text:
\section{Introduction}\label{Sec 1 Intro}

Let $f$ be a bounded function on $(0,\infty)$ and $u(f)$ the operator on $L^p(\mathbb{R}^d)$ defined by $[u(f)g]\hat{\phantom{i}}(\xi) = f(|\xi|^2) \hat{g}(\xi).$
H\"ormander's theorem on Fourier multipliers \cite[Theorem 2.5]{Hor} asserts that $u(f) : L^p(\mathbb{R}^d) \to L^p(\mathbb{R}^d)$ is bounded for any $p \in (1,\infty)$ provided that for some integer $N$ strictly larger than $\frac{d}{2}$
\begin{equation}\label{Equ Hormander original}
\sup_{R > 0} \int_{R/2}^{2R} \left|t^k f^{(k)}(t)\right|^2 \frac{dt}{t} < \infty \quad \left(k = 0,1,\ldots,N\right).
\end{equation}

This theorem has many refinements and generalisations to various similar contexts.
For $\alpha > \frac12,$ let $W^\alpha_2(\mathbb{R}) = \{ f \in L^2(\mathbb{R}):\: \|f\|_{W^\alpha_2(\mathbb{R})} = \| (1 + \xi^2)^{\alpha/2} \hat{f}(\xi)\|_{L^2(\mathbb{R})} < \infty \}$ denote the usual Sobolev space, and $W^\alpha = \{  f : (0,\infty) \to \mathbb{C}:\: f \circ \exp \in W^\alpha_2(\mathbb{R})\},$ which is a Banach algebra with respect to $\|f\|_{W^\alpha} = \|f \circ \exp\|_{W^\alpha_2(\mathbb{R})}.$
Let $\phi_0 \in C^\infty_c(\frac12,2).$
For $n \in \mathbb{Z},$ let $\phi_n = \phi_0(2^{-n} \cdot)$ and assume that $\sum_{n \in \mathbb{Z}} \phi_n(t) = 1$ for any $t > 0.$
Such a function exists \cite[Lemma 6.1.7]{BeL} and we call $(\phi_n)_{n \in \mathbb{Z}}$ a dyadic partition of unity.
We define the Banach algebra
\[ \Hor^\alpha = \left\{ f : (0,\infty) \to \mathbb{C} :\: \| f\|_{\Hor^\alpha} = \sup_{n \in \mathbb{Z}} \|\phi_n f\|_{W^\alpha} < \infty \right\}.\]
The definition of $\Hor^\alpha$ is independent of the dyadic partition of unity, different choices resulting in equivalent norms \cite[Section 4.2]{Kr}.
The space $\Hor^\alpha$ refines \eqref{Equ Hormander original}, more precisely, $f \in \Hor^\alpha$ implies that $f$ satisfies \eqref{Equ Hormander original} for $N \leq \alpha,$ and the converse holds for $N \geq \alpha$ \cite[Proposition 4.11]{Kr}.

Now if $A$ is a self-adjoint positive operator on some $L^2(\Omega,\mu),$ then its functional calculus assigns to any bounded measurable function $f$ on $(0,\infty)$ an operator $f(A)$ on $L^2(\Omega,\mu).$
In particular, if $A = - \Delta$ and $(\Omega,\mu) = (\mathbb{R}^d,dx),$ then $f(A)$ equals the above $u(f).$
A  theorem of H\"ormander type holds true for many elliptic differential operators $A,$ including sublaplacians on Lie groups of polynomial growth, Schr\"odinger operators and elliptic operators on Riemannian manifolds \cite{DuOS,Alex,Bluna,Chri,Duon}.
By this, we mean that
\begin{equation}\label{Equ Intro Hormander calculus}
u : \Hor^\alpha \to B(X),\, f \mapsto f(A)\text{ is a bounded homomorphism,}
\end{equation}
where $X = L^p(\Omega),\,p \in (1,\infty),$ $\alpha$ is the differentiation parameter typically larger than $\frac{d}{2},$ where $d$ is the dimension of $\Omega,$ and $f(A)$ is
(the unique bounded $L^p$-extension of) the self-adjoint functional calculus.\\

The aim of this article is to characterize the validity of the H\"ormander multiplier theorem for $A$ in terms of square function estimates.

The latter have been introduced in Stein's classical book \cite{Ste1} and have since then been used widely with applications to functional calculi and multiplier theorems.
Note that $\|(\cdot)^{it}\|_{\Hor^\alpha} \cong ( 1 + |t|^2)^{\alpha/2}$ \cite[Proposition 4.12 (4)]{Kr}, so that for this particular function, \eqref{Equ Intro Hormander calculus} implies $\|A^{it}\| \leq C ( 1 + |t|^2 )^{\alpha/2}.$
Then a natural square function estimate for our situation is
\begin{equation}\label{Equ Intro Square Function}
\| (1 + t^2)^{-\alpha/2} A^{it} x \|_{\gamma(\mathbb{R},X)} \leq C \| x\|_X ,
\end{equation}
where $\gamma(\mathbb{R},X)$ is given by
\[ \| x(t) \|_{\gamma(\mathbb{R},X)} \cong \left\| \left( \int_\mathbb{R} \left|x(t)\right|^2 dt \right)^{\frac12} \right\|_X \]
for $X = L^p(\Omega,\mu)$ and $p \in [1,\infty),$ which explains the name square function.
The general definition of the space $\gamma(\mathbb{R},X)$ involves Gaussian random sums in the Banach space $X,$ see Section \ref{Sec 2 Prelims}.

Our setting, developed in Section \ref{Sec 2 Prelims}, is as follows:
We let $X$ be a Banach space having Pisier's property $(\alpha),$ which a geometric property playing an important role for the theory of spectral multipliers.
It is natural to assume the operator $A$ to be $0$-sectorial
i.e. a negative generator of an analytic semigroup $(\exp(-zA))_{\Re z > 0}$ which is uniformly bounded on the sector $\Sigma_\omega = \{ z \in \mathbb{C} \backslash \{ 0 \} :\: | \arg z | < \omega \}$ for each $\omega < \frac{\pi}{2}.$
Indeed, $\exp(-z \cdot)$ belongs to $\Hor^\alpha$ with uniform norm bound on such sectors.
Further, for simplicity we assume throughout that $A$ has dense range.

We shall base the definition of $u$ in \eqref{Equ Intro Hormander calculus} on the well-known $H^\infty$ functional calculus \cite{CDMY,KuWe}.
This means that for $f$ belonging to $H^\infty_0(\Sigma_\omega) = \{ f \in H^\infty(\Sigma_\omega) :\: \exists\: \epsilon,\, C > 0 \text{ s.th. }|f(z)| \leq C \min ( |z|^{\epsilon}, |z|^{-\epsilon} ) \}$ which is a subclass of
$H^\infty(\Sigma_\omega) = \{ f : \Sigma_\omega \to \mathbb{C} :\: f\text{ is analytic, }\|f\|_{\infty,\omega} = \sup_{z \in \Sigma_\omega} | f(z) | < \infty \},$ $f(A) \in B(X)$ is defined by a certain Cauchy integral formula, see \eqref{Equ Cauchy integral formula}.
Secondly, under certain conditions, $A$ has a bounded $H^\infty$ calculus, which means that
there is an extension to a bounded homomorphism $H^\infty(\Sigma_\omega) \to B(X),\,f \mapsto f(A).$
Note that $H^\infty(\Sigma_\omega)$ is a subclass of $\Hor^\alpha.$
In Lemma \ref{Lem A u} it will be shown in particular that an extension of the $H^\infty$ calculus to a bounded homomorphism $u : \Hor^\alpha \to B(X)$ is unique.

For any such mapping $u$ and $n\in \mathbb{N},$ we now consider the linear tensor extension
\[ u_n : \begin{cases} M_n  \otimes \Hor^\alpha & \to M_n \otimes B(X) \\ a \otimes f & \mapsto a \otimes u(f) \end{cases},\]
where $M_n$ is the space of $n \times n$ scalar matrices.
We will equip both $M_n \otimes \Hor^\alpha$ and $M_n \otimes B(X)$ with suitable norms.
In fact, $\Hor^\alpha$ will become an operator space (see Section 4), $M_n \otimes B(X) \cong B(\ell^2_n \otimes_2 X)$ if $X$ is a Hilbert space,
and if $X$ is a Banach space, $M_n \otimes B(X) \cong B(\Gauss_n(X))$ carries the norm induced by an action on $X$-valued Gaussian random sums.
We call $u$ matricially $\gamma$-bounded in this article if
\begin{equation}\label{Equ Intro mat-gamma}
\|u\|_{\text{mat-}\gamma} = \sup_{n \in \mathbb{N}} \| u_n \| < \infty.
\end{equation}
This is in general strictly stronger than $\|u\| < \infty$ (see Proposition \ref{Prop bounded vs mat-gamma bounded}), and is related to the following two well-known boundedness notions, explained in Section \ref{Sec 2 Prelims}.
First, if $X$ is a Hilbert space, then \eqref{Equ Intro mat-gamma} is equivalent to the complete boundedness of $u,$ and
second, if $X$ is a Banach space, then \eqref{Equ Intro mat-gamma} entails that the set of spectral multipliers $\{ u(f) : \: \|f\|_{\Hor^\alpha} \leq 1\}$ is $\gamma$-bounded.

The main result reads as follows.

\begin{thm}\label{Thm Intro Main}
Let $X$ be a space with property $(\alpha).$
Let $A$ be a $0$-sectorial operator on $X$ with bounded $H^\infty$ calculus.
Let $\alpha > \frac12.$
Then the following are equivalent.
\begin{enumerate}
\item The square function estimate \eqref{Equ Intro Square Function} holds.
\item The $H^\infty$ calculus mapping $f\mapsto f(A)$ extends to a homomorphism $u : \Hor^\alpha \to B(X)$ which is matricially $\gamma$-bounded.
\end{enumerate}
\end{thm}

Theorem \ref{Thm Intro Main} entails a spectral multiplier theorem in the following situations:
The space $X = L^p(\Omega)$ for $p \in (1,\infty)$ has property $(\alpha).$
If $(\Omega,\mu)$ is a $d$-dimensional space of homogeneous type, e.g. a sufficiently regular open subset of $\mathbb{R}^d$ with Lebesgue measure $\mu,$ and $A$ is self-adjoint positive on $L^2(\Omega)$ such that the corresponding semigroup $\exp(-tA)$ has an integral kernel $k_t(x,y)$ that satisfies the Gaussian estimate for some $m \in \mathbb{N}$
\begin{equation}\label{Equ Intro GE}
|k_t(x,y)| \leq C \mu(B(x,t^{\frac1m}))^{-1} \exp\left( -c (\text{dist}(x,y)/t^{\frac1m})^{\frac{m}{m-1}} \right) \quad (x,y \in \Omega,\: t > 0),
\end{equation}
then $A$ has a bounded $H^\infty$ calculus on $X$
\cite[Theorem 3.4]{DR}, \cite[Corollary 2.3]{Blun}.
This is indeed the case for many operators listed before \eqref{Equ Intro Hormander calculus}
\cite[Section 2]{Bluna}.
Moreover, the mappings $u$ from \eqref{Equ Intro Hormander calculus} and Theorem \ref{Thm Intro Main} (2) are the same, so that we obtain as a corollary

\begin{cor}\label{Cor Intro}
Assume that $A$ is a self-adjoint positive operator on $L^2(\Omega)$ satisfying \eqref{Equ Intro GE}.
Let $\alpha > \frac12$ and $p \in (1,\infty).$
If $A$ satisfies the square function estimate
\begin{equation}\label{Equ Intro Cor}
\left\| \left( \int_\mathbb{R} \left|(1 + t^2 )^{-\alpha/2} A^{it} x \right|^2 dt \right)^{\frac12} \right\|_p \leq C \|x\|_p,
\end{equation}
then for any $f \in \Hor^\alpha,$ the spectral multiplier $f(A)$ is bounded $ L^p(\Omega) \to L^p(\Omega).$
\end{cor}
In Proposition \ref{Prop bounded vs mat-gamma bounded}, we will show a partial converse of Corollary \ref{Cor Intro}.
More precisely, \eqref{Equ Intro Hormander calculus} implies that a restriction to a smaller H\"ormander space $\mathcal{H}^\beta$ is matricially $\gamma$-bounded.

Let us close the introduction with an overview of the rest of the article.
In Section \ref{Sec 2 Prelims}, we give the necessary background of the above mentioned notions of matricial norms, square functions, Gaussian random sums and functional calculus.
Matricially $\gamma$-bounded mappings and the connection to square functions are explained in Section \ref{Sec 3 Main Thm}.
Section \ref{Sec 4 Hormander operator space} is devoted to homomorphisms $u : \Hor^\alpha \to B(X)$ and the connection to $H^\infty$ functional calculus.
Moreover Theorem \ref{Thm Intro Main} is proved.
A main ingredient is to deduce a spectral decomposition of Paley-Littlewood type, see \eqref{Equ Paley Littlewood}, under the hypotheses of Theorem \ref{Thm Intro Main}.
In Section \ref{Sec 5 Examples}, we discuss some extensions and applications.
Firstly, the square function estimate in terms of imaginary powers $A^{it}$ in Theorem \ref{Thm Intro Main} has several equivalent and almost equivalent rewritings in terms of other typical square functions, involving the analytic semigroup
\begin{align}
\left\| A^{\frac12} \exp(-te^{i\theta} A)x \right\|_{\gamma(\mathbb{R}_+,X)} & \leq C (\frac{\pi}{2} - |\theta| )^{-\beta} \|x\| & \quad \left(\theta \in (-{\pi}/{2},{\pi}/{2}) \right),
\label{Equ Intro Sgr Square Function} \\
\intertext{or resolvents}
\|A^{\frac12} R(e^{i\theta}t,A) x \|_{\gamma(\mathbb{R}_+,X)}  & \leq C |\theta|^{-\beta} \|x\| & \quad \left(\theta \in (-\pi,\pi) \backslash \{ 0 \} \right). \label{Equ Intro Res Square Function}
\end{align}
We have \eqref{Equ Intro Square Function} $\Rightarrow$ \eqref{Equ Intro Sgr Square Function} and \eqref{Equ Intro Res Square Function} for $\alpha \leq \beta,$ and conversely, \eqref{Equ Intro Sgr Square Function} or \eqref{Equ Intro Res Square Function} $\Rightarrow$ \eqref{Equ Intro Square Function} for $\alpha > \beta.$

Secondly, we discuss Theorem \ref{Thm Intro Main} in the presence of
generalized Gaussian estimates (see Assumption \ref{Ass Examples}), which in particular covers
semigroups satisfying \eqref{Equ Intro GE}.
This is a well-studied property in connection with (H\"ormander) functional calculus, see e.g. \cite{Duon,DuOS}.
In particular, we show the square function assumption of Corollary \ref{Cor Intro} in the form of \eqref{Equ Intro Sgr Square Function} and improve the derivation order of the H\"ormander theorem from $\alpha > \frac{d}{2} + \frac12$ as proved in \cite{Bluna} to $\alpha > d \left| \frac{1}{p_0} - \frac12 \right| + \frac12.$
We finally discuss the connections and differences between matricially $\gamma$-bounded H\"ormander calculus and bounded H\"ormander calculus.
The last Section \ref{Sec 6 Proofs Lemmas} contains some technical proofs of Section \ref{Sec 4 Hormander operator space}.

\section{Preliminaries on Operator spaces, Gaussian sums, Square functions and Functional calculus}\label{Sec 2 Prelims}

We will need in different contexts cross norms on a tensor product of two Banach spaces.

\subsection*{Operator spaces}

A Banach space $E$ is called operator space if it is isometrically embedded into $B(H),$ where $H$ is a Hilbert space.
Let $M_n$ denote the space of scalar $n \times n$ matrices.
What makes operator spaces different from mere Banach spaces is that there is a specific collection of norms on $M_n \otimes E$, the operator space structure of $E.$
Namely for all $n \in \mathbb{N},$ it is equipped with the norm
arising from the embedding $M_n \otimes E \hookrightarrow B(\ell^2_n(H)),\, [a_{ij}] \otimes x \mapsto \left( (h_i)_{i=1}^n \mapsto (\sum_{j=1}^n a_{ij}x(h_j))_{i=1}^n \right).$

Let $E$ and $F$ be operator spaces and $u : E \to F$ a linear mapping.
For any $n \in \mathbb{N},$ let $u_n$ be the linear mapping $M_n \otimes E \to M_n \otimes F,\, a \otimes x \mapsto a \otimes u(x).$
Then $u$ is called completely bounded (completely isometric) if $\|u\|_{cb} = \sup_{n \in \mathbb{N}} \|u_n\| < \infty$ (for any $n \in \mathbb{N},\:u_n$ is isometric).

