text:
\begin{document}
\maketitle

\begin{abstract}
	Let $X$ be a graph on $n$ vertices with with adjacency matrix $A$ and
	let $H(t)$ denote the matrix-valued function $\exp(iAt)$.  If $u$ and $v$ are distinct
	vertices in $X$, we say \textsl{perfect state transfer} from $u$ to $v$
	occurs if there is a time $\tau$ such that $|H(\tau)_{u,v}|=1$. If $u\in V(X)$ and
	there is a time $\sg$ such that $|H(\sg)_{u,u}|=1$, we say $X$ is
	\textsl{periodic at $u$} with period $\sg$. It not difficult to show that if
	the ratio of distinct non-zero eigenvalues of $X$ is always rational, then $X$ is
	periodic. We show that the converse holds, from which it follows that a regular graph
	is periodic if and only if its eigenvalues are distinct.
	For a class of graphs $X$ including all vertex-transitive graphs we prove that,
	if perfect state transfer occurs at time $\tau$, then $H(\tau)$ is a
	scalar multiple of a permutation matrix of order two with no fixed points.
	Using certain Hadamard matrices, we construct a new infinite family of graphs
	on which perfect state transfer occurs.
\end{abstract}

\section{Introduction}

Let $X$ be a graph with adjacency matrix $A$.  We define the matrix function $H(t)$
by
\[
H(t) :=\exp(itA) :=\sum_{n\ge0} i^nA^n\frac{t^n}{n!}.
\]
We note that $t$ is a real variable
\[
H(t)^* =\exp(-iAt) =H(t)^{-1}
\]
and therefore $H(t)$ is a unitary matrix.  We have $H(0)=I$ and
\[
H(s+t)=H(s)H(t).
\]
We say that graph is \textsl{periodic with respect to the vector $z$} if there is
a real number $\tau$ such that $H(\tau)z$ is a scalar multiple of $z$.  (Since
$H(t)$ is unitary, this scalar will have absolute value 1.)
We say that $X$ is \textsl{periodic relative to the vertex $u$} if it is
periodic relative
to the standard basis vector $e_u$ or, equivalently if there is a time $\tau$ such
that $|H(\tau)_{u,u}|=1$.  We say that $X$ itself is \textsl{periodic} if there
is a time $\tau$ such that $H(\tau)$ is diagonal.

If there are basis vectors $\seq z1n$ such that $X$ is periodic with period
$\tau_r$ relative to the vector $z_r$ for $r=1,\ldots,n$, then it follows that
$X$ is periodic,
with period dividing the product of the $\tau_i$'s.

We say we have \textsl{perfect state transfer} if there are distinct vertices $u$ and
$v$ in $X$ and a time $\tau$ such that
\[
|H(\tau)_{u,v}| = 1.
\]
(If this holds then $H(\tau)_{u,v}$ is the only non-zero entry in its row
and column.)
Christandl, Datta, Dorlas, Ekert, Kay and Landahl \cite{christandl-2005-71}
prove that
we have perfect state transfer between the end-vertices of paths of
length one and two, and between vertices at maximal distance in Cartesian powers
of these graphs.  A graph exhibiting perfect state transfer between two
vertices $u$ and $v$ models a network of quantum particles with fixed couplings, in
which the state of the particle in $u$ can be transferred to the
particle in $v$ without any information loss.

This paper is an attempt to extend some of the results in the above paper,
and in the
more recent work by Saxena, Severini and Shparlinski \cite{saxena-2007}.
We prove that
a regular graph is periodic if and only if its eigenvalues are integers.  We also
show for a wide class of regular graphs (including all vertex-transitive graphs
and all distance-regular graphs) that if perfect state transfer occurs, then there
is a time $\tau$ such that $H(\tau)$ is a scalar multiple of a permutation matrix
of order two
with zero diagonal.  (And hence the number of vertices in the graph must be even.)
Finally we present a new infinite class of antipodal distance-regular graphs where
perfect state transfer occurs.

\section{Eigenvalues}
\label{sec:evals}

We are going to derive consequences for the eigenvalues of $A$ when $X$ is periodic,
but first we show that periodicity and perfect state transfer are related.

\begin{lemma}\label{lem:}
    If perfect state transfer from $u$ to $v$ occurs at time $\tau$, then
    $X$ is periodic at $u$ and $v$ with period $2\tau$.
\end{lemma}

\proof
Assume $H=H(\tau)$ and suppose that
\[
    H(\tau)e_u = \ga e_v,
\]
where $\|\ga\|=1$. Then $H_{v,u}=\ga$ (because $H(t)$ is symmetric)
and therefore $He_v=\ga e_u$. Now
\[
    \ga^2 e_u =\ga He_v =H^2 e_u
\]
and similarly $H^2e_v=e_v$.\qed

Our main tool in this paper is the spectral decomposition of a symmetric matrix.
Suppose $A$ is a symmetric matrix with eigenvalues $\seq\th1d$, and let $E_r$
be the matrix that represents orthogonal projection on the eigenspace associated
with $\th_r$.  If $f$ a complex-valued function defined on the eigenvalues of $A$,
we have
\[
	f(A) =\sum_{r=1}^d f(\th_r)E_r.
\]
In particular,
\[
	H(t) =\sum_r \exp(it\th_r)E_r.
\]
Note that the scalars $f(\th_r)$ are the eigenvalues of $f(A)$ and that $\sum E_r=I$.
We will also make use of the fact that $f(A)$ is necessarily a polynomial in $A$
(because each projection $E_r$ is).

Assume now that $A$ is the adjacency matrix of a graph $X$ on $v$ vertices.
If $x$ is a vector in $\re^v$, we say that an eigenvalue $\th_r$ of $X$ is
in the \textsl{eigenvalue support} of $x$ if $E_rx\ne0$.
The case of interest to
us will be when $x$ is the characteristic vector of a subset $S$ of $V(X)$.
The eigenvalue support of a non-zero vector cannot be empty and, if $x\ne0$,
we define the \textsl{dual degree} of $x$ to be one less than the size of its
eigenvalue support.

\begin{theorem}
\label{thm:ratio}
	Let $X$ be a graph and let $u$ be a vertex in $X$ at which $X$ is periodic.
	If $\th_k$, $\th_\ell$, $\th_r$, $\th_s$ are eigenvalues in the support
	of $e_u$ and $\th_r\ne\th_s$,
	\[
		\frac{\th_k-\th_\ell}{\th_r-\th_s}\in\rats.
	\]
\end{theorem}

\proof
Suppose the period of $X$ at $1$ is $\tau$.  Then
\[
	H(\tau)_{1,1} =\ga
\]
where $|\ga|=1$.  Since $H(\tau)$ is unitary, this implies that we have
\[
	H(\tau) =\pmat{\ga&0 \\0&H_1}
\]
where $H_1$ is unitary.  Hence
\[
	\ga e_1 = H(\tau)e_1 =\sum_r \exp(i\tau\th_r) E_r e_1
\]
The non-zero vectors $E_re_1$ are pairwise orthogonal and so linearly
independent, whence we see that
\[
	\exp(i\tau\th_r) =\ga
\]
for each eigenvalue $\th_r$ in the support of $e_1$.

Consequently of $\th_r$ and $\th_s$ are in the eigenvalue support of $e_1$, then
\[
	\exp(i\tau(\th_r-\th_s)) =1
\]
and so $\tau(\th_r-\th_s)$ is an integral multiple of $2\pi$.  This proves the theorem.\qed

\begin{corollary}
	If $X$ is periodic at $u$ and there are two integer eigenvalues in the support
	of $e_u$, then all eigenvalues in its support are integers.
\end{corollary}

\proof
Suppose $k$ and $\ell$ are distinct integer eigenvalues in the support of $e_u$.  By
the theorem
\[
	\frac{\th_r-\ell}{k-\ell}\in\rats
\]
and therefore $\th_r$ is rational. Since $\th_r$ is an algebraic integer, it must
be an integer.\qed

The condition in the previous theorem is known as the \textsl{ratio condition}.
Christandl et al \cite{christandl-2005-71}
stated that if perfect state transfer takes place on a path, then its eigenvalues must
satisfy the ratio condition.  Similarly Saxena et al \cite{saxena-2007} proved that
if a circulant graph is periodic, its eigenvalues must satisfy the ratio condition.

