text:
\section{{\bf Introduction}}

A fundamental problem in complex analysis is the following: given bounded domains
$\Omega_1, \Omega_2\subset{\mathbb C}^n$ with smooth boundaries and a biholomorphic map $F:\Omega_1\longrightarrow
\Omega_2$, determine conditions on $\Omega_1, \Omega_2$ which guarantee that $F$ and $F^{-1}$
extend smoothly to the closures of the domains, $\overline{\Omega}_1, \overline{\Omega}_2$. When $n=1$, such maps always extend, without further conditions on $\Omega_1$ and $\Omega_2$. In several variables, however, it is unknown whether such maps universally extend or whether there are obstructions to extension. Furthermore, the extension problem takes on additional significance in several variables because the Riemann mapping theorem does not hold:
when $n>1$, the moduli space of biholomorphism classes of domains is infinite dimensional, even for the subclass of simply connected domains. Positive results about smooth extension to the boundary allows examination of a given equivalence class of domains by studying differential invariants on the boundary of the domains.

A groundbreaking result on this problem was obtained by Fefferman, \cite{fefferman74}, who showed that $F, F^{-1}$ extend smoothly to $\overline{\Omega}_1, \overline{\Omega}_2$ if both domains are strongly pseudoconvex. Fefferman's remarkable proof involved delicate estimates of the Bergman kernel, obtained by analyzing multiple error terms arising from locally approximating the boundaries of the domains, $b\Omega_1, b\Omega_2$ by euclidean balls, and used strong pseudoconvexity in several essential ways.

A subsequent, highly successful approach to this problem was initiated by Bell and Ligocka, \cite{BellLigocka}, and further developed by Bell, \cite{Bell81a}. The Bell-Ligocka program focused on a regularity property of the Bergman projection, rather than strong pseudoconvexity, and eventually led to showing the extension property holds on broad classes of weakly pseudoconvex domains. Let $B=B_\Omega$ denote the Bergman projection on $\Omega$, the orthogonal projection of $L^2(\Omega)$ onto its subspace of holomorphic functions. Say that $\Omega$ satisfies Condition R
if $B:C^\infty(\overline\Omega)\longrightarrow C^\infty(\overline\Omega)$. The main result in \cite{Bell81a} is the following: if $\Omega_1$ satisfies Condition R and $\Omega_2$ is pseudoconvex (both domains having smooth boundary), then $F$ and $F^{-1}$ extend smoothly to $\overline\Omega_1$ and $\overline\Omega_2$, respectively.

The question then arises: which smoothly bounded domains satisfy Condition R? There are many  hypotheses on $\Omega$ known to imply this condition, see \cite{Barrett86}, \cite{BellBoas81}, \cite{BoaStr91}, \cite{Catlin84b}, \cite{Catlin87}. For pseudoconvex domains, these results infer Condition R from a global regularity property of the $\bar\partial$-Neumann operator $N$, specifically that (*)
$N:\Lambda^{0,1}(\overline\Omega)\longrightarrow\Lambda^{0,1}(\overline\Omega)$, where
$\Lambda^{0,1}(\overline\Omega)$ denotes the $(0,1)$ forms with components in $C^\infty(\overline\Omega)$. The $\bar\partial$-Neumann operator is basic operator in complex analysis (see \cite{FolKoh72}) that inverts the $\bar\partial$-Laplacian with natural boundary conditions. The operator $\bar\partial^* N$ gives the special solution to the Cauchy-Riemann equations that is orthogonal to holomorphic functions on $\Omega$ --- this property establishes a relationship between $N$ and $B$. It was an open question, for many years, whether $N$ always satisfied (*) on a smoothly bounded, pseudoconvex domain. However, Christ  gave a negative answer to this question in  \cite{Christ96}, showing that $N$ does not satisfy property (*) on some pseudoconvex domains . Christ's theorem thus limits the applicability of Bell's theorem, though we emphasis that  \cite{Christ96} does not give a counterexample to the smooth extension of biholomorphic mappings.
The earlier works of Barrett, \cite{Barrett92}, and Kiselman \cite{Kiselman} were important precursors to the results in \cite{Christ96}.

The Bergman projection bears on the extension problem through its transformation formula:
\begin{equation}\label{E:bergtransform}
B_1\big(JF\cdot g\circ F\big)=JF\cdot\big[B_2(g)\big]\circ F,\qquad g\in L^2\left(\Omega_2\right)
\end{equation}
where $B_j$ denotes the Bergman projection on $\Omega_j$, $j=1,2$ and $JF$ is the determinant of the holomorphic Jacobian of $F$.
The crucial element in the Bell-Ligocka approach is the fact that $B_2$ has a large null space that is connected to a space of functions reproduced by $B_2$.
Consider an equivalence relation on functions $f,g\in L^2\left(\Omega_2\right)$ defined $f\sim g$ if $B_2f=B_2g$. Let $[f]$ denote the equivalence class of $f$. Then each class $[f]$ contains representatives that vanish to high order on $b\Omega_2$: if $g\in C^\infty\left(\overline{\Omega}_2\right)$ and $M\in{\mathbb Z}^+$ , there exists $g_M\in [g]$ such that $g_M=0$ on $b\Omega_2$ to order $M$.\footnote{This fact, which we refer to as Bell's Lemma, allows functions $g\in A^2\left(\Omega_2\right)$ on the right-hand side of \eqref{E:bergtransform} to be replaced by functions $g_M$ on the left-hand side of  \eqref{E:bergtransform}. But $JF\cdot g_M\circ F$ is essentially smooth up to $b\Omega_1$ if $M$ is large, by Cauchy's estimates. This lets one use \eqref{E:bergtransform} to gain control of $F$'s boundary behavior through Condition R.} The proof of Bell's Lemma hinges on two facts:

\begin{itemize}
\item[(a)] $B_2$ reproduces every function in $A^\infty\left(\Omega_2\right)={\mathcal O}\left(\Omega_2\right)\cap C^\infty\left(\overline{\Omega}_2\right)$ (as $A^\infty\left(\Omega_2\right)\subset A^2\left(\Omega_2\right)$ since $\Omega_2$ is bounded).
\item[(b)] $B_2$ annihilates a purely anti-holomorphic derivative of every function in $C^\infty\left(\overline{\Omega}_2\right)$ that also vanishes on $b\Omega_2$.
\end{itemize}
It follows that if $f\in C^\infty\left(\overline{\Omega}_2\right)$ is given, functions $\eta_\ell$ of the form $\frac{\partial}{\partial z_k}(r_2\cdot\sigma_\ell)$ with $\sigma_\ell\in C^\infty\left(\overline{\Omega}_2\right)$ can be chosen which have the same Taylor coefficients as $f$, up to order $\ell$, in the variable $r_2$ near $b\Omega_2$. Because of (b), it follows that $B_2(f-\eta_\ell)=B_2(f)$.

There are other solution operators to the Cauchy-Riemann equations besides $\bar\partial^* N$ and some of them are know to have good global regularity properties. Also, some of these operators are connected to projection operators.
Kohn \cite{Kohn73} produced such a solution operator on any smoothly bounded, pseudoconvex domain $\Omega$.
For $t >0$, let
$L^2_t(\Omega)$ be the Hilbert space of functions on $\Omega$ with inner product
$$\langle f,g\rangle_t=\int_\Omega f(z)\overline{g(z)} e^{-t|z|^2}\, dV(z).$$
Kohn showed that for any $s\in{\mathbb Z}^+$, there exists $t_0$ such that the weighted $\bar\partial$-Neumann operator $N_t$ maps $W^s(\Omega)\longrightarrow W^s(\Omega)$ boundedly, if $t >t_0$, where $W^s(\Omega)$ denotes the ordinary Sobolev norm of order $s$.
If $B_{t|z|^2}$ denotes the orthogonal projection of $L^2_t(\Omega)$ onto ${\mathcal O} (\Omega)$ in this inner product, then it follows that
$$B_{t|z|^2}:W^s(\Omega)\longrightarrow W^s(\Omega),\qquad\text{if } t>t_0, $$
boundedly. However, the weighted Bergman projection $B_{t|z|^2}$ does not exhibit the correct connection between the functions it reproduces and the functions it annihilates. This disconnection prevents an ``adjustment of Taylor jets" result of the type given by Bell's Lemma and thus the essential line of the Bell-Ligocka program is blocked.

To see this more explicitly, let $G=F^{-1}$. The transformation formula for the weighted Bergman projection is
$$ B_{t|z|^2}\left( JF\cdot\phi\circ F\right)=JF\cdot\left(  B_{t|G|^2}[\phi]\right)\circ F,\qquad \phi\in L^2\left(\Omega_2, e^{-t|G|^2}\right).$$
The Bergman projection $B_{t|G|^2}= B_2$ reproduces $f\in{\mathcal O} (\Omega_2)$. In order to find an operator $f\to Lf$ which satisfies
\begin{itemize}
\item[(i)] $B_2\left(Lf\right)=f$, and
\item[(ii)] $Lf$ vanishes to high order on $b\Omega_2$,
\end{itemize}
one is forced to consider
\begin{equation}\label{E:intro1}
e^{t|G|^2} T^p\left(e^{-t|G|^2}\cdot r^q\cdot\text{smooth}\right)
\end{equation}
where $p,q\in{\mathbb Z}^+$, $r$ defines $\Omega_2$, $T$ is an anti-holomorphic derivative, and smooth denotes a function in $C^\infty\left(\overline\Omega_2\right)$.
Obviously, derivatives land on $G$ in \eqref{E:intro1}---and these are the very quantities one wants to control.
The result is a vicious circle, with no boundary estimates on $F$ following from the known regularity of $B_{t|z|^2}$.

Observations of this kind seem to suggest that only estimates on the unweighted Bergman projection can be significantly connected to biholomorphic mappings. The situation changes, however, when the range of the projections are not restricted to holomorphic functions.
The purpose of this paper is to show that smoothness-to-the-boundary of $F$ can be obtained from regularity of a family of weighted, {\it non-holomorphic} projections. These projections are defined using two perturbation terms: one of them, $\tau$, shifts the space ${\mathcal O} (\Omega)$, and the other, $w$, weights the $L^2$ norms in the same manner as Kohn's weight mentioned above. The main result is

\begin{theorem} \label{T:main} Let $\Omega_1$ and $\Omega_2$ be smoothly bounded, pseudoconvex domains in ${\mathbb C}^n$ and suppose that $\Omega_1, \Omega_2$ satisfies Condition ${{\mathfrak R}}$. Let
 $F:\Omega_1\longrightarrow\Omega_2$ be a biholomorphic map.

Then the components of $F$ and $F^{-1}$ extend smoothly to $\overline\Omega_1$ and
 $\overline\Omega_2$, respectively.
 \end{theorem}

 Condition ${\mathfrak R}$ contains two separate features, compatibility and regularity, about a family of twist-weight factors $(\tau, w)$ on the domains $\Omega_j$.
 It is, first of all, essential that the pairs $(\tau, w)$ be {\it Bell compatible} (see Definition \ref{D:bellcompat2}). It is also necessary that the associated family
 of twisted-weighted Bergman projections
 ${\mathcal B}^{\tau,w}_\Omega$, defined in Section \ref{S:twist}, satisfy the regularity condition given in Definition \ref{D:frakR}.

We postpone addressing the question of which domains satisfy Condition ${\mathfrak R}$ here and simply prove Theorem \ref{T:main},  in order to expose the twisting-weighting idea clearly. This simple idea seems to open new avenues for studying other questions in complex analysis. We hope this justifies the inclusion of some routine proofs below, e.g., Propositions \eqref{P:berg_kern}--\eqref{P:transformation}.

\medskip

Without the encouragement of several colleagues, this paper might never have appeared in manuscript form. I am especially grateful to A.-K. Herbig for
her enthusiasm about these results and gentle prodding for the past eight years to write them down. I also thank D. Varolin for insisting the results were
worthwhile and his guarantee they would not go unread. Y. Zeytuncu made a valuable observation about Section \ref{S:compat} that I happily acknowledge. And I want to thank C. L. Fefferman, J. J. Kohn, E. M. Stein, and E. J. Straube for listening to many hours of lectures about this material and for freely offering their insight on the mathematics in and behind these results.
\medskip

This paper was greatly inspired by the work in \cite{Bell81a} and \cite{BellLigocka}. Much of the proof of Theorem \ref{T:main} amounts to modest modifications of Bell's ideas.