Clearly, any space $B(H)$ itself is an operator space, so in particular $M_m = B(\ell^2_m)$ is.
Further we will consider the Hilbert row space $\ell^2_r = \{h \mapsto \spr{h}{x}e :\:x \in \ell^2\} \subset B(\ell^2)$
where $e \in \ell^2$ is a fixed element of norm $1$ and $\spr{h}{x}$ is the scalar product.
Different choices of $e$ give isometric norms of $M_n \otimes \ell^2_r$ and $\ell^2_r$ is isometric to $\ell^2$ as a Banach space.
We shall also consider the $m$-dimensional subspaces $\ell^2_{m,r} \subset \ell^2_r.$
These are completely isometrically determined by the following embedding, which also explains the name of row space:

\begin{equation}\label{Equ i_m}
i_m : \ell^2_m \hookrightarrow M_m,\,(a_1,\ldots,a_m) \mapsto
\left(
  \begin{array}{ccc}
    a_1 & \ldots & a_m \\
    0 & \ldots & 0 \\
    \vdots & \ddots & \vdots \\
    0 & \ldots & 0 \\
  \end{array}
\right).
\end{equation}

We refer to the books \cite{ER,Pis2} for further information on operator spaces.

\subsection*{$\gamma$-bounded sets, property $(\alpha)$ and square functions}

We let $\Omega$ be a probability space and $(\gamma_k)_{k \in \mathbb{Z}}$ a sequence of independent standard Gaussian random variables on $\Omega.$
For a Banach space $X,$ we let $\Gauss(X)$ be the closure of $\vect\{ \gamma_k \otimes x_k :\: k \in \mathbb{Z}\}$ in $L^2(\Omega;X)$ with respect to the norm
\begin{equation}\label{Equ Def Gauss}
\bignorm{ \sum_k \gamma_k \otimes x_k }_{\Gauss(X)}  = \left( \int_\Omega \left\| \sum_{k=1}^n \gamma_k(\omega) x_k \right\|^2 d\omega \right)^{\frac12}.
\end{equation}
It will be convenient to denote $\Gauss_n(X)$ the subspace of $\Gauss(X)$ of elements of the form $\sum_{k=1}^n \gamma_k \otimes x_k.$

Note that if $X$ is a Hilbert space, then
\begin{equation}\label{Equ Hilbert Gaussian}
 \bignorm{ \sum_k \gamma_k \otimes x_k }_{\Gauss_n(X)}^2 = \sum_{k=1}^n \|x_k\|^2.
\end{equation}

A collection $\tau \subset B(X)$ is called $\gamma$-bounded if there exists $C > 0$ such that
\[ \left\| \sum_k \gamma_k \otimes T_k x_k \right\|_{\Gauss(X)} \leq C \left\| \sum_k \gamma_k \otimes x_k \right\|_{\Gauss(X)} \]

for any finite families $T_1,\ldots,T_n \in \tau$ and $x_1,\ldots, x_n \in X.$
The least admissible constant is denoted by $\gamma(\tau)$ (and $\gamma(\tau) := \infty$ if such a $C$ does not exist).
Note that a $\gamma$-bounded set is automatically uniformly norm bounded, since one has $\gamma(\tau) \geq \sup_{T \in \tau} \|T\|.$
For $\sigma,\tau \subset B(X)$ and $\sigma \circ \tau = \{S \circ T :\: S \in \sigma,\,T \in \tau\},$ one has $\gamma(\sigma\circ \tau) \leq \gamma(\sigma)\gamma(\tau).$
The set $\tau = \{ a \Id_X :\: a \in \mathbb{C},\, |a| \leq 1\}$ is $\gamma$-bounded with constant 1.

We say that $X$ has property $(\alpha)$ if there is a constant $C \geq 1$ such that for any finite family $(x_{ij})$ in $X$, we have
\begin{equation}\label{Equ Property Gauss alpha}
\frac1C \bignorm{ \sum_{i,j} \gamma_{ij} \otimes x_{ij} }_{\Gauss(X)} \leq \bignorm{\sum_{i,j} \gamma_i \otimes \gamma_j \otimes x_{ij} }_{\Gauss(\Gauss(X))} \leq C \bignorm{ \sum_{i,j} \gamma_{ij} \otimes x_{ij} }_{\Gauss(X)},
\end{equation}
where $\gamma_{ij}$ is a doubly indexed family of independent standard Gaussian variables.
Property $(\alpha)$ is inherited by closed subspaces and isomorphic spaces.
The $L^p$ spaces have property $(\alpha)$ for $1 \leq p < \infty$ and moreover, if $X$ has property $(\alpha),$ then also $L^p(\Omega;X)$ has.
Property $(\alpha)$ is usually defined in terms of independent Rademacher variables $\epsilon_i,$ i.e. $\Prob(\epsilon_i = \pm 1) = \frac12$ instead of Gaussian variables \cite{Pis}.
In analogy with \eqref{Equ Def Gauss}, we define $\Rad(X) \subset L^2(\Omega;X)$ by
\[\bignorm{ \sum_k \epsilon_k \otimes x_k }_{\Rad(X)}  = \left( \int_\Omega \left\| \sum_{k=1}^n \epsilon_k(\omega) x_k \right\|^2 d\omega \right)^{\frac12}.\]
It turns out that the two definitions are the same:

\begin{lem}\label{Rem Property Gauss alpha}
The property \eqref{Equ Property Gauss alpha} is equivalent to the following equivalence uniform in finite families $(x_{ij})$ in $X.$
\begin{equation}\label{Equ Property Rad alpha}
 \bignorm{ \sum_{i,j} \epsilon_i \otimes \epsilon_j \otimes x_{ij} }_{\Rad(\Rad(X))} \cong \bignorm{ \sum_{i,j} \epsilon_{ij} x_{ij} }_{\Rad(X)}.
\end{equation}
\end{lem}

\begin{proof}
First observe that the Schatten classes $S^p$ for $p \in (1,\infty) \backslash \{ 2 \}$ are spaces which do not satisfy \eqref{Equ Property Gauss alpha} nor \eqref{Equ Property Rad alpha}.
This is shown in \cite{Pis} for the Rademachers.
On the other hand, $S^p$ has finite cotype, which implies that on this space, Rademacher sums and Gaussian sums are equivalent \cite[Theorem 12.27]{DiJT}, i.e.
\begin{equation}\label{Equ Gauss Rad}
\|\sum_{k \in F} \gamma_k \otimes x_k\|_{\Gauss(X)} \cong \|\sum_{k \in F} \epsilon_k \otimes x_k\|_{\Rad(X)},
\end{equation}
uniformly in $F \subset \mathbb{Z}.$
From this one easily deduces that \eqref{Equ Property Gauss alpha} does not hold.

Next observe that by the Banach-Mazur theorem and \cite[Theorem 3.2]{DiJT}, $S^p$ (in fact any Banach space) has the property that all finite dimensional subspaces are isomorphic to a subspace of some $\ell^\infty_n,$
with one fixed isomorphism constant.
This implies that $\ell^\infty$ does not satisfy \eqref{Equ Property Gauss alpha} nor \eqref{Equ Property Rad alpha}.

Therefore, by the characterization of finite cotype in \cite[Theorem 14.1]{DiJT}, a space $X$ satisfying \eqref{Equ Property Gauss alpha} or \eqref{Equ Property Rad alpha} has finite cotype.
As cited above, Rademacher and Gaussian sums are then equivalent, so the corresponding expressions in \eqref{Equ Property Gauss alpha} and \eqref{Equ Property Rad alpha} are, which shows the lemma.
\end{proof}

We recall the construction of Gaussian function spaces from \cite{KaW2}, see also \cite[Section 1.3]{KNVW}.

Let $H$ be a separable Hilbert space.
We consider the tensor product $H \otimes X$ as a subspace of $B(H,X)$ in the usual way, i.e. by identifying
$\sum_{k=1}^n h_k \otimes x_k \in H \otimes X$ with the mapping $u : h \mapsto \sum_{k=1}^n \spr{h}{h_k} x_k$ for any finite
families $h_1,\ldots,h_n \in H$ and $x_1,\ldots,x_n \in X.$
Choose such families with corresponding $u$, where the $h_k$ shall be orthonormal.
Let $\gamma_1,\ldots,\gamma_n$ be independent standard Gaussian random variables over some probability space.
We equip $H \otimes X$ with the norm
\[ \bignorm{ u }_{\gamma(H,X)} = \bignorm{ \sum_k \gamma_k \otimes x_k }_{\Gauss(X)}.\]
By \cite[Corollary 12.17]{DiJT}, this expression is independent of the choice of the $h_k$ representing $u.$
We let $\gamma(H,X)$ be the completion of $H \otimes X$ in $B(H,X)$ with respect to that norm.
Then for $u \in \gamma(H,X),$ $\|u\|_{\gamma(H,X)} = \bignorm{\sum_k \gamma_k \otimes u(e_k)}_{\Gauss(X)},$ where the $e_k$ form an orthonormal basis of $H$ \cite[Remark 4.2]{KaW2}.\\

A particular subclass of $\gamma(H,X)$ will be important, which is obtained by the following procedure.
Assume that $(\Omega,\mu)$ is a $\sigma$-finite measure space and $H = L^2(\Omega).$
Denote $P_2(\Omega,X)$ the space of Bochner-measurable functions $f: \Omega \to X$ such that $x' \circ f \in L^2(\Omega)$ for all $x' \in X'.$
We identify $P_2(\Omega,X)$ with a subspace of $B(L^2(\Omega),X'')$ by assigning to $f$ the operator $u_f$ defined by
\begin{equation}\label{Equ u_f}
\spr{u_f h}{x'} = \int_\Omega \spr{f(t)}{x'} h(t) d\mu(t).
\end{equation}
An application of the uniform boundedness principle shows that, in fact, $u_f$ belongs to $B(L^2(\Omega),X)$ \cite[Section 4]{KaW2}, \cite[Section 5.5]{Frohl}.
Then we let
\[\gamma(\Omega,X) = \left\{ f \in P_2(\Omega,X):\: u_f \in \gamma(L^2(\Omega),X)\right\}\]
and set
\[\|f\|_{\gamma(\Omega,X)} = \|u_f\|_{\gamma(L^2(\Omega),X)}.\]
The space $\{u_f:\:f \in \gamma(\Omega,X)\}$ is a dense and in general proper subspace of $\gamma(L^2(\Omega),X).$
Resuming the above, we have the following embeddings of spaces, cf. also \cite[Section 3]{LM10}.
\[ L^2(\Omega)\otimes X \to \gamma(\Omega,X) \to \gamma(L^2(\Omega),X) \to B(L^2(\Omega),X).\]

In some cases, $\gamma(L^2(\Omega),X)$ and $\gamma(\Omega,X)$ can be identified with more classical spaces.
If $X$ is a Banach function space with finite cotype, e.g. an $L^p$ space for some $p \in [1,\infty),$ then for any
step function $f = \sum_{k=1}^n x_k \chi_{A_k}: \Omega \to X,$ where $x_k \in X$ and the $A_k$ are measurable and disjoint with $\mu(A_k) \in (0,\infty),$ we have
 (cf. \cite[Remark 3.6, Example 4.6]{KaW2})

\[\|f\|_{\gamma(\Omega,X)}  = \Bignorm{ \sum_k \gamma_k \otimes \mu(A_k)^{\frac12} x_k }_{\Gauss(X)} \!\!\!\!\!\!\!\!
\cong \Bignorm{ \left(\sum_k \mu(A_k) |x_k|^2 \right)^{\frac12}}_X \label{Equ classical square function}
= \Bignorm{ \left( \int_\Omega |f(t)(\cdot)|^2 d\mu(t) \right)^{\frac12} }_X.\]

The second equivalence follows from \cite[Theorem 16.18]{DiJT}.
The last expression above is a classical square function (see e.g. \cite[Section 6]{CDMY}),
whence for an arbitrary space $X,\,\|u\|_{\gamma(H,X)}$ is called (generalized) square function \cite[Section 4]{KaW2}.
In particular, if $X$ is a Hilbert space, then $\gamma(\Omega,X) = L^2(\Omega,X)$ with equal norms.

We have the following well-known properties of square functions.
\begin{lem}\label{Lem Folklore square functions}
Let $(\Omega,\mu)$ be a $\sigma$-finite measure space and $X$ a Banach space with property $(\alpha).$
\begin{enumerate}
\item Suppose that $f_n,f \in P_2(\Omega,X)$ and $f_n(t) \to f(t)$ for almost all $t \in \Omega.$
Then $\|f\|_{\gamma(\Omega,X)} \leq \liminf_n \|f_n\|_{\gamma(\Omega,X)}.$
\item Let $K \in B(H_2,H_1),$ where $H_1,H_2$ are separable Hilbert spaces.
Then for $u \in \gamma(H_1,X)$ we have $u \circ K \in \gamma(H_2,X)$ and $\|u \circ K\|_{\gamma(H_2,X)} \leq \|u\|_{\gamma(H_1,X)} \|K\|.$
\item If $\Omega \to B(X),\: t \mapsto N(t)$ is a strongly continuous map such that $\tau = \{ N(t) :\: t \in \Omega \}$ is $\gamma$-bounded,
and $f \in \gamma(\Omega,X),$ then $\|N \cdot f \|_{\gamma(\Omega,X)} \leq \gamma(\tau) \|f\|_{\gamma(\Omega,X)}.$
\end{enumerate}
\end{lem}

\begin{proof}
As $X$ has property $(\alpha),$ it does not contain $c_0$ isomorphically.
Using this fact, a proof of (1) can be found in \cite[Lemma 4.10]{KaW2}, or in \cite[Proposition 3.18]{vN}.
For (2), we refer to \cite[Proposition 4.3]{KaW2} or \cite[Corollary 6.3]{vN}.
Finally, (3) is proved in \cite[Proposition 4.11]{KaW2}, see also \cite[Theorem 5.2]{vN}.
\end{proof}

\subsection*{Sectorial operators and $H^\infty$ functional calculus}

Let $\theta \in (0,\pi)$ and $A :\: D(A) \subset X \to X$ a densely defined linear mapping on some Banach space $X.$
$A$ is called $\theta$-sectorial, if
\begin{enumerate}
\item The spectrum $\sigma(A)$ is contained in $\overline{\Sigma_\theta}.$
\item For all $\omega > \theta$ there is a $C_\omega > 0$ such that $\| \lambda (\lambda - A)^{-1} \| \leq C_\omega$ for all $\lambda \in \overline{\Sigma_\omega}^c.$
\item $R(A)$ is dense in $X.$
\end{enumerate}
We call $A$ $0$-sectorial if it is $\theta$-sectorial for all $\theta > 0.$
In the literature, property (3) is sometimes omitted.
It entails that $A$ is injective \cite[Proposition 15.2]{KuWe}.
For such an operator $A$ and $f \in H^\infty_0(\Sigma_\omega),\: \omega \in (\theta , \pi),$ one defines the operator

\begin{equation}\label{Equ Cauchy integral formula}
f(A) = \frac{1}{2\pi i} \int_{\partial \Sigma_{(\theta + \omega)/2}} f(\lambda) (\lambda - A)^{-1} d\lambda,
\end{equation}

where $\partial\Sigma_{(\theta + \omega)/2}$ is the sector boundary which is parametrized as usual counterclockwise.
It is easy to check that $f(A)$ is bounded and that $u : H^\infty_0(\Sigma_\omega) \to B(X)$ is a linear and multiplicative mapping.
Suppose that there exists $C > 0$ such that
\begin{equation}\label{Equ Bounded HI calculus}
\| f(A) \| \leq C \|f\|_{\infty,\omega} \quad (f \in H^\infty_0(\Sigma_\omega)).
\end{equation}
Then there exists an extension of $u$ to a bounded mapping $H^\infty(\Sigma_\omega) \to B(X),\,f \mapsto f(A),$ satisfying the so-called Convergence Lemma \cite[Lemma 2.1]{CDMY}.

\begin{lem}\label{Lem Convergence Lemma}
Let $(f_n)_{n \in \mathbb{N}}$ be a sequence in $H^\infty(\Sigma_\omega)$ such that $\sup_{n \in \mathbb{N}} \| f_n \|_{\infty,\omega} < \infty$ and $f_n(\lambda) \to f(\lambda)$ for all $\lambda \in \Sigma_\omega$ and some $f$
(which then necessarily belongs to $H^\infty(\Sigma_\omega)$).
Then $f(A)x = \lim_{n \to \infty} f_n(A) x$ for any $x \in X.$
\end{lem}

Note that the extension is uniquely determined by Lemma \ref{Lem Convergence Lemma} since for any $f \in H^\infty(\Sigma_\omega),$
\begin{equation}\label{Equ Convergence lemma}
f_n(\lambda) = f(\lambda) \left( \lambda / (1 + \lambda)^2 \right)^{\frac1n}
\end{equation}
is a sequence in $H^\infty_0(\Sigma_\omega)$ approximating $f$ in the sense of that lemma.
As a consequence, if \eqref{Equ Bounded HI calculus} is satisfied, then it also holds for any $f \in H^\infty(\Sigma_\omega).$
In this case, we say that $A$ has a bounded $H^\infty(\Sigma_\omega)$ calculus, or without precising the angle $\omega \in (\theta,\pi),$ a bounded $H^\infty$ calculus.