We can derive these results from ours, taking the second first.  A circulant
graph is vertex transitive and so if $E_r e_u=0$ for some vertex $u$, then
for all vertices $u$, we have $E_r e_u=0$.  But this implies that $E_r=0$,
and so we conclude that the support of $e_u$ is the set of all eigenvalues of $X$.
Hence the eigenvalues of a periodic circulant satisfy the ratio condition.

Now paths.  From the spectral decomposition we have
\[
	(tI-A)^{-1}_{u,u} =\sum_r (t-\th_r)^{-1}(E_r)_{u,u}
\]
while standard matrix theory yields that
\[
	(tI-A)^{-1}_{u,u} = \frac{\phi(X\diff u,t)}{\phi(X,t)}.
\]
Since
\[
	(E_r)_{u,u} = e_u^T E_r e_u =e_u^T E_r^2e_u =e_u^TE_r^TE_re_u,
\]
we see that $(E_r)_{u,u}$ is not zero if and only if $E_re_u$ is not zero, that
is, if and only if $\th_r$ is in the support of $e_u$.
Therefore the dual degree of $e_u$ is
equal to the number of poles of the rational function $\phi(X\diff u,t)/\phi(X,t)$,
less 1.  The characteristic polynomial of the path $P_n$ on $n$ vertices
satisfies the recurrence
\[
	\phi(P_{n+1},t) =\phi(P_n,t) - \phi(P_{n-1},t),\qquad (n\ge1);
\]
since $\phi(P_0,t)=1$ and $\phi(P_1,t)=t$, it follows that $\phi(P_n,t)$ and
$\phi(P_{n-1},t)$ are coprime.  So our rational function has $n$ distinct zeros,
and again we find that the eigenvalue support of a vertex of degree one in a path
is the set of all eigenvalues of the path.

\section{Periodicity}

It follows from the spectral decomposition
\[
	H(\tau) =\sum_r \exp(i\tau\th_r) E_r
\]
that if the eigenvalues of $X$ are integers, then
\[
	H(2\pi) =\sum_r E_r =I
\]
and therefore $X$ is periodic with period $2\pi$.  This shows that the path $P_2$
is periodic.  However, as Christandl et al noted, $P_3$ is periodic and its
eigenvalues are
\[
	-\sqrt2,0,\sqrt2.
\]
Thus integrality is sufficient, but not necessary.  There is a more general
condition which is necessary and sufficient.

\begin{theorem}\label{thm:period}
	Suppose that the eigenvalues of $X$ are $\seq\th1d$.  Then $X$ is periodic
	if and only if the ratio of any two non-zero eigenvalues is rational.
\end{theorem}

\proof
We first show that the given condition is sufficient.  If it holds and $\th_s\ne0$,
there are rational numbers $\seq{q}1r$ such that $\th_r= q_r\th_s$ and
therefore there is an integer $N$ and integers $\seq{k}1r$ such that $N\th_r =k_r\th_s$.
Hence if we take
\[
	\tau = \frac{2N\pi}{\th_s}
\]
we have
\[
	\exp(i\tau\th_r) =\exp(2ik_r\pi) =1
\]
for all $r$, and consequently $X$ is periodic.

Conversely, suppose $X$ is periodic with period $\tau$.
Then $H(\tau)$ is diagonal and commutes with $A$. If we view the diagonal
of $H(\tau)$ as a function on $V(X)$, it must be constant on the connected
components of $X$.

If $X$ is connected it follows that there is a complex
number $\ga$ such that $|\ga|=1$ and $H(\tau)=\ga I$.
Since the sum of the eigenvalues of $X$ is zero,
the product of the eigenvalues of $H(\tau)$ is one, and thus if
$v=|V(X)|$, then
\[
	1 = \det(H(\tau)) =\ga^v.
\]
Therefore $H(v\tau)=I$, and so all eigenvalues of $H(v\tau)$ are equal to one.
If $X$ is not connected, we can apply this argument to deduce that each
diagonal entry of $H(\tau)$ is an $m$-th root of unity, for some integer $m$,
and therefore if $w$ is the least common multiple of the sizes of the
connected components of $X$, then $H(w\tau)=I.$

Hence
\[
	\exp(iw\tau\th_r) =1
\]
for all $r$, and there are integers $\seq\ell1r$ such that
\[
	w\tau\th_r =2\ell_r\pi.
\]
This implies that, for all $r$ and $s$,
\[
	\frac{\th_r}{\th_s} = \frac{\ell_r}{\ell_s} \in\rats.\qed
\]

\begin{lemma}\label{lem:}
    If the ratios of the non-zero eigenvalues of $X$ are rational, then the
    square of an eigenvalue is an integer.
\end{lemma}

\proof
If the condition of the lemma holds and $\th_s\ne0$, there are rationals
$\seq{q}1r$ such that $\th_r =q_r\th_s$.
Since the sum of the squares of the eigenvalues of $A$
is equal to twice the number of edges, we see that $\th_s^2$ is rational.
Since $\th_s$ is an algebraic integer, $\th_s^2$ is thus a rational algebraic
integer, and accordingly it is actually an integer.\qed

\begin{corollary}\label{cor:period}
    A graph is $X$ is periodic if and only if either:
    \begin{enumerate}[(a)]
        \item
        The eigenvalues of $X$ are integers, or
        \item
        The eigenvalues of $X$ are rational multiples
        of $\sqrt\De$, for some square-free integer $\De$.
    \end{enumerate}
    If the second alternative holds, $X$ is bipartite.
\end{corollary}

\proof
The stated conditions are sufficient, and so we show they are necessary.

Suppose $k$ is a non-zero integer eigenvalue of $X$.  Then $\th_r/k\in\rats$
and so  $\th_r\in\rats$.  Since $\th_r$ is an algebraic integer, it is therefore
an integer. If no non-zero eigenvalue of $X$ is an integer, then
by the theorem each eigenvalue
is a rational multiple of the spectral radius $\th_1$, say $\th_i = m_i\th_1$.

Also by the theorem $\th_1^2\in\ints$, whence it follows that $t^2-\th_1^2$ is
the minimal polynomial of $\th_1$ over the rationals, and therefore $-\th_1$
is an eigenvalue of $X$. By the Perron-Frobenius theory, this implies that $X$
is bipartite.\qed

\section{Coherent Algebras}

A \textsl{coherent algebra} is a real or complex algebra of matrices that
contains the all-ones matrix $J$ and is closed under Schur multiplication,
transpose and complex conjugation.  (The \textsl{Schur product} $A\circ B$
of two matrices $A$ and $B$ with same order is defined by
\[
(A\circ B)_{i,j} = A_{i,j}B_{i,j}.
\]
It has been referred to as the ``bad student's product''.)
A coherent algebra always has a basis of $01$-matrices and this is unique, given
that its elements are $01$-matrices.  This set of matrices determines a set of
directed graphs and the combinatorial structure they form is known as a
\textsl{coherent configuration}.  When we say that a graph $X$ is a
``graph in a coherent algebra'', we mean that $A(X)$ is a sum of distinct
elements of the $01$-basis.

A coherent algebra is \textsl{homogeneous} if the identity matrix is an element
of its $01$-basis. If $M$ belongs to a homogeneous coherent algebra, then
\[
M\circ I = \mu I
\]
for some scalar $\mu$. Hence the diagonal of any matrix in the algebra is
constant. If $A$ is a $01$-matrix in the algebra, the diagonal entries of $AA^T$
are the row sums of $A$. Therefore all row sums and all column sums of any matrix
in the $01$-basis are the same, and therefore this holds for each matrix in the
algebra. In particular we can view the non-identity matrices as adjacency
matrices of regular directed graphs. Any directed graph in a homogeneous coherent
algebra must be regular.