\section{{\bf Twisted Bergman projections}}\label{S:twist}

Let $\Omega\subset{\mathbb C}^n$ be a bounded domain with $C^\infty$ smooth boundary, shortened to a \textit{smoothly bounded domain} below, and $r$ a smooth defining function:
$\Omega=\left\{ z\in{\mathbb C}^n: r(z) < 0\right\}$ and $dr\neq 0\text{ on } b\Omega=\left\{ z: r(z)=0\right\}$.
Let ${\mathcal O}(\Omega)$ denote the set of holomorphic functions on $\Omega$ and $C^k(\Omega)$ the $k$-times continuously differentiable functions on $\Omega$. We consider $L^2$ projections onto cosets of the form
$C^1(\Omega)\cdot{\mathcal O}(\Omega)$ with respect to weighted $L^2$ inner products.

To begin, if $w:\Omega\longrightarrow{\mathbb R}$ is a function such that $e^{-w}\in L^1_{\text{loc}}(\Omega)$, define the weighted $L^2$ function space
\begin{equation}\label{E:weightedL2}
L^2\left(\Omega, e^{-w}\right)=\left\{ f \text{ measurable on } \Omega: \int_\Omega |f|^2\, e^{-w}\, dV <\infty\right\},
\end{equation}
where $dV$ stands for the euclidean volume element. Call $w$ a weight factor.
We denote the dependence of the inner product
and norm on $w$ by a subscript: if $f, g\in L^2\left(\Omega, e^{-w}\right)$
\begin{equation}\label{E:weightedinner}
(f,g)_w=\int_\Omega f\cdot \bar g\, e^{-w}\, dV, \qquad\text{and}\qquad ||f||^2_w=(f,f)_w.
\end{equation}

Next, if $\tau:\Omega\longrightarrow{\mathbb R}^+$ is a positive function, belonging to $C^1(\Omega)$, define the space
\begin{equation}\label{E:twistedholo}
{\mathcal O}^\tau(\Omega)=\left\{ f\in C^1(\Omega):\frac{\partial}{\partial \bar z_k}\left(\sqrt\tau\cdot f\right)=0,\quad k=1,\dots, n\right\},
\end{equation}
where $\frac{\partial}{\partial\bar z_k}=\frac 12\left(\frac{\partial}{\partial x_k} +i\frac{\partial}{\partial y_k}\right)$
are the Cauchy-Riemann operators with respect to the standard coordinates $(z_1,\dots, z_n)$,
$z_k=x_k+iy_k$, on ${\mathbb C}^n$. Call $\tau$ a twist factor and write $\bar\partial\left(\sqrt\tau\cdot f\right)=0$ to express the vanishing of the $n$ equations in \eqref{E:twistedholo}. Clearly, ${\mathcal O}^\tau(\Omega)=\frac 1{\sqrt\tau}\cdot {\mathcal O}(\Omega)$ as sets; these are the sets we will project $L^2\left(\Omega, e^{-w}\right)$ onto. The set ${\mathcal O}^\tau(\Omega)$ is called the set of $\tau$-{\it twisted holomorphic functions} on $\Omega$.

\subsection{The basic inequality}\label{SS:tau_ineq}
\bigskip

For a general twist-weight pair $(\tau, w)$, let $A^2_{\tau,w}(\Omega)$ denote the $\tau$-twisted holomorphic functions in $L^2\left(\Omega, e^{-w}\right)$. If $w$ satisfies a mild integrability condition near $b\Omega$, an inequality of Bergman type holds for functions in $A^2_{\tau,w}(\Omega)$. For $\delta >0$, let $S_\delta=\left\{ z\in \Omega: |r(z)|<\delta\right\}$.

\begin{proposition}\label{T:bergineq} Let $\Omega$ be a smoothly bounded domain and $(\tau, w)$ a twist-weight pair on $\Omega$.
Suppose that $w\in L^1_{\text loc}\left(S_\delta\right)$, for some $\delta >0$.

Then, for any compact $K\subset\Omega$, there exists a constant $C_K$ such that
$$\sup_{z\in K}|f(z)|\leq C_K\, ||f||_w,\qquad f\in A^2_{\tau,w}(\Omega).$$
\end{proposition}

\begin{remark} The constant $C_K$ also depends on the functions $\tau$ and $w$. Crucially, it is independent of
$f\in A^2_{\tau,w}(\Omega)$.
\end{remark}

\medskip

\begin{proof} If $ f\in{\mathcal O}^\tau(\Omega)$, then $f=\frac 1{\sqrt\tau}\cdot h$ for some $h \in{\mathcal O}(\Omega)$.

Choose a compact set $K_1$ such that $K\subset K_1$ and $bK_1\subset S_\delta$. Then

\begin{align*}
 \sup_{z\in K}|f(z)|&\leq \sup_{z\in K_1}\frac 1{\sqrt{\tau(z)}}\cdot\sup_{z\in K_1} |h(z)| \\
&\leq C(K_1, \tau)\cdot \sup_{z\in bK_1} |h(z)|,
\end{align*}
by the maximum principle for holomorphic functions.

Let $B(p,\eta)$ denote the euclidean ball centered at $p$ of radius $\eta$ and let $V(p,\eta)$ denote the volume of $B(p,\eta)$.
Choose $\rho >0$ such that
$B(z,\rho)\subset S_\delta$ for all $z\in bK_1$. Let $z\in bK_1$ be temporarily fixed. Since $\log |h|$ is subharmonic and $w$ is integrable on $B(z,\rho)$, we have

\begin{align*}
 2\log |h(z)|&\leq \frac 1{V(z,\rho)}\, \int_{B(z,\rho)} 2\log |h|\, dV \\
&=\int_{B(z,\rho)}\log\left(|h|^2\cdot e^{-w}\right)\frac {dV}{V(z,\rho)} +\frac 1{V(z,\rho)}\,\int_{B(z,\rho)} w\, dV.
\end{align*}
Exponentiating both sides and applying Jensen's inequality yields

\begin{align*}
 |h(z)|^2&\leq \exp\left[C(\rho, n)\int_{B(z,\rho)} w\, dV\right]
\cdot\int_{B(z,\rho)} |h|^2\, e^{-w}\,  \frac {dV}{V(z,\rho)} \\ &\leq C \left\{\sup_{x\in B(z,\rho)} \tau(x)\right\}\cdot
\int_{B(z,\rho)} |f|^2\,
e^{-w}\, dV \\ &\leq C' ||f||^2_w.
\end{align*}
Now cover $bK_1$ by balls $B(z_1, \rho), \dots B(z_m, \rho)$. It follows that
$$ \sup_{z\in bK_1} |h(z)|\leq C''\, ||f||_w,$$
which completes the proof.

\end{proof}

\begin{remark}  The hypothesis on $w$ in Proposition \ref{T:bergineq} can be weakened.
For example, if $K_1\subset K_2\subset\dots$ are compact sets which exhaust $\Omega$ and
$\left\{ U_j\right\}$, $j\in{\mathbb Z}^+$, are open subsets of $\Omega$ such that $bK_j\subset U_j$, then
we need only require that $w\in L^1_{\text loc}\left(U_j\right)$, for $j\geq J$, in order to conclude
that the inequality in Proposition \ref{T:bergineq} holds. This observation shows that
Proposition \ref{T:bergineq} holds for weights $w$ that are identically $= +\infty$ on ``rings'' accumulating to $b\Omega$ as long as there are complementary ``rings'' accumulating to the boundary where $w$ is locally integrable.
\end{remark}

\subsection{The kernel function}\label{SS:kernel}

\bigskip

From now on, consider twist-weight pairs $(\tau, w)$ with $w\in L^1_{\text loc}\left(S_\delta\right)$, for some $\delta >0$.
Proposition \ref{T:bergineq} implies that $A^2_{\tau, w}(\Omega)$ is a closed subset of $L^2\left(\Omega, e^{-w}\right)$. It also implies that for any fixed $a\in\Omega$, the evaluation functional
$$f\longrightarrow f(a),\qquad f\in A^2_{\tau, w}(\Omega)$$
is continuous in the $||\cdot||_w$ norm. The Riesz representation theorem gives, for each fixed $a\in \Omega$, a function $R_a\in A^2_{\tau, w}(\Omega)$ such that $f(a)=\left( f, R_a\right)_w$. Rewriting this, we obtain
\begin{equation}\label{E:repro}
f(a)=\int_\Omega{\mathcal B}^{\tau, w}_\Omega (a,b) f(b)\, e^{-w(b)}\, dV(b), \qquad  f\in A^2_{\tau, w}(\Omega),
\end{equation}
where ${\mathcal B}^{\tau, w}_\Omega (a,b)=:\overline{R_a(b)}$. This function is the $(\tau, w)$-{\it Bergman kernel} associated to $\Omega$. When the parameters $\tau, w$, and $\Omega$ are clear, we drop the super and subscripts on the kernel.

\begin{proposition}\label{P:berg_kern}The $(\tau, w)$-Bergman kernel associated to $\Omega$,
${\mathcal B}^{\tau, w}_\Omega (a,b)={\mathcal B}(a,b)$, satisfies

\begin{itemize}
\item[(i)] \eqref{E:repro} holds
\item[(ii)] $\overline{{\mathcal B}(a,\cdot)}\in A^2_{\tau, w}(\Omega)$ for each $a\in \Omega$,
\item[(iii)] $\overline{{\mathcal B}(a,b)}={\mathcal B}(b,a)$.
\end{itemize}
Moreover the properties (i)-(iii) uniquely determine ${\mathcal B}(a,b)$.
\end{proposition}

\begin{proof} Properties (i) and (ii) hold by definition. To see property (iii), apply \eqref{E:repro} to
$\overline{{\mathcal B}(a,\cdot)}\in H^2_{\tau, w}(\Omega)$:
\begin{align*}
 \overline{{\mathcal B}(a,b)}&=\int_\Omega {\mathcal B}(b,s)\overline{{\mathcal B}(a,s)} e^{-w(s)}\, dV(s) \\
&=\overline{\int_\Omega \overline{{\mathcal B}(b,s)} {\mathcal B}(a,s) e^{-w(s)}\, dV(s)} \\
&=\overline{\overline{{\mathcal B}(b,a)}} = {\mathcal B}(b,a).
\end{align*}

To verify uniqueness, suppose $K(a,b)$ is another kernel satisfying (i)-(iii). Then
\begin{align*}
{\mathcal B}(a,b)&=\overline{{\mathcal B}(b,a)}=\int_\Omega K(a,s)\overline{{\mathcal B}(b,s)} e^{-w(s)}\, dV(s)
\\ &=\overline{\int_\Omega K(s,a){\mathcal B}(b,s) e^{-w(s)}\, dV(s)} \\ &=\overline{K(b,a)} = K(a,b).
\end{align*}

\end{proof}

\subsection{Transformation formula}\label{SS:trans}
\bigskip

Let $\Omega_1, \Omega_2$ be smoothly bounded domains in ${\mathbb C}^n$ and suppose $F:\Omega_1\longrightarrow\Omega_2$ is a biholomorphic map. If $\tau$ and $w$ are functions defined on $\Omega_1$, let $\sigma =\tau\circ F^{-1}$ and $v=w\circ F^{-1}$ be the corresponding functions defined on $\Omega_2$. We want to express the relationship between the $(\tau,w)$-Bergman kernel
on $\Omega_1$ and the $(\sigma, v)$-Bergman kernel on $\Omega_2$.

