Riesz transform}\label{sec:RT}
Consider a second order divergence form elliptic operator $L$ which is defined as
\begin{align*}
L f
=
-\mathop{\rm div}(A\,\nabla f)
\end{align*}
and
is understood in the standard weak sense as a maximal-accretive operator on $L^2( \mathbb{R} ^n,dx)$ with domain $\mathcal{D}(L)$ by means of a
sesquilinear form, and
where $A$ is an $n\times n$ matrix of complex and
$L^\infty$-valued coefficients defined on $ \mathbb{R} ^n$. We assume that
this matrix satisfies the following ellipticity (or \lq\lq
accretivity\rq\rq) condition: there exist
$0<\lambda\le\Lambda<\infty$ such that
$$
\lambda\,|\xi|^2
\le
\Re A(x)\,\xi\cdot\bar{\xi}
\quad\qquad\mbox{and}\qquad\quad
|A(x)\,\xi\cdot \bar{\zeta}|
\le
\Lambda\,|\xi|\,|\zeta|,
$$
for all $\xi,\zeta\in\mathbb{C}^n$ and almost every $x\in  \mathbb{R} ^n$. We have used the notation
$\xi\cdot\zeta=\xi_1\,\zeta_1+\cdots+\xi_n\,\zeta_n$ and therefore
$\xi\cdot\bar{\zeta}$ is the usual inner product in $\mathbb{C}^n$. Note
that then
$A(x)\,\xi\cdot\bar{\zeta}=\sum_{j,k}a_{j,k}(x)\,\xi_k\,\bar{\zeta_j}$.

We recall some facts regarding the operator $-L$. This operator generates a $C^0$-semigroup $\{e^{-t L}\}_{t>0}$ of contractions on $L^2( \mathbb{R} ^n)$ which is called the heat semigroup. As in \cite{Auscher} and \cite{AuscherMartell:II}, we denote by $(p_-(L),p_+(L))$ the maximal open interval on which this semigroup $\{e^{-tL}\}_{t>0}$ is uniformly bounded on $L^p( \mathbb{R} ^n)$:
\begin{align}\label{p-}
p_-(L) &:= \inf\left\{p\in(1,\infty): \sup_{t>0} \|e^{-tL}\|_{L^p( \mathbb{R} ^n)\rightarrow L^p( \mathbb{R} ^n)}< \infty\right\},
\\[4pt]
p_+(L)& := \sup\left\{p\in(1,\infty) : \sup_{t>0} \|e^{-tL}\|_{L^p( \mathbb{R} ^n)\rightarrow L^p( \mathbb{R} ^n)}< \infty\right\}.
\label{p+}
\end{align}
Moreover, we denote by $(q_-(L),q_+(L))$ the maximal open interval on which the gradient of the heat semigroup, i.e.  $\{t\nabla_y e^{-t^2L}\}_{t>0}$, is uniformly bounded on $L^p( \mathbb{R} ^n)$:
\begin{align}\label{q-q+}
q_-(L) &:= \inf\left\{p\in(1,\infty): \sup_{t>0} \|t\nabla_y e^{-t^2L} \|_{L^p( \mathbb{R} ^n)\rightarrow L^p( \mathbb{R} ^n)}< \infty\right\},
\\[4pt]
q_+(L)& := \sup\left\{p\in(1,\infty) : \sup_{t>0} \|t\nabla_y e^{-t^2L} \|_{L^p( \mathbb{R} ^n)\rightarrow L^p( \mathbb{R} ^n)}< \infty\right\}.
\end{align}
From \cite{Auscher} (see also \cite{AuscherMartell:II}) we know that $p_-(L)=1$ and  $p_+(L)=\infty$ if $n=1,2$; and if $n\ge 3$ then $p_-(L)<\frac{2\,n}{n+2}$ and $p_+(L)>\frac{2\,n}{n-2}$. Moreover, $q_-(L)=p_-(L)$, $ q_+(L)\le p_+(L)$, and we always have $q_+(L)>2$, with $q_+(L)=\infty$ if $n=1$.

We shall obtain a pointwise inequality for the Riesz transform taking a generalized version of two inequalities that appear in \cite[Lemma 4.8 and (4.6)]{Auscher}. These are:
\begin{lemma}\label{lemma1:riesz}
For every ball $B$, with radius $r_B$, and $q_-(L)<r<q_{+}(L)$,
\begin{align*}
\|\nabla L^{-\frac{1}{2}}(I-e^{-r_B^2L})^Mh\|_{L^r(B)}
\leq
|B|^{\frac{1}{r}}\sum_{j\geq 1}g(j)\left(\Xint-_{2^{j+1}B}|h|^r\right)^{\frac{1}{r}},
\end{align*}
with $g(j)=C2^{j\frac{n}{2}}4^{-jM}$, where $M\in  \mathbb{N} $ is arbitrary and $C$ depends on $M$.
\end{lemma}
\begin{lemma}\label{lemma2:riesz}
For every ball $B$, with radius $r_B$, any constant $k>0$, and $q_-(L)<p_0\leq r<q_+(L)$,
\begin{align*}
\left(\Xint-_{B}|\nabla e^{-kr_B^2L} h|^{r}\right)^{\frac{1}{r}}
\leq
\sum_{j\geq 1}g(j)\left(\Xint-_{2^{j+1}B}|\nabla h|^{p_0}\right)^{\frac{1}{p_0}},
\end{align*}
with $\sum_{j\geq 1}g(j)<\infty$.
\end{lemma}
It is in the first inequality that was used the integral representation $\nabla L^{-\frac{1}{2}}h = \pi^{-1/2}\int_{0}^\infty \nabla e^{-tL}h \frac{dt}{\sqrt t}$ for appropriate $h$ (to replace for  the kernel representation in the case of Calder\'on-Zygmund operators).