We consider two classes of examples.  First, if $\cP$ is a set of permutation
matrices of order $n\times n$, then the commutant of $\cP$ in
$\textrm{Mat}_{n\times n}(\cx)$
is Schur-closed.
Therefore it is a coherent algebra, and this algebra is homogeneous
if and only the permutation group generated by $\cP$ is transitive.
Thus any graph whose automorphism group acts transitively on its vertices
belongs to a coherent algebra

For second class of examples, if $X$ is a graph we define the \textsl{$r$-th
distance graph} $X_r$ of $X$ to be the graph with $V(X_r)=V(X)$, where vertices
$u$ and $v$ are adjacent in $X_r$ if and only if they are at distance $r$ in $X$.
So if $X$ has diameter $d$, we have distance graphs $\seq X1d$ with corresponding
adjacency matrices $\seq A1d$. If we use $A_0$ to denote $I$, then the graph $X$
is \textsl{distance regular} if the matrices $\seq A0d$ are the $01$-basis of a
homogeneous coherent algebra. (It must be admitted that this is not the standard
definition.) We will refer to the matrices $A_i$ as \textsl{distance matrices}.

A commutative coherent algebra is the same thing as a Bose-Mesner algebra of an
association scheme. We will not go into this here, but we do note that the
coherent algebra belonging to a distance-regular graph is commutative.

\begin{theorem}\label{thm:}
	If $X$ is a graph in a coherent algebra with vertices $u$ and $v$, and
	perfect
	state transfer from $u$ to $v$ occurs at time $\tau$, then $H(\tau)$ is a
	scalar multiple of a permutation
	matrix with order two and no fixed points that lies in the centre of the
	automorphism group of $X$.
\end{theorem}

\proof
First, if $A=A(X)$ where $X$ is a graph in a homogeneous coherent algebra then,
because it is a polynomial in $A$, the matrix $H(t)$ lines in the algebra for all
$t$.
Hence if
\[
|H(\tau)_{u,v}| = 1
\]
it follows that $H(\tau) =\xi P$ for some complex number $\xi$ such that $|\xi|=1$
and some permutation matrix $P$.  Since $A$ is symmetric, so is $H(t)$ for any $t$,
and therefore $P$ is symmetric.  So
\[
P^2 =PP^T =I
\]
and $P$ has order two.  Since $P$ has constant diagonal, its diagonal is zero
and it has no fixed points.  As $P$ is a polynomial in $A$, it commutes with
any automorphism of $X$ and hence is central.\qed

\begin{corollary}\label{cor:evncoh}
	If $X$ is a graph in a coherent algebra with vertices $u$ and $v$ and perfect
	state transfer from $u$ to $v$ occurs at some time, then the number of vertices
	of $X$ is even.
\end{corollary}

\proof
Since $P^2=I$ and the diagonal of $P$ is zero, the number of columns of $P$ is even.\qed

Saxena et al \cite{saxena-2007} proved this corollary for circulant graphs.
A homogeneous coherent
algebra is \textsl{imprimitive} if there is a non-identity matrix in its $01$-basis
whose graph is no connected, otherwise it is \textsl{primitive}.  If the algebra
is the commutant of a transitive permutation group, it is imprimitive if and
only the group is imprimitive as a permutation group.  The above corollary implies that
if perfect state transfer takes place on a graph from a homogeneous coherent algebra,
the algebra is imprimitive.

Note that our corollary holds for any vertex-transitive graph, and for any distance-regular
graph.

\section{Walk-Regular Graphs}

A graph is \textsl{walk regular} if its vertex-deleted subgraphs $X\diff u$
are all cospectral.
We have the following result which implies that any graph in a coherent
configuration is walk regular. It follows immediately from Theorem~4.1
in \cite{MR568777}.

\begin{lemma}\label{lem:wr-diag}
	A graph $X$ with adjacency matrix $A$ is \textsl{walk regular}
	if and only if the diagonal
	entries of $A^k$ are constant for all non-negative integers $k$.
\end{lemma}

Any connected regular graph with at most four distinct eigenvalues is walk
regular (see Van Dam \cite{MR1344560}). Note that a walk-regular graph is
necessarily regular.
The graph in \fref{wr12a} is walk regular but not vertex transitive.
This graph does not lie in a homogeneous
coherent algebra---the row sums of the Schur product
\[
A\circ (A^2-4I)\circ (A^2-4I-J)
\]
are not all the same.

\begin{figure}[htbp]
	\centering
		\includegraphics[scale=0.5]{wr12a.pdf}
	\caption{A Walk-Regular Graph}
	\label{fig:wr12a}
\end{figure}

Suppose perfect state transfer takes place from $u$ to $v$ at time $\tau$.
Then there is a complex number $\ga$ where $|\ga|=1$ such that
\[
	H(2\tau)_{u,v} =H(2\tau)_{v,u} =\ga.
\]
As $|\ga|=1$, we see that $H_{u,u}=0$ and, as $X$ is walk-regular, all diagonal
entries of $H$ are zero and in particular, $\tr(H(\tau))=0$. Since
\[
	H(2\tau)_{u,u} =\ga^2
\]
we also see that that $H(2\tau)=\ga^2I$. We conclude that the eigenvalues of
$H(\tau)$ are all $\pm\ga$ and, since $\tr(H(\tau))=0$, both $\ga$ and $-\ga$
have multiplicity $|V(X)|/2$. (Accordingly $|V(H)|$ must be even.)

If $m_\th$ denotes the multiplicity of the eigenvalue $\th$,
then
\[
\tr(H(t)) =\sum_\th m_\th \exp(it\th)
\]
and thus we may define the Laurent polynomial $\mu(z)$ by
\[
\mu(z) := \sum_\th m_\th z^\th.
\]
Here $z=\exp(it)$ and we have made use of the fact that $X$ is periodic and hence
its eigenvalues are integers.  We call $\mu(z)$ the \textsl{multiplicity enumerator} of $X$.

\begin{lemma}\label{lem:}
	Let $X$ be a walk-regular graph with integer eigenvalues.  If perfect state
	transfer occurs on $X$, then $\mu(z)$ has an eigenvalue on the unit circle
	of the complex plane.
\end{lemma}

\proof
The trace of $H(t)$ is zero if and only if $\mu(\exp(it))=0$.\qed

The characteristic polynomial of the graph in \fref{wr12a} is
\[
(t-4)(t-2)^3t^3(t+2)^5
\]
and its multiplicity enumerator is
\[
\mu(z) =z^{-2}(z^6+3z^4+3z^2+5).
\]
This polynomial has no roots on the unit circle, and we conclude that perfect
state transfer does not occur.

We see no reason to believe that, if perfect state transfer occurs on a
walk-regular graph at time $\tau$, then $H(\tau)$ must be a multiple of
a permutation matrix. (But we do not have an example.)

For more information on walk-regular graphs, see \cite{MR568777}. The
computations in this section were carried out in sage \cite{sage}. Van Dam
\cite{MR1344560} studies regular graphs with four eigenvalues, as we noted these
provide examples of walk-regular graphs.

\section{Hypercubes}

The $n$-cube was one of the first cases where perfect state transfer was shown to
occur (see \cite{christandl-2005-71}). The $n$-cube is an example of a
distance-regular graph, and in this section we establish some of the consequences
of our work for distance-regular graphs.

Suppose $X$ is a distance-regular graph with diameter $d$ and distance matrices
$\seq A0d$. It has long been known that if $X$ is imprimitive then either $X_2$
is not connected or $X_d$ is not connected (see \cite[Chapter~22]{nbagt} or
\cite[Section~4.2]{bcn}). For the $n$-cube, neither $X_2$ nor $X_n$ are
connected. If $X_2$ is not connected then $X$ is bipartite and $X_m$ is not
connected if $m$ is even. Here we will be concerned with the case where $X_d$ is
not connected. In this case it follows from the work just cited that the
components of $X_d$ must be complete graphs of the same size, and that two
vertices of $X$ lie in the same component of $X_d$ if and only if they are at
distance $d$ in $X$. The components are called the \textsl{antipodal classes} of
$X$; note these may have size greater than two. If $X_d$ is not connected, we say
that $X$ is an \textsl{antipodal} distance regular graph.

As already noted the $n$-cubes provide natural examples of distance-regular
graphs. The $n$-cube has diameter $n$, and its vertices can be partitioned into
pairs such that two vertices in the same pair are at distance $n$, while vertices
in different pairs are at distance less than $n$, hence it is antipodal.