\section{Square function estimate and matricially bounded homomorphism}\label{Sec 3 Main Thm}

Throughout the section, we let $X$ be a Banach space.
For any $n \in \mathbb{N},$ we identify $M_n \otimes B(X)$ with $B(\Gauss_n(X))$ by associating $[a_{ij}] \otimes T \in M_n \otimes B(X)$ with the operator
\begin{equation}\label{Equ M_n otimes B(X)}
\sum_{k=1}^n \gamma_k \otimes x_k \mapsto \sum_{k,j=1}^n \gamma_k \otimes a_{kj} T(x_j).
\end{equation}
Via this identification, we get a norm on the tensor product space, which we note by $M_n \otimes_\gamma B(X).$

\begin{defi}\label{Def matricially gamma bounded}
Let $E$ be an operator space.
Let further $u : E \to B(X)$ be a linear mapping.
We call $u$ matricially $\gamma$-bounded, if $\Id_{M_n} \otimes u : M_n\otimes E \to M_n \otimes_\gamma B(X)$ is bounded uniformly in $n \in \mathbb{N},$
i.e. if there is a constant $C > 0$ such that for any $n \in \mathbb{N},$
\[ \bignorm{\sum_{i,j = 1}^n \gamma_i \otimes u(f_{ij}) x_j }_{\Gauss_n(X)} \leq C \bignorm{ \left[f_{i,j}\right]}_{M_n \otimes E} \bignorm{\sum_{i=1}^n \gamma_i \otimes x_i}_{\Gauss_n(X)} . \]
We denote the least admissible constant $C$ by $\|u\|_{{\text{mat-}\gamma}}.$
\end{defi}

\begin{rem}\label{Rem mat}~
\begin{enumerate}
\item If $u : E \to B(X)$ is matricially $\gamma$-bounded then
\begin{equation}\label{Equ u gamma bounded}
 \{ u(x) : \: \|x\|_E \leq 1\}
\end{equation}
is $\gamma$-bounded.
Indeed, from the definition of $\gamma$-boundedness in Section \ref{Sec 2 Prelims}, we immediately deduce that
\eqref{Equ u gamma bounded} is satisfied if and only if $\Id_{M_n} \otimes u|_{D_n \otimes E}$ is bounded uniformly in $n \in \mathbb{N},$
where $D_n \subset M_n$ denotes the subspace of diagonal matrices.
We call a linear mapping $u$ $\gamma$-bounded if \eqref{Equ u gamma bounded} holds.
\item Assume that $X$ is a Hilbert space.
Then by \eqref{Equ Hilbert Gaussian}, $u$ is matricially $\gamma$-bounded if and only if $u$ is completely bounded, and in this case, $\|u\|_{cb} = \|u\|_{\text{mat-}\gamma}.$
\end{enumerate}
\end{rem}

A first example for Definition \ref{Def matricially gamma bounded} is given by
\begin{prop}\label{Prop sigma(m,X)}
For a given space $X$ and $m \in \mathbb{N},$ consider
\[
\sigma_{m,X} : M_m \to M_m \otimes_\gamma B(X),\,[a_{ij}] \mapsto [a_{ij}\Id_X].
\]
Then $\sigma_{m,X}$ is matricially $\gamma$-bounded with $\sup_{m \in \mathbb{N}} \|\sigma_{m,X}\|_{\text{mat-}\gamma} < \infty$
if and only if $X$ has property $(\alpha).$
\end{prop}

It is shown in \cite[Lemma 4.3]{KrLM} that the $\sigma_{m,X}$ are $\gamma$-bounded uniformly in $m \in \mathbb{N}$ if and only if $X$ has property $(\alpha)$ (with Rademachers in place of Gaussians).
Actually the same proof applies to Proposition \ref{Prop sigma(m,X)}.

Mappings which are $\gamma$-bounded or matricially $\gamma$-bounded have been studied so far in connection with functional calculi and unconditional decompositions \cite{KrLM,dPR} where $E$ is a $C(K)$-space
and representations of amenable groups \cite{LM10}, where $E$ is a nuclear $C^*$-algebra.
We shall focus in this section on the row Hilbert space $E = \ell^2_r.$

\begin{thm}\label{Thm Main 1}
Let $u : \ell^2 \to B(X)$ be a bounded linear mapping.
Assume that $X$ has Pisier's property $(\alpha).$
For $n \in \mathbb{N},$ denote by $C_n \subset M_n$ the subspace of matrices vanishing outside the first column.
Then the following conditions are equivalent:
\begin{enumerate}
\item $\bignorm{u(\cdot)x}_{\gamma(\ell^2,X)} \leq C \| x \|.$
\item $u : \ell^2_r \to B(X)$ is matricially $\gamma$-bounded.
\item The restriction $\Id \otimes u: C_n \otimes \ell^2_r \to M_n \otimes_\gamma B(X)$ is bounded uniformly in $n \in \mathbb{N}.$
\end{enumerate}
\end{thm}

\begin{proof}
We fix an orthonormal basis $(e_k)_k$ of $\ell^2.$
Write $T_k = u(e_k).$
Then condition (1) of the statement rewrites
\begin{equation}\label{Equ 1 Thm}
\bignorm{ u(\cdot)x }_{\gamma} = \sup_{n \in \mathbb{N}} \bignorm{ \sum_{k=1}^n \gamma_k \otimes T_k x }_{\Gauss_n(X)} \leq C \| x \|.
\end{equation}
On the other hand, for $[f_{ij}] \in M_n \otimes \ell^2,$ we have with $f^{(k)}_{ij} = \spr{f_{ij}}{e_k},$
\[\bignorm{(\Id_{M_n}\otimes u)[f_{ij}]}_{M_n \otimes_\gamma B(X)} = \lim_m \bignorm{ \left[ \sum_{k=1}^m f^{(k)}_{ij} T_k \right] }_{M_n \otimes_\gamma B(X)}.\]
Thus, condition (2) is equivalent to
\begin{equation}\label{Equ 2 Thm}
\bignorm{ \sum_{i=1}^n \sum_{k=1}^m \gamma_i \otimes f_{ij}^{(k)} T_k x_j }_{\Gauss_n(X)} \leq C \|[f_{ij}]\|_{M_n \otimes \ell^2_{m,r}} \bignorm{ \sum_{i=1}^n \gamma_i \otimes x_i }_{\Gauss_n(X)} ,
\end{equation}
where $f_{ij} = (f_{ij}^{(k)})_k \in \ell^2_m,$ and $C$ is independent of $n$ and $m.$
Denote the linear bounded mapping $\Gauss_n(X) \to \Gauss_n(X)$ arising from \eqref{Equ 2 Thm} by $u_{n,m}([f_{ij}]).$
Finally, condition (3) is equivalent to \eqref{Equ 2 Thm} with $f_{ij} = 0$ for $j \geq 2.$\\

\noindent
(1) $\Longrightarrow$ (2)

For $m \in \mathbb{N}$ fixed, let $Y = \Gauss_m(X)$ and define the operators
\[ V : X \to Y, \, x\mapsto \sum_{k=1}^m \gamma_k \otimes T_k x,\quad W : Y \to X,\, \sum_{k=1}^m \gamma_k \otimes x_k \mapsto x_1.\]
By assumption \eqref{Equ 1 Thm}, $V$ is bounded with constant $C$ independent of $m.$
Further, $W$ is bounded (see e.g. \cite[(2.13)]{Kr} for a simple proof).
For $n \in \mathbb{N},$ denote
\[V_n = \Id_{\ell^2_n} \otimes V : \Gauss_n(X) \to \Gauss_n(Y),\, \sum_{k=1}^n \gamma_k \otimes x_k \mapsto \sum_{k=1}^n \gamma_k \otimes V(x_k).\]
It is easy to check that $\|V_n\| = \|V\|.$
Similarly, defining $W_n = \Id_{\ell^2_n} \otimes W : \Gauss_n(Y) \to \Gauss_n(X),$ one has $\|W_n\| = \|W\|.$
Let $i_m : \ell^2_{m,r} \to M_m$ be the first row identification as in \eqref{Equ i_m} which is completely bounded of $cb$-norm 1.
Then by Remark \ref{Rem mat} and Proposition \ref{Prop sigma(m,X)}, along with property $(\alpha),$ $\pi_m = \sigma_{m,X} \circ i_m : \ell^2_{m,r} \to B(M_m \otimes X)$ is a matricially $\gamma$-bounded mapping
and $\sup_{m \in \mathbb{N}} \|\pi_m\|_{\text{mat-}\gamma} < \infty.$
For $f = [f_{ij}]_{ij} \in M_n \otimes \ell^2_{m,r},$ we have the identity
\[ u_{n,m}(f) = \left[\sum_{k=1}^m f_{ij}^{(k)} T_k \right] = [W \pi_m(f_{ij}) V] = W_n [\pi_m(f_{ij})] V_n.\]
Therefore,
\[\|u_{n,m}(f)\| \leq \|W_n\| \, \|V_n\| \, \| [\pi_m(f_{ij})] \| \leq \|W\| \, \|V\| \, \|\pi_m\|_{\text{mat-}\gamma} \|[f_{ij}]\|_{M_n \otimes \ell^2_{m,r}},\]
so \eqref{Equ 2 Thm} follows.\\

\noindent
(2) $\Longrightarrow$ (3)

This is clear, since (3) is an obvious restriction of (2).\\

\noindent
(3) $\Longrightarrow$ (1)

Choose $n = m \in \mathbb{N}$ and $f = [f_{ij}] \in M_n \otimes \ell^2_{n,r}$ with $f_{ij} = \delta_{j1} e_i,$ where $(e_i)$ is the standard basis of $\ell^2_n.$
By definition of the row norm, we have
\begin{align*}
\|f\|_{M_n \otimes \ell^2_{n,r}} & = \bignorm{ \left[ \sum_k \spr{f_{ik}}{f_{jk}} \right] }^{\frac12}_{M_n} \\
& = \bignorm{ \left[ \sum_k \delta_{k1} \spr{e_i}{e_j} \right] }_{M_n}^{\frac12} \\
& = \bignorm{ [\delta_{ij}] }_{M_n}^{\frac12} \\
& = 1.
\end{align*}
As $f$ is supported by the first column, by assumption \eqref{Equ 2 Thm} there is some $C < \infty$ such that
\begin{align*}
C & \geq \|u_{n,n}(f)\|_{M_n \otimes_\gamma B(X)} \\
& = \bignorm{ \left(
                                                         \begin{array}{cccc}
                                                           T_1 & 0 & \ldots & 0 \\
                                                           \vdots & \vdots & \ddots & \vdots \\
                                                           T_n & 0 & \ldots & 0 \\
                                                         \end{array}
                                                       \right)}_{M_n \otimes_\gamma B(X)}
\\
& = \sup \left\{ \bignorm{ \sum_k \gamma_k \otimes T_k x_1 }_{\Gauss_n(X)} :\: \bignorm{ \sum_k \gamma_k \otimes x_k }_{\Gauss_n(X)} \leq 1 \right\} \\
& = \sup \left\{ \bignorm{ \sum_k \gamma_k \otimes T_k x   }_{\Gauss_n(X)} :\: \|x\| \leq 1 \right\}.
\end{align*}
Letting $n \to \infty$ shows that \eqref{Equ 1 Thm} holds.
\end{proof}

\begin{rem}
Theorem \ref{Thm Main 1} is a generalization of \cite[Proposition 3.3]{LM04}, where $X$ is an $L^p$-space, and \cite[Corollary 3.19]{HaKu}, where more generally $X$ has property $(\alpha).$
There it is shown that condition (1) of the theorem implies that $u : \ell^2 \to B(X)$ is $\gamma$-bounded.
(In these two references, $u$ maps to $B(Y,X)$ instead of $B(X).$
A corresponding version of Theorem \ref{Thm Main 1} with $B(Y,X)$ in place of $B(X)$ holds with the same proof).
\end{rem}

\section{The H\"ormander functional calculus}\label{Sec 4 Hormander operator space}

Recall the spaces $W^\alpha$ and $\Hor^\alpha$ and the dyadic partition of unity $(\phi_n)_{n \in \mathbb{Z}}$ from the introduction.
Clearly the space $W^\alpha$ is a Hilbert space.
We equip $\Hor^\alpha$ with an operator space structure by putting
\begin{equation}\label{Equ Ha OSS}
\|[f_{ij}]\|_{M_n \otimes \Hor^\alpha} = \sup_{k \in \mathbb{Z}} \| [\phi_k f_{ij}] \|_{M_n \otimes W^\alpha_r},
\end{equation}
where the index $r$ refers to the row space structure.
It is easy to check that \eqref{Equ Ha OSS} indeed defines an operator space, arising from the embedding
\[ \Hor^\alpha \hookrightarrow B(\bigoplus^2_{k \in \mathbb{Z}} W^\alpha),\, f \mapsto \left((g_k)_{k \in \mathbb{Z}} \mapsto (\spr{g_k}{\phi_k f} e)_{k \in \mathbb{Z}} \right),\]
where $\displaystyle \bigoplus^2_{k \in \mathbb{Z}} W^\alpha$ is the Hilbert sum, and $e$ is some fixed element in $W^\alpha$ of norm 1.

In this section we focus on (unital) homomorphisms
\begin{equation}\label{Equ u} u : \Hor^\alpha \to B(X) .
\end{equation}
We give a characterization when such mappings are matricially $\gamma$-bounded.
An example of a bounded homomorphism of this type is given by H\"ormander's classical theorem mentioned in the introduction,
which states that for $\alpha > \frac{d}{2},\: X = L^p(\mathbb{R}^d)$ and $p \in (1,\infty),$ the radial Fourier multiplier representation
$u_{-\Delta} : \Hor^\alpha \to B(X) $
given by
\begin{equation}\label{Equ u Delta}
u_{-\Delta}(f)g = \left[ f(|\cdot|^2) \hat{g} \right]\check{\phantom{i}} = f(-\Delta) g
\end{equation}
is bounded.
In fact, by means of our characterization, we will show in Section \ref{Sec 5 Examples} that $u_{-\Delta}$ is even matricially $\gamma$-bounded provided that $\alpha > \frac{d+1}{2}.$

For $n \in \mathbb{N},$ let $\M^n$ be the space consisting of $n$-times continuously differentiable functions $f$ defined on $(0,\infty)$ such that
$\|f\|_{\M^n} = \sum_{k=0}^n \sup_{t > 0} |t^k f^{(k)}(t)|$ is finite.
Let us record how $W^\alpha,\,\Hor^\alpha,\,H^\infty(\Sigma_\omega)$ and the auxiliary space $\M^n$ compare.
The proof is an easy verification, see also \cite[Lemma 4.15, Proof of Proposition 4.22, Proposition 4.9]{Kr}.

\begin{lem}\label{Lem Function spaces}
Let $\omega \in (0,\pi)$ and $\alpha > \frac12.$
\begin{enumerate}
\item $H^\infty(\Sigma_\omega) \hookrightarrow \Hor^\alpha.$
\item $W^\alpha \hookrightarrow \Hor^\alpha,$ where the embedding is completely bounded.
\item For any $\omega \in (0,\pi),$ $H^\infty(\Sigma_\omega) \cap W^\alpha$ is a dense subset of $W^\alpha.$
\item $\M^n \hookrightarrow \Hor^\alpha$ for $n > \alpha.$
\item $H^\infty(\Sigma_\omega)$ is a dense subset of $\M^n.$
Moreover, any $f \in W^\alpha \cap \M^n$ can be simultaneously approximated by a sequence $(f_k)_{k \in \mathbb{N}} \subset H^\infty(\Sigma_\omega) \cap W^\alpha \cap \M^n.$
\end{enumerate}
\end{lem}

The main interest of $\M^n$ is the following convergence lemma, which is proved in \cite[Section 4.2.4]{Kr}.

\begin{lem}\label{Lem Convergence Mn}
Let $u : \M^n \to B(X)$ be bounded such that $u(f) = f(A)$ for some $0$-sectorial operator $A$ and any $f \in \bigcup_{\theta \in (0,\pi)} H^\infty(\Sigma_\theta).$
Let $(\phi_n)_{n \in \mathbb{Z}}$ be a dyadic partition of unity and $(a_n)_{n \in \mathbb{Z}}$ a bounded sequence.
Then $\sum_{n \in \mathbb{Z}} a_n \phi_n$ belongs to $\M^n$ and
\begin{equation}\label{Equ Lem A u}\sum_{n \in \mathbb{Z}} a_n u(\phi_n) x = u \left( \sum_{n \in \mathbb{Z}} a_n \phi_n \right)x \quad (x \in X). \end{equation}
\end{lem}

Many spectral multiplier theorems for Laplace type operators $A$ consist in the boundedness of $u$ in \eqref{Equ u}, which in turn is the functional calculus $u_A$ of some $0$-sectorial operator.
For example, in the case of \eqref{Equ u Delta} one has $A = - \Delta.$
In the sequel we will only consider homomorphisms of the form $u = u_A.$
The next lemma gives a criterion when this is the case.