From these two results we have the following corollary.

\begin{corollary}\label{cor:riesz} Let $q_-(L)<p_0< r<q_+(L)$.
For every $x\in  \mathbb{R} ^n$ and $t>0$ and $f\in L^r( \mathbb{R} ^n)$.
\begin{multline*}
\left(\Xint-_{B(x,t)}|\nabla L^{-\frac{1}{2}}(f)(y)|^r\,dy\right)^{\frac{1}{r}}
\\
\lesssim
\sum_{j\geq 1}4^{-jM}\left(\int_{B(x,2^{j+1}t)}|f(y)|^r\,\frac{dy}{t^n}\right)^{\frac{1}{r}}
+
\sum_{k=1}^{M}C_{k,M}
\mathcal{M}_{p_0}\left(\nabla L^{-\frac{1}{2}}e^{-\frac{kt^2}{2}L}(f)\right)(x),
\end{multline*}
where $M\in  \mathbb{N} $ is arbitrarily large and $\mathcal{M}_{p_0}(f):=(\mathcal{M}(|f|^{p_0}))^{\frac{1}{p_0}}$.
\end{corollary}
\begin{proof}
Fix $x\in  \mathbb{R} ^n$, $t>0$ and  $M\in  \mathbb{N} $ arbitrarily large. We have that
\begin{multline*}
\left(\Xint-_{B(x,t)}|\nabla L^{-\frac{1}{2}}(f)(y)|^r\,dy\right)^{\frac{1}{r}}
\\
\lesssim
\left(\Xint-_{B(x,t)}|\nabla L^{-\frac{1}{2}}(I-e^{-t^2L})^M(f)(y)|^r\,dy\right)^{\frac{1}{r}}
+
\left(\Xint-_{B(x,t)}|\nabla L^{-\frac{1}{2}}A_{t,M}(f)(y)|^r\,dy\right)^{\frac{1}{r}}
=:I+II,
\end{multline*}
 where $A_{t,M}:=I-(I-e^{-t^2L})^M$. Then, applying Lemma \ref{lemma1:riesz} for $B=B(x,t)$ and $h=f$, we obtain that
\begin{align*}
I
\lesssim
\sum_{j\geq 1}4^{-jM}
\left(\int_{B(x,2^{j+1}t)}|f(y)|^r\frac{dy}{t^n}\right)^{\frac{1}{r}}.
\end{align*}
As for the estimate of $II$, note that expanding the binomial expression, we have that $A_{t,M}=\sum_{k=1}^{M}C_{k,M}e^{-kt^2L}$. Then,
 applying Lemma \ref{lemma2:riesz} for $B=B(x,t)$ and $h=L^{-\frac{1}{2}}e^{-\frac{kt^2}{2}L}f$,
\begin{multline*}
II\lesssim \sum_{k=1}^{M}C_{k,M}\left(\Xint-_{B(x,t)}|\nabla e^{-\frac{kt^2L}{2}} L^{-\frac{1}{2}}e^{-\frac{kt^2L}{2}}(f)(y)|^r\,dy\right)^{\frac{1}{r}}
\\
\lesssim
\sum_{k=1}^{M}C_{k,M}
\sum_{j\geq 1}g(j)
\left(\Xint-_{B(x,2^{j+1}t)}|\nabla L^{-\frac{1}{2}}e^{-\frac{kt^2}{2}L}(f)(y)|^{p_0}dy\right)^{\frac{1}{p_0}}
\\
\lesssim
\sum_{k=1}^{M}C_{k,M}
\mathcal{M}_{p_0}\left(\nabla L^{-\frac{1}{2}}e^{-\frac{kt^2}{2}L}(f)\right)(x).
\end{multline*}
\end{proof}
\subsection{Proof of Theorem \ref{thm:riesztransform}}
Recall that  the Riesz transform associated with this operator $L$, acting over a function $F\in T^r_r$ (so that $F(\cdot, t)\in L^r( \mathbb{R} ^n)$ for almost every $t>0$), is defined by
$\nabla L^{-\frac{1}{2}}(F(\cdot,t))(x)$
for almost every $t>0$.
Applying Corollary \ref{cor:riesz}, we obtain, for all $F\in T^r_r$, \begin{multline*}
\|\nabla L^{-\frac{1}{2}}(F)\|_{T^q_r}
\lesssim
\sum_{j\geq 1}4^{-jM}\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\int_{B(x,2^{j+1}t)}|F(y,t)|^r\frac{dy\,dt}{t^{n+1}}\right)^{\frac{q}{r}}dx\right)^{\frac{1}{q}}
\\
+\sum_{k=1}^{M}C_{k,M}
\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\left|\mathcal{M}_{p_0}\left(\nabla L^{-\frac{1}{2}}e^{-\frac{kt^2}{2}L}(F(\cdot,t))\right)(x)\right|^r\frac{dt}{t}\right)^{\frac{q}{r}}dx\right)^{\frac{1}{q}} =:I+\sum_{k=0}^{M}C_{k,M}II.
\end{multline*}
Applying \cite[Section 3, Proposition 4]{CoifmanMeyerStein} or \cite{Auscherangles}, but taking $r$ in place of $2$ (the proof is the same), and  taking $M>\frac{n}{\min\{q,r\}}$, we have that
\begin{align*}
I\lesssim
\sum_{j\geq 1}4^{-j\left(M-\frac{n}{\min\{q,r\}}\right)}\left(\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\int_{B(x,t)}|F(y,t)|^r\frac{dy\,dt}{t^{n+1}}\right)^{\frac{q}{r}}dx\right)^{\frac{1}{q}}
\lesssim \|F\|_{T^q_r}.
\end{align*}
Finally the estimate of $II$ follows by extrapolation. For all weights $w\in A_{\frac{r}{q_-(L)}}\cap RH_{\left(\frac{q_+(L)}{r}\right)'}$ we have that $\nabla L^{-\frac{1}{2}}:L^r(w)\rightarrow L^r(w)$ (\cite[Theorem 5.2]{AuscherMartell:III}) and that $\mathcal{M}_{p_0}:L^r(w)\rightarrow L^r(w)$, for some $p_0>q_-(L)$
close enough to $q_-(L)$ so that $w\in A_{\frac{r}{p_0}}$. Besides, we  can also take $r<q_0<q_+(L)$ so that
$w\in RH_{\left(\frac{q_0}{r}\right)'}$. Using these three facts, applying H\"older's inequality for $\frac{q_0}{r}$, the $L^r( \mathbb{R} ^n)-L^{q_0}( \mathbb{R} ^n)$ off-diagonal estimates that the semigroup $\{e^{-t^2L}\}_{t>0}$ satisfies (see \cite{Auscher}), and Fubini's theorem, we have that
\begin{align*}
\Bigg(\int_{ \mathbb{R} ^n}&\int_0^{\infty}
|\mathcal{M}_{p_0}(\nabla L^{-\frac{1}{2}}e^{-\frac{kt^2}{2}L}(F(\cdot,t)))(x)|^{r}
\frac{dt}{t}w(x)dx\Bigg)^{\frac{1}{r}}
\\
&
\lesssim
\left(\int_0^{\infty}\int_{ \mathbb{R} ^n}
|e^{-\frac{kt^2}{2}L}(F(\cdot,t))(x)|^{r}w(x)dx
\frac{dt}{t}\right)^{\frac{1}{r}}
\\
&
=
\left(\int_0^{\infty}\int_{ \mathbb{R} ^n}
|e^{-\frac{kt^2}{2}L}(F(\cdot,t))(x)|^{r}
\int_{B(x,t)}w(y)\frac{dy}{w(B(x,t))}
w(x)dx
\frac{dt}{t}\right)^{\frac{1}{r}}
\\
&
=
\left(\int_{ \mathbb{R} ^n}\int_0^{\infty}\int_{B(y,t)}
|e^{-\frac{kt^2}{2}L}(F(\cdot,t))(x)|^{r}
w(x)\frac{dx}{w(B(x,t))}
\frac{dt}{t}w(y)dy\right)^{\frac{1}{r}}
\\
&
\lesssim
\left(\int_{ \mathbb{R} ^n}\int_0^{\infty}\int_{B(y,t)}
|e^{-\frac{kt^2}{2}L}(F(\cdot,t))(x)|^{r}
w(x)dx
\frac{dt}{tw(B(y,t))}w(y)dy\right)^{\frac{1}{r}}
\\
&
\lesssim
\left(\int_{ \mathbb{R} ^n}\int_0^{\infty}\left(\int_{B(y,t)}
|e^{-\frac{kt^2}{2}L}(F(\cdot,t)(x)|^{q_0}
dx\right)^{\frac{r}{q_0}}
\left(\int_{B(y,t)}w(x)^{\left(\frac{q_0}{r}\right)'}dx\right)^{\frac{q_0-r}{q_0}}
\frac{dt}{tw(B(y,t))}w(y)dy\right)^{\frac{1}{r}}
\\
&
\lesssim
\sum_{j\geq 1}e^{-c4^{j}}
\left(\int_{ \mathbb{R} ^n}\int_0^{\infty}\int_{B(y,2^{j+1}t)}
|F(x,t)|^{r}
dx
\left(\Xint-_{B(y,t)}w(x)^{\left(\frac{q_0}{r}\right)'}dx\right)^{\frac{q_0-r}{q_0}}
\frac{dt}{tw(B(y,t))}w(y)dy\right)^{\frac{1}{r}}
\\
&
\lesssim
\sum_{j\geq 1}e^{-c4^{j}}
\left(\int_{ \mathbb{R} ^n}\int_0^{\infty}\int_{B(y,2^{j+1}t)}
|F(x,t)|^{r}
\frac{dx\,dt}{t^{n+1}}w(y)dy\right)^{\frac{1}{r}}
\\
&
\lesssim
\left(\int_{ \mathbb{R} ^n}\int_0^{\infty}\int_{B(y,t)}
|F(x,t)|^{r}
\frac{dx\,dt}{t^{n+1}}w(y)dy\right)^{\frac{1}{r}}.
\end{align*}
The second inequality follows from the fact that $B(y,t)\subset B(x,2t)$ if $x\in B(y,t)$ and from the doubling property of the weight. Then,
$w(B(y,t))\leq w(B(x,2t)\leq 2^{nc_w}w(B(x,t))$.