Our next lemma is more or less a specialization of \cref{evncoh} to the case of
distance-regular graphs.

\begin{lemma}\label{lem:pfsdrg}
	Suppose $X$ is a distance-regular graph with diameter $d$.  If perfect state
	transfer from $u$ to $v$ occurs at time $\tau$, then $X$ is antipodal with
	antipodal classes of size two and $H(\tau)=X_d$.\qed
\end{lemma}

We outline a proof that perfect state transfer occurs on the $n$-cube.  In the following
section we will use much the same argument to provide a second infinite class of examples.

The $n$-cube gives rise to an association scheme, that is, a commutative coherent
configuration as follows. Let $X$ be the graph of the $n$-cube, and view its
vertices as binary vectors of length $n$. Let $X_i$ be the graph with the same
vertex set as $X$, where two vertices are adjacent if and only if they differ in
exactly $i$ positions. Thus $X_1=X$ and two vertices are adjacent in $X_i$ if and
only they are at distance $i$ in $X$. Let $A_i$ be the adjacency matrix of $X_i$
and set $A_0=I$. Then $\seq A0n$ are symmetric $01$-matrices and
\[
\sm r0n A_r =J.
\]
It is known that these matrices commute and that their span is closed under
matrix multiplication. Hence they form a basis for a commutative coherent algebra
of dimension $d+1$. Since this algebra is commutative and semisimple, it has a
second basis
\[
\seq E0n
\]
of matrix idempotents.  These are real symmetric matrices such that
$E_iE_j =\de_{i,j}$ and
\[
 \sm i0n E_i =I.
\]
These idempotents represent orthogonal projection onto the eigenspaces of
the algebra and hence there are scalars $p_i(j)$ such that
$A_iE_j =p_i(j)E_j$.  Consequently
\[
A_i =A_i I =A_i\sm r0n E_r =\sm r0n p_i(r)E_r.
\]
The scalars $p_i(j)$ are known as the \textsl{eigenvalues} of the scheme.

On the other hand, the matrices $\seq A0n$ are a basis for the algebra and thus
there are scalars $q_i(j)$ such that
\[
E_j =2^{-n}\sm r0n q_j(r)A_r.
\]
The scalars $q_i(j)$ are the \textsl{dual eigenvalues} of the scheme.

If $f$ is any function defined on the eigenvalues of the scheme then
\[
f(A_1) =\sm r0n f(p_1(r))E_r.
\]
For the $n$-cube, $p_1(r)=n-2r$ and accordingly
\[
H(t) = \sm r0n \exp(i(n-2r)t)\,E_r =\exp(int) \sm r0n \exp(-2irt)E_r.
\]
If we set $t=\pi$, we then have
\[
H(\pi) =(-1)^n \sm r0n E_r = (-1)^nI.
\]
Thus the $n$-cube is periodic with period $\pi$.  Now try $n=\pi/2$.  Then
\[
H(\pi/2) =i^n \sm r0n (-1)^r E_r
\]
From, for example, \cite[Section~21.3]{MacSlo} we know that
\[
\sm r0n (-1)^r E_r =A_d
\]
and we conclude that perfect state transfer occurs at time $\pi/2$.

In \cite{jafarizadeh-2007}, Jafarizadeh and Sufiani give examples of perfect state transfer
on distance-regular graphs that arise when a weighted adjacency matrix is used
in place of the ordinary adjacency matrix.

\section{Hadamard Matrices}

We now show how to use a special class of Hadamard matrices to construct
distance-regular graphs of diameter three with perfect state transfer.

The Hadamard matrices $H$ we want must be as follows:
\begin{enumerate}[(a)]
	\item
	Symmetric: $H=H^T$.
	\item
	Regular: all row and column sums are equal.
	\item
	Constant diagonal: all entries on the diagonal are equal.
\end{enumerate}
Suppose $H$ is $n\times n$ and satisfies this constellation of conditions.
If its rows sum to $c$, then $HJ=cJ$ and $H^2J=c^2J$. Since
\[
    H^2 I =HH^T =nI,
\]
whence $c=\pm\sqrt{n}$.  It follows that there are two possibilities:
either the diagonal entries and the rows sums of $H$ have the same
sign, or they do not.  To handle this we assign a parameter $\eps=\eps(H)$
to $H$, by defining $\eps$ to be the sign of the product of the row sum
of $H$ with a diagonal entry.  Whatever the value of $\eps(H)$, the order of
$H$ must be a square.  For a recent useful paper on these Hadamard matrices,
see Haemers \cite{wh-maxen}.

We remark that if the row sums of a Hadamard matrix are all equal then its columns
sums must all be equal as well, and again the order of the matrix must be a square.
Similarly, a symmetric Hadamard matrix with constant diagonal must have square order.
If $H$ and $K$ are regular symmetric Hadamard matrices with constant diagonal,
then so is their Kronecker product $H\otimes K$ and
\[
\eps(H\otimes K) =\eps(H)\eps(H).
\]
Finally if $H$ is a regular Hadamard matrix of order $n\times n$ with constant diagonal,
and $P$ is the permutation matrix on $\re^n\otimes\re^n$ defined by
\[
P(u\otimes v) =v\otimes u,
\]
then $P(H\otimes H^T)$ is a symmetric regular Hadamard matrix with constant diagonal.

Suppose $H$ is a regular symmetric $n\times n$ Hadamard matrix with constant diagonal.
Replacing $H$ by $-H$ is necessary, we may assume that the diagonal entries
of $H$ are equal to $1$, and then the matrix
\[
\frac12(J+H)-I
\]
is the adjacency matrix of a regular graph (in fact a strongly regular graph).
Unfortunately the graphs that result are not the ones we need.
Our aim rather is to construct an antipodal distance-regular graph of diameter three
on $2n$ vertices from $H$.  (These graphs are member of a class known as
\textsl{regular two-graphs}.  An introductory treatment is offered
in \cite[Chapter~11]{MR1829620})

We construct our graphs as follows.  Let $H$ be a symmetric Hadamard matrix
of order $n\times n$ with constant row sum and with all diagonal entries
equal to 1.  We construct a graph $X(H)$ with vertex set
\[
\{1,\ldots,n\} \times\{0,1\},
\]
where $(i,a)$ is adjacent to $(j,b)$ if and only if $i\ne j$ and
\[
H_{i,j} = (-1)^{a+b}
\]
Clearly this gives us a regular graph $X(H)$ on $2n$ vertices with valency $n-1$.
It is not hard to check that its diameter is three, and two vertices are
at distance three if and only if they are of the form
\[
(i,0),\ (i,1)
\]
for some $i$.  Suppose $A$ is the adjacency matrix of $X(H)$.  The space
$S$ of vectors that are constant on the antipodal classes of $X(H)$ has dimension $n$
and the matrix representing the action of $A$ on $S$ is the $n\times n$ matrix $J_I$.
Its eigenvalues are $n-1$ and $-1$ with respective multiplicities 1 and $n-1$.
The orthogonal complement of $S$ is also $A$-invariant and the matrix representing
the action of $A$ is $H$.