Let $JF(s)=\det \left[F'(s)\right]$ denote the determinant of the holomorphic Jacobian matrix of $F$ and use the symbol $J_{\Bbb R} F(s)$ to denote the determinant of the real Jacobian matrix of $F$, i.e., where $F$ is viewed as a diffeomorphism from ${\Bbb R}^{2n}$ to ${\Bbb R}^{2n}$. Two elementary facts are used in the proof below:
\begin{equation}\label{E:jacob}
JF\left(F^{-1}(\beta)\right)=\frac 1{JF^{-1}(\beta)},\qquad J_{\Bbb R} F^{-1}(\beta) =
JF^{-1}(\beta)\cdot\overline{JF^{-1}(\beta)}.
\end{equation}
Also, write ${\mathcal B}^{*, *}_{\Omega_j} (\cdot, \cdot)={\mathcal B}^{*, *}_{j} (\cdot, \cdot)$, for $j=1,2$.

\begin{proposition}\label{P:kern_trans}
If $F:\Omega_1\longrightarrow\Omega_2$ is a biholomorphic mapping between smoothly bounded domains in ${\mathbb C}^n$, $(\tau,w)$ a twist-weight pair defined on $\Omega_1$, and $(\sigma,v)$ the corresponding twist-weight pair on $\Omega_2$, then
\begin{equation}\label{E:kern_trans}
{\mathcal B}^{\tau, w}_{1} (a,b)={\mathcal B}^{\sigma ,v}_{2} \left(F(a),F(b)\right)\cdot JF(a)\cdot\overline{JF(b)}
\end{equation}
for all $(a,b)\in \Omega_1\times\Omega_1$.
\end{proposition}

\begin{proof} Let $K(a,b)$ denote the function on the right hand side of \eqref{E:kern_trans}. We use Proposition \ref{P:berg_kern} to show that $K(a,b)={\mathcal B}^{\tau, w}_1(a,b)$. Property (iii) of Proposition \ref{P:berg_kern} holds for $K(a,b)$ because of the corresponding property for ${\mathcal B}^{\sigma, v}_2(x,y)$. For property (ii), note first that
$$0=\bar\partial\left(\sqrt{\sigma(\cdot)}\cdot{\mathcal B}^{\sigma, v}_2\left(\cdot, F(b)\right)\right) \implies
0=\bar\partial\left(\sqrt{\tau(a)}\cdot{\mathcal B}^{\sigma, v}_2\left(F(a), F(b)\right)\right).$$
Since $\phi\in{\mathcal O}^\tau(\Omega_1)$, $\psi\in{\mathcal O}(\Omega_1)$ implies that $\phi\cdot\psi\in
{\mathcal O}^\tau(\Omega_1)$, it follows that $K(\cdot, b)\in{\mathcal O}^\tau(\Omega_1)$.

For property (i), let $g\in A^2_{\tau, w}(\Omega_1)$. Applying the change of variables $b=F^{-1}(\beta)$, we have
\begin{align*}
 \int_{\Omega_1}K(a,b) g(b) e^{-w(b)}\, dV(b) = JF(a)\, \int_{\Omega_2}{\mathcal B}^{\sigma, v}_2\left( F(a), \beta\right) \, &\overline{JF\left(F^{-1}(\beta)\right)}\, g\left(F^{-1}(\beta)\right)  \\
&\cdot e^{-v(\beta)}\, J_{\Bbb R}F^{-1}(\beta)\, dV(\beta).\end{align*}
However, \eqref{E:jacob} shows that the
right-hand side is
\begin{equation}\label{E:cv}
JF(a)\, \int_{\Omega_2}{\mathcal B}^{\sigma, v}_2\left( F(a), \beta\right) \, \left[
 JF^{-1}(\beta)\,g\left(F^{-1}(\beta)\right) \right]
\cdot e^{-v(\beta)}\,  dV(\beta)=M.
\end{equation}
Since the quantity $[\dots]$ in \eqref{E:cv} is in $A^2_{\sigma, v}(\Omega_2)$ --- by the change of variables theorem --- it follows from the reproducing property of ${\mathcal B}^{\sigma, v}_2$ that
\begin{align*}
 M&= JF(a)\cdot JF^{-1}\left(F(a)\right)\cdot g(a) \\ &= g(a).\end{align*}
Here \eqref{E:jacob} has been used. The uniqueness statement in Proposition \ref{P:berg_kern} now completes the proof.
\end{proof}

The transformation formula \eqref{E:kern_trans} may also be written at the operator level. First, extend the operator in \eqref{E:repro} to all of $L^2\left(\Omega_1, e^{-w}\right)$. The $(\tau, w)$-Bergman projection is defined
$${\mathcal B}^{\tau, w}_{\Omega_1}[g](a)=\int_{\Omega_1}{\mathcal B}^{\tau, w}_{\Omega_1}(a,b)g(b) e^{-w(b)}
\, dV(b),\qquad g\in L^2\left(\Omega_1, e^{-w}\right).$$
It follows from Proposition \ref{P:berg_kern} that ${\mathcal B}^{\tau, w}_{\Omega_1}: L^2\left(\Omega_1, e^{-w}\right)\longrightarrow A^2_{\tau, w}(\Omega_1)$ is the orthogonal projection of $ L^2\left(\Omega_1, e^{-w}\right)$ onto $ A^2_{\tau, w}(\Omega_1)$.

\begin{proposition}\label{P:transformation} Let $F:\Omega_1\longrightarrow\Omega_2$ be biholomorphic, $(\tau, w)$ a twist-weight pair on $\Omega_1$, and
$(\sigma ,v)$ the corresponding pair on $\Omega_2$.

Then, denoting ${\mathcal B}^{\tau, w}_{\Omega_1}$ as ${\mathcal B}_1$ and ${\mathcal B}^{\sigma, v}_{\Omega_2}$ as ${\mathcal B}_2$,
\begin{equation}\label{E:transform}
{\mathcal B}_1\left( JF\cdot\phi\circ F\right)=JF\cdot\left( {\mathcal B}_2[\phi]\right)\circ F,\qquad \phi\in L^2\left(\Omega_2, e^{-v}\right).
\end{equation}
\end{proposition}

\begin{proof} Making the change of variables $\zeta= F^{-1}(\xi)$,
\begin{align*}
{\mathcal B}_1\left( JF\cdot\phi\circ F\right(z))=\int_{\Omega_1}{\mathcal B}_1(z,\zeta) JF(\zeta)\cdot &\phi\left(F(\zeta)\right) e^{-w(\zeta)}\, dV(\zeta) \\ = \int_{\Omega_2}
{\mathcal B}_1\left(z, F^{-1}(\xi)\right) J&F\left(F^{-1}(\xi)\right)\cdot \phi(\xi) e^{-v(\xi)} \\
&\cdot JF^{-1}(\xi)\overline{JF^{-1}(\xi)}\, dV(\xi).\end{align*}

By Proposition \ref{P:kern_trans}, this
\begin{align*}
= \int_{\Omega_2}{\mathcal B}_2\left(F(z), \xi\right)&\cdot JF(z)
\cdot\overline{JF\left(F^{-1}(\xi)\right)} \\
&\cdot JF\left(F^{-1}(\xi)\right)JF^{-1}(\xi)\overline{JF^{-1}(\xi)}\, \phi(\xi) e^{-v(\xi)}\, dV(\xi)\end{align*}

\begin{align*}
=JF(z)\, \int_{\Omega_2}{\mathcal B}_2&\left(F(z), \xi\right)\phi(\xi) e^{-v(\xi)} \\
&\cdot\left[\overline{JF\left(F^{-1}(\xi)\right)} JF\left(F^{-1}(\xi)\right) JF^{-1}(\xi)\overline{JF^{-1}(\xi)}\right]\, dV(\xi) \end{align*}

$$= JF(z)\cdot \left({\mathcal B}_2[\phi]\circ F\right) (z).$$
For the last equality, \eqref{E:jacob} is used to show that $[\dots]=1$. This completes the proof.
\end{proof}

\section{{\bf The role of pseudoconvexity}}\label{S:psc}

We shall use pseudoconvexity through the following result of Diederich and Forn\ae ss, \cite{DieFor77-2}: if $\Omega$ is a smoothly bounded, pseudoconvex domain in ${\mathbb C}^n$, there exists a smooth defining function $\rho$ for $\Omega$ and a positive exponent $\eta$, $1\geq\eta >0$, such that $-(-\rho)^\eta$ is plurisubharmonic on $\Omega$. It is known that there is no strictly positive lower bound on $\eta$, over the class of all smoothly bounded pseudoconvex domains, for which this theorem holds; see \cite{DieFor77-2} and \cite{DieFor77-1}.

This result of Diederich-Forn\ae ss implies that, if $r$ is a defining function for a smoothly bounded pseudoconvex $\Omega$, $\log (-r)$ is quasi-invariant under biholomorphic mappings of $\Omega$. This corollary of \cite{DieFor77-2} was obtained independently by Range and Forn\ae ss:

\begin{proposition}[\cite{Fornaess79} and \cite{Range78}]\label{P:distancecomparison}
Let $\Omega_1, \Omega_2$ be smoothly bounded, pseudoconvex domains in ${\mathbb C}^n$ and $F:\Omega_1\longrightarrow\Omega_2$ a biholomorphic map.

There exists an $d\in{\mathbb Z}^+$ such that if $r_1, r_2$ are defining functions for $\Omega_1, \Omega_2$ respectively, there are constants $C_1, C_2$ such that
\begin{equation}\label{E:distancecomparison}
C_1\cdot \left|r_1(z)\right|^d\leq \left| r_2\big(F(z)\big)\right| \leq C_2\cdot \left| r_1(z)\right|^{\frac 1d},\qquad\text{for all } z\in\Omega_1.
\end{equation}

The constants in \eqref{E:distancecomparison} depend on $F$ and the defining functions $r_1, r_2$, but are independent of $z\in\Omega_1$.
\end{proposition}

Proposition \ref{P:distancecomparison} says in particular that if $g\in C^\infty\left(\overline{\Omega_2}\right)$ vanishes on $b\Omega_2$, then $g\circ F$ must vanish on $b\Omega_1$ (though perhaps to lesser order). We will use this to estimate Sobolev norms on $\Omega_1$ by shifted Sobolev norms on $\Omega_2$. For $s\in{\mathbb Z}^+$, let
$$\| u\|^2_{(s)} =\sum_{|\alpha|\leq s}\int_\Omega \left|D^\alpha u\right|^2\, dV,\qquad u\in C^\infty(\Omega),$$
denote the $L^2$ Sobolev norm of order $s$. If $h\in{\mathcal O}(\Omega)$, this norm can be expressed using only anti-holomorphic derivatives:
\begin{equation}\label{E:holoSob}
\| h\|^2_{(s)} =\sum_{|\alpha|\leq s}\int_\Omega \left|
\frac{\partial^\alpha h}{\partial z^\alpha}\right|^2\, dV, \qquad h\in{\mathcal O}(\Omega).
\end{equation}
Let $W^s(\Omega)$ denote the closure of $C^\infty\left(\overline{\Omega}\right)$ in the norm $\|\cdot\|_{(s)}$ and $W^s_0(\Omega)$ denote the closure of $C^\infty_0(\Omega)$ in this norm.

A class of multipliers on the spaces $W^s_0(\Omega)$ arises naturally.

\begin{definition}\label{D:Gtm}
Let $\Omega\subset{\mathbb C}^n$ be a smoothly bounded domain. For $t\in{\mathbb R}$, $m\in{\mathbb Z}^+$, and $r$ a defining function for $\Omega$, define
$${\mathscr G}_m^t(\Omega)=\left\{\mu\in C^\infty(\Omega): r^{-t+|\alpha|}\, D^\alpha\mu\in L^\infty(\Omega),\text{ for }|\alpha|\leq m\right\}.$$
Also set
$${\mathscr G}_\infty^t(\Omega)=\left\{\mu\in C^\infty(\Omega): r^{-t+|\alpha|}\, D^\alpha\mu\in L^\infty(\Omega),\forall\,\alpha\right\}.$$
Call elements in ${\mathscr G}_\infty^t(\Omega)$ good $W_0^*(\Omega)$ multipliers of shift $t$.
\end{definition}

\begin{remark}
The spaces ${\mathscr G}_m^t(\Omega)$ do not depend on the choice of $r$ in Definition \ref{D:Gtm}.
\end{remark}

The notation $A\lesssim B$ will henceforth express the inequality $A\leq\kappa\cdot B$ for some constant $\kappa$. The constant $\kappa$ will be independent of certain parameters, made clear in context.

\begin{lemma}\label{L:mult} Let $\mu\in{\mathscr G}_m^t(\Omega)$.

\begin{itemize}
\item[(i)] If $t\in{\mathbb Z}^+$ and $m\geq t$, then $\mu\in W^t_0(\Omega)$.
\item[(ii)] If $s\in{\mathbb Z}^+$ and $s \leq m$,
the multiplication operator $g\longrightarrow\mu\cdot g$ maps $W_0^{s-t}(\Omega)$ to $W_0^s(\Omega)$ boundedly.
\end{itemize}
\end{lemma}

\begin{proof} The condition $\mu\in{\mathscr G}_m^t(\Omega)$ says
\begin{equation}\label{E:3.6a}
\left|D^\alpha \mu(z)\right|\lesssim |r(z)|^{t-|\alpha|},\quad |\alpha|\leq m.
\end{equation}
For (i), if $t\leq m$, \eqref{E:3.6a} implies
\begin{equation*}
\left|D^\alpha \mu(z)\right|\lesssim |r(z)|^{t-|\alpha|},\quad |\alpha|\leq t-1.
\end{equation*}
and $D^\alpha\mu\in L^\infty(\Omega)$ if $|\alpha|=t$. Thus, $\mu\in W^t(\Omega)$ and the trace of $D^\alpha\mu$ on $b\Omega$ vanishes for $|\alpha|\leq t-1$.
Theorem 11.5 in \cite{LionsMagenes} implies that $\mu\in W_0^t(\Omega)$.