\begin{lem}\label{Lem A u}
Let $\omega \in (0,\pi).$
\begin{enumerate}
\item
Let $u$ be a bounded homomorphism $u : \Hor^\alpha \to B(X).$
There exists a $0$-sectorial operator $A$ such that
\begin{equation}\label{Equ Lem A u 2}
u(f) = f(A) \quad (f \in H^\infty(\Sigma_\omega)),
\end{equation}
if and only if the restriction of $u$ to $H^\infty(\Sigma_\omega)$ satisfies the Convergence Lemma \ref{Lem Convergence Lemma},
i.e. for any $f_n,\: f \in H^\infty(\Sigma_\omega)$ with $\sup_n \|f_n\|_{\infty,\omega} < \infty$ and $f_n(\lambda) \to f(\lambda)$ pointwise, we have $u(f)x = \lim_n u(f_n)x$ for any $x \in X.$
In this case, we write $f(A)$ in place of $u(f)$ for any $f \in \Hor^\alpha.$
\item
Let $A$ be a $0$-sectorial operator.
Then there exists a bounded homomorphism $u : \Hor^\alpha \to B(X)$ satisfying \eqref{Equ Lem A u 2} if and only if $\|f(A)\| \leq C \|f\|_{\Hor^\alpha}\quad (f \in H^\infty(\Sigma_\omega)).$
Moreover, $u$ is uniquely determined by \eqref{Equ Lem A u 2}.
\end{enumerate}
\end{lem}

\begin{proof}
(1) By Lemma \ref{Lem Function spaces} (1), only the ``if'' part has to be shown.
Suppose that $u : \Hor^\alpha \to B(X)$ satisfies the Convergence Lemma \ref{Lem Convergence Lemma}.
Since $u$ is a homomorphism, one has $u((\lambda - \cdot)^{-1}) - u((\mu - \cdot)^{-1}) = (\mu - \lambda) u((\lambda - \cdot)^{-1}) u((\mu - \cdot)^{-1})$
for any $\lambda,\mu \in \mathbb{C} \backslash [0,\infty),$ i.e. $u((\lambda - \cdot)^{-1})$ is a pseudo resolvent \cite[Definition 9.1]{Paz}.
By Lemma \ref{Lem Convergence Lemma}, $u(\lambda (\lambda - \cdot)^{-1} )x \to x$ for any $x \in X$ and $|\lambda|\to \infty.$
Thus, by \cite[Corollary 9.5]{Paz}, there exists a densely defined operator $A$ such that $u((\lambda - \cdot)^{-1}) = (\lambda-A)^{-1}$ for $\lambda \in \mathbb{C} \backslash [0,\infty).$
Again by Lemma \ref{Lem Convergence Lemma}, $-\frac1n (\frac1n + A)^{-1}x \to 0$ for any $x \in X$ and $n \to \infty.$
Thus, $A$ has dense range \cite[Proposition 15.2]{KuWe}.
As $u$ is a bounded homomorphism, it now follows that $A$ is $0$-sectorial and for any rational function $r \in \Hor^\alpha,$
$\|r(A)\| \lesssim \|r\|_{\Hor^\alpha} \lesssim \|r\|_{\infty,\omega}.$

We claim that $A$ has an $H^\infty$ calculus coinciding with $u.$
Indeed, a given $f \in H^\infty_0(\Sigma_\omega),$ we write
\[f = \frac{1}{2\pi i} \int_{\partial\Sigma_{\omega/2}} f(\lambda) (\lambda - \cdot)^{-1} d\lambda.\]
As $f(\lambda) (\lambda - \cdot)^{-1} : \partial\Sigma_{\omega/2} \to H^\infty(\Sigma_{\omega/4})$ is continuous, we find a sequence
$r_n = \sum_{k=1}^K c_k f(\lambda_k) (\lambda_k-\cdot)^{-1}$ such that $r_n \to f$ in $H^\infty(\Sigma_{\omega/4}),$ so in particular in $\Hor^\alpha.$
Clearly, $r_n$ are rational functions.
Inserting formally $(\cdot) = A$ in the Cauchy integral, the same arguments apply, and $r_n(A) \to f(A).$
We conclude $u(f) = \lim_n u(r_n) = \lim_n r_n(A) = f(A).$

We have shown that $u(f) = f(A)$ for any $f \in H^\infty_0(\Sigma_\omega).$
For a general $f \in H^\infty(\Sigma_\omega)$ we use the approximation \eqref{Equ Convergence lemma}.\\

(2) The ``only if'' part is clear and  the ``if'' part is shown in \cite[Remark 4.27]{Kr}.
Using density and Lemma \ref{Lem Function spaces}, $u$ is uniquely determined on $W^\alpha$ and $\M^n$ for any $n > \alpha.$
Thus $u$ satisfies the decomposition \eqref{Equ Lem A u}.
Then for $f \in \Hor^\alpha,$ we have $u(f)x  = u(f) \sum_{k \in \mathbb{Z}} u(\phi_k) x = \sum_{k \in \mathbb{Z}} u(f \phi_k) x.$
As $f \phi_k \in W^\alpha,$ we conclude the uniqueness of $u.$
\end{proof}

The strategy to prove matricial $\gamma$-boundedness of a mapping from $\Hor^\alpha$ to $B(X)$ will be to show the matricial $\gamma$-boundedness from $W^\alpha$ to $B(X),$
and then to pass to $\Hor^\alpha$ by means of a spectral decomposition, given by \eqref{Equ Paley Littlewood} in the following theorem.
The restriction of the $H^\infty$ calculus angle $\omega$ to $(0,\pi/4)$ is only for technical reasons.

\begin{thm}\label{Thm Paley Littlewood}
Let $X$ be a Banach space with property $(\alpha).$
Let $\alpha > \frac12,\,\omega \in (0,\pi/4)$ and $A$ be a $0$-sectorial operator on $X$ having a bounded $H^\infty(\Sigma_\omega)$ calculus.
Assume that \[\|f(A)\| \leq C \|f\|_{W^\alpha} \quad (f \in H^\infty(\Sigma_\omega) \cap W^\alpha) \]
and that the extension resulting from density $u : W^\alpha \to B(X),\, f \mapsto f(A)$ is matricially $\gamma$-bounded.
Let $(\phi_n)_{n \in \mathbb{Z}}$ be a dyadic partition of unity.
Then
\begin{equation}\label{Equ Paley Littlewood}
\|x\| \cong \| \sum_{n \in \mathbb{Z}} \gamma_n \otimes \phi_n(A)x \|_{\Gauss(X)},
\end{equation}
the sum on the right hand side being convergent in $\Gauss(X).$
\end{thm}

As the proof is rather long we separate four preliminary lemmas, whose proofs are annexed in Section \ref{Sec 6 Proofs Lemmas}.

\begin{lem}\label{Lem 1}
Let $X$ have property $(\alpha)$, let $\alpha > \frac12.$
Let $A$ be as in Theorem \ref{Thm Paley Littlewood}.
Then for $\beta \in \mathbb{N},\,\beta > \alpha,$
\begin{equation}\label{Equ NT}
\left\{\exp(-2^k z A) :\: k \in \mathbb{Z} \right\} \text{ is }\gamma\text{-bounded with constant }\lesssim \left| \frac{z}{\Re z}\right|^\beta \quad (\Re z > 0).
\end{equation}
\end{lem}

\begin{lem}\label{Lem 2}
Let $A$ be a $0$-sectorial operator on some Banach space $X$ such that for some $\beta > 0,$ \eqref{Equ NT} holds.
Then for $\gamma = \beta + 1,$ we have
\begin{equation}\label{Equ Res}
\left\{ \lambda^{\frac12} (2^k A)^{\frac12}(\lambda - 2^k A)^{-1} :\: k \in \mathbb{Z} \right\}\text{ is }\gamma\text{-bounded with constant }\lesssim
|\arg \lambda|^{-\gamma} \quad (\Re \lambda > 0).
\end{equation}
\end{lem}

\begin{lem}\label{Lem 3}
Let $A$ be a $0$-sectorial operator on some space $X$ with property $(\alpha)$ having a bounded $H^\infty(\Sigma_\omega)$ calculus.
Assume that $A$ satisfies \eqref{Equ Res} for some $\gamma > 0.$
Then for any $n > \gamma,$
\[ \|f(A)\| \leq C \|f\|_{\M^n} \quad \left(f \in H^\infty(\Sigma_\omega)\right).\]
\end{lem}

\begin{lem}\label{Lem 4}
Let $n \in \mathbb{N}.$
Let $(g_k)_{k \in \mathbb{Z}}$ satisfy $\sup_{k \in \mathbb{Z}} \|g_k\|_{\M^n} < \infty.$
Suppose that the supports of $g_k$ satisfy the following overlapping condition
\[ \sup_{x > 0}\#\{ k \in \mathbb{Z}:\: \supp g_k \cap [\frac12 x,2x] \not= \emptyset \} < \infty.\]
Then $\sum_{k \in \mathbb{Z}} g_k,$ which is consequently pointwise a finite sum belongs to $\M^n,$ and
\[\|\sum_{k \in \mathbb{Z}} g_k\|_{\M^n} \lesssim \sup_{k\in\mathbb{Z}} \|g_k\|_{\M^n} < \infty.\]
\end{lem}

\begin{proof}[Proof of Theorem \ref{Thm Paley Littlewood}]
Using Lemmas \ref{Lem 1}, \ref{Lem 2} and \ref{Lem 3} one after another shows that $\|f(A)\| \lesssim \|f\|_{\M^n}$ for $n$ sufficiently large $(n > \lfloor \alpha \rfloor + 2).$
For any $k \in \mathbb{Z},$ let $a_k \in \{ 1 , - 1\}.$
Apply Lemma \ref{Lem 4} with $g_k = a_k \phi_k.$
It is easy to check that $\|g_k\|_{\M^n}$ is independent of $k \in \mathbb{Z}.$
Further, the overlapping condition is clearly satisfied with constant 2.
Thus we have, for any finite $F \subset \mathbb{Z},$
\[ \| \sum_{k \in F} a_k \phi_k(A) x\| \lesssim \|\sum_{k \in F} a_k \phi_k \|_{\M^n} \|x\| \lesssim \|x\|.\]
Replacing $a_k$ by independent Rademacher variables $\epsilon_k$ and taking expectation gives
\[\|\sum_{k \in F} \epsilon_k \otimes \phi_k(A) x\|_{\Rad(X)} \lesssim \|x\|.\]
Since $X$ has property $(\alpha),$ the equivalence of Gaussian and Rademacher sums \eqref{Equ Gauss Rad} holds.
By \eqref{Equ Lem A u}, $\sum_{k \in \mathbb{Z}} a_k \phi_k(A)x$ converges in $X.$
By dominated convergence (resp. \eqref{Equ Gauss Rad}), convergence holds also in $\Rad(X)$ (resp. $\Gauss(X)$), when $a_k$ is replaced by $\epsilon_k$ (resp. $\gamma_k$).

We have shown
\[ \| \sum_{k \in \mathbb{Z}} \gamma_k \otimes \phi_k(A) x \|_{\Gauss(X)} \lesssim \|x\|. \]

For the reverse inequality, we argue by duality.
Let $x' \in X',$ write $\mathbb{E}$ for expectation and $\widetilde\phi_l = \sum_{k=l-1}^{l+1}\phi_k.$
By the support condition on the $\phi_k,$ $\widetilde\phi_l \phi_l = \phi_l.$
Then using the independence of the $\gamma_k,$ we have
\begin{align*}
\left|\spr{x}{x'}\right| & = \left|\mathbb{E} \spr{\sum_{k \in \mathbb{Z}} \gamma_k \phi_k(A)x}{\sum_{l \in \mathbb{Z}} \gamma_l \widetilde\phi_l(A)'x'}\right| \\
& \leq \|\sum_{k \in \mathbb{Z}} \gamma_k \otimes \phi_k(A) x\|_{\Gauss(X)} \| \sum_{l \in \mathbb{Z}} \gamma_l \otimes \widetilde\phi_l(A)'x'\|_{\Gauss(X')}.
\end{align*}
We conclude the proof by the same argument as above which shows that
\[\| \sum_{l \in \mathbb{Z}} \gamma_l \otimes \widetilde\phi_l(A)' x'\|_{\Gauss(X')} \lesssim \|x'\|.\]
\end{proof}

The main result of this section reads as follows.

\begin{thm}\label{Thm Ha}
Let $A$ be a $0$-sectorial operator with bounded $H^\infty(\Sigma_\theta)$ calculus for some $\theta \in (0,\pi)$ on a space $X$ with property $(\alpha).$
Let $\alpha > \frac12.$
Then the following are equivalent.
\begin{enumerate}
\item For any $x \in X,$ $(1+t^2)^{-\alpha/2} A^{it}x$ belongs to $\gamma(\mathbb{R},X)$ and
\[ \| (1 + t^2)^{-\alpha/2} A^{it} x\|_{\gamma(\mathbb{R},X)} \leq C \|x\|.\]
\item The $H^\infty$ calculus of $A$ extends to a matricially $\gamma$-bounded mapping $u : \Hor^\alpha \to B(X).$
\end{enumerate}
\end{thm}

\begin{proof}
Assume first that the theorem is shown under the additional assumption that $\theta < \frac{\pi}{4}.$
For a general $\theta \in (0,\pi),$ we can reduce to this case by considering $B = A^{\frac14}.$
Namely, by \cite[Theorem 2.4.2]{Haa}, $B$ has a bounded $H^\infty(\Sigma_\omega)$ calculus for some $\omega \leq \frac{\theta}{4} < \frac{\pi}{4}.$
Moreover,
\[ (1 + t^2)^{-\alpha/2} B^{it} x  = \left( \frac{1 + (\frac{t}{4})^2}{1 + t^2}\right)^{\alpha/2} \cdot \left( 1 + \frac{t}{4} \right)^{-\frac{\alpha}{2}} A^{i\frac{t}{4}} x.\]
The first factor is bounded, so its multiplication with an $L^2(\mathbb{R})$ function is a bounded operation on $L^2(\mathbb{R}).$
The same holds for its inverse, and also for the change of variables $f \mapsto f(\frac{\cdot}{4}),$ and its inverse.
Thus, by Lemma \ref{Lem Folklore square functions}, if $A$ satisfies (1) then so does $B,$ so $B$ satisfies (2).
As $H^\infty(\Sigma_\omega) \hookrightarrow \Hor^\alpha,$ $B$ has an $H^\infty(\Sigma_\omega)$ calculus actually for any $\omega >0,$ so by \cite[Theorem 2.4.2]{Haa}, $A$ has an $H^\infty(\Sigma_\theta)$ calculus for some $\theta < \frac{\pi}{4}.$
The same is true, provided that $A$ satisfies (2).
Thus (1) or (2) imply that the assumption of the theorem actually holds with $\theta < \frac{\pi}{4}.$
We suppose from now on that $\theta < \frac{\pi}{4}.$\\

\noindent
$(1) \Longrightarrow (2).$\\
Fix an arbitrary orthonormal basis $(f_k)_k$ of $L^2(\mathbb{R}).$
Let $T_k \in B(X)$ be defined by
\[\spr{T_k x}{x'} = \frac{1}{2 \pi} \int_\mathbb{R} f_k(t) (1 + t^2)^{-\alpha/2} \spr{A^{it} x}{x'} dt\]
and $v : \ell^2 \to B(X)$ the linear mapping given by $e_k \mapsto T_k.$
Then $(1)$ implies
\[\|v(\cdot)x\|_{\gamma(\ell^2,X)} = \| \sum_k \gamma_k \otimes v(e_k)x \|_{\Gauss(X)} = \| \sum_k \gamma_k \otimes T_k x \|_{\Gauss(X)} = \|(1 + t^2)^{-\frac{\alpha}{2}} A^{it}x\|_\gamma \lesssim \|x\|,\] so that by Theorem \ref{Thm Main 1}, $v : \ell^2_r \to B(X)$ is matricially $\gamma$-bounded.
Consider the mapping $w : W^\alpha \to \ell^2,\, f \mapsto \left( \frac{1}{2 \pi} \int_\mathbb{R} (f\circ \exp)\hat{\phantom{i}}(t) (1 + t^2)^{\alpha/2} \overline{f_k}(t) dt \right)_k.$
It is easy to check that $w$ is unitary and consequently, $\tilde{u} = v \circ w : W^\alpha_r \to B(X)$ is also matricially $\gamma$-bounded.
On the other hand, $\tilde{u}(f) = f(A)$ for any $f \in H^\infty(\Sigma_\theta) \cap W^\alpha.$
Indeed, by the representation formula in \cite[Proposition 4.22]{Kr},
\begin{align*}
2 \pi \spr{f(A)x}{x'} & = \int_\mathbb{R} (f \circ \exp)\hat{\phantom{i}}(t) \spr{A^{it}x}{x'} dt\\
& = \int_\mathbb{R} (f \circ \exp)\hat{\phantom{i}}(t) (1 + t^2)^{\alpha/2} (1 + t^2)^{-\alpha/2} \spr{A^{it}x}{x'} dt\\
& = \sum_{k} \int_\mathbb{R} (f\circ\exp)\hat{\phantom{i}}(t) (1 + t^2)^{\alpha/2} \overline{f_k}(t) dt \spr{T_k x}{x'}\\
& = 2 \pi \spr{\tilde{u}(f)x}{x'}.
\end{align*}