If we write $H$ as $I+C-D$ where $C$ and $D$ are 01-matrices, then
\[
A(X(H)) =C\otimes\pmat{1&0\\0&1} +D\otimes\pmat{0&1\\1&0}
\]
and the eigenvalues of $A$ are the eigenvalues of the matrices
\[
C+D =J-I,\quad C-D =H-I.
\]
Therefore the eigenvalues of $A$ are $n-1$, $-1$ and the eigenvalues
of $H-I$. Since $H^2=nI$, we see that the eigenvalues of $A$
are
\[
n-1,\ -1,\ \sqrt{n}-1,\ -\sqrt{n}-1.
\]
If $a$ is the multiplicity of $\sqrt{n}$ as an eigenvalue of $F$, then
\[
n =\tr(A) =a\sqrt{n} -(n-a)\sqrt{n}
\]
whence
\[
a = \frac12(n+\sqrt{n}).
\]
it follows that
\begin{align*}
\mu(z) &= z^{n-1} +(n-1)z^{-1} +\frac12(n+\sqrt{n}) z^{\sqrt{n}-1}
									+\frac12(n-\sqrt{n}) z^{-\sqrt{n}-1}\\
       &=z^{-\sqrt{n-1}} (z^{n+\sqrt{n}} +(n-1)z^{\sqrt{n}}
				+\frac12(n-\sqrt{n})z^{2\sqrt{n}}+\frac12(n-\sqrt{n})).
\end{align*}
If $z^{\sqrt{n}}=-1$, then $z^{2\sqrt{n}}=1$ and, since $\sqrt{n}$ is even,
\[
    z^{n-\sqrt{n}} =(z^{\sqrt{n}})^{\sqrt{n}} z^{-\sqrt{n}} =-1.
\]
Therefore if $z^{\sqrt{n}}=-1$, then $\mu(z)=0$ and if $z=e^{it}$ then
\[
    H(t) = -E_{n-1} -E_{(-1)} +E_{(-1+\sqrt{n})} +E_{(-1-\sqrt{n})}.
\]

Let $F$ be the direct sum of $n$ copies of the matrix
\[
    \frac12\pmat{1&1\\ 1&1}.
\]
Then $\rk(F)=n$ and $\col(F)$ is the space of functions constant on the antipodal classes
of $X(H)$.  Since $F^2=F$, we find that $E_{(-1)}=F-E_{n-1}$ and consequently
\[
    H(t) =I -2F.
\]
So $H(t)$ is a permutation matrix and we have perfect state transfer.

Infinitely many examples of regular symmetric Hadamard matrices with constant diagonal
are known, in particular they exist whenever the order $n$ is a power of four.
Haemers \cite{wh-maxen} provides a summary of the state of our knowledge.

\section{Questions and Comments}

The most obvious problem remaining is to characterize the pairs of vertices
where perfect state transfer takes place.

Angeles-Canul et al \cite{acnoprt} et al give examples of graphs where perfect state
transfer occurs but the underlying graph is not periodic; this answered a question
we raised in an earlier version of this paper.

Facer, Twamley and Cresser \cite{facer-2007} construct certain Cayley
graphs for $\ints_2^d$ on which perfect state transfer; the resulting
family of graphs include the $n$-cubes. In \cite{bgs}, it is shown
that if $C$ is the connection set
for a Cayley graph for $\ints_2^d$ and $\sg$ is the sum of the elements of $C$
and $\sg\ne0$, then we have perfect state transfer from $v$ to $v+\sg$ for
each element $v$ of $\ints_2^d$.
The $d$-cube has $2^d$ edges. It would be interesting to have a good estimate
for the minimum number of edges in a graph where perfect state transfer between
two vertices at distance $d$ occurs.

I would like to acknowledge a number of very useful discussions with
Simone Severini  on the topics of this paper.


text:
\begin{document}

\title[Bilinear paraproducts]{Some endpoint estimates for bilinear paraproducts and applications}
\keywords{}
\subjclass[2000]{}
\author[S.~Rodr\'iguez-L\'opez]{Salvador Rodr\'iguez-L\'opez}
\address{Department of Mathematics, Uppsala University, Uppsala, SE 75106,  Sweden}
\email{salvador@math.uu.se}
\urladdr{http://www.math.uu.se/~salvador}
\author[W.~Staubach]{Wolfgang Staubach}
\email{wulf@math.uu.se}
\urladdr{http://www.math.uu.se/~wulf}

\thanks{The first author has been partially supported by the Grant MTM2010-14946.}

\subjclass[2010]{Primary 35S05, 35S30, 35S50, 42B35; Secondary 42B15, 42B20, 42B25}

\keywords{Bilinear paraproducts, BMO-type spaces, Bilinear Fourier integral operators, Bilinear Multipliers, Kato-Ponce estimates}

\maketitle

\begin{abstract}
In this paper we establish the boundedness of bilinear paraproducts on local BMO spaces. As applications, we also investigate the boundedness of bilinear Fourier integral operators and bilinear Coifman-Meyer multipliers on these spaces and also obtain a certain end-point result concerning Kato-Ponce type estimates.
\end{abstract}

\section{Introduction}
This paper is mainly concerned with end-point estimates for bilinear paraproducts of the form
\begin{equation}\label{intro:const coeff para}
 \Pi(f,g)(x):= \int_0^\infty Q_t f(x)\, P_tg(x) m(t)\frac{\dd t}{t}
\end{equation}
or
\begin{equation}\label{intro:var coeff para}
\PP(f,g)(x):= \int_0^1 Q_t f(x)\, P_tg(x) m(t,x)\frac{\dd t}{t},
\end{equation}
where $P_t$ and $Q_t$ are standard frequency localisation operators. More specifically, we are interested in studying the behaviour of these paraproducts when the functions $f$ and $g$ belong to various local or global BMO classes. In this connection the basic results are due to L. Grafakos and R. Torres \cite{GT} which encompass the main end-point estimates regarding the boundedness of multilinear paraproducts. In particular Grafakos and Torres show the $L^\infty\times L^\infty \to \BMO$ boundedness of bilinear Calder\'on-Zygmund operators, but the case where the functions $f$ and $g$ belong to BMO-type spaces is not covered by \cite{GT}, or any other investigations that we are aware of.
As a matter of fact, this paper stems from our investigation of the problem of boundedness of bilinear Fourier integral operators \cite{RRS}, where it was shown that there exist amplitudes $\sigma(x,\xi,\eta)\in S^{0}_{1,0}(1,2)$ and non-degenerate phase functions $\phase_1$ and $\phase_2$ (see Definitions \ref{defn of hormander symbols} and \ref{defn:phase}) for which the associated one dimensional bilinear Fourier integral operator given by
\begin{equation*}
T^{\phase_1,\phase_2}_{\sigma}(f,g)(x) = \iint \sigma(x,\xi,\eta)\widehat{f}(\xi)\widehat{g}(\eta)e^{i\phase_1(x,\xi)+i \phase_2(x,\eta)} \ddd\xi \ddd\eta,
\end{equation*}
 fails to be bounded from $L^\infty\times L^\infty\to \BMO$. This rather surprising fact prompted us to search for alternative spaces for which a modification of the aforementioned negative result is valid. However, in doing so, we soon entered a rather unexplored territory which included at one end, the study of certain endpoint estimates for bilinear paraproducts about which little was known, and at the other end, the study of exotic function spaces which didn't exist in the literature.

Therefore, we had to deal with various issues which were not a-priori related to the study of multilinear operators. However as a bi-product, our investigation yields a new characterisation for the local BMO space (Theorem \ref{cor:bmo}). The definition of the new function spaces (see e.g. Definitions \ref {defn: BMO_w} and \ref{defn: X_w}) enable us to prove the boundedness of bilinear paraproducts of the form \eqref{intro:const coeff para} and \eqref{intro:var coeff para} which have been established in Theorem \ref{thm:main} and Theorem \ref{xdependent version} respectively. Since we needed the boundedness of linear Fourier integral operators on local BMO and we were not able to locate such a result anywhere in the literature, a proof was provided in Theorem \ref{eq:thm_bmo}. This theorem is also used in proving one of our main results concerning the boundedness of one dimensional bilinear Fourier integral operators (Theorem \ref{thm:1d_FIO}).\\

The paper also deals with two other issues, the first is the bilinear Coifman-Meyer multipliers, and the second is the problem of end-point Kato-Ponce estimates. From the point of view of this paper, these two problems are intimately connected. The problem of finding the least number of derivatives required for the validity of boundedness of the bilinear multipliers on one hand, and that of the Kato-Ponce estimate on the other, has been intensively investigated by many authors. Here we only mention those that have been of particular importance and interest to us, which are the papers by N. Tomita \cite{Tomita}, A. Miyachi and N. Tomita \cite{MT} and those of L. Grafakos, D. Maldonado and V. Naibo \cite{GMN}, and L. Grafakos and S. Oh \cite{GOh}. We also got the opportunity of applying our results to the $\bmo$ related end-point estimates for bilinear Coifman-Meyer multipliers (Theorem \ref{thm:H-M}) and Kato-Ponce estimates (Theorem \ref{thm:kato_ponce}).\\

The structure of the paper is as follows. Section \ref{prem} introduces some notations and recalls the definition of some standard function spaces. In Section \ref{functionspaces} we introduce our new function spaces, which are used in the formulation of our main results. Section \ref{local estimates} contains estimates for localization operators, where to our knowledge, the estimate for the operator $P_t$ in Proposition \ref{eq:pt_qt} is new. In Section \ref{paraproducts} we state and prove our main result concerning bilinear paraproducts and also obtain a variable coefficient version thereof in Subsection \ref{xdependent}. In Section \ref{FIOs} the boundedness of one dimensional bilinear Fourier integral operators on BMO-type spaces is proven. Section \ref{kato-ponce} is devoted to the study of bilinear Coifman-Meyer multipliers and our version of the end-point Kato-Ponce estimate.