For (ii), let $g\in C^\infty\left(\overline\Omega\right)$ satisfy
\begin{equation*}
\left|D^\alpha g(z)\right|\lesssim \left\{ \begin{array}{ll} \left|r(z)\right|^{s-t-|\alpha|} & \mbox{$|\alpha|\leq s-t$} \\
1 & \mbox{$|\alpha| >s-t$}
\end{array} \right.
\end{equation*}
with constant independent of $z$ in a fixed neighborhood of $b\Omega$, i.e., $g$ vanishes to order $s-t$ on $b\Omega$.
For a fixed multi-index $\beta$, with $|\beta|\leq s-1$, it follows that
\begin{align*}
\left|D^\beta\left[\mu\cdot g\right](z)\right|&=\left|\sum_{\gamma+\delta=\beta}\frac{\beta !}{\gamma !\,\delta !}D^\gamma\mu\cdot D^\delta g\right| \\
&\lesssim \sum_{\gamma+\delta=\beta \atop |\delta|\leq s-t} |r(z)|^{t-|\gamma|}\cdot |r(z)|^{s-t-|\delta|} + \sum_{\gamma+\delta=\beta \atop |\delta| > s-t} |r(z)|^{t-|\gamma|}\\
&=|r(z)|^{s-|\beta|}.
\end{align*}
Thus $\mu\cdot g$ vanishes to order $s$ on $b\Omega$. Theorem 11.5 in \cite{LionsMagenes} implies that $\mu\cdot g\in W_0^s(\Omega)$. Since
the set of $C^\infty\left(\overline\Omega\right)$ functions vanishing to order $s-t$ on $b\Omega$ is dense in $W^{s-t}_0(\Omega)$, the proof is complete.

\end{proof}

If $h$ is a bounded holomorphic function on $\Omega$, Cauchy's estimates imply
\begin{equation}\label{E:cauchy}
\left|D^\gamma h(z)\right|\lesssim |r(z)|^{-|\gamma|}
\end{equation}
where the constant depends only on $\Omega$, the differentiation order $\gamma$, and $\sup_\Omega |h|$. Thus $h\in{\mathscr G}_\infty^0(\Omega)$. The product rule shows that $g_1\in{\mathscr G}_\infty^{t_1}(\Omega),
g_2\in{\mathscr G}_\infty^{t_2}(\Omega)\implies g_1\cdot g_2\in{\mathscr G}_\infty^{t_1+t_2}(\Omega)$. In particular, each component of the biholomorphic map $F=\left(f^1,\dots ,f^n\right):
\Omega_1\longrightarrow\Omega_2$ is an element of ${\mathscr G}_\infty^0\left(\Omega_1\right)$ and $JF\in{\mathscr G}_\infty^{-n}\left(\Omega_1\right)$.

Because of the distortion exponent $d$ in Proposition \ref{P:distancecomparison}, pullbacks of functions in ${\mathscr G}_m^t\left(\Omega_2\right)$ can only be asserted to belong to shifted spaces ${\mathscr G}_{\tilde m}^{\tilde t}\left(\Omega_1\right)$.

\begin{lemma}\label{L:r^p}
Let $\Omega_1, \Omega_2\subset{\mathbb C}^n$ be smoothly bounded, pseudoconvex domains, $F:\Omega_1\longrightarrow\Omega_2$ a biholomorphic map, and $r_2$ a defining function for $\Omega_2$. Let $d=d(F,\Omega_1,\Omega_2)$ be associated to $F$ by Proposition \ref{P:distancecomparison}.

For any $k\in{\mathbb Z}^+$,
\begin{itemize}
\item[(i)] $\big( r_2\circ F\big)^{2k}\in W_0^{\frac{2k}{1+d}}(\Omega_1)$.
\item[(ii)] $\big( r_2\circ F\big)^{2k}\in {\mathscr G}_k^{k/d}\left(\Omega_1\right)$.
\end{itemize}
\end{lemma}

\begin{proof} If $\beta$ is a multi-index in ${\mathbb N}^n$, let $D^\beta F$ denote the derivative of order $\beta$ of an (unspecified) component of $F$.
The product and chain rules imply
\begin{equation*}
D^\alpha\left(r_2^{2k}\circ F\right)=\sum_{j=1}^{|\alpha|}\sum_{\sum |\beta_j|=|\alpha|} C_{k\alpha j\beta}\big(r_2\circ F\big)^{2k-j}\cdot D^{\beta_1}F\dots D^{\beta_j}F.
\end{equation*}
for some combinatorial constants $C_*$. It follows from \eqref{E:cauchy} and Proposition \ref{P:distancecomparison}
\begin{align}\label{E:r^p1}
\left|D^\alpha\left(r_2\circ F\right)^{2k}(z)\right|&\lesssim \left| r_2\circ F\right|^{2k-|\alpha|}\cdot \left| r_1(z)\right|^{-|\alpha|} \notag \\
&\lesssim \left| r_1(z)\right|^{\frac {2k}d -|\alpha|\left(1+\frac 1d\right)}\qquad\text{for } |\alpha|\leq 2k
\end{align}
and
\begin{equation*}
\lesssim |r_1(z)|^{-|\alpha|}\qquad\text{for }|\alpha| >2k.
\end{equation*}

If $|\alpha|\leq\frac{2k}{1+d}$, this implies $\left|D^\alpha\left(r_2\circ F\right)^{2k}(z)\right|\in L^\infty(\Omega_1)$ and that $r_2\circ F$ vanishes to order at least $\frac{2k}{1+d}$, which is (i).
If $|\alpha|\leq k$, \eqref{E:r^p1} implies $\left|D^\alpha\left(r_2\circ F\right)^{2k}(z)\right|\lesssim |r_1(z)|^{\frac kd -|\alpha|}$, which is the claimed result (ii).
\end{proof}

\begin{proposition}\label{P:bdedSob}
Suppose $\Omega_1, \Omega_2$ are smoothly bounded, pseudoconvex domains in ${\mathbb C}^n$ and $F:\Omega_1\longrightarrow\Omega_2$ is a biholomorphic map.
Let $\mu\in{\mathscr G}_\infty^k\left(\Omega_1\right)$.

For each $s\in{\mathbb Z}^+$, there exists $T(s)\in{\mathbb Z}^+$ such that the operator $$g\longrightarrow\mu\cdot g\circ F$$ maps $W_0^{s+T(s)}\left(\Omega_2\right)$ to
$W_0^s\left(\Omega_1\right)$ boundedly.
\end{proposition}

\begin{proof} Fix $s\in{\mathbb Z}^+$ and defining functions $r_1, r_2$ for $\Omega_1$ and $\Omega_2$. Let $d$ be the exponent associated to $F$ by Proposition \ref{P:distancecomparison} and let $w=F(z)$.

Let $g\in C^\infty\left(\Omega_2\right)$ vanish to order $Q$, $Q>s$ to be determined, on $b\Omega_2$:
\begin{equation}\label{E:bS1}
\left| D^\delta g(w)\right|\lesssim \left|r_2(w)\right|^{Q-|\delta|}, \quad |\delta|\leq Q,\qquad w\text{ near }b\Omega_2.
\end{equation}

If $F=\left(f^1,\dots ,f^n\right)$, each $f^k$ satisfies \eqref{E:cauchy}. The chain rule gives, for $|alpha|\leq Q$,
\begin{equation*}
\left| D^\alpha\left[g\circ f\right](z)\right|\lesssim\sum_{\beta: |\beta|=|\alpha|}\left|D^\beta g\big(F(z)\big)\right|\cdot a_1\cdot\dots\cdot a_{|\beta|},
\end{equation*}
where each factor $a_l\in{\mathscr G}_\infty^{-l}\left(\Omega_1\right), l=1,\dots ,|\beta|$, by \eqref{E:cauchy}. Combining this with \eqref{E:bS1} and using \eqref{E:distancecomparison} yields
\begin{align*}
\left| D^\alpha\left[g\circ F\right](z)\right|&\lesssim \left|r_2(w)\right|^{Q-|\alpha|}\cdot \left|r_1(z)\right|^{-|\alpha|} \\
&\lesssim \left| r_1(z)\right|^{\frac Qd-\left(1+\frac 1d\right)|\alpha|}.
\end{align*}
Leibniz's rule thus implies
\begin{align}\label{E:bS2}
\left|D^\beta\left[\mu\cdot g\circ F\right](z)\right|&=\left|\sum_{\gamma+\delta=\beta}\frac{\beta !}{\gamma !\,\delta !}D^\gamma\mu (z)\cdot
 D^\delta\left[ g\circ F\right](z)\right| \notag\\
 &\lesssim \sum_{\gamma+\delta=\beta}\left| r_1(z)\right|^{-k-|\gamma|}\cdot\left| r_1(z)\right|^{\frac Qd -|\delta|-\frac{|\delta|}d}.
\end{align}

If $Q\geq dk +(d+1)s+1$, it follows from \eqref{E:bS2} that $\mu\cdot g\circ F$ vanishes to order $\geq s$ on $b\Omega_1$ and, consequently,
$\mu\cdot g\circ F\in W_0^s\left(\Omega_1\right)$ by Theorem 11.5 of \cite{LionsMagenes}. If $T(s)\geq dk+(d-2)s+1$, the conclusion follows from the fact that functions in $C^\infty\left(\Omega_2\right)$ satisfying \eqref{E:bS1} are dense in $W_0^Q\left(\Omega_2\right)$.

\end{proof}

\section{{\bf A variation on Bell's operator}}\label{S:bell}

Constructions in this section occur on a single domain, so notation is simplified. Let $\Omega$ be a smoothly bounded domain in ${\mathbb C}^n$, given by a smooth defining function $r$.
The twist factor will be denoted by $T$ and the weight factor by $W$: $f\in{\mathcal O}^T(\Omega)$ if $\bar\partial\left(\sqrt T\cdot f\right) =0$ and $f\in L^2\left(\Omega, e^{-W}\right)$ if $||f||_W<\infty$ where $||\cdot||_W$ is defined in \eqref{E:weightedinner}. The $(T,W)$-Bergman projection on $\Omega$, ${\mathcal B}^{T,W}_\Omega$, is denoted ${\mathcal B}$.

Several spaces of functions that ``vanish on $b\Omega$'' arise in the analysis. For $L\in{\mathbb Z}^+$, let
$$V^L(\Omega)=\left\{ g: g=b\cdot r^L, \text{for some } b\in L^\infty(\Omega)\right\}$$
denote the bounded functions that vanish to order $L$ on $b\Omega$.  For $\ell\in{\Bbb Z}^+\cup\{0\}$, let
$${\mathscr B}_\ell(\Omega)=\big\{b\in C^\infty(\Omega): D^\alpha b\in L^\infty(\Omega),\quad |\alpha|\leq\ell\big\}.$$
Then, for $L\in{\mathbb Z}^+$, define subsets of $V^L$ by
$${\mathcal V}_\ell^L(\Omega)=\big\{ g: g=b\cdot r^L,\, b\in{\mathscr B}_\ell(\Omega)\big\}.$$
The spaces ${\mathcal V}_\ell^L(\Omega)$ are sometimes written ${\mathcal V}_\ell^L$ below, when $\Omega$ is fixed.
Clearly, $V^L={\mathcal V}^L_0\supset{\mathcal V}^L_1\supset\dots$.

Define the differential operators
$$D_k= e^W\, \sqrt T\, \frac\partial{\partial\bar z_k},\qquad k=1,\dots, n.$$
The initial observation is that the image of ${\mathcal V}^1_1$ under the operators $D_k$
is orthogonal to $\overline{{\mathcal O}^T(\Omega)}$ in $L^2\left(\Omega, e^{-W}\right)$.

\begin{lemma}\label{L:var1} Let $\Omega$ be a smoothly bounded domain and $(T,W)$ a twist-weight pair on $\Omega$ with $T\in L^\infty(\Omega)$.