Let $n \in \mathbb{N}$ and $F = [f_{ij}] \in M_n \otimes H^\infty(\Sigma_\theta).$
We show that $\| [ f_{ij}(A) ] \|_{M_n \otimes_\gamma B(X)} \lesssim \|[f_{ij}]\|_{M_n \otimes \Hor^\alpha}.$
For $N \in \mathbb{N}$ consider
\[ F_N = \left[
           \begin{array}{cccc}
             [\phi_{-N} f_{ij}] & 0 & \ldots & 0 \\
             0 & [\phi_{-N + 1} f_{ij}] &  & \vdots \\
             \vdots &  & \ddots & 0 \\
             0 & \ldots & 0 & [\phi_N f_{ij}] \\
           \end{array}
         \right] \in M_{(2N + 1)n} \otimes W^\alpha.
\]
By \eqref{Equ Ha OSS}, we have $\sup_N \|F_N\|_{M_{(2N+1)n} \otimes W^\alpha} = \|F\|_{M_n \otimes \Hor^\alpha}.$
Observe first that for any scalars $g_1,\ldots,g_n,$ by Theorem \ref{Thm Paley Littlewood}, with $\tilde\phi_k = \sum_{l = k - 1}^{k + 1} \phi_l,$
\[\| \sum_{i,j = 1}^n g_i f_{ij}(A)x_j \| \cong \| \sum_{i,j = 1}^n \sum_{k \in \mathbb{Z}} \gamma_k \otimes g_i \tilde{u}(f_{ij} \phi_k) x_j \|
\cong \| \sum_{i,j = 1}^n \sum_{k \in \mathbb{Z}} \gamma_k \otimes g_i \tilde{u}(f_{ij} \phi_k) \tilde{u}(\widetilde\phi_k) x_j \|.\]
Replacing $g_i$ by Gaussian variables and taking expectations shows that
\begin{equation}\label{Equ Proof Main Thm 1}
\|\sum_{i,j=1}^n \sum_{k \in \mathbb{Z}} \gamma_i \otimes \tilde{u}(f_{ij}) x_j\|_{\Gauss_n(X)} \cong \| \sum_{i,j=1}^n \sum_{k \in \mathbb{Z}} \gamma_i \otimes \gamma_k \otimes
\tilde{u}(f_{ij}\phi_k) \tilde{u}( \widetilde\phi_k) x_j\|_{\Gauss_n(\Gauss(X))}.
\end{equation}
Further we have
\begin{align}
& \| \sum_{i,j=1}^n \sum_{k=-N}^N \gamma_i \otimes \gamma_k \otimes \tilde{u}(f_{ij} \phi_k) \tilde{u}(\widetilde\phi_k) x_j \|_{\Gauss_n(\Gauss(X))} \nonumber \\
& \cong \| \sum_{i,j,k} \gamma_{ik} \otimes \tilde{u}(f_{ij} \phi_k) \tilde{u}(\widetilde\phi_k) x_j\|_{\Gauss(X)} \nonumber \\
& \lesssim \| F_N \|_{M_{(2N+1)n} \otimes W^\alpha} \| \sum_{i,k} \gamma_{i,k} \otimes \tilde{u}(\widetilde\phi_k) x_i \|_{\Gauss(X)} \nonumber \\
& \lesssim \|F\|_{M_n \otimes \Hor^\alpha} \|\sum_i \gamma_i \otimes x_i \|_{\Gauss_n(X)}. \label{Equ Proof Main Thm 2}
\end{align}
Finally taking the supremum over $N \in \mathbb{N},$ \eqref{Equ Proof Main Thm 1} and \eqref{Equ Proof Main Thm 2} give
\[ \| [f_{ij}(A)] \|_{M_n \otimes_\gamma B(X)} \lesssim \| [f_{ij}] \|_{M_n \otimes \Hor^\alpha} .\]
In particular, $\|f(A)\| \lesssim \|f\|_{\Hor^\alpha},$ so that by Lemma \ref{Lem A u}, there exists a bounded mapping $u : \Hor^\alpha \to B(X)$ extending the $H^\infty$ calculus in the sense of \eqref{Equ Lem A u 2}.
Now repeat the above argument with an arbitrary $F = [f_{ij}] \in M_n \otimes \Hor^\alpha$ and $u(f_{ij})$ in place of $f_{ij}(A),$ to deduce that $u$ is matricially $\gamma$-bounded.\\

\noindent
$(2) \Longrightarrow (1).$
Denote $\tilde{u}$ the restriction of $u$ to $W^\alpha$ which by Lemma \ref{Lem Function spaces} is again matricially $\gamma$-bounded.
Thus also the mapping $v= \tilde{u} \circ w^{-1}$ from the first part of the proof is matricially $\gamma$-bounded and by Theorem \ref{Thm Main 1},
$\| (1 + t^2)^{-\alpha/2} A^{it} x \|_{\gamma(\mathbb{R},X)} \leq C \|x\|.$
\end{proof}

\section{Extensions and Applications}\label{Sec 5 Examples}

We have characterized in Theorem \ref{Thm Ha} the matricially $\gamma$-bounded H\"ormander calculus in terms of square functions of $A.$
In fact, the imaginary powers $A^{is}$ appearing in these square functions can be replaced by several other typical operator families associated with $A,$
such as resolvents $R(\lambda,A)$ for $\lambda \in \mathbb{C} \backslash [0,\infty)$ and the semigroup generated by $-A,$ $T(z) = \exp(-zA)$ for $\Re z > 0.$
This gives (almost) equivalent conditions, see Proposition \ref{Prop BIP Sgr Res} below.
Subsequently, we use the semigroup condition of this proposition to apply Theorem \ref{Thm Ha} to some examples.
The starting point for us will be semigroups that satisfy (generalized) Gaussian estimates (see \eqref{Equ Gaussian estimate}).

The following lemma serves as a preparation for Proposition \ref{Prop BIP Sgr Res}.

\begin{lem}\label{Lem square function integral transform}
For $i = 1,2,$ let $(\Omega_i,\mu_i)$ be $\sigma$-finite measure spaces and $K \in B(L^2(\Omega_1),L^2(\Omega_2)).$
\begin{enumerate}
\item
Assume that $f \in \gamma(\Omega_1,X)$ and that there exists a Bochner-measurable $g : \Omega_2 \to X$ such that
\[ \spr{g(\cdot)}{x'} = K(\spr{f(\cdot)}{x'}) \quad (x' \in X'). \]
Then $g \in \gamma(\Omega_2,X)$ and
\[ \|g\|_{\gamma(\Omega_2,X)} \leq \|K\|\,\|f\|_{\gamma(\Omega_1,X)}. \]
\item
Let $\Omega_1 \to B(X),\,t \mapsto N(t)$ and $\Omega_2 \to B(X),\,t \mapsto M(t)$ be weakly measurable.
Assume that $\|N(\cdot)x\|_\gamma \leq C \|x\|$ and that there is $K \in B(L^2(\Omega_1),L^2(\Omega_2))$ such that
$K \left[ \spr{N(\cdot)x}{x'} \right] = \spr{M(\cdot)x}{x'}$ for $x \in D,$ where $D$ is some dense subset of $X.$
Then $M(\cdot)x \in \gamma(\Omega_2,X)$ for any $x \in X$ and $\|M(\cdot)x\|_\gamma \lesssim \|N(\cdot)x\|_\gamma.$
\item
Let $(\Omega,\mu)$ be a measure space and $g :\Omega \to X$ measurable.
For $n \in \mathbb{N},$ let $\varphi_n:\Omega \to [0,1]$ measurable with $\sum_{n = 1}^\infty \varphi_n(t) = 1$ for all $t \in \Omega.$
Then
\[ \|g\|_{\gamma(\Omega,X)} \leq \sum_{n = 1}^\infty \| \varphi_n g \|_{\gamma(\Omega,X)}. \]
\end{enumerate}
\end{lem}

\begin{proof}
(1) By assumption, $g \in P_2(\Omega_2,X).$
Consider the associated operator $u_g : H \to X$ as in \eqref{Equ u_f}.
We have $u_g = u_f \circ K'.$
Thus, by Lemma \ref{Lem Folklore square functions}, $\|u_g\|_{\gamma(L^2(\Omega_2),X)} \leq \|K'\|\,\|u_f\|_{\gamma(L^2(\Omega_1),X)},$ which proves (1).\\

\noindent
(2) We show first that $M(\cdot)x$ belongs to $P_2(\Omega_2,X)$ for any $x \in X.$
For $x \in D,$ this follows immediately from the assumption.
For $x \in X,$ we let $x_n \in D$ such that $x_n \to x$ $(n \to \infty).$
Then $\spr{M(t)x}{x'} = \lim_n \spr{M(t)x_n}{x'}$ for any $t \in \Omega_2.$
On the other hand, $\spr{M(\cdot)x_n}{x'}$ is convergent in $L^2(\Omega_2).$
Indeed,
\begin{align*}
\| \spr{M(\cdot)(x_n - x_m)}{x'} \|_{L^2(\Omega_2)} & = \|K\left[ \spr{N(\cdot)(x_n - x_m)}{x'} \right] \|_{L^2(\Omega_1)} \\
& \lesssim \| N(\cdot)(x_n - x_m)\|_\gamma \, \|x'\| \lesssim \|x_n - x_m\| \, \|x'\|,
\end{align*}
which converges to $0\: (n,m \to \infty).$
Thus, $\spr{M(\cdot)x}{x'}$ the pointwise limit, so necessarily equal to the $L^2$ limit, belongs to $L^2(\Omega_2).$
Consequently, by (1) and Lemma \ref{Lem Folklore square functions}, $\|M(\cdot)x\|_\gamma = \lim_n \|M(\cdot)x_n\|_\gamma \leq \|K\| \lim_n \|N(\cdot)x_n\|_\gamma = \|K\| \,\|N(\cdot)x\|_\gamma,$ which shows (2).\\

\noindent
(3) For $n \in \mathbb{N},$ put $\phi_n = \sum_{k=1}^n \varphi_k.$
Then $\phi_n: \Omega \to [0,1]$ and $\phi_n(t) \to 1$ monotonically for all $t \in \Omega.$
Then $\sup_n \|\phi_n g\|_\gamma \leq \sup_n \sum_{k=1}^n \| \varphi_k g\|_\gamma = \sum_{k=1}^\infty \|\varphi_k g \|_\gamma.$
It remains to show $\|g\|_\gamma \leq \sup_n \| \phi_n g\|_\gamma.$
Let us show first that $g \in P_2(\Omega,X),$ i.e. for any $x' \in X',\, \spr{g(\cdot)}{x'} \in L^2(\Omega).$
By assumption, we have $|\spr{g(t)}{x'}| = \lim_n \phi_n(t) |\spr{g(t)}{x'}|$ for any $t \in \Omega,$ and this convergence is monotone.
Then by Beppo Levi's theorem,
\[\|\spr{g(\cdot)}{x'}\|_{L^2(\Omega)} = \lim_n \|\spr{\phi_n(\cdot) g(\cdot)}{x'}\|_{L^2(\Omega)}
\leq \limsup_n \|\phi_n \cdot g\|_{\gamma(\Omega,X)} \|x'\|.\]
where we have used that $\|\spr{f(\cdot)}{x'}\|_{L^2(\Omega)} \leq \|f\|_{\gamma(\Omega,X)} \, \|x'\|$ for any $f \in \gamma(\Omega,X).$
Thus we have shown that $g \in P_2(\Omega,X).$
Then by Lemma \ref{Lem Folklore square functions},
\[\|g\|_{\gamma(\Omega,X)} \leq \liminf_n \|\phi_n \cdot g\|_{\gamma(\Omega,X)} \leq \sup_n \|\phi_n \cdot g\|_{\gamma(\Omega,X)}.\]
\end{proof}

\begin{prop}\label{Prop BIP Sgr Res}
Let $A$ be a $0$-sectorial operator having a bounded $H^\infty$ calculus on some space $X$ with property $(\alpha).$
Let $\alpha > \frac12.$
Consider the following conditions.
\[
\begin{array}{ll}
& \textit{H\"ormander functional calculus}\\
(1) & \text{The }H^\infty\text{ calculus of }A\text{ extends to a matricially }\gamma\text{-bounded mapping }\Hor^\alpha \to B(X).\vspace{0.2cm}\\
& \textit{Imaginary powers}\\
(2) &\| (1 + t^2)^{-\alpha/2} A^{it} x \|_{\gamma(\mathbb{R},X)} \leq C \|x\|.\vspace{0.2cm}\\
& \textit{Resolvents}\\
(3)&\text{For some }\beta \in (0,1)\text{ and }\theta \in (-\pi,\pi) \backslash \{ 0 \} :\: \| t^\beta A^{1 - \beta} R(e^{i\theta} t,A) x \|_{\gamma(\mathbb{R}_+,dt/t,X)} \lesssim |\theta|^{-\alpha} \|x\|.\\
(4)&\text{For some }\beta \in (0,1),\:\theta_0 \in (0,\pi] :\: \| \, \|\theta|^{\alpha - \frac12} t^\beta A^{1 - \beta}R(e^{i \theta} t,A)x \|_{\gamma(\mathbb{R}_+ \times [-\theta_0,\theta_0], dt/t d\theta,X)} \lesssim \|x\|.\vspace{0.2cm}\\
& \textit{Analytic semigroup}\\
(5)&\text{For }\theta \in (-\frac{\pi}{2},\frac{\pi}{2}),\:\|A^{1/2}T(e^{i\theta}t)x\|_{\gamma(\mathbb{R}_+,dt,X)} \lesssim (\frac{\pi}{2}-|\theta|)^{-\alpha} \|x\|.\\
(6)& \|( 1+ \left| \frac{a}{b} \right|^2)^{\alpha/2} |a|^{-\frac12} A^{1/2} T(a+ib)x\|_{\gamma(\mathbb{R}_+\times \mathbb{R},da db,X)} \lesssim \|x\|.
\end{array}
\]
Then the conditions (1), (2), (4), (6) are equivalent.
Further these conditions imply the remaining ones (3), (5), which conversely imply that the $H^\infty$ calculus of $A$ extends to a matricially $\gamma$-bounded homomorphism $\Hor^{\alpha + \epsilon} \to B(X)$ for any $\epsilon > 0.$
\end{prop}

\begin{proof}
$(1) \Longleftrightarrow (2).$\\
This is Theorem \ref{Thm Ha}.\\

\noindent
$(2) \Longleftrightarrow (4).$\\
Consider
\begin{equation}\label{Equ Proof Charact K}
 K : L^2(\mathbb{R},ds) \to L^2(\mathbb{R} \times (-\pi,\pi),ds d\theta),\,
f(s) \mapsto (\pi-|\theta|)^{\alpha-\frac12}\frac{1}{\sin \pi(\beta + is)} e^{\theta s} \langle s \rangle^{\alpha} f(s),
\end{equation}
where we write in short
\[ \langle s \rangle = (1 + |s|^2)^{\frac12}.\]
Note that $|\sin \pi(\beta + is)| \cong \cosh(\pi s)$ for $\beta \in (0,1)$ fixed.
$K$ is an isomorphic embedding.
Indeed,
\[ \|Kf\|_2^2 =
\int_\mathbb{R} \int_{-\pi}^\pi \left((\pi-|\theta|)^{\alpha-\frac12} e^{\theta s}\right)^2 d\theta \frac{1}{|\sin^2(\pi(\beta + i s))|} \langle s \rangle^{2\alpha} |f(s)|^2 ds\]
and
\begin{align}
\int_{-\pi}^\pi (\pi-|\theta|)^{2\alpha -1} e^{2\theta s} d\theta
& \cong \int_0^\pi \theta^{2\alpha -1} e^{2(\pi - \theta) |s|} d\theta
\nonumber \\
& \cong \cosh^2(\pi s) \int_0^\pi \theta^{2\alpha-1} e^{2\theta |s|} d\theta.
\nonumber
\end{align}
The last integral is bounded from below uniformly in $s \in \mathbb{R},$ and for $|s|\geq 1,$
\[\int_0^\pi \theta^{2\alpha -1} e^{2\theta |s|} d\theta = (2|s|)^{-2\alpha} \int_0^{2|s|\pi} \theta^{2\alpha -1} e^{\theta}d\theta \cong |s|^{-2\alpha}.\]
This clearly implies that $\|Kf\|_2 \cong \|f\|_2.$
Applying Lemma \ref{Lem square function integral transform}, we get
\[ \| \langle s \rangle^{-\alpha} A^{is}x \|_{\gamma(\mathbb{R},ds,X)}
\cong \left\|(\pi-|\theta|)^{\alpha-\frac12} \frac{1}{\cosh (\pi s)} e^{\theta s}A^{is}x \right\|_{\gamma(\mathbb{R} \times (-\pi,\pi),ds d\theta,X)}.\]
In \cite[p. 228 and Theorem 15.18]{KuWe}, the following formula is derived for $x \in A(D(A^2))$ and $|\theta| < \pi:$
\begin{equation}\label{Equ Res A Mellin BIP}
\frac{\pi}{\sin \pi (\beta + i s)} e^{\theta s} A^{is} x =
\int_0^\infty t^{is}\left[t^{\beta} e^{i\theta \beta} A^{1 - \beta}(e^{i\theta}t+A)^{-1}x\right] \frac{dt}{t}.
\end{equation}
Note that $A(D(A^2))$ is a dense subset of $X.$
As the Mellin transform $f(s) \mapsto \int_0^\infty t^{is} f(s) \frac{ds}{s}$ is an isometry $L^2(\mathbb{R}_+,\frac{ds}s)\to L^2(\mathbb{R},dt),$ we get by Lemma \ref{Lem square function integral transform} (2)
\begin{align}
\| \langle s \rangle^{-\alpha} A^{is} x\|_{\gamma(\mathbb{R},X)}
& \cong \|(\pi -|\theta|)^{\alpha-\frac12} t^{\beta}A^{1-\beta}(e^{i\theta}t+A)^{-1} x\|_{\gamma(\mathbb{R}_+\times(-\pi,\pi),\frac{dt}{t}d\theta,X)}
\nonumber \\
& \cong \| |\theta|^{\alpha-\frac12} t^\beta A^{1 - \beta} R(e^{i\theta}t,A) x\|_{\gamma(\mathbb{R}_+ \times (0,2\pi),dt/t d\theta,X)}.
\nonumber
\end{align}

so that (2) $\Leftrightarrow$ (4) for $\theta_0 = \pi.$

For a general $\theta_0 \in (0,\pi],$ consider $K$ from \eqref{Equ Proof Charact K} with restricted image, i.e.
\[K : L^2(\mathbb{R},ds) \to L^2(\mathbb{R} \times (-\pi,-(\pi-\theta_0)] \cup [\pi-\theta_0,\pi),ds d\theta).\]
Then argue as in the case $\theta_0 = \pi.$\\