Therefore, we have that, for all $w\in A_{\frac{r}{q_-(L)}}\cap RH_{\left(\frac{q_+(L)}{r}\right)'}$ and $F\in T^r_{r}$,
\begin{align*}
\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}
|\mathcal{M}_{p_0}(\nabla L^{-\frac{1}{2}}e^{-\frac{kt^2}{2}L}(F(\cdot,t)))(x)|^{r}
\frac{dt}{t}\right)^{\frac{r}{r}}w(x)dx
\lesssim
\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\int_{B(x,t)}
|F(y,t)|^{r}
\frac{dy\,dt}{t^{n+1}}\right)^{\frac{r}{r}}w(x)dx.
\end{align*}
Recall that $p_{0}$ depended on $w$. But if we now fix $p_{0}>q_{-}(L)$, we have this inequality for  all $w\in A_{\frac{r}{p_{0}}}\cap RH_{\left(\frac{q_+(L)}{r}\right)'}$.
Then, applying \cite[Theorem 3.31]{CruzMartellPerez}, we obtain that, for all $p_{0}<q<q_+(L)$, $w_0\in A_{\frac{q}{p_{0}}}\cap RH_{\left(\frac{q_+(L)}{q}\right)'}$, and all $F\in T^r_{r}$,
\begin{align*}
\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}
|\mathcal{M}_{p_0}(\nabla L^{-\frac{1}{2}}e^{-\frac{kt^2}{2}L}(F(\cdot,t)))(x)|^{r}
\frac{dt}{t}\right)^{\frac{q}{r}}w_0(x)dx
\lesssim
\int_{ \mathbb{R} ^n}\left(\int_0^{\infty}\int_{B(x,t)}
|F(y,t)|^{r}
\frac{dy\,dt}{t^{n+1}}\right)^{\frac{q}{r}}w_0(x)dx.
\end{align*}
In particular, if we take $w_0\equiv 1$, we have that $w_0\in A_{\frac{q}{p_{0}}}\cap RH_{\left(\frac{q_+(L)}{q}\right)'}$. Then, for all $p_{0}<r,q<q_+(L)$ and $1\leq k\leq M$, we finally conclude that
$$
II
\lesssim \|F\|_{T^q_r}.
$$
In conclusion, we obtain $\|\nabla L^{-\frac{1}{2}}(F)\|_{T^q_r} \lesssim  \|F\|_{T^q_r}$ for all
$p_{0}<r,q<q_+(L)$ and all $q_{-}(L)<p_{0}<q_{+}(L)$, and for all $F\in T^r_{r}$. The density of $T^r_{r}\cap T^q_{r}$ in $T^q_{r}$  finishes the proof.
\qed