\section{Preliminaries}\label{prem}

In what follows, we use the notation
\[
	a(tD)f(x):=\int_{\R^n} a(t\xi) \widehat{f}(\xi)e^{ix\cdot \xi} \, \ddd\xi,
\]
for $t>0$ and $a$ in a suitable symbol class, where $ \ddd\xi$ denotes the Lebesgue measure in $\mathbb{R}^n$ normalised by $(2\pi)^{-n}$ . Whenever $t=1$ we shall simply write $a(D)$.

Here and in the rest of the paper, the notation $A\lesssim B$ means that there exist a constant $C$ such that $A\leq C B .$ The notation $A\thickapprox B$ stands for $A\lesssim B$ and $B\lesssim A$.

Given a bump function $\widehat{\Theta},$ such that $\widehat{\Theta}=1$ in a neighbourhood of the origin (where $\widehat{\;}$ denotes the Fourier transform),  the Hardy space $H^1$ is the class of tempered distributions $f$ such that
\begin{equation}\label{eq:H1}
	\norm{f}_{H^1}\thickapprox \int \sup_{t>0}\sup_{\abs{x-y}<t} \abs{\widehat{\Theta}(tD) f(y)}\dd x,
\end{equation}
see e.g. \cites{FS}.
The local Hardy space $h^1$ can be defined in a similar way as $H^1$ (taking the supremum on $0<t<1$ instead), but we will use the following characterisation of $h^1$;\\
A function $f\in h^1(\R^n)$ if and only if $f\in L^1(\R^n)$ and $r_j(f)\in L^1(\R^n)$ for any $j=1,\ldots, n.$ Here $r_j$ denotes the local Riesz transform, which is defined by
\begin{equation}\label{eq:local_Riesz}
	r_j(f)(x)=-i\int \frac{\xi_j}{\abs{\xi}}(1-\widehat{\Theta}(\xi)) \widehat{f}(\xi) e^{i x\cdot\xi}\ddd \xi.
\end{equation}
Moreover
\begin{equation}\label{eq:norm_h1}
	\norm{f}_{h^1(\R^n)}\thickapprox \norm{f}_{L^1}+\sum_{j=1}^n \norm{r_j( f)}_{L^1}.
\end{equation}

We refer the reader to the work of D.~Goldberg \cite{Gol} and the paper of C.~Fefferman and E.M.~Stein \cite{FS} for further properties of $h^1$ and $H^1$ respectively.

The dual of $H^1$ is the space of functions in $\BMO$ (see e.g \cite{FS} or \cite{G}). The dual of $h^1$ is the space $\bmo$ which is the class of locally integrable functions for which
\begin{equation}\label{eq:bmo}
	\begin{split}
	\norm{f}_{\bmo} &:=\norm{f}_{\BMO}+\norm{\widehat{\Theta}(D) f}_{L^\infty}\thickapprox\norm{(1-\widehat{\Theta}(D)) f}_{\BMO}+\norm{\widehat{\Theta}(D) f}_{L^\infty}<+\infty
	\end{split}
\end{equation}
see e.g.~\cites{Gol}. It is worth mentioning that the definitions above do not depend on the choice of the function $\Theta$ (i.e. different choices yield equivalent norms).
\section{Admissible weights and related function spaces}\label{functionspaces}
In our investigation, we will introduce some classes of function spaces depending on certain weights which are required to satisfy certain properties. To this end we define the following class of weights.
\begin{defn}
A positive weight function $w$ defined on $(0,\infty)$ is {\em admissible}, if it has all of the following properties:
\begin{enumerate}[itemindent=0pt,label=\rm\bfseries (A\arabic*), fullwidth]
\item\label{A1} For every $s>0$, $0< \mathfrak{i}(s):=\inf_{t>0} \frac{w(st)}{w(t)}\leq \sup_{t>0} \frac{w(st)}{w(t)}=:\mathfrak{s}(s)<\infty.$
\item\label{A2} For every $t>0$, $w(t)\geq 1$
\item\label{A3} For some $N>0$, $\sup_{t>0} w(t) \brkt{1+ \frac{1}{t}}^{-N}<\infty$.
\item\label{A4} For any closed interval $I\subset (0,\infty)$, $0<\inf_{s\in I} \mathfrak{i}(s)\leq \sup_{s\in I}\mathfrak{s}(s)<\infty.$

\end{enumerate}
\end{defn}

\subsection*{Examples of admissible weights}

\begin{enumerate}[fullwidth]
	\item It is trivial to see that $w(t)=1$ is admissible.
	\item The function $w(t)=1+\log_+ 1/t$ is admissible. Property \ref{A2} is clear from the definition. Moreover, since for any $\epsilon>0$, $\sup _{0<t<1} t^\epsilon w(t)$ is finite and $w$ is bounded for $t>1$, it follows that it satisfies \ref{A3}. Furthermore, it is easy to see that, for any $s\geq 1$ and $t>0$ one has  $(1+\log s)^{-1}\leq \frac{w(st)}{w(t)}\leq 1$.
	Then, if $s\leq 1$, since this particular weight is decreasing, $w(st)\geq w(t)=w((st)/s)\geq (1-\log s)^{-1} w(st)$,
	which yields $\mathfrak{i}(s)\geq  \chi_{(0,1)}(s)+(1+\log s)^{-1}\chi_{[1,\infty)}(s)$ and $\mathfrak{s}(s)\leq 1-\log s \chi_{(0,1)}(s)$. Hence \ref{A1} and \ref{A4} are also satisfied.
\item Given any $\alpha>0$, $w_\alpha(t)= \brkt{1+\log_+ 1/t}^\alpha$ is also admissible.
\end{enumerate}

\begin{cor}\label{cor:equivalence} For any admissible weight $w$, one has that if $t\approx s$ then
$w(t)\approx w(s)$.
\end{cor}
\begin{proof}   The result  follows by combining properties \ref{A1} and \ref{A4}.
 \end{proof}

Let $\psi$ be a function such that $\widehat{\psi}\in \mathcal{C}^\infty_c(\R^n)$ such that $0\not\in \supp \widehat {\psi}$ and such that
\begin{equation}\label{eq:normalisation}
	\int_0^\infty \abs{\widehat{\psi}(t\xi)}^2 \frac{\dd t}{t}\thickapprox 1.
\end{equation}
Given an admissible weight $w$ and $\psi$ as above, let us define
\begin{equation}\label{eq:sigma}
	\sigma_w(\xi)=\brkt{\int_0^\infty \abs{\widehat{\psi}(t\xi)}^2 w^2(t)\frac{\dd t}{t}}^{\frac{1}{2}}.
\end{equation}
For simplicity of notation, we omit the explicit dependency of $\sigma_w$ on $\psi$.