\noindent
(4) $\Longleftrightarrow$ (6).\\
The proof of (2) $\Leftrightarrow$ (4) above shows that condition (4) is independent of $\theta_0 \in (0,\pi]$ and $\beta \in (0,1).$
Put $\theta_0 = \pi$ and $\beta = \frac12.$
The equivalence follows again from Lemma \ref{Lem square function integral transform},
using the fact that for $\theta \in (-\frac{\pi}{2},\frac{\pi}{2})$ and $\mu > 0,$
\begin{equation}\label{Equ Proof BIP Sgr Res}
(e^{i\theta} \mu + it)^{-1} = K[\exp(-(\cdot)e^{i\theta} \mu)\chi_{(0,\infty)}(\cdot)](t),
\end{equation}
where $K : L^2(\mathbb{R},ds) \to L^2(\mathbb{R},dt)$ is the Fourier transform.\\

\noindent
(3) $\Longleftrightarrow$ (5) for $\beta = \frac12.$\\
We use the same argument as right above.\\

\noindent
(2) $\Longrightarrow$ (3).\\
We use a similar $K_\theta$ as in the proof of $(2) \Leftrightarrow (4),$ fixing $\theta \in (-\pi,\pi):$
\[ K_\theta:  L^2(\mathbb{R},ds) \to L^2(\mathbb{R},ds),\, f(s) \mapsto (\pi - |\theta|)^{\alpha} \frac{1}{\sin\pi (\beta + is)}  e^{\theta s} \langle s \rangle^{\alpha} f(s). \]
We have
\[\sup_{|\theta| < \pi} \|K_\theta\| = \sup_{|\theta| < \pi,\,s\in\mathbb{R}} \langle s \rangle^\alpha (\pi-|\theta|)^\alpha
\frac{e^{\theta s}}{|\sin \pi (\beta + is)|} \lesssim \sup_{\theta,s} \langle s(\pi-|\theta|) \rangle^\alpha e^{-|s|(\pi-|\theta|)} < \infty.\]
Thus, by \eqref{Equ Res A Mellin BIP},
\begin{align}
\sup_{0 < |\theta| \leq \pi} |\theta|^{\alpha} \|t^\beta A^{1-\beta}R(te^{i\theta},A)x\|_{\gamma(\mathbb{R}_+,dt/t,X)}
& = \sup_{|\theta| < \pi} (\pi-|\theta|)^{\alpha} \|t^{\beta} A^{1 - \beta} (e^{i\theta} t +A)^{-1}x\|_{\gamma(\mathbb{R}_+,\frac{dt}{t},X)}
\nonumber \\
& = \sup_{|\theta| < \pi} (\pi-|\theta|)^{\alpha} \| \frac{\pi}{\sin\pi (\beta + is)}e^{\theta s}A^{is}x \|_{\gamma(\mathbb{R},ds,X)}
\label{Equ Proof Thm Charact} \\
& \lesssim \|\langle s \rangle^{-\alpha} A^{is}x\|_{\gamma(\mathbb{R},ds,X)}.
\nonumber
\end{align}

\noindent
(3), $\alpha$ $\Longrightarrow$ (2), $\alpha + \epsilon.$\\
First we consider $\langle s \rangle^{-(\alpha+\epsilon)}A^{is}x$ for $s\geq 1.$
\begin{align}
\|\langle s \rangle^{-(\alpha+\epsilon)}A^{is}x\|_{\gamma([1,\infty),X)}
& \leq \sum_{n = 0}^\infty 2^{-n\epsilon} \| \langle s \rangle^{-\alpha} A^{is}x\|_{\gamma([2^n,2^{n+1}],X)}.
\label{Equ Proof Thm Charact 2}
\end{align}
For $s \in [2^n,2^{n+1}],$ we have
\[\langle s \rangle^{-\alpha} \lesssim 2^{-n\alpha} \lesssim 2^{-n\alpha} e^{-2^{-n}s} \lesssim (\pi-\theta_n)^\alpha \frac{e^{\theta_n s}}{\sin \pi(\beta + i s)} ,\]
where $\theta_n = \pi - 2^{-n}.$
Therefore
\begin{align}
\| \langle s \rangle^{-\alpha} A^{is}x\|_{\gamma([2^n,2^{n+1}],X)}
& \lesssim (\pi-\theta_n)^{\alpha} \| \frac{\pi}{\sin\pi (\beta + is)}e^{\theta_n s}A^{is}x \|_{\gamma(\mathbb{R},X)} \nonumber \\
& \overset{\eqref{Equ Proof Thm Charact}}{\lesssim} \sup_{0<|\theta| \leq \pi} |\theta|^{\alpha} \| t^\beta A^{1 - \beta}R(te^{i\theta},A) x\|_{\gamma(\mathbb{R}_+,dt/t,X)} < \infty.
\nonumber
\end{align}
Thus, the sum in \eqref{Equ Proof Thm Charact 2} is finite.

The part $\langle s \rangle^{-(\alpha + \epsilon)}A^{is}x$ for $s \leq -1$ is treated similarly,
whereas $\|\langle s \rangle^{-\alpha}A^{is}x\|_{\gamma((-1,1),X)} \cong \|A^{is}x\|_{\gamma((-1,1),X)}.$
It remains to show that the last expression is finite.
We have assumed that $X$ has property $(\alpha).$
Then the fact that $A$ has an $H^\infty$ calculus implies that $\{A^{is}:\: |s| < 1\}$ is $\gamma$-bounded \cite[Corollary 6.6]{KaW2}.
Then by Lemma \ref{Lem Folklore square functions} (3), we have $\|A^{is}x\|_{\gamma((-1,1),X)} \leq \gamma\left(\left\{ A^{is} : |s| < 1 \right\}\right) \|1\|_{L^2(-1,1)} \|x\|.$
\end{proof}

Condition (5) of the preceding proposition can be checked in the following way.

\begin{lem}\label{Lem Sgr}
Let $A$ be a $0$-sectorial operator on a space $X$ with property $(\alpha)$ having an $H^\infty$ calculus.
If for some $\beta > 0$
\begin{equation}\label{Equ Lem Sgr}
\left\{ T \left(te^{\pm i(\frac{\pi}{2} - \theta)} \right) :\: t > 0\right\}\text{ is }\gamma\text{-bounded with constant }\lesssim \theta^{-\beta},
\end{equation}
then $\| A^{\frac12} T(e^{\pm i (\frac{\pi}{2} - \theta)}t) x \|_{\gamma(\mathbb{R}_+,X)} \lesssim \theta^{-\alpha} \|x\|$ with $\alpha = \beta + \frac12.$
\end{lem}

\begin{proof}
Decompose
\[ t \exp \left(\pm i (\frac{\pi}{2} - \theta)\right) = s(t,\theta) + r(t,\theta) \exp\left(\pm i (\frac{\pi}{2} - \frac{\theta}{2})\right) \]
where the reals $s$ and $r$ are uniquely determined by $t$ and $\theta.$
We have $s(t,\theta) = \kappa(\theta) t$ with $\kappa(\theta) \cong \theta.$
Then by Lemma \ref{Lem Folklore square functions},
\begin{align*}
\| A^{\frac12} T(t e^{\pm i (\frac{\pi}{2} - \theta)}) x \|_{\gamma(\mathbb{R}_+,X)} & = \| T(r e^{\pm i (\frac{\pi}{2} - \frac{\theta}{2})}) A^{\frac12} T(s) x\|_{\gamma(\mathbb{R}_+,X)} \\
& \leq \gamma\left(\left\{ T(r e^{\pm i (\frac{\pi}{2} - \frac{\theta}{2})}) :\: r > 0 \right\} \right) \| A^{\frac12} T(s(t,\theta)) x \|_{\gamma(\mathbb{R}_+,X)} \\
& \lesssim (\theta/2)^{-\beta} \theta^{-\frac12} \| A^{\frac12} T(t) x \|_{\gamma(\mathbb{R}_+,X)}.
\end{align*}
By \eqref{Equ Proof BIP Sgr Res} and \cite[Theorem 7.2]{KaW2}, $\| A^{\frac12} T(t) x\|_{\gamma(\mathbb{R}_+,X)} = \| A^{\frac12} (it - A)^{-1} x\|_{\gamma(\mathbb{R},X)} \leq C \|x\|,$ which finishes the proof.
\end{proof}

Let us now turn to some examples.

\begin{defi}
Let $\Omega$ be a topological space which is equipped with a distance $\rho$ and a Borel measure $\mu.$
Let $d \geq 1$ be an integer.
$\Omega$ is called a homogeneous space\index{homogeneous space} of dimension $d$ if there exists $C > 0$ such that for any $x \in \Omega,\,r> 0$ and $\lambda \geq 1:$
\[\mu(B(x,\lambda r)) \leq C \lambda^d \mu(B(x,r)).\]
\end{defi}

Typical cases of homogeneous spaces are open subsets of $\mathbb{R}^d$ with Lipschitz boundary and Lie groups with polynomial volume growth, in particular stratified nilpotent Lie groups (see e.g. \cite{FoSt}).

We will consider operators satisfying the following assumption.

\begin{assumption}\label{Ass Examples}
$A$ is a self-adjoint positive (injective) operator on $L^2(\Omega),$ where $\Omega$ is a homogeneous space of a certain dimension $d.$
Further, there exists some $p_0 \in [1,2)$ such that the semigroup generated by $-A$ satisfies the so-called generalized Gaussian estimate
(see e.g. \cite[(GGE)]{Bluna}):
\begin{equation}\tag{GGE}\label{Equ generalized Gaussian estimate}
\| \chi_{B(x,r_t)} e^{-tA} \chi_{B(y,r_t)} \|_{p_0 \to p_0'} \leq C \mu(B(x,r_t))^{\frac{1}{p_0'}-\frac{1}{p_0}} \exp \left(-c \left(\rho(x,y)/r_t \right)^\frac{m}{m-1} \right) \quad (x,y \in \Omega,\,t>0).
\end{equation}
Here, $p_0'$ is the conjugated exponent to $p_0,\,C,c > 0,\,m \geq 2$ and $r_t = t^{\frac1m},$ $\chi_B$ denotes the characteristic function of $B,$ $B(x,r)$ is the ball $\{y \in \Omega:\: \rho(y,x) < r\}$
and $\|\chi_{B_1} T \chi_{B_2}\|_{p_0 \to p_0'} = \sup_{\|f\|_{p_0} \leq 1} \|\chi_{B_1} \cdot T(\chi_{B_2} f)\|_{p_0'}.$
\end{assumption}

If $p_0 = 1,$ then it is proved in \cite{BlKub} that \eqref{Equ generalized Gaussian estimate} is equivalent to the usual Gaussian estimate,
i.e. $e^{-tA}$ has an integral kernel $k_t(x,y)$ satisfying the pointwise estimate (cf. e.g. \cite[Assumption 2.2]{DuOS})
\begin{equation}\tag{GE}\label{Equ Gaussian estimate}
|k_t(x,y)| \lesssim \mu(B(x,t^{\frac1m}))^{-1} \exp\left(-c \left(\rho(x,y)/t^{\frac1m}\right)^{\frac{m}{m-1}}\right)\quad (x,y \in \Omega,\,t>0).
\end{equation}
This is satisfied in particular by sublaplacian operators on Lie groups of polynomial growth \cite{Varo} as considered e.g. in
\cite{MaMe,Chri, Alex, MSt,Duon}, or by more general elliptic and sub-elliptic operators
\cite{Davia, Ouha}, and Schr\"odinger operators \cite{Ouhaa}.
It is also satisfied by all the operators in \cite[Section 2]{DuOS}.

Examples of operators satisfying a generalized Gaussian estimate for $p_0 > 1$ are higher order operators with bounded coefficients and Dirichlet boundary conditions on domains of $\mathbb{R}^d,$ Schr\"odinger operators with singular potentials on $\mathbb{R}^d$ and elliptic operators on Riemannian manifolds as listed in \cite[Section 2]{Bluna} and the references therein.

\begin{thm}\label{Thm R-bounded Blunck Hormander thm}
Let Assumption \ref{Ass Examples} hold.
Then for any $p \in (p_0,p_0'),$ the $H^\infty$ calculus of $A$ extends to a matricially $\gamma$-bounded homomorphism $\Hor^\alpha \to B(L^p(\Omega))$ with
\[\alpha > d\left| \frac1{p_0} - \frac12\right|+\frac12.\]
\end{thm}

\begin{proof}
We show that \eqref{Equ Lem Sgr} holds with $\beta = d (\frac{1}{p_0} - \frac12).$
By \cite[Proposition 2.1]{BlKua}, the assumption \eqref{Equ generalized Gaussian estimate} implies that
\[ \| \chi_{B(x,r_t)} e^{-tA} \chi_{B(y,r_t)}\|_{p_0\to 2} \leq C_1 \mu(B(x,r_t))^{\frac{1}{2}-\frac{1}{p_0}} \exp(-c_1 (\rho(x,y)/r_t)^{\frac{m}{m-1}}) \quad (x,y \in \Omega,\,t>0)\]
for some $C_1,c_1>0.$
By \cite[Theorem 2.1]{Blun}, this can be extended from real $t$ to complex $z = te^{i\theta}$ with $\theta \in (-\frac{\pi}{2},\frac{\pi}{2}):$
\[ \|\chi_{B(x,r_z)} e^{-zA} \chi_{B(y,r_z)} \|_{p_0 \to 2} \leq C_2 \mu(B(x,r_z))^{\frac{1}{2}-\frac{1}{p_0}} (\cos \theta )^{-d(\frac{1}{p_0} -\frac{1}{2})} \exp(-c_2 (\rho(x,y)/r_z)^{\frac{m}{m-1}}),\]
for $r_z = (\cos \theta )^{-{\frac{m-1}{m}}} t^{\frac1m},$ and some $C_2,c_2 >0.$
By \cite[Proposition 2.1 (i) (1) $\Rightarrow$ (3) with $R = e^{-zA},\,\gamma =\alpha = \frac{1}{p_0} - \frac12,\,\beta = 0,\,r = r_z,\,u=p_0$ and $v = 2$]{BlKua}, this gives for any $x \in \Omega,\,\Re z > 0$ and $k \in \mathbb{N}_0$
\[ \|\chi_{B(x,r_z)} e^{-zA} \chi_{A(x,r_z,k)}\|_{p_0 \to 2} \leq C_3 \mu(B(x,r_z))^{\frac12 - \frac{1}{p_0}} (\cos \theta )^{-d(\frac{1}{p_0} -\frac{1}{2})} \exp(-c_3 k^{\frac{m}{m-1}}),\]
where $A(x,r_z,k)$ denotes the annular set $B(x,(k+1)r_z)\backslash B(x,k r_z).$
By \cite[Theorem 2.2 with $q_0 = p_0,\,q_1 = s = 2,\rho(z) = r_z$ and $S(z) = (\cos \theta)^{d(\frac{1}{p_0}-\frac{1}{2})} e^{-zA}$]{Kuns} and property $(\alpha),$
we deduce that
\[ \{ (\cos \theta)^{d(\frac{1}{p_0}-\frac{1}{2})} e^{-zA} :\: \Re z > 0\} \]
is $\gamma$-bounded.
Now apply Lemma \ref{Lem Sgr} and Proposition \ref{Prop BIP Sgr Res}, noting that $A$ has an $H^\infty$ calculus on $L^p(\Omega)$ \cite[Corollary 2.3]{Blun}.
\end{proof}