\section{Amalgam spaces and generalization}\label{section:amalgam}

Amalgam spaces were first defined by
Norbert Wiener in 1926, in the formulation of his generalized harmonic analysis. Although,  he considered the particular cases $W(L^1,\ell^2)$ and $W(L^2,L^{\infty})$ in \cite{Wiener}, and $W(L^1,L^{\infty})$ and $W(L^{\infty}, L^1)$ in \cite{Wiener2},
in general,
for $1\leq p,q<\infty$, the amalgam space $W(L^p,L^q)$ is defined as
$$
W(L^p,L^q):=\left\{f\in L^{p}_{loc}( \mathbb{R} ):\left(\sum_{n\in  \mathbb{Z} }\left(\int_{n}^{n+1}|f(t)|^pdt\right)^{\frac{q}{p}}\right)^{\frac{1}{q}}<\infty\right\}.
$$
A significant difference in considering amalgam spaces instead of $L^p$ spaces is that
amalgam spaces give information about the local, $L^q$, and global, $L^p$, properties of the functions, while $L^p$ spaces do not make that distinction.

A generalization of the definition of amalgam spaces for Banach function spaces was done by Feichtinger (see for instance \cite{Feichtinger} and \cite{Feichtinger2}). For $B$ and $C$ Banach function spaces on a locally compact group $G$, satisfying certain conditions, he defined  spaces $W(B,C)$ of distributions. The important thing is that we have equivalence of continuous and discrete norms on those spaces. This has been an important tool in applications. We refer to \cite{Heil} for a deeper discussion on amalgam spaces in the real line.

 Going on in the historical background of amalgam spaces, we highlight the paper of Holland, \cite{Holland}, that appears to be the first methodical study on amalgam spaces. After that there were important studies on amalgam spaces, for example, by Bertrandias, Datry, and Dupuis \cite{BertrandiasDatryDupuis}, Stewart \cite{Stewart}, and Busby and Smith \cite{BusbySmith}.
For a complete survey on amalgam spaces see \cite{FournierStewart}.

A natural definition of amalgam spaces in dimension $n\geq 2$ is
$$
(L^p,L^q)( \mathbb{R} ^n):=\left\{f\in L^{p}_{loc}( \mathbb{R} ^n):\left(\int_{ \mathbb{R} ^n}\|\chi_{B(x,1)}f\|_{L^p}^q\,dy\right)^{\frac{1}{q}}<\infty\right\}.
$$
Beside, for $1\leq \alpha\leq \infty$,  the subspace $(L^p,L^q)^{\alpha}( \mathbb{R} ^n)$ of $(L^p,L^q)( \mathbb{R} ^n)$ is defined in \cite{Fofana} by
$$
(L^p,L^q)^{\alpha}( \mathbb{R} ^n):=\left\{f\in L^{p}_{loc}( \mathbb{R} ^n):\|f\|_{(L^p,L^q)^{\alpha}( \mathbb{R} ^n)}<\infty\right\},
$$
where
$$
\|f\|_{(L^p,L^q)^{\alpha}( \mathbb{R} ^n)}:=\sup_{r>0}
\left(\int_{ \mathbb{R} ^n}\left(|B(y,r)|^{\frac{1}{\alpha}-\frac{1}{p}-\frac{1}{q}}\|\chi_{B(x,r)}f\|_{L^p}\right)^q\,dy\right)^{\frac{1}{q}}.
$$
In \cite{AuscherMourgoglou} retracts of tent spaces called slice-spaces, are used. It turns out that they are closely related with amalgam spaces. Let us generalize their definition. For each $t>0$ and $0<p,r<\infty$, the slice-space $(E^p_{r})_{t}$ is defined as the following set:
$$
(E^p_{r})_{t}:=\left\{f\in L^{r}_{loc}( \mathbb{R} ^n):\left(\Xint-_{B(x,t)}|f(y)|^rdy\right)^{\frac{1}{r}} \in L^p( \mathbb{R} ^n)\right\}
$$
with
$$
\|f\|_{(E^p_{r})_{t}}= \left(\left(\Xint-_{B(x,t)}|f(y)|^rdy\right)^{\frac{p}{r}} \, dx\right)^{\frac{1}{p}}.
$$
Besides, consider the weak slice-spaces
$$
(wE_r^{p})_{t}:=\left\{f\in L^{r}_{loc}( \mathbb{R} ^n):\left(\Xint-_{B(x,t)}|f(y)|^rdy\right)^{\frac{1}{r}} \in L^{p,\infty}( \mathbb{R} ^n)\right\}.
$$
with
$$
\|f\|_{(wE^p_{r})_{t}}= \left\|\left(\Xint-_{B(x,t)}|f(y)|^rdy\right)^{\frac{p}{r}} \, dx\right\|_{L^{p,\infty}( \mathbb{R} ^n)}.
$$
For $1\le r,p<\infty$, note that, for $n=1$, $(E_r^p)_1=W(L^r,L^p)$, and for $n\geq 2$,  $(E_r^p)_1=(L^r,L^p)( \mathbb{R} ^n)$. Furthermore, for $p\in [r,\infty)$, since
$
\|f\|_{(E^p_r)_t}\leq \|f\|_{L^p}$, for all  $t>0$,
$(E_r^p)_t=(L^r,L^p)^p( \mathbb{R} ^n)$.