If $h\in\overline{{\mathcal O}^T(\Omega)}$ and $f\in{\mathcal V}^1_1(\Omega)$, then
\begin{equation}\label{E:ortho1}
\big(h, D_k(f)\big)_W=0\qquad\text{for } k=1,\dots, n.
\end{equation}
\end{lemma}

\begin{proof} Let $h\in\overline{{\mathcal O}^T(\Omega)}$, i.e., $\partial\left(\sqrt{T}h\right)=0$. Assume, temporarily, that $h\in L^\infty(\Omega)$ as well. If $f\in{\mathcal V}^1_1$, integration by parts yields
\begin{align}\label{E:ortho2}
\big( h,D_k(f)\big)_W &=\int_\Omega h\cdot\left(e^W\sqrt{T}\frac{\partial\bar f}{\partial z_k}\right)\, e^{-W}\, dV \notag \\
&=\, -\int_\Omega\frac{\partial}{\partial z_k}\left(\sqrt{T}h\right)\cdot\bar f =0.
\end{align}
No boundary term occurs because $f\in{\mathcal V}^1_1$ and $\sqrt{T}h\in L^\infty(\Omega)$; the integrand in \eqref{E:ortho2} vanishes since $h\in\overline{{\mathcal O}^T(\Omega)}$.

For general $h\in\overline{{\mathcal O}^T(\Omega)}$, a limiting argument is used. A partition of unity reduces the problem to showing
\eqref{E:ortho1} holds for $f\in V$ supported near some $p\in b\Omega$. Let $\nu$ denote the outward unit normal to $b\Omega$ at $p$.
For $h\in\overline{{\mathcal O}^T(\Omega)}$ given, set $h_\epsilon(z) = h(z-\epsilon \,\nu)$. If $\epsilon>0$ is small, $h_\epsilon$ is well-defined and belongs to
$\overline{{\mathcal O}^T(\Omega)}\cap L^\infty(\Omega)$. If the support of $f$ is sufficiently small, the above integration by parts argument applies, giving $\big(h_\epsilon, D_kf\big)_W=0$.
However since $\sqrt{T},\frac{\partial\bar f}{\partial z_k}\in L^\infty(\Omega)$,

$$\big(h, D_k(f)\big)_W=\lim_{\epsilon\to 0}\big(h_\epsilon,  D_k(f)\big)_W$$
by dominated convergence. Thus \eqref{E:ortho1} holds under the stated hypothesis.
\end{proof}

A reformulation in terms of the null space of the $(T,W)$-Bergman projection on $\Omega$ is convenient:

\begin{corollary}\label{C:var2} Under the hypothesis of Lemma \ref{L:var1},

\begin{equation}\label{E:ortho3}
{\mathcal B}\left[ \bar D_k f\right] =0,\quad k=1,\dots, n,\qquad\text{for any }f\in{\mathcal V}^1_1(\Omega).
\end{equation}
\end{corollary}

\begin{proof} Recall that the $(T,W)$-Bergman kernel satisfies ${\mathcal B}(a,\cdot)\in\overline{{\mathcal O}^T(\Omega)}$ and
$${\mathcal B}\left[ \bar D_k f\right](a) =\left({\mathcal B} (a,\cdot), D_k\bar f\right)_W.$$
\eqref{E:ortho3} now follows from Lemma \ref{L:var1}.
\end{proof}

The main result of this section generalizes Lemma 2 in \cite{Bell81a} (see, also, \cite{Bell79} Lemma 3.1).

\begin{proposition}\label{P:mainbell} Let $\Omega$ be a smoothly bounded domain and $(T,W)$ a twist-weight pair on $\Omega$ with $T\in L^\infty(\Omega)$.
\smallskip

Let $M\in{\mathbb Z}^+$ and consider functions $\phi$ of the form
$$\phi=e^W\sqrt T\cdot \left(b\cdot s\right), \qquad\text{for } b\in{\mathscr B}_M(\Omega)\text{ and } s\in C^\infty(\overline\Omega).$$
For any $m\in{\mathbb Z}^+$, $m\leq M$,
there exists a function $\psi^m$ such that

\begin{itemize}
\item[(i)] ${\mathcal B}\left[\phi\right]= {\mathcal B}\left[\psi^m\right],$ and
\smallskip
\item[(ii)] $e^{-W}\frac 1{\sqrt T}\cdot \psi^m\in{\mathcal V}_{M-m}^m$.
\end{itemize}
\end{proposition}

\begin{proof}

The analysis occurs near $b\Omega$, where the defining function $r$ can be used as a coordinate. The function $s$, the ``smooth part'' of $\phi$, then locally has a Taylor expansion in powers of $r$ up to $b\Omega$. $\psi^m$ is constructed by removing terms of order $< m$ from this expansion; Corollary \ref{C:var2} implies the removed terms are in
$\text{Null}({\mathcal B})$.

For any $p\in b\Omega$, there exists a neighborhood $U$ and at least one anti-holomorphic derivative, say $\frac{\partial r}{\partial z_{(p)}}= r_{z_{(p)}}$, that is
non-vanishing on $U$, since $dr\neq 0$ on $b\Omega$. Fix $p\in b\Omega$ and such a neighborhood $U$. Initially, suppose the smooth part of $\phi$ is supported in $U$,
 i.e., $\phi =e^{W}\sqrt T\cdot\big( b\cdot s\big)$ with $s\in C^\infty(\overline\Omega)\cap C^\infty_0(U)$. The function
$\psi^m$ will be of the form
\begin{equation}\label{E:bell1}
\psi^m=\phi -e^W\sqrt T\cdot\frac\partial{\partial z_{(p)}}\left(\sum_{j=1}^m g_j\cdot r^j\right),
\end{equation}
for functions $g_1,\dots, g_m$ to be chosen. Corollary \ref{C:var2} says that ${\mathcal B}\left[\psi^m\right]={\mathcal B}\left[\phi\right]$, for any choices of $g_l\in{\mathscr B}_1(\Omega)$. The goal is to chose $g_j$ such that $e^{-W}\frac 1{\sqrt T}\cdot \psi^m\in{\mathcal V}^m_{M-m}$; the proposed form \eqref{E:bell1} reduces this to a two-term recursion relation. Choose $g_1$ satisfying
\begin{equation}\label{E:bell2}
b\cdot s-g_1\cdot r_{z_{(p)}} =0
\end{equation}
to force the $r^0$ term in \eqref{E:bell1} to vanish
(throughout, $r^t$ denotes $r$ raised to the $t$-power). It then follows that
\begin{equation*}
g_1(z)=\begin{cases}\frac{b\cdot s}{r_{z_{(p)}}}(z) &\text{ if $z\in U$} \\
0 & \text{if $z\notin U$}
\end{cases}
\end{equation*}
determines $g_1$ as an element in ${\mathscr B}_M(\Omega)$, since $r_{z_{(p)}}\neq 0$ in $U$, $s$ is supported in $U$, and $b\in{\mathscr B}_M(\Omega)$.

Next, choose $g_2$ satisfying
$$\frac{\partial g_1}{\partial z_{(p)}}+ 2g_2\cdot r_{z_{(p)}}=0,$$
in order to annihilate the $r^1$ term in \eqref{E:bell1}. This equation says $g_2\sim \frac{\partial g_1}{\partial z_{(p)}}\sim \frac{\partial b}{\partial z_{(p)}}$ in $U$, where $\sim$ denotes equality up to factors in $C^\infty\left(\overline{\Omega}\right)$.
As before, setting $g_2=0$ outside $U$ determines $g_2$ as an element of ${\mathscr B}_{M-1}(\Omega)$.
Continue by choosing $g_j$, $j=3,\dots m$, so that
$$\frac{\partial g_{j-1}}{\partial z_{(p)}}+ j\, g_j\cdot r_{z_{(p)}}=0.$$
The result is that
\begin{equation}\label{E:bell3}
\psi^m=-e^W\sqrt T\cdot\left(\frac{\partial g_m}{\partial z_{(p)}}\right)\cdot r^m,
\end{equation}
with
\begin{equation*}
\frac{\partial g_m}{\partial z_{(p)}}\sim \frac{\partial^m b}{\partial z_{(p)}^m}\in{\mathscr B}_{M-m}(\Omega).
\end{equation*}

To globalize this, let $U_1,\dots, U_N$ be neighborhoods of $p_1,\dots, p_N\in b\Omega$ such that

\begin{itemize}
\item[(i)] $r_{z_{(k)}}\neq 0$ on $U_k$, and
\item[(ii)] $b\Omega\subset\cup_{\ell =1}^N U_l$.
\end{itemize}
Let $\left\{\chi_\ell\right\}_{\ell =1}^N$ be a partition of unity subordinate to $\left\{U_\ell\right\}_{\ell =1}^N$.
On each $U_k$, the above construction yields functions $g_1^{(k)},\dots, g_m^{(k)}$, $g_\ell\in{\mathscr B}_{M-\ell+1}(\Omega)$ such that
$\chi_k\phi-e^W\sqrt T\cdot\frac{\partial}{\partial z_{(k)}}\left(\sum_{\ell=1}^m g_\ell^{(k)}\cdot r^\ell\right)\,\,\in e^W\sqrt T\cdot{\mathcal V}^m_{M-m}.$
 Thus
$$\psi^m =\phi -e^W\sqrt T\cdot\sum_{k=1}^N
\frac{\partial}{\partial z_{(k)}}\left(\sum_{\ell=1}^m g_\ell^{(k)}\cdot r^\ell\right)$$
gives the desired function.
\end{proof}

\section{{\bf Compatibility of twist-weight pair}}\label{S:compat}

We first introduce some auxiliary classes of functions:

\begin{definition}
Let $\Omega$ be a smoothly bounded domain. For $t\in{\mathbb R}$ and $m\in{\mathbb Z}^+$ define
$${\mathscr A}_m^t(\Omega)=\left\{f\in C^\infty(\Omega): f\in L^\infty(\Omega),\text{ and } \left|D^\alpha f(z)\right|\lesssim |r(z)|^t, 0<|\alpha|\leq 2m\right\}.$$

If $f\in{\mathscr A}_m^t$ for all $t\in{\mathbb R}^+$, write $f\in{\mathscr A}_m^\infty(\Omega)$.
\end{definition}

\begin{remark}
Clearly, $\{\text{constants}\}\subset {\mathscr A}_m^\infty(\Omega)$ for all $m\in{\mathbb Z}^+$. However, ${\mathscr A}_m^\infty(\Omega)$ contains other functions, e.g., any function of the form $s\cdot e^{\frac 1r}$ for $s\in C^\infty\left(\overline\Omega\right)$.

If $\Omega$ is pseudoconvex, the classes ${\mathscr A}_m^t(\Omega)$ are quasi-invariant under biholomorphic maps. If $F:\Omega_1\longrightarrow\Omega_2$ is a biholomorphism between smoothly bounded, pseudoconvex domains $\Omega_1, \Omega_2$ and $f\in {\mathscr A}_m^t(\Omega_1)$, then $f\circ F^{-1}\in{\mathscr A}^m_{\tilde t}(\Omega_2)$ for some $\tilde t$.\footnote{This argument gives $\tilde t << t$, since the estimate \eqref{E:cauchy} is used. After Theorem \ref{T:main} is known, it follows that the classes ${\mathscr A}_m^t(\Omega)$ are biholomorphic invariants.} In particular, if $f\in {\mathscr A}_m^t(\Omega_1)$
\textit{for large enough} $t$ (depending on $F$), then $f\circ F^{-1}\in {\mathscr A}_m^0(\Omega_2)$.
\end{remark}

Return to the set-up in Section \ref{S:twist}: $\Omega_1, \Omega_2$ are smoothly bounded domains,
$F:\Omega_1\longrightarrow\Omega_2$ is a biholomorphic map, $(\tau,w)$ a twist-weight pair on $\Omega_1$, and $\sigma=\tau\circ F^{-1}$, $v=w\circ F^{-1}$ the corresponding pair on $\Omega_2$. Denote the twisted-weighted Bergman projections ${\mathcal B}^{\tau, w}_{\Omega_1}$, ${\mathcal B}^{\sigma, v}_{\Omega_2}$ by ${\mathcal B}_1$ and ${\mathcal B}_2$.

The initial goal is to formulate conditions on the pair $(\tau, w)$ on $\Omega_1$ that ensure $\frac{e^{-v}}{\sigma}\in {\mathscr B}_*(\Omega_2)$.

\begin{definition}\label{D:bellcompat1}
Let $\Omega\subset {\mathbb C}^n$ be a smoothly bounded domain, $t\in{\mathbb R}$, and $m\in{\mathbb Z}^+\cup \{0\}$.
A  twist-weight pair $(\tau, w)$ is called ${\mathscr B}$-compatible to index $(m,t)$  if $\frac{e^{-w}}{\tau}\in{\mathscr A}_m^t(\Omega)$.
\end{definition}

\begin{lemma}\label{L:bellcompat} Let $\Omega_1, \Omega_2\subset{\mathbb C}^n$ be smoothly bounded, pseudoconvex domains, $F:\Omega_1\longrightarrow\Omega_2$ a biholomorphic map, and $d=d(F,\Omega_1,\Omega_2)$ be the distortion exponent given by Proposition \ref{P:distancecomparison}.