\begin{lem}\label{lem:regularity_sigma} For any $\sigma_w$ defined as in \eqref{eq:sigma} one has that $\sigma_w,\sigma_w^{-1}\in \mathcal{C}^{\infty}(\R^n\setminus\{0\})$ and $\sigma_w^{-1}$ is bounded. Moreover, for any multiindex $\alpha$
\begin{enumerate}[itemindent=0pt,label=\rm\bfseries (\arabic*), fullwidth]
	\item\label{eq:one} $\abs{\partial^\alpha_\xi \brkt{\sigma_w(\xi)}}\lesssim w\brkt{1/\abs{\xi}} \abs{\xi}^{-\abs{\alpha}}$;
	\item\label{eq:two}\label{eq:sima_inv} $\abs{\partial^\alpha_\xi \brkt{1/\sigma_w(\xi)}}\lesssim w\brkt{1/\abs{\xi}}^{-1} \abs{\xi}^{-\abs{\alpha}}$.
\end{enumerate}
In particular, $1/\sigma_w$ is a H\"ormander-Mikhlin multiplier.
\end{lem}
\begin{proof}
Let $\Psi(\xi):=\abs{\widehat{\psi}(\xi)}^2\in \mathcal{C}^\infty_c(\R^n)$, which is supported in an annulus. Substituting this in \eqref{eq:sigma} and changing variables yield
\[
	\begin{split}
	\d^\alpha_\xi \sigma_w^2(\xi)
=\abs{\xi}^{-\abs{\alpha}}\int_0^\infty \d^\alpha_\xi \Psi(t{\rm sgn}(\xi)) t^{\abs{\alpha}} w^2(t/\abs{\xi})\frac{\dd t}{t},
	\end{split}
\]
where ${\rm sgn}(\xi):=\xi/\abs{\xi}$. Since $\Psi$ is supported in an annulus, one has that
$t\thickapprox 1$ in the integral above and therefore Corollary \ref{cor:equivalence} and the boundedness of $\d^\alpha \Psi$ yield
\begin{equation}\label{eq:square}
	\abs{\d^\alpha_\xi \sigma_w^2(\xi) }\lesssim \abs{\xi}^{-\abs{\alpha}} w^2(1/\abs{\xi}).
\end{equation}
Observe that a change of variables, {Corollary \ref{cor:equivalence}} and \eqref{eq:normalisation} imply
\begin{equation}\label{eq:sigma equiv weight}
	\sigma_w(\xi)=\brkt{\int_0^\infty \Psi(t{\rm sgn}(\xi)) w^2(t/\abs{\xi})\frac{\dd t}{t}}^{1/2}\thickapprox w(1/\abs{\xi}).
\end{equation}
By Leibniz rule we have
\[
	\d^{\alpha} \sigma_w(\xi)=\frac{1}{2\sigma_w(\xi)}\brkt{\d^\alpha(\sigma_w^2(\xi))-\sum_{\overset{\alpha=\beta+\gamma}{ \beta\neq 0\neq \gamma}} c_{\beta,\gamma}\,\d^\beta \sigma_w(\xi) \d^{\gamma}\sigma_w(\xi)}.
\]
Therefore, \ref{eq:one} follows by using this and \eqref{eq:square}-\eqref{eq:sigma equiv weight}, followed by an inductive argument.

 To prove \ref{eq:two} we use \ref{eq:one}, \eqref{eq:sigma equiv weight} and apply the Leibniz rule to the function $\sigma_w(\xi)/\sigma_w(\xi)$. Observe also that \ref{A2} and \eqref{eq:sigma equiv weight} imply that $\sigma_w^{-1}$ is bounded, which yields that $\sigma_w^{-1}$ is a H\"ormander-Mikhlin multiplier. The verification of the details are left to the interested reader.
\end{proof}

\begin{prop}\label{prop:independency} Let $\psi_1$ and $\psi_2$ be as above, and let $\sigma_w^1$ and $\sigma_w^2$ be the corresponding symbols defined as in \eqref{eq:sigma}. Then
\[
	\norm{\brkt{\frac{\sigma_w^1}{\sigma_w^2}}(D) f}_{H^1}\thickapprox \norm{f}_{H^1},
 \]
 for any $f\in H^1(\R^n)$.
\end{prop}
\begin{proof}  Let us first observe that from Lemma \ref{lem:regularity_sigma} and the Leibniz formula, it follows that
 for any multiindex $\alpha$
 $$\abs{\partial^\alpha_\xi \brkt{\frac{\sigma_w^1(\xi)}{\sigma_w^2(\xi)}}}\lesssim  \abs{\xi}^{-\abs{\alpha}}.$$
In particular, this implies that ${\sigma_w^1}/{\sigma_w^2}$ is a H\"ormander-Mikhlin multiplier. Therefore, \cite{MR807149}*{Thm. III.7.30} yields that
\[
	\norm{\brkt{\frac{\sigma_w^1}{\sigma_w^2}}(D) f}_{H^1}\lesssim\norm{f}_{H^1},
\]
for any $f\in H^1(\R^n)$. A similar estimate also holds for the operator with symbol  ${\sigma_w^2}/{\sigma_w^1}$. Hence the result follows since
\[
	\norm{f}_{H^1}=\norm{\brkt{\frac{\sigma_w^2}{\sigma_w^1}}(D)\brkt{\brkt{\frac{\sigma_w^1}{\sigma_w^2}}(D) f}}_{H^1}\lesssim \norm{\brkt{\frac{\sigma_w^1}{\sigma_w^2}}(D) f}_{H^1}.
\]
\end{proof}

Let $Z(\R^n)$ be the the complete locally convex space defined by
\[
	Z(\R^n):=\set{f\in \mathcal{S}(\R^n):\, \partial^\alpha \widehat{f}(0)=0 \quad \mbox{for every multi-index $\alpha$}},
\]
as a topological subspace of $\mathcal{S}(\R^n)$. Let $Z'(\R^n)$ be its topological dual, which can be identified as the factor space $\mathcal{S}'(\R^n)/\mathcal{P}(\R^n)$, where $\mathcal{P}(\R^n)$ is the space of polynomials in $\R^n$. That is, $Z'(\R^n)$ consists of the class of distributions of the form $f+P$ where $f\in \mathcal{S}'(\R^n)$ and $P\in \mathcal{P}(\R^n)$.

\begin{prop} Let $m(\xi)$ be either $\sigma_w$ or $1/\sigma_w$. Then the functional given by
\begin{equation*}\label{eq:multiplier}
	I_{m}f(x)=m(D)f
\end{equation*}
defines a continuous linear operator on $Z(\R^n)$. Moreover
\[
	I_ {\sigma_w}I_ {1/\sigma_w}f=I_ {1/\sigma_w}I_ {\sigma_w}f=f,\quad \mbox{for any $f\in Z(\R^n)$}.
\]
\end{prop}
\begin{proof}
We will only show the result in the case $m=\sigma_w$, since the case of $m=1/\sigma_w$ is done similarly. It suffices to show that for any $f\in Z(\R^n)$, $F(\xi)=m(\xi)\widehat{f}(\xi) \in \mathcal{S}(\R^n)$ and $\partial^\alpha F(0)=0$ for any multiindex $\alpha$. To this end, observe first that trivially $F\in\mathcal{C}^{\infty}(\R^n\setminus\{0\})$. So, it remains to study what happens at the origin. Since $f\in Z(\R^n)$,
for any $L>0$ and any $\alpha$, $\lim_{\xi \to 0} |{\partial^\alpha\widehat{f}(\xi)}|{\abs{\xi}^{-L}}=0$. Moreover, for any $\xi\neq 0$,  Leibniz's rule, Lemma \ref{lem:regularity_sigma} and \ref{A3} yield
\begin{equation}\label{eq:F}
	\begin{split}
	|{\partial^\alpha F(\xi)}| &\lesssim w(1/\abs{\xi}) \sum\abs{\xi}^{-\abs{\alpha_1}} \abs{\partial^{\alpha_2} \widehat{f}(\xi)}
	\\
	&\lesssim (1+\abs{\xi})^{N} \sum\abs{\xi}^{-\abs{\alpha_1}} \abs{\partial^{\alpha_2} \widehat{f}(\xi)} ,
	\end{split}
\end{equation}
where the sum runs over all the multiindices $\alpha_1+\alpha_2=\alpha$.
This yields $\lim_{\xi\to 0} |{\partial^\alpha F(\xi)}|=0$. Hence, for any $\alpha$, $\partial^\alpha F(\xi)$ can be extend continuously to $0$ by setting $\partial^\alpha F(0):=0$.
Now \eqref{eq:F} yields $\sup_{\abs{\xi}\geq 1}|{\xi^\beta \partial^\alpha F(\xi)}|<+\infty.$\\
On the other hand, the fact that $\sup_{\abs{\xi}\leq 1}|\xi^\beta \partial^\alpha F(\xi)|$ is finite is a consequence of continuity and the compactness of the unit ball.