Boundedness on amalgam spaces of the Hardy Littlewood maximal operator,
of Calder\'on-Zygmund operators, of maximal fractional operators, Riesz potentials, etc,  has been studied. See for instance \cite{CLHH}, \cite{CowlingMP}, \cite{FeutoFK}, \cite{Kikuchi}.
From Lemmas \ref{lemma:C_Z}, \ref{lemma:H-Lmaximal} and \ref{lemma:rieszpotential}, and Corollary \ref{cor:riesz}, we can obtain easily boundedness of those operators on slice-spaces, and, therefore,  on amalgam spaces. This  significantly simplifies the previous proofs on this issue.

Let $1\le r<\infty$. Let $t>0$. Consider the applications $i_{t}$ and $\pi_{t}$ in \cite{AuscherMourgoglou}: for $f: \mathbb{R} ^n\rightarrow  \mathbb{C} $,
$$
i_{t}(f)(x,s)=f(x)\chi_{[t,et]}(s),
$$
and for $G: \mathbb{R} ^{n+1}_+\rightarrow  \mathbb{C} $,
$$
\pi_{t}(G)(x)=\int_{t}^{et}G(x,s)\frac{ds}{s}.
$$
It is easy to see that
$$\pi_{t}\circ i_{t}(f)=f.$$
\begin{lemma} Let $0<p<\infty$ and $1\le r<\infty$. Then $i_{t}:(E_r^p)_t\rightarrow T_r^p$ and $\pi_{t}:T_r^p\rightarrow (E_r^p)_t$ are bounded with the norms being uniform with respect to $t$. In particular, the slice-spaces  $(E_r^p)_t$ are retracts of $T_r^p$. The same happens for the weak slice-spaces, they are retracts of the weak tent spaces.
\end{lemma}

\begin{proof}
For the slice-spaces when $r=2$, this is observed without proof in  \cite{AuscherMourgoglou}. The proof is the same for all (weak) slice-spaces. It suffices to note that
$$
\left(\Xint-_{B(x,t)}|\pi_{t}(G)(y)|^rdy\right)^{\frac{1}{r}}  \le C \mathcal{A}_r(G)(x)
$$
and
$$\mathcal{A}_r(i_{t}(f))(x) \le C \left(\Xint-_{B(x,et)}|f(y)|^rdy\right)^{\frac{1}{r}}$$
for some dimensional constants $C$,
and to use the norm comparison below for the slice-spaces and similarly for the weak slice-spaces.
\end{proof}

\begin{lemma}\label{lemma:changeofnorms}
If $0<t,s<\infty$ with $t\sim s$,  $1\le r<\infty$ and $p\in (0,\infty)$, then $(E_r^p)_t=(E_r^p)_s$ with
$$
\|f\|_{(E_r^p)_t}\sim_{n,p}\|f\|_{(E_r^p)_s}
$$
\end{lemma}

For any  linear operator $T$ on functions on $ \mathbb{R} ^n$, if $\mathcal{T}$ is its extension to functions on $ \mathbb{R} ^{n+1}$ by tensorisation, then we have
$T =\pi_{t} \circ \mathcal{T} \circ i_{t}$.  Hence the boundedness of $\mathcal{T}$ carries to $T$ (In the previous theorems, we used the opposite direction: boundedness of $T$ yields boundedness of $\mathcal{T}$. But it was not that immediate). This also applies to maximal operators with easy modifications. So immediate corollaries of our results on tent spaces are the followings.
\begin{proposition}
Let $\mathcal{M}$ be the centered Hardy-Littlewood maximal operator. We have, for all $1<r<\infty$,
\begin{list}{$(\theenumi)$}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm\itemsep=0.2cm\topsep=.0cm \renewcommand{\theenumi}{\alph{enumi}}}

\item $\mathcal{M}:(E_r^p)_t\rightarrow (E_r^p)_t$ for all $1<p<\infty.$

\item $\mathcal{M}:(E_r^{1})_t\rightarrow (wE_{r}^{1})_t$\ .

\end{list}
\end{proposition}

\begin{proposition}
Let $\mathcal{T}$ be a Calder\'on-Zygmund operator of order $\delta \in (0,1]$. We have, for all $1<r<\infty$,
\begin{list}{$(\theenumi)$}{\usecounter{enumi}\leftmargin=1cm \labelwidth=1cm\itemsep=0.2cm\topsep=.0cm \renewcommand{\theenumi}{\alph{enumi}}}

\item $\mathcal{T}:(E_r^p)_t\rightarrow (E_r^p)_t$,\, for all $1<p<\infty.$

\item $\mathcal{T}:(E_r^{1})_t\rightarrow (wE_{r}^{1})_t$.

\item $\mathcal{T}:(\mathfrak{E}_r^p)_t\rightarrow (E_r^p)_t$,\, for all $\frac{n}{n+\delta}<p\leq 1.$

\item $\mathcal{T}:(\mathfrak{E}_r^p)_t\rightarrow (\mathfrak{E}_r^p)_t$,\, for all $\frac{n}{n+\delta}<p\leq 1.$ if $\mathcal{T}^*(1)=0$.

\end{list}
\end{proposition}
Here,
$
(\mathfrak{E}_r^{p})_t
$
is the space of functions in $(E^p_r)_t$ such that there exists an atomic decomposition  $\sum_{i=1}^{\infty}{\lambda_i}a_i$ with $\int_{ \mathbb{R} ^n}a_i(x)\,dx=0$, for all $i\in  \mathbb{N} $. The atoms are defined in \cite{AuscherMourgoglou} for $r=2$ and this adapts here. It suffices for understanding the statement to remark that $(\mathfrak{E}_r^{p})_t= \pi(\mathfrak{T}_r^p)$.

\begin{proposition}
Let $\mathcal{M}_{\alpha}$ be the maximal fractional and $\mathcal{I}_{\alpha}$ the Riesz potential of order $\alpha\in (0,n)$. We have, for all $\frac{n}{n-\alpha}<r<\infty$ and $1<p<q<\infty$ with $\frac{1}{p}-\frac{1}{q}=\frac{\alpha}{n}$,
$$
\mathcal{M}_{\alpha},\mathcal{I}_{\alpha}:(E_r^q)_t\rightarrow (E_r^p)_t.
$$
\end{proposition}

\begin{proposition}
Let $\nabla L^{-\frac{1}{2}}$ be the Riesz transform associated to $L$. We have, for $q_-(L)<p,r<q_+(L)$,
$$
\nabla L^{-\frac{1}{2}}:(E_r^p)_t\rightarrow (E_r^p)_t.
$$
\end{proposition}
\section{Concluding remarks}

We note that all the arguments using extrapolation prove much more than what we stated.