\begin{rem}\label{Rem R-bounded Blunck Hormander thm}
\begin{enumerate}
\item Theorem \ref{Thm R-bounded Blunck Hormander thm} improves on \cite[Theorem 1.1]{Bluna} in that it includes the matricial $\gamma$-boundedness of the H\"ormander calculus.
Note that \cite{Bluna} obtains also a weak-type result for $p = p_0.$
If $p_0$ is strictly larger than $1,$ then Theorem \ref{Thm R-bounded Blunck Hormander thm} improves the order of derivation $\alpha$ of the calculus from $\displaystyle \frac{d}{2} + \frac{1}{2} +\epsilon$ in \cite{Bluna} to $\displaystyle d \left|\frac{1}{p_0} - \frac{1}{2} \right| + \frac12 + \epsilon.$
In \cite[Theorem 6.4 a)]{Uhl}, under the assumptions of Theorem \ref{Thm R-bounded Blunck Hormander thm}, a $\mathcal{H}^\beta_r$ calculus with
with $\beta > (d+1) | \frac{1}{p_0} - \frac{1}{2} |$ and $r > | \frac12 - \frac1p |^{-1}$ is derived.
Here $\mathcal{H}^\beta_r$ is defined similarly to $\Hor^\alpha$ by
\[ \mathcal{H}^\beta_{r} = \{ f : (0,\infty)\to \mathbb{C}:\: \sup_{k \in \mathbb{Z}} \|(f \circ \exp )\phi_k\|_{W^\beta_r} < \infty\}.\]
Note that $\mathcal{H}^\beta_r$ is larger than $\mathcal{H}^\alpha.$
In the classical case of Gaussian estimates, i.e. $p_0 = 1,$ \cite{DuOS} yields a $\mathcal{H}^{\alpha_2}_\infty$ calculus under Assumption \ref{Ass Examples}
and even a $\mathcal{H}^{\alpha_2}$ calculus for many examples, e.g. homogeneous operators, with the better derivation order $\alpha_2 > \frac{d}{2}.$
\item The theorem also holds for the weaker assumption that $\Omega$ is an open subset of a homogeneous space $\tilde{\Omega}.$
In that case, the ball $B(x,r_t)$ on the right hand side in \eqref{Equ generalized Gaussian estimate} is the one in $\tilde{\Omega}.$
This variant can be applied to elliptic operators on irregular domains $\Omega \subset \mathbb{R}^d$ as discussed in \cite[Section 2]{Bluna}.
\end{enumerate}
\end{rem}

In Theorem \ref{Thm R-bounded Blunck Hormander thm}, the operator $A$ was assumed to be self-adjoint, and thus, admits a functional calculus $L^\infty \to B(L^2(\Omega)).$
The space $L^\infty = L^\infty((0,\infty);d\mu_A)$ is larger than $\Hor^\alpha,$ and one can use this fact to ameliorate the functional calculus of $A$ on $L^q(\Omega)$ by complex interpolation.

\begin{prop}
Let $A$ satisfy Assumption \ref{Ass Examples}.
Then for $q \in (p_0,p_0'),\, \alpha > d \left| \frac{1}{p_0}- \frac12 \right| + \frac12$ and $\theta \in (0,1)$ with $\theta > \left| \frac1q -\frac{1}{p_0}\right| / \left|{\frac12 - \frac1{p_0}} \right|,$
the functional calculus of $A$ on $L^q$
\begin{equation}\label{Equ Proof Rem Blunck}
u_{L^q} : (L^\infty,\Hor^\alpha)_\theta \to B(L^q(\Omega)) \text{ is matricially }\gamma\text{-bounded.}
\end{equation}
Here $(L^\infty,\Hor^\alpha)_\theta$ is the complex interpolation space which is given an operator space structure \cite[p.~56]{Pis2} by
\[ M_n \otimes (L^\infty,\Hor^\alpha)_{\theta} = (M_n \otimes L^\infty,M_n \otimes \Hor^\alpha)_\theta. \]
\end{prop}

\begin{proof}
The self-adjoint calculus $u_{L^2} : L^\infty \to B(L^2(\Omega))$ is completely bounded since it is a $\ast$-representation \cite[Proposition 1.5]{Pis2}, so by Remark \ref{Rem mat}, $u_{L^2}$ is matricially $\gamma$-bounded.
Moreover, we have $(\Gauss(L^p),\Gauss(L^2))_\theta = \Gauss((L^p,L^2)_\theta)$ \cite[Proposition 3.7]{KaKW}.
Then by bilinear interpolation between
\[ M_n \otimes L^\infty \times \Gauss_n(L^2) \to \Gauss_n(L^2),\, ([a_{ij}] \otimes f ,\sum_k \gamma_k \otimes x_k) \mapsto \sum_{k,j} \gamma_k a_{kj} u_{L^2}(f)x_j \]
and, with the mapping $u_{L^p}$ resulting from Theorem \ref{Thm R-bounded Blunck Hormander thm}, $p$ given by $\theta = |\frac1q - \frac1p| / |\frac12 - \frac1p|,$
\[ M_n \otimes \Hor^\alpha \times \Gauss_n(L^p) \to \Gauss_n(L^p),\, ([a_{ij}] \otimes f , \sum_k \gamma_k \otimes x_k) \mapsto \sum_{k,j} \gamma_k a_{kj} u_{L^p}(f)x_j, \]
one deduces \eqref{Equ Proof Rem Blunck}.
\end{proof}

Note that the space $(L^\infty,\Hor^\alpha)_{\theta}$ contains $\mathcal{H}^\beta_{r,0},$ where $\frac1r > \frac{\theta}{2},$ $\beta > \alpha \theta + (\frac1r - \frac{\theta}{2}).$
Here $\mathcal{H}^\beta_{r,0} = \left\{ f \in \mathcal{H}^{\beta}_r :\: \|(f \circ \exp) \phi_k \|_{W^\beta_r} \to 0\text{ for }|k| \to \infty \right\}.$
Then \eqref{Equ Proof Rem Blunck} implies that in particular, $\mathcal{H}^\beta_{r,0} \to B(L^q),\,f \mapsto f(A)$ is (norm) bounded and by \cite[Section 4.6.1]{Kr}, this extends moreover boundedly to $\mathcal{H}^\beta_r \to B(L^q).$

In \cite[Section 7]{DuOS}, for many examples of operators $A$ satisfying \eqref{Equ Gaussian estimate}, it is shown that the functional calculus
\begin{equation}\label{Equ DuOS}
u: \Hor^\alpha \to B(L^p(\Omega)),\,f \mapsto f(A)\text{ is bounded for }1 < p < \infty\text{ and }\alpha > \frac{d}{2}.
\end{equation}
Moreover, for the fundamental example $A = - \Delta$ on $L^p(\mathbb{R}^d),$ the critical order $\frac{d}{2}$ in \eqref{Equ DuOS} is optimal \cite[IV.7.4]{Ste},\cite[Proposition 4.12 (2)]{Kr}.
Note that the derivation order for the matricially $\gamma$-bounded calculus obtained in Theorem \ref{Thm R-bounded Blunck Hormander thm} under the assumption \eqref{Equ Gaussian estimate} (i.e. $p_0 = 1$)
is only $\frac{d+1}{2},$ and therefore gives a weaker result
in the derivation order compared to \eqref{Equ DuOS}.

Thus the question arises if an arbitrary $A$ that has a norm-bounded H\"ormander calculus also has a matricially $\gamma$-bounded H\"ormander calculus.
In contrast to the self-adjoint $L^\infty$ calculus on Hilbert space, which is always matricially $\gamma$-bounded (see the proof above), we have the following result.

\begin{prop}\label{Prop bounded vs mat-gamma bounded}
Let $A$ be a $0$-sectorial operator on a space $X$ with property $(\alpha).$
Let $\alpha > \frac12$ and $\beta > \alpha + 1.$
Suppose that its functional calculus $f \mapsto f(A)$ is bounded $u_\alpha : \Hor^\alpha \to B(X),$ and denote $u_\beta$ the restriction of $u_\alpha$ to $\mathcal{H}^\beta.$
Then $u_\beta : \mathcal{H}^\beta \to B(X)$ is matricially $\gamma$-bounded.

On the other hand, for any $\alpha > 0,$ there exists some $A$ on a Hilbert space $X$ such that
$u_\alpha : \Hor^\alpha \to B(X)$ is bounded (even $\gamma$-bounded because of \eqref{Equ Hilbert Gaussian}),
but its restriction $u_{\alpha + \frac12} : \mathcal{H}^{\alpha + \frac12} \to B(X)$ is not matricially $\gamma$-bounded.
\end{prop}

\begin{proof}
For $t \in \mathbb{R},$ let $f_t(\lambda) = \lambda^{it}.$
It is easy to check that $\|f_t\|_{\Hor^\alpha} \lesssim \langle t \rangle^{\alpha}$ \cite[Lemma 4.12 (4)]{Kr}.
By Lemma \ref{Lem Function spaces} (1), $A$ has an $H^\infty$ calculus.
By \cite[Corollary 6.3]{Kr1}, the set $\left\{ T(te^{\pm i (\frac{\pi}{2} - \theta)}) :\: t > 0 \right\}$ is $\gamma$-bounded with constant $\leq C \theta^{-\alpha - \frac12}\quad (\theta \in (0,\frac{\pi}{2})).$
By Lemma \ref{Lem Sgr}, condition (5) of Proposition \ref{Prop BIP Sgr Res} is satisfied with $\alpha + 1$ in place of $\alpha$ and therefore, $u_{\beta} : \mathcal{H}^{\beta} \to B(X)$ is matricially $\gamma$-bounded.

For the second statement, let $\alpha > \frac12.$
Consider $X = W^\alpha$ and the group $U(t) g = (\cdot)^{it} g$ on $X.$
Note that
\[ \| (\cdot)^{it} g \|_X = \| (g \circ \exp)\hat{\phantom{i}}(\cdot - t) \langle \cdot \rangle^\alpha \|_2 = \| (g \circ \exp)\hat{\phantom{i}} (\cdot) \langle (\cdot) + t \rangle^\alpha \|_2 \cong \langle t \rangle^\alpha \|g\|_X. \]
In particular, $\|U(t)\| \cong \langle t \rangle^\alpha.$
It is easy to check that $U(t) = A^{it}$ are the imaginary powers of a $0$-sectorial operator $A$ and that $f(A) g = f g$ for any $g \in X$ and $f \in \bigcup_{\omega > 0} H^\infty(\Sigma_\omega).$
By \cite{Str}, one has $\|fg\|_{W^\alpha} \lesssim \|f\|_{\Hor^\alpha} \|g\|_{W^\alpha}.$
Thus, $A$ has a bounded $\Hor^\alpha$ calculus.

On the other hand, since $X$ is a Hilbert space, the square function condition of Theorem 4.10 reads
\[ \| \langle t \rangle^{-\beta} A^{it} x \|_{\gamma(\mathbb{R},X)} = \| \langle t \rangle^{-\beta} A^{it} x\|_{L^2(\mathbb{R},X)} \cong \left( \int_\mathbb{R} \langle t \rangle^{-2 \beta + 2 \alpha} dt \right)^{\frac12} \|x\|, \]
which is finite if and only if $\beta > \alpha + \frac12.$
\end{proof}

\section{Proofs of Lemmas \ref{Lem 1} - \ref{Lem 4}}\label{Sec 6 Proofs Lemmas}

\begin{proof}[Proof of Lemma \ref{Lem 1}]
Since $X$ has property $(\alpha),$ the fact that $A$ has a bounded $H^\infty(\Sigma_\omega)$ calculus implies \cite[Theorem 12.8]{KuWe} that for any $\theta > \omega,$
\[ \left\{ g(A) :\: \|g\|_{\infty,\theta} \leq 1 \right\}\text{ is }\gamma\text{-bounded}.\]
We fix some $\theta \in (\omega, \frac{\pi}{4}).$
As the mapping $u : \: W^\alpha \to B(X)$ is matricially $\gamma$-bounded, by Remark \ref{Rem mat},
\[ \left\{ h(A) :\: \|h\|_{W^\alpha} \leq 1 \right\}\text{ is }\gamma\text{-bounded}.\]
The lemma stated that
$\gamma\left(\left\{ f_{2^k z}(A) :\: k \in \mathbb{Z} \right\}\right) \lesssim \left| z /\Re z\right|^\beta,$ where $f_{2^k z}(\lambda) = \exp(-2^k z \lambda).$
Thus it suffices to decompose $f_{2^k z} = g + h,$ where $\|g\|_{\infty,\theta},\,\|h\|_{W^\alpha} \lesssim \left| z / \Re z \right|^\beta.$

As $\Psi : f \mapsto f(r \cdot)$ is an isomorphism $H^\infty(\Sigma_\theta) \to H^\infty(\Sigma_\theta)$ and $W^\alpha \to W^\alpha,$ with $\|\Psi \| \cdot \|\Psi^{-1}\| \leq C,\:C$ independent of $r > 0,$ it suffices to have the above decomposition for $|z| = 1$ and $k = 0.$
We choose $g(\lambda) = \exp(-(z+1) \lambda)$ and $h(\lambda) = \exp(-z\lambda) (1 - e^{-\lambda}).$
As $|\arg(z+1)| + \theta \leq \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2},$ we actually have $\|g\|_{\infty,\theta} \leq 1 \lesssim \left |\Re z \right|^{-\beta}.$
Further it is a simple matter to check that $\|h\|_{W^\alpha} \lesssim \left |\Re z \right|^{-\beta}$
for any $\beta > \alpha.$
\end{proof}

\begin{proof}[Proof of Lemma \ref{Lem 2}]
The assumption of the lemma was
\begin{equation}\label{Equ Proof Lem 2}
\gamma\left(\left\{ \exp(-2^k z A) :\: k \in \mathbb{Z} \right\}\right) \lesssim \left| \frac{z}{\Re z} \right|^\beta \quad (\Re z > 0).
\end{equation}
We first show that
\begin{equation}\label{Equ 2 Proof Lem 2}
\gamma \left( \left\{ (2^k t A)^{\frac12} \exp(-2^k t e^{\pm i (\frac{\pi}{2} - \omega) } A ) :\: k \in \mathbb{Z} \right\} \right) \lesssim \omega^{-(\beta + \frac12)} \quad (\omega \in (0,\frac{\pi}{2})).
\end{equation}
Decompose
\[ e^{\pm i (\frac{\pi}{2} - \omega)} t  = s + e^{\pm i (\frac{\pi}{2} - \frac{\omega}{2})} r,\]
where $s,r > 0$ are uniquely determined by $t$ and $\omega.$
Then
\[ (2^k t A)^{\frac12} \exp(-e^{\pm i (\frac{\pi}{2} - \omega)} 2^k t A) = \left( \frac{t}{s} \right)^{\frac12} (2^k s A)^{\frac12} \exp(-2^k s A) \exp(- 2^k r e^{\pm i (\frac{\pi}{2} - \frac{\omega}{2})}A),\]
and consequently,
\begin{align}
\gamma\left(\left\{(2^k t A)^{\frac12} \exp(-e^{\pm i (\frac{\pi}{2} - \omega)}2^k t A) :\: k \in \mathbb{Z} \right\} \right) & \leq \sup_t (t/s)^{\frac12} \times \gamma\left(\left\{ (2^k  s A)^{\frac12} \exp(-2^k s A) :\: k \in \mathbb{Z} \right\} \right) \nonumber \\
& \times \gamma \left(\left\{ \exp(-2^k r e^{\pm i (\frac{\pi}{2} - \frac{\omega}{2})}A):\: k \in \mathbb{Z}\right\} \right).
\label{Equ 3 Proof Lem 2}
\end{align}
We will show that the right hand side of \eqref{Equ 3 Proof Lem 2} can be estimated by $\lesssim \omega^{-\frac12} \times 1 \times \omega^{-\beta}.$
The estimate for the first factor follows from the law of sines
\[ t/s  = \sin (\frac{\pi}{2} + \omega/2)/\sin(\omega/2) \cong \omega^{-1}. \]
For the second estimate, note that by \cite[Example 2.16]{KuWe}, \eqref{Equ Proof Lem 2} implies that $\{ \exp(-zA) :\: z \in \Sigma_\delta \}$ is $\gamma$-bounded for any $\delta < \frac{\pi}{2}$ and consequently, by \cite[Theorem 2.20, $(iii) \Longrightarrow (i)$]{KuWe},
$\{ \lambda (\lambda - A)^{-1} :\: - \lambda \in \Sigma_\theta \}$ is $\gamma$-bounded for any $\theta \in (\frac{\pi}{2},\pi).$
Then with $f(\lambda) = \lambda^{\frac12} e^{-\lambda},$ the Cauchy integral formula \eqref{Equ Cauchy integral formula} gives
\begin{align*}
(2^k t A)^{\frac12} \exp(-2^k t A) & = \frac{1}{2\pi i} \int_{\partial \Sigma_{\pi - \theta}} f(\lambda) (\lambda - 2^ktA)^{-1} d\lambda \\
& = \frac{1}{2\pi i} \int_{\partial \Sigma_{\pi - \theta}} \frac{f(\lambda)}{\lambda} \times \frac{\lambda}{2^k t} (\frac{\lambda}{2^k t} - A)^{-1} d\lambda
\end{align*}
The first factor in the last integral belongs to $L^1(\partial \Sigma_{\pi - \theta},|d\lambda|)$
and the second factor is $\gamma$-bounded by the above for any $\theta < \pi.$
Thus by the well-known integral lemma for $\gamma$-bounds \cite[Corollary 2.14]{KuWe},
the second factor in \eqref{Equ 3 Proof Lem 2} is finite.