For $L\in{\mathbb Z}^+$, if $(\tau,w)$ is ${\mathscr B}$-compatible to index $(L,2Ld)$ on $\Omega_1$, then $(\sigma,v)$ is ${\mathscr B}$-compatible to index $(L,0)$ on $\Omega_2$. In particular,
$\frac{e^{-v}}\sigma\in{\mathscr B}_{2L}(\Omega_2)$ if $(\tau,w)$ is ${\mathscr B}$-compatible to index $(L,2Ld)$ on $\Omega_1$.
\end{lemma}

\begin{proof}
This follows directly from \eqref{E:cauchy}, Proposition \ref{P:distancecomparison}, and the definition of the spaces ${\mathscr A}_m^t(\Omega)$.
\end{proof}

The condition of ${\mathscr B}$-compatibility connects the type of functions reproduced by ${\mathcal B}_j$, $j=1,2$, and the multiplier in front of $b\cdot s$ in Proposition \ref{P:mainbell}. This gives

\begin{proposition}\label{P:mainstep}
Let $\Omega_1, \Omega_2\subset{\mathbb C}^n$ be smoothly bounded, pseudoconvex domains and $F:\Omega_1\longrightarrow\Omega_2$ a biholomorphic map with distortion exponent $d=d(F,\Omega_1,\Omega_2)$.

Let $t\in{\mathbb Z}^+$ be given and suppose
$(\tau, w)$ are ${\mathscr B}$-compatible to index $(L,2Ld)$, with $L=(2n+t)(1+d)$, on $\Omega_1$.

For any $g\in A^\infty\left(\Omega_2\right)$, there exists a function $G\in W^t_0\left(\Omega_1\right)$ such that

\begin{equation}\label{E:mainstep}
JF\cdot g\circ F=\sqrt\tau\,{\mathcal B}_1\left[e^w\sqrt\tau\,\cdot G\right].
\end{equation}
\end{proposition}

\begin{proof} For given $g\in A^\infty\left(\Omega_2\right)$, set $\phi=\frac 1{\sqrt\sigma}\cdot g$. Note that $\phi\in A^2_{\sigma, v}\left(\Omega_2\right)$
since $(\sigma, v)$ are ${\mathscr B}$-compatible to index $(*,0)$.
Rewrite
$\phi$ as
\begin{equation}\label{E:naturalB_L}
\phi= e^v\sqrt\sigma\,\left(\frac{e^{-v}}\sigma\cdot g\right).
\end{equation}
Since $(\tau, w)$ are ${\mathscr B}$-compatible to index $(L,2Ld)$, it follows from Lemma \ref{L:bellcompat} that $\frac{e^{-v}}\sigma\in{\mathscr B}_{2L}(\Omega_2)$.
Proposition \ref{P:mainbell} applied for $M=2L, m=L$ gives $\psi^L$ such that ${\mathcal B}_2\left[\psi^L\right]={\mathcal B}_2[\phi]$ and
\begin{equation}\label{E:mainstep1}
e^{-v}\frac 1{\sqrt\sigma}\,\cdot \psi^L\in{\mathcal V}^{L}_{L}\left(\Omega_2\right).
\end{equation}
Since ${\mathcal B}_2$ reproduces $\phi$, \eqref{E:transform} gives
\begin{align*}
JF\cdot\frac 1{\sqrt\tau}\cdot\left[ g\circ F\right]=&JF\cdot\left( {\mathcal B}_2[\phi]\circ F\right) \\
=& JF\cdot\left( {\mathcal B}_2\left[\psi^L\right]\circ F\right) \\
=&{\mathcal B}_1\left( JF\cdot\left(\psi^L\circ F\right)\right).
\end{align*}

\eqref{E:mainstep1} says that $\psi^L=e^v\sqrt\sigma\left(b\cdot r_2^L\right)$ where $b\in{\mathscr B}_L\left(\Omega_2\right)$. Inserting this above and simplifying, we obtain
\begin{equation*}
JF\cdot g\circ F=\sqrt\tau\,{\mathcal B}_1\left[e^w\sqrt\tau\,\cdot G\right],
\end{equation*}
where $G=JF\cdot (b\circ F)\cdot (r_2^L\circ F)=:g_1\cdot g_2\cdot g_3$. It remains to verify that $G\in W_0^{t}\left(\Omega_1\right)$. However, $g_3\in W_0^{\frac L{1+d}}\left(\Omega_1\right)$ by Lemma \ref{L:r^p}. Also,
$g_1\in{\mathscr G}_\infty^{-n}\left(\Omega_1\right)$ and $g_2\in{\mathscr G}_L^{-n}(\Omega_1)$, as noted above Lemma \ref{L:r^p}. It follows from Lemma \ref{L:mult} that
$G\in W_0^{t}\left(\Omega_1\right)$ as claimed.
\end{proof}

We shall also require that the multiplier in front of $G$, on the right-hand side of \eqref{E:mainstep}, be a good $W^*_0(\Omega_1)$ multiplier.
This is a second, separate notion of compatibility on $(\tau,w)$. Both compatibility notions are combined in the next definition, formulated on a family of twist-weight pairs in order that the conclusion of Proposition \ref{P:mainstep} holds as the biholomorphism $F$ varies. Note that the second compatibility condition is required to hold uniformly in the family.

\begin{definition}\label{D:bellcompat2}
Let $\Omega\subset {\mathbb C}^n$ be a smoothly bounded domain. A family of twist-weight pairs
${\mathcal F}=\left\{(\tau_i, w_i), i\in{\mathbb Z}^+\right\}$, on $\Omega$ is Bell compatible if

\begin{itemize}
\item[(i)] for any $(A,B)\in{\mathbb Z}^+\times{\mathbb Z}^+$, there exists $j(A,B)$ such that $(\tau_j,w_j)\in{\mathcal F}$ are ${\mathscr B}$-compatible to index $(A,B)$ when $j\geq j(A,B)$,
\item[(ii)] there exists $K$ such that $e^{w_i}\,\sqrt{\tau_i}\in{\mathscr G}_\infty^K(\Omega)$ for all $i\in{\mathbb Z}^+$.
\end{itemize}
\end{definition}

\begin{corollary}\label{C:mainstep}
Let $\Omega_1$ be a smoothly bounded pseudoconvex domain in ${\mathbb C}^n$ and ${\mathcal F}=\left\{(\tau_i,w_i), i\in{\mathbb Z}^+\right\}$ be a Bell compatible family on $\Omega_1$.

Let $s\in{\mathbb Z}^+$. Then for any biholomorphic map $F:\Omega_1\longrightarrow\Omega_2$ onto a smoothly bounded pseudoconvex domain $\Omega_2$, there exists $j(s)\in{\mathbb Z}^+$ such that: for any $h\in A^\infty\left(\Omega_2\right)$, there exists a function $H\in W^s_0\left(\Omega_1\right)$ such that

\begin{equation}\label{E:mainstep2}
JF\cdot h\circ F=\sqrt\tau\,{\mathcal B}^{\tau_j,w_j}_1\left[H\right],
\end{equation}
for any $(\tau_j,w_j)\in{\mathcal F}$ with $j\geq j(s)$.
\end{corollary}

\begin{proof} Let $K$ be the constant given by Definition \ref{D:bellcompat2} (i) for the family ${\mathcal F}$.
By Proposition \ref{P:mainstep}, any twist-weight pair $(\tau,w)$ that are ${\mathscr B}$-compatible to sufficiently high order
cause \eqref{E:mainstep} to hold $G\in W^{s+K}_0(\Omega)$.
Set $H=e^w\sqrt\tau\cdot G$.
Lemma \ref{L:mult} implies that $H$ belongs to $W^{s}_0(\Omega)$, which completes the proof.

\end{proof}

The trivial family ${\mathcal F}=\left\{(\tau_i,w_i)=(1,0), \text{ for all }i\in{\mathbb Z}^+\right\}$ is obviously Bell compatible, since the constant function $1=e^0\sqrt{1}$ belongs ${\mathscr G}_\infty^K$ for any $K\in{\mathbb R}$ and $1=\frac{e^{-0}}{1}$ is ${\mathscr B}$-compatible to any index in ${\mathbb Z}^+\times{\mathbb Z}^+$. Other Bell compatible families may be obtained, e.g., by taking a fixed function $\tau$ such that $\frac 1{\sqrt\tau}\in{\mathscr G}_\infty^0(\Omega)$ and setting $w_i=-\log\left(\tau\left(1+(-r)^i\right)\right)$, $i\in{\mathbb Z}^+$.

\section{{\bf Proof of main theorem}}\label{S:proof}

The regularity hypothesis in Theorem \ref{T:main} can now be stated:

\begin{definition}\label{D:frakR} A smoothly bounded domain $\Omega$ satisfies \textit{Condition} ${\mathfrak R}$ if there exists a Bell compatible family of twist-weight pairs $\left\{(\tau_j,w_j),\,\, j\in{\mathbb Z}^+\right\}={\mathcal F}$ satisfying the following:

for each $s\in{\mathbb Z}^+$, there exist $m(s), \ell(s)\in{\mathbb Z}^+$ such that for all $m\geq m(s)$

\begin{equation}\label{E:frakR_reg}
\sqrt{\tau_m}\cdot{\mathcal B}^m:W^{s+\ell(s)}_0(\Omega)\longrightarrow W^s(\Omega),
\end{equation}
where ${\mathcal B}^m={\mathcal B}^{\tau_m,w_m}_\Omega$.
\end{definition}

Definition \ref{D:frakR}  reduces  to Condition R of Bell-Ligocka when ${\mathcal F}$ is taken to be the single pair $(1,0)$.

\begin{proof}[Proof of Theorem \ref{T:main}] Let $F:\Omega_1\longrightarrow\Omega_2$ be a fixed biholomorphism and ${\mathcal F}=\left\{(\tau_j,w_j)\right\}$ the Bell compatible
family on $\Omega_1$.

Let $I\in{\mathbb Z}^+$ be arbitrary. First apply Corollary \ref{C:mainstep} for $s=I+\ell(I)$ where $\ell(I)$ is given by Condition ${\mathfrak R}$.
Set $N=\max(m(I),j(I))$, $m(I)$ given by Condition ${\mathfrak R}$ and $j(I)$ given by Corollary \ref{C:mainstep}. Take any $(\tau_j,w_j)\in{\mathcal F}$ with $j\geq N$. It follows from  \eqref{E:frakR_reg} that

\begin{equation}\label{E:proof1}
\left\|JF\cdot h\circ F\right\|_{(I)} <\infty
\end{equation}
for any $h\in A^\infty(\Omega_2)$. Since $I$ was arbitrary, Sobolev's lemma implies $JF\cdot h\circ F\in C^\infty\left(\overline{\Omega}_1\right)$. Choosing $h\equiv 1$ yields $JF\in C^\infty\left(\overline{\Omega}_1\right)$.

The same argument on $F^{-1}:\Omega_2\longrightarrow\Omega$, using the Bell compatible family $\tilde{\mathcal F} =\left\{(\tilde\tau_j,\tilde w_j)\right\}$ on $\Omega_2$, shows that $JF^{-1}\in C^\infty\left(\overline{\Omega}_2\right)$. Note that $\tilde{\mathcal F}$ is not necessarily the family $\left\{(\sigma_j,v_j)\right\}$ with $(\sigma,v)$ associated to $(\tau, w)$ as previously. Since $JF^{-1}$ is smooth up to $b\Omega_2$, it follows that $JF\neq 0$ on $\overline{\Omega}_1$.

Apply \eqref{E:proof1} to the coordinate functions, $h_k(w_1,\dots , w_n)=w_k, k=1,\dots , n$, to obtain $JF\cdot f_k \in C^\infty\left(\overline{\Omega}_1\right)$,
where $F=\left(f_1,\dots , f_n\right)$. Dividing out the non-vanishing factor $JF$ yields $f_k \in C^\infty\left(\overline{\Omega}_1\right)$, $k=1,\dots , n$. Since the argument is reversible, we also obtain $\left(F^{-1}\right)_k \in C^\infty\left(\overline{\Omega}_2\right)$, which completes the proof.