For $\frac{n}{n+1}<q<\infty$ and $1<r<\infty$, one can show  that  the set
$$
E:=\left\{\varphi \in C^{\infty}_0( \mathbb{R} ^{n+1}_{+}):\, \int_{ \mathbb{R} ^n}\varphi(x,t)\,dx=0\, \textrm{for all}\,t>0\right\}
$$
is dense in $\mathfrak{T}^q_r$ when $q\le 1$ and in ${T}^q_r$ when $q>1$. For $q\le 1$, it suffices to do that on $\mathfrak{T}^q_r$ atoms and for $q>1$, we already know that the space of compactly supported smooth functions in $ \mathbb{R} ^{n+1}_{+}$ is dense and  those functions can be approximated in $L^r$ norm imposing the  mean value condition   using $r>1$. So the fact that there is a common dense subspace is an indication that the space $\mathfrak{T}^q_r$ is not to small.

It is clear one can push Theorem \ref{thm:C-Zoperator}, part (c) and (d), to any  Calder\'on-Zygmund operator on $ \mathbb{R} ^n$ of order $\delta\ge 1$ (see \cite{Duo}, \cite{Grafakos} for definition) imposing more vanishing moments in the definition of $\mathfrak{T}^q_r$ atoms when $q\le \frac{n}{n+1}$ and more cancellation conditions on the adjoint. Similarly, we can play the same game on slice-spaces. These slice-spaces will be subspaces of the classical real Hardy spaces as one can show. We do not insist.

Consider a standard Littlewood-Paley decomposition of $ \mathbb{R} ^n$ given from a pair of
$C^\infty_{0}$  functions $\psi,\tilde\psi$ with all vanishing moments and such that
$$
\int_{0}^\infty Q_{t}\tilde Q_{t} f\frac{dt}{t}=f
$$
on appropriate distributions $f$, where $Q_{t}$ and
$\tilde Q_{t}$ are convolutions with $\psi_{t}$ and $\tilde \psi_{t}$ respectively. We have set $\psi_{t}(x)= t^{-n}\psi(x/t)$ and likewise for $\tilde \psi_{t}$.
One can show that $f\in H^q( \mathbb{R} ^n)$ implies $F(x,t)= \tilde Q_{t}f(x)$ belongs to $\mathfrak{T}^q_2$ and the action is bounded. Conversely, $F\in \mathfrak{T}^q_2$ implies that $f=\int_{0}^\infty Q_{t}F(\cdot, t)\, \frac{dt}{t}$ belongs to $H^q( \mathbb{R} ^n)$ and the action is bounded. This is fairly easy to show using atoms and molecules. This can be done for $0<q\le 1$. Thus, $H^q( \mathbb{R} ^n)$ can be seen as a  retract of the space  $\mathfrak{T}^q_2$. It is also the case using ${T}^q_2$ instead as shown in \cite{CoifmanMeyerStein}. Nevertheless, the spaces $\mathfrak{T}^q_2$ are preserved by the singular integrals (of convolution) while the ${T}^q_2$ are not. It would be interesting to explore further these spaces (interpolation, etc) and their applications. In particular, one could recover boundedness for Calder\'on-Zygmund operators on tent spaces from interpolation.

redpajama_set_name: RedPajamaArXiv


Watch this week's multimedia videos HIPAA Privacy Laws and HIPPA- Healthy Wilmington Update- February 2013. In 250-300 words discuss what a medical practice or hospital can do to ensure that the patient and the entity are in compliance with disclosures of private health information. How does the hospital or medical practice cover themselves when a HIPAA violation takes place via an unsecured laptop or an unencrypted memory stick with patient health information on it? Respond to two of your classmates' posts.
Hospitals have come up with better methods for ensuring that patient's information is safe. Some electrical methods for recording data have been invented in different hospitals. For example, the Electronic Medical Records (EMR) for keeping patients medical records has been invented with the new technology. This type of keeping patients medical……………….


text:
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Member Preview - A Tale of Today: Yinka Shonibare CBE
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A special evening for Members to celebrate the launch of A Tale of Today and this highly anticipated exhibition with light refreshments and live music. Admission is included with current Driehaus Museum membership.
View A Tale of Today, Yinka Shonibare CBE before the general public at two events open to Driehaus Museum members.
Preview Reception
Friday, March 1, 6 - 8 p.m.
Enjoy drinks, bites, and live entertainment at an evening reception celebrating the opening of this highly anticipated exhibition.
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Vegan on The Magnificent Mile
The rise of veganism is swift. In the past year, it is estimated that U.S consumers spent $1.9 billion on plant based milks and $3.3 billion on plant based foods. It is no surprise that the food and restaurant industry has taken notice and is being more accommodating. Read more
Twenty Things to Do in Chicago in Your Twenties
What I've learned from living in Chicago as a young adult is that your twenties are your "selfish years". While living in the city you must travel, explore, and try new things. After all, it's important to love yourself and the place that you're in! Read more
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text:
The New 42nd Street is a not-for-profit organization based in Manhattan, New York City. In 1990, the New 42nd Street was formed to oversee the redevelopment of seven neglected and historic theatres on 42nd Street between Seventh and Eighth Avenues, and to restore the block to a desirable tourist destination in Manhattan. The theatres were the Apollo Theatre, the Empire Theatre, the Liberty Theatre, the Lyric Theatre, the Selwyn Theatre, the Times Square Theatre, and the Victory Theater.