The estimate for the third factor in \eqref{Equ 3 Proof Lem 2} follows from the assumption \eqref{Equ Proof Lem 2}, so that we have shown \eqref{Equ 2 Proof Lem 2}.

Now we will write the expression in \eqref{Equ Res} as an integral of the expression in \eqref{Equ 2 Proof Lem 2}.
Let $\theta \in (0,\frac{\pi}{2}),\,\lambda = t e^{i \theta}$ and set $\phi  = \frac{\pi}{2} - \frac{\theta}{2},$ so that $\Re ( e^{i \phi} \lambda ) < 0.$
Then
\begin{align*}
\lambda^{\frac12} (2^k A)^{\frac12} ( \lambda  - 2^k A)^{-1} & = \lambda^{\frac12} (2^k A)^{\frac12} e^{i \phi} (e^{i \phi} \lambda - e^{i \phi} 2^k A)^{-1} \\
& = \int_0^\infty  -e^{i \phi} s^{-\frac12} \lambda^{\frac12} \exp( e^{i \phi} \lambda s) \times (2^k s A)^{\frac12} \exp(-2^k e^{i \phi} s A) ds.
\end{align*}
The second factor of the integral is $\gamma$-bounded by \eqref{Equ 2 Proof Lem 2}
and the first factor is integrable, as the following lines show.
\begin{align*}
\int_0^\infty s^{-\frac12} |\lambda^{\frac12} \exp(e^{i\phi} \lambda s)| ds & = \int_0^\infty s^{-\frac12} | \exp(e^{i \phi} e^{i\theta} s)| ds \\
& = \int_0^\infty s^{-\frac12} \exp( \cos(\frac{\pi}{2} + \frac{\theta}{2}) s) ds \\
& = \int_0^\infty s^{-\frac12} \exp(-s) ds \,|\cos(\frac{\pi}{2} + \frac{\theta}{2})|^{-\frac12} \\
& \lesssim \theta^{-\frac12}
\end{align*}
Then $\tau = \left\{ \lambda^{\frac12}(2^k A)^{\frac12} (\lambda - 2^k A)^{-1} :\: k \in \mathbb{Z} \right\}$ is $\gamma$-bounded since by \cite[Proposition 2.6 (5)]{Kr}, we have
\begin{align*}
\gamma(\tau) & \leq \int_0^\infty s^{-\frac12} | \lambda^{\frac12} \exp(e^{i\phi} \lambda s) | ds \times
\sup_{ t  > 0} \gamma\left( \left\{ (2^k t A)^{\frac12} \exp(-2^k t e^{\pm i (\frac{\pi}{2} - \omega)} A) :\: k \in \mathbb{Z} \right\} \right) \\
& \lesssim |\arg \lambda |^{- \frac12} \times |\arg \lambda |^{-\beta - \frac12}
\end{align*}
The same reasoning applies for $\lambda = t e^{i\theta}$ and $\theta \in (-\frac{\pi}{2},0).$
\end{proof}

\begin{proof}[Proof of Lemma \ref{Lem 3}]
By \cite[Proposition 4.18]{Kr} and \cite[p. 73]{CDMY}, it suffices to show that for some $\delta \in (\gamma,n),$
\begin{equation}\label{Equ 1 Lem Intermediate bad Mihlin calculus}
\|f(A)\| \lesssim \theta^{-\delta} \|f\|_{\infty,\theta}\text{ for any }f \in \bigcup_{\theta > 0} H^\infty_0(\Sigma_\theta).
\end{equation}
To show \eqref{Equ 1 Lem Intermediate bad Mihlin calculus}, we use the Kalton-Weis characterization of the bounded $H^\infty(\Sigma_\theta)$ calculus in terms of $\gamma$-bounded operator families (\cite{KaW}, see also \cite[Theorem 12.7]{KuWe}).
More precisely, we follow that characterization in the form of the proof of \cite[Theorem 12.7]{KuWe} and keep track of the dependence of appearing constants on the angle $\theta$.
It is shown there that for $f \in H^\infty_0(\Sigma_{2\theta}),\,x \in X$ and $x' \in X',$
\begin{align}
|\spr{f(A)x}{x'}| & = |\frac{1}{2\pi i} \int_{\partial\Sigma_\theta} \spr{\lambda^{-\frac12}f(\lambda) A^{\frac12} (\lambda- A)^{-1}x}{x'} d\lambda|
\nonumber \\
& \leq \frac{1}{2\pi} \sum_{j = \pm 1} \int_0^\infty |\spr{f(te^{ij\theta}) (tA)^{\frac12}(e^{ij\theta}- tA)^{-1} x}{x'}|\frac{dt}{t}
\nonumber \\
& = (*).
\nonumber
\end{align}
We put
\[\phi_{j \theta}(\lambda) = \frac{\lambda^{\frac14}(1+\lambda)^{\frac12}}{e^{ij\theta}-\lambda}
\text{ and }\psi(\lambda)= \left(\frac{\lambda}{(1+\lambda)^2}\right)^{\frac18},
\]
so that $(tA)^{\frac12} (e^{ij\theta}-tA)^{-1} = \phi_{j\theta}(tA)\psi(tA)\psi(tA).$
By \cite[Lemma 12.6]{KuWe}, the integral $(*)$ can be controlled by $\Gauss$-norms.
More precisely, we have
\begin{align}
(*) & \lesssim \sup_{j=\pm 1} \sup_{t > 0} \sup_N  \| \sum_{k=-N}^N \gamma_k \otimes f(2^k t e^{ij \theta}) \phi_{j \theta}(2^k t A) \psi(2^k t A) x\|_{\Gauss(X)}
\label{Equ 2 Lem Intermediate bad Mihlin calculus} \\
& \cdot \|\sum_{k=-N}^N \gamma_k \otimes \psi(2^k t A)'x'\|_{\Gauss(X')}
\nonumber \\
& \lesssim \|f\|_{\infty,\theta}  \sup_{j,t} \gamma\left(\{ \phi_{j \theta}(2^k t A) :\: k \in \mathbb{Z} \}\right) \sup_{N,t} \| \sum_{k=-N}^N \gamma_k \otimes \psi(2^k t A)x\|_{\Gauss(X)}
\nonumber \\
& \cdot \sup_{N,t}\| \sum_{k = -N}^N \gamma_k \otimes \psi(2^k t A)'x'\|_{\Gauss(X')}.
\nonumber
\end{align}
By \cite[Theorem 12.2]{KuWe}, the fact that $A$ has a bounded $H^\infty$ calculus implies that
$\sup_{N,t} \| \sum_{k=-N}^N \gamma_k \otimes \psi(2^k t A) x\|_{\Gauss(X)} \lesssim \|x\|$ and
$\sup_{N,t} \| \sum_{k=-N}^N \gamma_k \otimes \psi(2^k t A)' x'\|_{\Gauss(X')} \lesssim \|x'\|.$
Note that there is no dependence on $\theta$ in these two inequalities.
It remains to show that
\begin{equation}\label{Equ R-bound of phi_theta}
 \sup_{j = \pm 1,t> 0}\gamma(\{ \phi_{j \theta}(2^k t A) :\: k \in \mathbb{Z} \}) \lesssim \theta^{-\delta}.
\end{equation}
We have
\begin{align}
 \phi_{j \theta}(2^ktA) & = \frac{1}{2\pi i} \int_{\partial\Sigma_{\frac{\theta}{2}}} \phi_{j\theta}(\lambda) \lambda^{\frac12} (2^ktA)^{\frac12} (\lambda-2^ktA)^{-1} \frac{d\lambda}{\lambda}
\nonumber \\
& = \frac{1}{2\pi i} \int_{\partial\Sigma_{\frac{\theta}{2}}} \phi_{j\theta}(t\lambda) \lambda^{\frac12} (2^kA)^{\frac12} (\lambda-2^kA)^{-1} \frac{d\lambda}{\lambda}.
\nonumber
\end{align}
By \cite[Proposition 2.6 (5)]{Kr},
\begin{align*}
\sup_{j= \pm 1,t>0}\gamma(\{\phi_{j\theta}(2^ktA):\:k \in \mathbb{Z}\}) & \lesssim \sup_{j = \pm 1,t > 0} \|\phi_{j \theta}(t\lambda)\|_{L^1(\partial\Sigma_{\frac{\theta}{2}},|\frac{d\lambda}{\lambda}|)} \\
& \times
\sup_{\lambda \in \partial\Sigma_{\theta/2} \backslash \{ 0 \} }\gamma\left(\{ \lambda^{\frac12} (2^kA)^{\frac12} (\lambda - 2^kA)^{-1} :\:  k \in \mathbb{Z} \}\right).
\end{align*}
By assumption, it suffices to show that for any $\epsilon > 0$
\[\sup_{t > 0} \|\phi_{j \theta}(t\lambda)\|_{L^1(\partial\Sigma_{\frac{\theta}{2}},|\frac{d\lambda}{\lambda}|)} \leq C_\epsilon \theta^{-\epsilon}.\]
\begin{align}
\int_{\partial\Sigma_{\frac{\theta}{2}}} |\phi_{j \theta}(t\lambda)| \left|\frac{d\lambda}{\lambda}\right|
& = \int_{\partial\Sigma_{\frac{\theta}{2}}} | \phi_{j \theta}(\lambda)| \left| \frac{d\lambda}{\lambda}\right|
= \sum_{l = \pm 1}\int_0^\infty \left|\frac{s^\frac14 (1 + e^{i l \frac{\theta}{2}} s)^{\frac12}}{e^{i j\theta} - e^{il\frac{\theta}{2}} s}\right| \frac{ds}{s}.
\nonumber
\end{align}
The denominator is estimated from below by
\begin{align}
|e^{ij \theta} - e^{il \frac{\theta}{2}} s | & = |e^{i\theta (j - \frac{l}{2})} - s|
 \gtrsim |\cos(\theta (j- \frac{l}{2})) - s| + |\sin(\theta (j-\frac{l}{2}))|
\nonumber \\
& \gtrsim |1-s| - |\cos(\theta (j-\frac{l}{2}))-1| + \theta
\nonumber \\
& \gtrsim |1-s| - \theta^2 + \theta
\gtrsim |1-s| + \theta
\nonumber
\end{align}
for the crucial case of small $\theta.$
Thus
\[ \int_{\partial\Sigma_{\frac{\theta}{2}}} |\phi_{j \theta}(\lambda)| \left|\frac{d\lambda}{\lambda}\right|
 \lesssim \int_0^\infty \frac{s^{\frac14} (1+s)^{\frac12}}{\theta + |1-s|} \, \frac{ds}{s}.
\]
We split the integral into the parts $\int_0^\infty = \int_0^{\frac12} + \int_{\frac12}^{1-\theta} + \int_{1-\theta}^{1+\theta} + \int_{1+\theta}^2 + \int_2^\infty.$
\[ \int_0^{\frac12} \frac{s^{\frac14}(1+s)^{\frac12}}{\theta + |1-s|} \, \frac{ds}{s} \leq \int_0^{\frac12} \frac{s^{\frac14}(1+s)^{\frac12}}{|1-s|} \, \frac{ds}{s} < \infty\]
is independent of $\theta.$
The same estimate applies to $\int_2^\infty.$
\[ \int_{\frac12}^{1-\theta} \frac{s^{\frac14}(1+s)^{\frac12}}{\theta + |1 - s|} \, \frac{ds}{s} \lesssim \int_{\frac12}^{1-\theta} \frac{1}{\theta + |1 -s|} ds \leq \int_{\frac12}^{1-\theta}  \frac{1}{1-s} ds \lesssim | \log \theta |.\]
Similarly,
\[ \int_{1 + \theta}^2 \frac{s^{\frac14}(1+s)^{\frac12}}{\theta + |1 - s|} \, \frac{ds}{s} \lesssim \int_{1 + \theta}^2 \frac{1}{s-1} ds \lesssim |\log \theta|.\]
Finally,
\[ \int_{1-\theta}^{1 + \theta} \frac{s^{\frac14}(1+s)^{\frac12}}{\theta + |1 - s|} \, \frac{ds}{s} \lesssim \int_{1-\theta}^{1 + \theta} \frac{1}{\theta} ds \lesssim 1.\]
Since $1 + |\log \theta| \leq C_\epsilon \theta^{-\epsilon},$ the lemma is shown.
\end{proof}

\begin{proof}[Proof of Lemma \ref{Lem 4}]
Denote $N = \sup_{x > 0} \# \{ k \in \mathbb{Z}: \: \supp g_k \cap [\frac12 x, 2x] \neq \emptyset \} < \infty.$
Fix $x > 0$ and $j \in \{ 0, 1 ,\ldots, n\}.$
Then almost all $g_k$ vanish in a neighborhood of $x,$ and thus
\[ \left|x^j \frac{d^j}{dx^j} \left(\sum_{k \in \mathbb{Z}} g_k\right)(x)\right| = \left|\sum_{k \in \mathbb{Z}} x^j \frac{d^j}{dx^j}g_k(x)\right| \leq N \sup_{k \in \mathbb{Z}} |x^j \frac{d^j}{dx^j}g_k(x)| \leq N \sup_{k \in \mathbb{Z}} \|g_k\|_{\M^n} .\]
Taking the supremum over $x$ and $j$ gives $\|\sum_{k \in \mathbb{Z}} g_k\|_{\M^n} \leq N \sup_{k \in \mathbb{Z}} \|g_k\|_{\M^n}.$
\end{proof}

\section*{Acknowledgment}
I would like to thank Lutz Weis for several discussions on the subject of this article.
Further, I acknowledge financial support from the Karlsruhe House of Young Scientists KHYS and the Franco-German University DFH-UFA.

redpajama_set_name: RedPajamaArXiv


New York: Carl Fischer, 1915. Plate 18492-11.
Original images: ca.300dpi, grayscale jp2 files approx. 2710 by 3610 pixels. Editing: re-sampled to 600dpi, converted to black and white tif files, de-skewed, and set uniform margins.
New York: Carl Fischer, 1915. Plate 18493-12.


Even though every year the Lutz Independence Day Run is ridiculously hot, I always love this patriotic race. Apparently, my love for this small town 5K and wearing tutus to races has rubbed off on Allana.
I told Allana that I really wanted to PR this race and she was fine to run it with her friends: Matthew, Jared and Dylan.
We had tons of friends at the race, so no one would ever be alone. Lots of Suncoast Striders were there to support us too.
After all the usual pre-race photos and fun with friends, it was time to get serious. From the start, I was running about 7:30. I tried my best to maintain that pace until the first water station. My legs were burning, nagging and crying but I hung onto that pace like I was following behind the Boston Strong Suncoast Striders.
In past years, the residents in the community turn on sprinklers for the runners to cool us down under the hot 4th of July sun, however, this year lacked the usual spectator sprinklers. As much as I missed the sprinklers, the weather didn't feel as hot as previous years.
Thankfully, the course was as flat as I remembered it. Funny enough, a woman remembered me from last year's race, but I think it may have been the tutu she remembered. She reminisced about our chance meeting last year as she passed me.
Around the mile two mark, Larisa and Marie scooted by me. As they ran ahead, I spotted our other marathon mama, Daffodil, around the bend approaching in the opposite direction with her neighbor and her son, Sebastian. Shortly after Daffodil and Sebastian, I passed Mica, Faith, Allana, Matthew and Jared.
With only a mile left to go, I tried my best to keep my pace slightly under 9:00. Once I hit that last half mile, I pushed out an 800m sprint that my legs have learned to love.
I mentioned before about my Lutz Independence Day 5K a-ha moment. It was my running epiphany, so to speak. In the Lutz Independence Day Run, I learned to embrace the burning in my legs. I ignored my tired legs and just pushed myself to get that PR. I definitely earned a PR and surprisingly, I didn't feel like I would vomit. So, maybe I can push a little harder next time.
Allana did great too! She finished around her usual 5K time.
For me, the beauty of this particular PR is that I beat my time from the same race four years ago. And I did it in a tutu and a tiara while taking photos.
This entry was posted in fitness, fun, Life, running, sports and tagged 5K, Florida, friends, humor, Life, miscellaneous, musings, photographs, race, recap, Run DMT, running, thoughts.
I love Allana's finishing photo! That's how Alice was built before cross country. but she never finishes with a smile like that!
PR in a tutu?! NICE!