\end{proof}

 \vskip .5cm
\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


I have been in the Spa Industry for 12+ years. I opened up Beneath The Surface Spa 4 years ago and we are in our fifth year. Which is very exciting! We do spa services not just massage, but we do massage extremely well and have lots of repeat business. We have wonderful massage therapists working here and everyone of them have a thriving business and have been with me for years. If you have any issues that you need attention to, we really try to find an answer for you, and we always get the result you need.


text:
BSW Environmental Blog
The Clean Water Rule Gets A Well-Needed Review
The EPA continues to implement the de-regulatory agenda of President Trump. One rule that is now being reviewed for repeal and revision is the Clean Water Rule. Many felt that the Rule went well beyond the scope of the Clean Water Act and regulated private property that had not been previously subject to regulation. As a result, they support EPA's and the Corps' current efforts to repeal the Rule and revise it to a more traditional view of the Clean Water Act's jurisdictional reach.
The Clean Water Rule was published in June, 2015, expanding the definition of 'navigable waters' and 'waters of the United States' in several subtle but important ways. For example, although traditional navigable waterways were always included, EPA stated in the preamble to the Rule that these waterways included all waters used for commercial waterborne recreation, such as guided fishing trips or water ski tournaments. Additionally, it defined tributaries as any water that contributes flow, directly or indirectly, to a traditional navigable waterway and that is characterized by the presence of a bed and banks and an ordinary high water mark. Finally, it added a definition of adjacent as bordering, contiguous, or neighboring.
The Rule increased the jurisdictional reach of the EPA and the Corps. Now, waterways, and their adjacent wetlands, that were not previously subject to regulation would be subject to permitting under the Clean Water Act. Obtaining a permit is itself expensive and, for every impact in wetlands, expensive mitigation is also required. Additionally, a landowner or developer could find themselves the recipient of an enforcement action from the agencies.
The Sixth Circuit stayed the rule almost immediately, finding that the petitioners were likely to succeed on their claims that the rule exceeded EPA's and the Corps' authority under the Clean Water Act. The rule, as published, never went into effect.
An executive order was issued on February 28, 2017 which required that EPA and the Corps review the rule. Among other things, EPA and the Corps were required to consider defining the scope of jurisdiction in a manner similar to the approach announced by Justice Antonin Scalia in Rapanos case. In that case, Justice Scalia stated that jurisdiction extended to waters that are navigable in the traditional sense, relatively permanent, standing, or continuously flowing bodies of water, and wetlands with a continuous surface connection to those relatively permanent waters.
The EPA and the Corps announced a two-step process to implement the executive order. In step one, the agencies would re-institute the rules as they existed prior to issuance of the Rule. In step two, the agencies will propose a new definition that would replace the approach in the Rule, taking into consideration the principles that Justice Scalia outlined in Rapanos. The EPA and the Corps published a proposed rule on July 27 to implement step one.
Due to its expansive reach, the Rule would have allowed the EPA and the Corps to control development in many areas that were never intended by the Clean Water Act and were not part of the original definition of 'waters of the United States.' In fact, as the placement of any fill material in a wetlands requires a permit and mitigation, the EPA and the Corps would essentially control development in even the most innocuous of places, like your own backyard. Hopefully, by seeking input and public comment, a balance can be reached which protects our rivers, streams, and adjacent wetlands while allowing normal development of areas that were never intended to be regulated under the Clean Water Act.
Posted on August 8, 2017 August 8, 2017 by BSWPosted in Clean Water Act, EPA, Trump Administration
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text:
Behaviour and Welfare Specialist
St Marylebone CE School
City of Westminster, London (Greater)
NJC points 10 – 21, depending on experience: FTE £25,613 - £31,168
School support, Caretaker, Schools
The St Marylebone CE School welcomes applications from people of all backgrounds, races, cultures, faiths and genders. The School actively promotes diversity and inclusion across its community and our students come from a wide range of backgrounds: 40% of our students speak English as an additional language; 59% of our students come from backgrounds other than white British.
Salary: NJC points 10 – 21, depending on experience: FTE £25,613 - £31,168. Equates to £21,553 - £26,227 for term-time only working.
Start: ASAP
We are seeking to appoint an organised and proactive Behaviour and Welfare Specialist. This role works alongside strong pastoral and SLT teams. Supporting positive behaviour and improving attendance of our more vulnerable and disaffected students.
Ideal candidates will be resilient, have effective communication skills and will demonstrate a commitment to developing potential and supporting young people experiencing difficulties.
Founded in 1791, The St Marylebone CE School is an outstanding comprehensive school for girls with a mixed sixth form, serving students of all faiths or no faith. St Marylebone takes pride in being a brilliant place to work and learn: rated consistently 'outstanding' by Ofsted and SIAMS, the School is one of the best performing state schools nationally. We value highly the role of the arts and creativity in education, across the curriculum. We lead the regional Maths Hub and the St Marylebone Teaching School Alliance.
Closing date: 12pm, Friday 21st January 2022
Interview date: w/c 24th January 2022
Please fill out an application form available to download from our website. Completed application forms should be emailed to: recruitment@stmaryleboneschool.com.
Please note that applicants may be called earlier so we recommend submitting your application as soon as possible.
The School is committed to safeguarding and promoting the welfare of children & vulnerable adults and it is our expectation that all staff and volunteers share this commitment. Any offers of employment will be subject to satisfactory references and an Enhanced Disclosure and Barring Service (DBS) check.


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text:
After Uber, Tesla incidents, can artificial intelligence be trusted?
In light of recent accidents with autonomous and semi-autonomous vehicles, will people put their trust in artificial intelligence? Missouri S&T researchers are digging for answers.
Given the choice of riding in an Uber driven by a human or a self-driving version, which would you choose?
Considering last month's fatal crash of a self-driving Uber that took the life of a woman in Tempe, Arizona, and the recent death of a test-driver of a semi-autonomous vehicle being developed by Tesla, peoples' trust in the technology behind autonomous vehicles may also have taken a hit. The reliability of self-driving cars and other forms of artificial intelligence is one of several factors that affect humans' trust in AI, machine learning and other technological advances, write two Missouri University of Science and Technology researchers in a recent journal article.
"Trust is the cornerstone of humanity's relationship with artificial intelligence," write Dr. Keng Siau, professor and chair of business and information technology at Missouri S&T, and Weiyu Wang, a Missouri S&T graduate student in information science and technology. "Like any type of trust, trust in AI takes time to build, seconds to break and forever to repair once it is broken."
The Uber and Tesla incidents point to the need to rethink the way AI applications such as autonomous driving systems are developed, and for designers and manufacturers of these systems to take certain steps to build greater trust in their products, Siau says.
Despite these recent incidents, Siau sees a strong future for AI, but one fraught with trust issues that must be resolved.
'A dynamic process'
"Trust building is a dynamic process, involving movement from initial trust to continuous trust development," Siau and Wang write in "Building Trust in Artificial Intelligence, Machine Learning, and Robotics," published in the February 2018 issue of Cutter Business Technology Journal.
In their article, Siau and Wang examine prevailing concepts of trust in general and in the context of AI applications and human-computer interaction. They discuss the three types of characteristics that determine trust in this area – human, environment and technology – and outline ways to engender trust in AI applications.
Siau and Wang point to five areas that can help build initial trust in artificial intelligence systems:
Representation. The more "human" a technology is, the more likely humans are to trust it. "That is why humanoid robots are so popular," Siau says, adding that it is easier to "establish an emotional connection" with a robot that looks and acts more like a human or a robotic dog that acts more like a canine. Perhaps first-generation autonomous vehicles should have a humanoid "chauffeur" behind the wheel to help ease concerns.
Image or perception. Science fiction books and movies have given AI a bad image, Siau says. People tend to think of AI in dystopian terms, as colored by Terminator or Bladerunner movies or Isaac Asimov and Philip K. Dick novels. "This image and perception will affect people's initial trust in AI," Siau and Wang write.
Reviews from other users. People tend to rely on online product reviews, and "a positive review leads to greater initial trust."
Transparency and "explainability." When a technology's inner workings are hidden in a "black box," that opacity can hinder trust. "To trust AI applications, we need to understand how they are programmed and what function will be performed in certain conditions," Siau says.
Trialability. The ability to test a new AI application before being asked to adapt it leads to greater acceptance, Siau says.
How to maintain trust in AI
Beyond developing initial trust, however, creators of AI also must work to maintain that trust. Siau and Wang suggest seven ways of "developing continuous trust" beyond the initial phases of product development:
Usability and reliability. AI "should be designed to operate easily and intuitively," Siau and Wang write. "There should be no unexpected downtime or crashes."
Collaboration and communication. AI developers want to create systems that perform autonomously, without human involvement. Developers must focus on creating AI applications that smoothly and easily collaborate and communicate with humans.
Sociability and bonding. Building social activities into AI applications is one way to strengthen trust. A robotic dog that can recognize its owner and show affection is one example, Siau and Wang write.
Security and privacy protection. AI applications rely on large data sets, so ensuring privacy and security will be crucial to establishing trust in the applications.
Interpretability. Just as transparency is instrumental in building initial trust, interpretability – or the ability for a machine to explain its conclusions or actions – will help sustain trust.
Job placement. As concerns about AI replacing humans on the job continue to grow, policies must be put in place to provide retraining and education to those affected by this trend.
Goal congruence. "Since artificial intelligence has the potential to demonstrate and even surpass human intelligence, it is understandable that people treat it as a threat," Siau and Wang write. "Making sure that AI's goals are congruent with human goals is a precursor in maintaining continuous trust." Policies to govern how AI should be used will be important as technology advances, the authors add.
"The AI age is going to be unsettling, transformative and revolutionary," Siau writes in another recent article ("How Will Technology Shape Learning?" published in the March 2018 issue of The Global Analyst). But in this unsettling environment, higher education can play a significant role.
"Higher education must rise to the challenge to prepare students for the AI revolution and enable students to successfully surf in the AI age," Siau writes.
Already, Siau is working to prepare MBA students at Missouri S&T for the AI age through Artificial Intelligence, Robotics, and Information Systems Management, a course he introduced in 2017. As part of the coursework, Siau asks each student to present an article on a new artificial intelligence or machine learning technology or application.
On April 9, 2018. Posted in CASB Research, College of Arts, Sciences, and Business, Department of Business and Information Technology, email, Research
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A long time ago in a place where all was not understood, not a thing made sense and I was very much smaller than everyone else.
Where landscapes disintegrate and bodies fragment, where the surface is shinier, a distressed narrator struggles to form a sense of self. If i were a dog... but i'm a fish. MacCormack's surreal, lush imagery and fragmented text reflect a harrowing account of trauma. He lives in the Dream of a Fantasy and breaks off pieces for us. As terror and alienation wash up on every shore, the child's desire for belonging and human connection is undone by her critical eye, and an inability to accept that her perception of reality is unsound. All these awakenings are only small gasps of air sucked into the lungs. Then we go back under. Before the child drowns, the body transforms into creatures of the sea, while the Missing Ones still can't be found anywhere. When we have long ago fallen off the places that might have made sense, where there are many fish, many directions, we are led to ask what it might have looked like if anyone was looking.
The author reveals a heart-wrenching portrayal of survival.
Jessica MacCormack's interdisciplinary practice engages with the intersection of institutional violence and the socio-political reality of personal trauma. Her recent works integrate animation, video, painting and collage in social interventions and community productions that explore issues of criminalization, HIV/AIDS, racism, transphobia, sexual assault and mental health. Working with communities and individuals affected by stigma and oppression, MacCormack uses cultural platforms and distribution networks to facilitate collaborations which position art as a tool to engender personal and political agency. She has an MFA in Public Art and New Artistic Strategies from the Bauhaus University (2008). Currently she is an Assistant Professor at Concordia University and lives in Montréal, Québec.


text:
WEBINAR: Mental Health in Third Spaces
by James White | Nov 19, 2019 | Blog, Webinars
For those who ask: "What do children's museums and mental health have in common?", we would answer, "more than you might think."
For those of us lucky enough to have a children's museum in our communities, we may be aware of the important role they play not only in educating children but also in meeting a variety of family and community needs.
In this webinar, we explored the potential of children's museums and other "third spaces" to help meet the emotional and mental health needs of children and families. Mental health services can be hard to access for many families, especially in communities with fewer resources. Expanding access through children's museums and other non-traditional partners is one way to help build the health and well-being of our communities.
Learn about efforts in Topeka, Kansas and southeast Washington, DC to expand mental health services for families through creative partnerships with communities and families.
Dr. Matthew Biel, Ascend Fellow and Division Chief and Associate Professor, Division of Child and Adolescent Psychiatry and Pediatrics, Georgetown University School of Medicine/MedStar Georgetown University Hospital
Laura Huerta Migus, Ascend Fellow and Executive Director, Association of Children's Museums
Dene Mosier, Executive Director, Kansas Children's Discovery Center
Travis Spencer, Executive Director, Institute of African American Mindfulness
Lori Severens, Assistant Director for Leadership and Design, Ascend at the Aspen Institute