The Victory Theater was the first theater on the block to be restored, and reopened as the off-Broadway New Victory Theater in 1995.  The New Victory Theater is programmed by the New 42nd Street with a focus on family entertainment, including international productions of theater, circus, puppetry, opera and dance for kids of all ages. The theater's programming is complemented by an award-winning educational program in New York City schools.
The Apollo and Lyric theatres were demolished, but sections were preserved for incorporation into a new 1,900-seat Broadway musical venue. On December 26, 1997, it opened as The Ford Center for the Performing Arts with the New York premiere of Ragtime. Subsequently, it was renamed the Hilton Theatre and later the Foxwoods Theatre. Following a takeover by the Ambassador Theatre Group, it has taken the Lyric Theatre name.
The Empire and Liberty became parts of an entertainment complex built by Forest City Ratner which includes the New York branch of Madame Tussauds Wax Museum and the Ripley's Believe It or Not! Odditorium. The shell of the Empire was physically lifted and moved closer to Eighth Avenue, becoming the lobby of an AMC Theatres cinema, which opened in 2000.
The Selwyn Theatre became the 750-seat American Airlines Theatre, reopening on July 27, 2000, following renovations, and is currently one of Roundabout Theatre Company's Broadway venues.
In 2011, Broadway 4D Theaters, LLC leased the Times Square Theater for a new multimedia Broadway-themed 4-D attraction; however, the project was cancelled. In 2018, developers announced the venue would be converted to retail space that would retain the proscenium, boxes, and many elements from the original structure. The work would take approximately two years at a cost of $100 million.

The New 42nd Street also operates the New 42nd Street Building at 229 West 42nd Street, designed by the firm of Platt Byard Dovell, which opened in 2000 and is home to the New 42nd Street Studios as well as The Duke on 42nd Street – a 199-seat black box theater named for Doris Duke – and three floors of office spaces used by seven non-profit performing arts organizations, including the New 42nd Street.

References

External links

 The New 42nd Street website
 The New Victory Theater website

Theatre in New York City
American theatre managers and producers
Companies based in New York City
42nd Street (Manhattan)
1990 establishments in New York City

redpajama_set_name: RedPajamaWikipedia


15 Things You Didn't Know about Your Paycheque | Unions Yeah!
1. Canada's economy has doubled in size since 1980.
2. But all that extra cash isn't flowing into your wallet.
3. And taxes haven't adjusted as they should.
4. Today, the top 1% pay a lower total tax rate than the bottom 10%.
5. In 2009, 3.8% of the population controlled $1.78 trillion of the wealth — that's 67% of the total wealth in the country!
9. And governments have been cutting corporate taxes dramatically.
10. In theory, this is supposed to lead to job creation, but most corporations keep the money for themselves.
11. Between 2006 and 2013 alone, the Harper government tax cuts saved corporations $220 billion!
12. That's money that could have been used to fund education, child care, health care, or a whole lot of other public services.
13. Instead, corporations kept the money and the cost of living keeps going up.
14. Personal debt has reached 171.1 % — meaning for every dollar earned, Canadians owe $1.71.
15. All of this is bad news for Canadians, but there are ways to help.
The National Union of Public and General Employees (NUPGE) has relaunched the All Together Now! campaign.
Focused on raising awareness around growing income inequality, NUPGE will do everything it can to protect the future of all Canadians. Because if it matters to you, we'll fight for it.
We'll give you the tools you need to help us make a fairer Canada for everyone. Read our fact sheets. Use our map to find allies against income inequality in your area. Talk to your friends, neighbours, and family. Make your voice heard!
And if you're reading this thinking, "This is garbage, I've never been helped by a union," why not try The Union Challenge?
What are you waiting for? Go to UnionsYeah.ca right now and start the conversation!


Tbilisi, Georgia. Three policemen arrest Checkie, 45, and tell him to introduce his friend's son to drugs so that they can blackmail his father. If he refuses to cooperate, he will end his life in prison. Unemployed and addicted to heroin, Checkie is in no position to refuse — but, faced with such a stark choice, he desperately tries to find a way out.
The director said about film: For me this film is about a strange and a very sad generation. These people were born and grew up in the Soviet Union, but after this country collapsed in front of their eyes, they were forced to adapt to a different reality. (…) Many of them are no longer alive, and those who survived are called 'the lost generation'. This film is very personal, because it tells the story of a generation which I remember, which I love, and which I pity — the generation I feel I belong to.
Levan Koghuashvili was born in Georgia in 1973. After finishing school, he began to study at the State Institute of Film and Theatre in Tbilisi, but one year later a civil war broke out in Georgia and he began to work as a journalist and later as a director. He graduated from New York Film School.


While the work with children and young people is overseen by Somerset County Council.
Please follow the links below for further support and services.
Change4Life can help children and families get healthy, with tips on healthy eating, games and activities.
The Family Counselling Trust offers a grant-supported service for children, adolescents and families where there are emotional, behavioural or other mental health problems. We see children, adolescents and families where there are children up to their 18th birthday.
Look Out For Your Mates is about giving people information so that they are able to make safer choices when drinking alcohol. It is aimed at people aged 16-24 who live in Somerset.
If you, or a young person close to you has a concern or problem with alcohol or drugs, there are a range of services in Somerset to offer support and help.
FRANK helps you find out everything you might want to know about drugs (and some stuff you don't). For friendly, confidential advice, Talk to FRANK.


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redpajama_set_name: RedPajamaGithub


text:
Luis Federico de Wurtemberg-Montbéliard (Montbéliard, 29 de enero de 1586-ibidem, 26 de enero de 1631) fue el fundador de una rama dinástica lateral de la Casa de Wurtemberg conocida como duques de Wurtemberg-Montbéliard.

Luis Federico de Wurtemberg nació en el castillo de Montbéliard como el quinto vástago y segundo hijo varón del duque Federico I de Wurtemberg y de la princesa Sibila de Anhalt.

Biografía
El 28 de mayo de 1617 Luis Federico acordó con su hermano mayor, el duque Juan Federico dividir la posesión de Wurtemberg. Luis Federico recibió las posesiones en la margen izquierda del Rin (Montbéliard, Riquewihr y Horbourg) con plena soberanía. Así, Luis Federico se hizo fundador de la línea menor de Wurtemberg-Montbéliard.

Durante la Guerra de los Treinta Años, su territorio sufrió gravemente por el hambre y la peste. Después de la muerte de su hermano Juan Federico el 28 de julio de 1628, Luis Federico se trasladó a Stuttgart, para actuar como tutor y regente en nombre de su primo menor de edad Everardo III. Su mandato como regente fue dominado por los conflictos con el emperador Fernando II, con los que Wurtemberg perdió un tercio de su territorio debido al Edicto de Restitución. En noviembre de 1630, Luis Federico retornó a Montbéliard, donde murió el 26 de enero de 1631.

Matrimonio e hijos

Luis Federico contrajo matrimonio el 14 de julio de 1617 con Isabel Magdalena de Hesse-Darmstadt, hija del landgrave Luis V de Hesse-Darmstadt. El matrimonio produjo tres hijos:
 Cristóbal (24 de diciembre de 1620 - 1 de enero de 1621)
 Enriqueta Luisa (20 de junio de 1623 - 24 de agosto de 1650), desposó el 21 de agosto de 1642 al Margrave Alberto II de Brandeburgo-Ansbach.
 Leopoldo Federico (30 de mayo de 1624 - 15 de junio de 1662), su sucesor.

Después de la muerte de Isabel Magdalena el 9 de junio de 1624, Luis Federico contrajo matrimonio el 15 de mayo de 1625 con Ana Leonor de Nassau-Weilburg-Saarbrücken. Este matrimonio también produjo tres hijos:
 Jorge II (5 de octubre de 1626 - 1 de junio de 1699)
 Enrique (19 de diciembre de 1627 - enero de 1628)
 Georgia Ludovica (1 de febrero de 1630 - 11 de abril de 1630)

Ancestros

Referencias
 Jean-Marc Debard: Ludwig Friedrich. In: Sönke Lorenz, Dieter Mertens, Volker Press (Hrsg.): Das Haus Württemberg. Ein biographisches Lexikon. Kohlhammer, Stuttgart 1997, ISBN 3-17-013605-4, pp. 178-180.
 Gerhard Raff: Hie gut Wirtemberg allewege. Vol. 2: Das Haus Württemberg von Herzog Friedrich I. bis Herzog Eberhard III. Mit den Linien Stuttgart, Mömpelgard, Weiltingen, Neuenstadt am Kocher, Neuenbürg und Oels in Schlesien. 4. Auflage. Landhege, Schwaigern 2014, ISBN 978-3-943066-12-8, pp. 159-184.

Enlaces externos

Duques de Wurtemberg del siglo XVII
Nacidos en Montbéliard


This is a simple test structure (with low build height) to test the handling of implicit intersections between solid lumps.
This is a simple ladder structure with 10 rungs between two double rails. The rungs stick into the rails, and the double rails intersect length-wise. The path was generated by hand using the Berkeley UniGrafix language.
This part was specifically designed to test the handling of such implicit intersections by different slicing programs.
The orientation, scale or strength of this model is not critical; it can be scaled so as to fit optimally a particular process or some remaining space in a run of a larger piece. The .STL files describe a part that 2" wide (in x), 4" long (in y), and about 0.3" thick (in z).
Berkeley UniGrafix description of the knot (ladder.ug).


If you find that your lower back is aching after sitting down for long periods, it could be because there is too much pressure on your lower spinal discs and tailbone (coccyx). This fully portable, uniquely designed Cushion can help relieve that pressure. Made fromhardwearing and comfortable polyurethane foam the Cushion has a removable zip-up cover and a convenient carry handle.
Wipe clean only as it is leather.


*Evelyn Gibson Lowery, founder and chair of SCLC/Women's Organizational Movement for Equality Now, Inc. announced today that former Georgia State Director of USDA Rural Development, Shirley Sherrod; actress Jasmine Guy and entertainer Fonzworth Bentley are among the honorees during the 32nd annual Drum Major for Justice Awards Dinner on Monday April 4.
Held in honor of Dr. Martin Luther King Jr. on the anniversary of his assassination, the dinner will take place at the Hyatt Regency Hotel in downtown Atlanta. For ticket information call (404) 584-0303.
Dr. Bill Releford, a leader in the field of diabetic limb salvage and founder of The Black Barbershop Health Outreach Program, will also be honored.
Other honorees include Real Men Cook for Charity, a national Father's Day event held to positively change the way the world views black men; and the first black all-female flight crew: Captain Rachelle Jones, First Officer Stephanie Grant, Flight Attendant Diana Galloway, and Flight Attendant Robin Rogers.
Recipients of the Presidential Medal of Freedom in Civil Rights will also be awarded including, civil rights icon, Dr. Joseph E. Lowery; former Atlanta mayor and U.S. Ambassador, Andrew Young; and Congressman John Lewis.
Entertainment will include a performance by 10-year-old Kirlisa Broxton, winner of a talent contest held during the recent Evelyn Gibson Lowery Civil Rights Heritage Tour.


Grupo Gondi has more than 58 years experience in the paper and packaging industry and is one of the largest packaging companies in Mexico. Consisting of seven companies and a labour force of more than 3,800 their multi-plant operation includes 5 paper mills and over 40 converting machines. Their customer base consists of many global companies who are household names in the UK. They include customers from the food and beverage, personal care, appliances and produce industries. Their customers expect high quality products, backed by excellent service, expertise and support.
Faced with a highly competitive global marketplace, continual and rapid changes in the industry alongside the usual cost pressures, Grupo Gondi continue to strive for business efficiency. When Grupo Gondi approached Onesys they had identified a need for much greater uniformity of systems within their sales operation. They were looking for an integrated system with a centralised database in order to improve on quote lead times.
OrderlineBOX was chosen as the ideal solution. Grupo Gondi now have a total of 7 staff across 4 different Mexican cities using the Spanish version of OrderlineBOX. This number is planned to increase to 15 in the near future as they roll it out across their various operations.
*Our users are really happy with OrderlineBOX because it is really user-friendly and easy to use."