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Last Wednesday evening I was just about to settle down to do some more of my ongoing work when the phone rang.
As an exception to my rule, I had given my home number to one of my adult clients with whom I was planning to do some early-morning speech sessions before the rest of the group arrived.
This man, the newest member of the stroke group, has in the last few months unfortunately developed epilepsy. He hasn't been feeling well since he started taking medicine for this and, as an added problem, the balance has not yet been found. This has meant that he has had to cancel some of our sessions. His wife phoned me on Wednesday to cancel another one as he still feels quite poorly.
As this man cannot speak spontaneously it is difficult to know how the medicine is affecting him. But we can see that he is wobbly on his feet and very sleepy. He confirms that he cannot concentrate very well and feels very insecure when moving around on his own. He is anxious about leaving the house. He is worried that his wife will be driving him and that he may have a seizure on the way.
His wife had just phoned to say that they were sorry that they wouldn't be able to make it again in the morning. She said that she wanted me to know so that I do not leave home specially early to meet them. She went on to tell me a lot more than just this. How glad I was that she had my phone number, and not because I could leave for work in the morning thirty minutes later.
This family is one of the best adult conductive-upbringing families that I have ever met. The wife is amazing, I would welcome her any day to work with me in my group. In fact. she is now actually present in the stroke group most of the time. She drives her husband the sixty minutes on the motorway to us and then watches or joins in when the weather is too cold and dreary to take a walk outside.
This winter the sessions have been quite lively with partners and carers joining in, not wanting to be outside in the freezing cold and snow. However inviting the countryside looks, they have found our company more inviting than a tramp through deep snow!
Over the past week I have had a lot of feedback from these people. I hear that the partners, carers, wives, sons and daughters have not only been observing their own family member but also the rest of the group, and they report back on their progress too. It is very motivating for us all to hear this.
The wife I had on the phone did just the same. She began to tell me about everything that she had observed in our group, both when I work directly with her husband and also with the other clients. She told me how she tries to adapt all that she learns, to use it in their life at home.
Copy me. Look, this is how you do it!
She told me so many things, like how she saw that her husband was standing without shoes on in the group and could hardly believe it when he lifted first one foot from the ground and then the other. When they were at home the same day she suggested that because her husband could do this unaided he could perhaps take his trousers off standing up rather than sitting down, which would be so much easier for him. He now does this always.
She told me how she observed that her husband, even though it is still less than a year since the stroke, walks better than most of the people she sees at the clinic when they go for appointments with their doctor.
She says that it is because I told them at the beginning, about six months ago now, that he must try to prevent the rotating movement from his hip as he walks. I told them that, as he can bend and stretch both hip and knee, he must learn to use this movement again when walking as soon as possible, and must be continuously reminded to do so.
He has done so. He has learnt it and now he walks without a stick indoors, and sometimes outdoors too.
This observant wife told me how she tells her husband that she will be his eyes, looking on the floor for him and will tell him if there is anything in his path. She tells him that, while she does this, he must concentrate on walking upright and looking further ahead than his feet, thereby stretching his body with every step he makes.
He does it very well. When we work in the group I always ask him to look at his wife as we walk. He is very proud to be able to do so.
She told me too that sometimes he comes into the kitchen and asks her to bring a glass of juice into the other room for him. She says to him "You can do it yourself, look this is how". She shows him how, she knows how, having observed him carry his walking stick in his hand in the group, then she walks beside him as he does it until he feels confident enough to do it alone.
She tells me that she never tells him simply that he can do it himself. She always shows him how. She says that she has read so much about stroke and aphasia and she realises that she can never be really sure whether he remembers how to do it himself.
What a wonderful lady, wife and mum she is.
Not surprisingly the information that she has gathered over the past six months, apart from what she has learnt in our group, comes from the sort of books that I always recommend. Books written by fellow suffers, or by carers and partners of stroke victims. It is from these books that we learn most.
Not only is she there living conductively with her husband, this wonderful lady is educating not only her two sons but also their extended family and friends.
"Oh no, you must not stop coming, my husband has got to learn to speak again. How will he do that with no one to talk to? And me, what about me? I need someone to talk to, too."
Of course huge crowds can be exhausting for stroke victims, especially those suffering from aphasia, but small groups of dear friends can provide the key to opening up new means of communication.
We went on to discuss that the reason for planned these early morning speech sessions for her husband had been because we were not only uncertain where the key is, but also where the lock is that is preventing us getting in, and of course preventing her husband getting his voice out.
I need a lot more time with this man alone to be able to determine the best methods to use to prevent frustration and to encourage communication. His wife is doing all that she can, using all that she has learnt and is also searching for this lock and its key.
The members of our group are also doing their bit. They are determined, as they always are with new members, to get him talking again. I encourage a form of questioning that requires simple answers and I realise how difficult this is for even some of the more fluent speakers. The wife observes and teaches this to her sons.
We have two more early morning speech sessions planned for this week. I hope that we can make a good start then in our hunt for a way through.
This conductive wife explains to me how disappointing it is when family members say: "But he isn't making any progress". This lady is tough though, undeterred, and she immediately begins to list for her visitors all the small and almost invisible steps that he has made. Things that happen when they are not there, like cleaning teeth standing up, carrying a drink to his place, walking with his head held high, spending thirty minutes at a go, and not ten as before, in the office sorting out their papers.
She tells me about how their two teenage boys long to have chats with their dad again. About how she has to comfort them and teach them what she has learnt in our group, so that they too learn how to encourage their Dad to speak.
She told me on the phone about her ambitions as a young woman. She wanted to be a physiotherapist. I told her she that is already a better conductor than many. She understood what our group is aiming for from the first moment she joined us, and introduces everything step by step into her family's lives.
She doesn't tell her husband to blow his nose, or to carry the glass or to comb his hair. She says: "Do it like this, watch, you can do this, copy me". She tells me that he does it, he copies her.
She also told me that she has learnt to sit down on the sofa beside her husband, leave the washing or the cleaning till tomorrow, and just sit there beside him and read a book. She told me how much she appreciates it when in the group I ask the partners how they are. I inquire how their aches and pains are, how often they do something for themselves, suggest massages and relaxing times with friends, and I remind them that they are still themselves, still a wife and a mother, or a husband, a father or a worker.
She tells me that she has had to learn this and that it helps her to remember to think about herself when I ask her each time we meet how she is.
It is knowing people like this wonderful lady, this amazingly loving wife and mother, this intelligent gatherer of information and transformer of all that she learns into something useful for their daily life, that makes me cry with joy that I do the job I do.
It is conversations like this one last Wednesday that make me so angry with myself because sometimes I let the world get too much for me, even when, in comparison to the life described above, my life is so easy and so enjoyable.
When I put the phone down I made a note for myself to suggest to this lovely, articulate lady that she keeps a diary. I will suggest that she records what is happening in their life, if possible going back to the early days.
I am sure that reading what she has to say about "life as a family after a stroke" would be a great help to her and her family, and also provide much motivation and information to other families in a similar situation.
We will be swapping lists of biographies and autobiographies in the group this week, I will list any new ones here later.


We're continuing our polishing of Opera 10.10, so here's a new build with some important fixes.
Opera Link should be working without having to log in every time you start Opera. We have also fixed a crash in the Unite setup wizard, and fixed a bug where the Unite password is remembered even if "remember password" is unchecked.
The default message in Opera Mail is now translated again.


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National Cold War Museum
The National Cold War Museum in Blytheville, Arkansas, has announced formation of its primary building and design team. Clark Contractors is the construction manager/general contractor, Polk Stanley Wilcox is the architect and Gallagher and Associates of Washington, D.C. is the exhibit design company. More information about the museum is available at their website (thenationalcoldwarmuseum.org) and the Memphis Business Journal.
"We could not be more excited about this team," said museum chair, Mary Gay Shipley. "This is not just any group. These are highly touted professionals who are already engaged and captivated by our concept—absolute top-of-the-line pros. We have every reason to believe these people will bring the dream to a beautiful and meaningful reality for our entire country."
The National Cold War Museum is planned for construction on the former Eaker Air Force Base in Blytheville. Once known as Blytheville Air Force Base, the facility was a Strategic Air Command base throughout the Cold War. Its "alert shack," the compound that housed air crews while they awaited a call to nuclear warfare, is one of the only intact such facilities in the nation and will be a central exhibit of the museum. "Perhaps a Cold War museum could be built anywhere," said Shipley. "But here in Blytheville, when you visit the museum, you will be standing on a very real piece of Cold War history".
Polk Stanley WIlcox


text:
76 Percent of Parents Concerned For Children's Online Safety
A recent PCMag survey of 2,500 US consumers found that a majority of parents are worried about their kids' online safety—and many respondents have never been properly educated on cybersecurity.
By Rob Marvin
It's a dangerous online world for kids, and parental controls can only do so much. According to PCMag's 2018 Consumer Cybersecurity Trends survey of 2,500 US consumers, 76 percent of parents have some level of concern for their child's safety online.
The survey found that while 24 percent of the respondents had no worries, 51 percent of parents harbored significant or major concern over potential dangers lurking online for their kids. Tech companies have created child-safe apps and social-media environments such as Facebook Messenger Kids and YouTube Kids, but both of the latter apps have seen their share of controversy and scandal as the companies roll out new features and fixes to mitigate problems.
We also surveyed respondents on how informed they are about cybersecurity, and we found that many have never received a proper education on the subject. Thirty-six percent of respondents said they've never been given adequate cybersecurity training.
The rest of the respondents said they've gained some level of cybersecurity knowledge over time. Twenty-one percent said they received online safety education when they entered the workforce, and 15 percent said they were educated on cybersecurity after college, 12 percent in college, 10 percent in high school, and 6 percent of respondents received some level of cybersecurity education in grade school.
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As for whether respondents felt they knew enough to effectively prevent common cyber attacks such as credit card fraud, malware, and ransomware, only 52 percent said yes. Check out some more data from PCMag's 2018 Consumer Cybersecurity Trends survey to find out what measures people take to protect themselves online.
Verizon Introduces Smartphone Plan Designed for Kids
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About Rob Marvin
Rob Marvin is PCMag's Associate Features Editor. He writes features, news, and trend stories on all manner of emerging technologies. Beats include: startups, business and venture capital, blockchain and cryptocurrencies, AI, augmented and virtual reality, IoT and automation, legal cannabis tech, social media, streaming, security, mobile commerce, M&A, and entertainment. Rob was previously Assistant Editor and Associate Editor in PCMag's Business section. Prior to that, he served as an editor at SD Times. He graduated from Syracuse University's S.I. Newhouse School of Public Communications. You can also find his business and tech coverage on Entrepreneur and Fox Business. Rob is also an unabashed nerd who does occasional entertainment writing for Geek.com on movies, TV, and culture. Once a year you can find him on a couch with friends marathoning The Lord of the Rings trilogy--extended editions. Follow Rob on Twitter at @rjmarvin1.
Read the latest from Rob Marvin
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Intelli Heat and EcoDesign, pushing the electric heating industry towards compliance since 2013.
Through the Cali Sense eco-design range IntelliHeat have been pioneering the integration of Intelligent heating controls on direct acting electric heating systems since 2013. Indeed we have worked tirelessly to deliver an intelligent, highly efficient and fully controllable heating system to the UK market. Indeed we have promoted the EcoDesign legislations key drivers in each of our ranges since the first promotion of the 2016 standards.
What is EcoDesign Lot 20?
What does LOT 20 this mean?
Very basically… From 1st January 2018 all local space electric heaters manufactured in the EU must meet a set of minimum efficiency standards, this essentially means all heaters will have to incorporate intelligent room temperature controls to minimise wasted energy.
What are intelligent heating controls?
Intelligent controls managing the running times of your heater will incorporate three key features.
24 hour or Timer Control. This could be an integral programmer or wireless through a smartphone app with WIFI connectivity.
Thermostatic Control. Accurate room temperature measurement by an electronic thermostat.
Smart Start Control. V27, Open door or window recognition, presence detection technology reduces wasted energy when a thermostat detects a drop in temperature the heater will choose when to provide heat.
Does Electric Central Heating just supply the radiators?