The last assertion is a direct consequence of the definition of the operators.
\end{proof}

\begin{cor}\label{cor:technic_1} Let $m(\xi)$ be either $\sigma_w$ or $1/\sigma_w$. The functional $I_{m}(u)$ defined by
\[
	I_{m}(u)(f):=u(I_{m} f),\quad \mbox{for any $u\in Z'(\R^n)$, $f\in Z(\R^n)$},
\]
is a continuous, linear, one-to-one mapping of $Z'(\R^n)$ onto itself. Moreover,
\[
	I_ {\sigma_w}I_ {1/\sigma_w}u=I_ {1/\sigma_w}I_ {\sigma_w}u=u,\quad \mbox{for any $u\in Z'(\R^n)$}.
\]
\end{cor}

\begin{prop}\label{prop:boundedness_H1_BMO} The operator $I_ {1/\sigma_w}$ maps $H^1(\R^n)\to H^1(\R^n)$ and $\mathrm{BMO}(\R^n)\to \mathrm{BMO}(\R^n)$ continuously, and
${\rm Ker }_{H^1}\, I_ {1/\sigma_w}\equiv\{0\}.$
\end{prop}
\begin{proof} The boundedness of the operator $I_ {1/\sigma_w}$ on $H^1$ is a consequence of the boundedness of H\"ormander-Mikhlin multipliers on these spaces. More specifically, the boundedness on $H^1$ can be found in \cite{MR807149}*{Thm. III.7.30}, and using that together with the self-adjointness of $I_ {1/\sigma_w}$ and duality, we obtain the boundedness on BMO.

On the other hand,  since $I_ {1/\sigma_w}$ is bijective in $Z'(\R^n)$ and $H^1(\R^n)\subset Z'(\R^n)$, Corollary \ref{cor:technic_1} and the characterisation of $Z'(\R^n)$ yield that if for $f\in H^1(\R^n)$, ${ I_ {1/\sigma_w} f}=0,$ then $f=0$ (modulo polynomials) i.e. $f=P$ where $P$ is a polynomial. Now since $H^1\subset L^1$, it follows that $P=0$ and therefore  $I_ {1/\sigma_w}$ restricted to $H^1(\R^n)$ is injective.
\end{proof}

The previous proposition allows us to define the following Hardy-Sobolev-type space, $H^1_{1/\sigma_w}(\R^n)$.
\begin{defn}\label{defn:hardy_sobolev} We shall denote by $H^1_{1/\sigma_w}(\R^n)$ the space
\[
	H^1_{1/\sigma_w}(\R^n):=\set{f\in H^1(\R^n):\, f=I_{1/\sigma_w} h,\, h\in H^1(\R^n)},
\]
endowed with the norm
\[
	\norm{f}_{H^1_{1/\sigma_w}(\R^n)}=\norm{h}_{H^1(\R^n)}.
\]
\end{defn}

\begin{prop} The space $(H^1_{1/\sigma_w},\norm{\cdot}_{H^1_{1/\sigma_w}})$ is a Banach space.
\end{prop}
\begin{proof} It is easy to see that $\norm{\cdot}_{H^1_{1/\sigma_w}}$ is a norm. For proving the completeness one observes that for any Cauchy-sequence $\{f_n\}_n\subset H^1_{1/\sigma_w}$, there exists a Cauchy-sequence $\{h_n\}_n \subset H^1(\R^n)$ such that, for any $n\geq 1$, $I_{1/\sigma_w} h_n=f_n$. By completeness of $H^1(\R^n)$, there exists $h\in H^1(\R^n)$ such that $\lim_n h_n=h$. Then, defining $f=I_{\sigma_w} h\in H^1_{1/\sigma_w}$ and, by the linearity of $I_{1/\sigma_w}$ one has $\lim_n \norm{f_n-f}_{H^1_{1/\sigma_w}}=0$.
\end{proof}

The previous result and the fact that $I_ {1/\sigma_w}$ is an isomorphism in $Z(\R^n)$ implies that
$$
	Z(\R^n)\subset H^1_{1/\sigma_w}(\R^n)\subset H^1(\R^n).
$$

 In the following proposition, we show that the space $H^1_{1/\sigma_w}(\R^n)$ is well-defined. By this we mean that the definition \ref{defn:hardy_sobolev} above depends only on $w$ and not on the underlying function $\psi$, in the sense that different choices of $\psi$ induce equivalent norms.
\begin{prop}
Let $\psi_1$ and $\psi_2$ be as above, and let $\sigma_w^1$ and $\sigma_w^2$ be the corresponding symbols defined as in \eqref{eq:sigma}. Then $H^1_{1/\sigma_w^1}(\R^n)=H^1_{1/\sigma_w^2}(\R^n)$ and for any $f\in H^1_{1/\sigma_w^1}(\R^n)$
\[
	\norm{f}_{H^1_{1/\sigma_w^1}(\R^n)}\thickapprox
	\norm{f}_{H^1_{1/\sigma_w^2}(\R^n)},
 \]
 with constants depending on $\psi_1$ and $\psi_2$.
\end{prop}
\begin{proof} Let $f\in H^1_{1/\sigma_w^1}(\R^n)$ and let $h\in H^1(\R^n)$ such that  $f=I_{1/\sigma_w^1} h$. Then
\[
	f=I_{1/\sigma_w^2} (I_{\sigma_w^2/\sigma_w^1} h),
\]
 and by  Definition \ref{defn:hardy_sobolev} and Proposition
\ref{prop:independency} one has
 \[
 	\norm{f}_{H^1_{1/\sigma_w^2}(\R^n)}=\norm{I_{\sigma_w^2/\sigma_w^1} h}_{H^1(\R^n)}\thickapprox \norm{h}_{H^1(\R^n)}=
 	\norm{f}_{H^1_{1/\sigma_w^1}(\R^n)}.
 \]
\end{proof}
From the definition of the space $H^1_{1/\sigma_w}(\R^n)$, it follows that $I_{1/\sigma_w}$ is an isometric isomorphism between $H^1(\R^n)$ and $H^{1}_{1/\sigma_w}(\R^n)$. The inverse of $I_{1/\sigma_w}$ is obtained by its restriction to $H^{1}_{1/\sigma_w}(\R^n)$ and is denoted by $S_{\sigma_w}$. This implies that the corresponding dual spaces are isomorphic. More precisely, for any $\Lambda\in (H^1_{1/\sigma_w}(\R^n))^*$ there exists a unique (modulo constants) $b_\Lambda \in \BMO$ given by  $b_\Lambda=I_{1/\sigma_w}\Lambda$, such that for any $h\in H^1(\R^n)$
 \[
 	 \esc{\Lambda, I_{1/\sigma_w} h}=\esc{b_\Lambda,h},
 \]
 and conversely, for any $b\in \BMO$ there exists $S_{\sigma_w}(b):=\Lambda_b\in  (H^1_{1/\sigma_w}(\R^n))^*$ such that
  \[
 	 \esc{\Lambda_b, I_{1/\sigma_w} h}=\esc{b,h}.
 \]
 This motivates the following definition:
 \begin{defn}\label{defn: BMO_w} We define the space $\BMO_{\sigma_w}$ as
\begin{equation*}
	\BMO_{{\sigma_w}}:=\set{\Lambda\in \mathcal{S}'(\R^n):\quad \norm{I_{1/\sigma_w}\Lambda}_{\BMO}<+\infty},
\end{equation*}
and equip it with the norm
\[
	\norm{\Lambda}_{\BMO_{{\sigma_w}}}:=
	\norm{I_{1/\sigma_w}\Lambda}_{\BMO}.
\]
\end{defn}
As such, one has that $(H^1_{1/\sigma_w}(\R^n))^*=\BMO_{\sigma_w}$. { Moreover, Proposition \ref{prop:boundedness_H1_BMO} yields that $\BMO\subset \BMO_{\sigma_w}$. }

\section{Estimates for localisation operators }\label{local estimates}
Let $\phi, \psi$ be Schwartz class functions, with spectrum included  in a ball and an annulus around $0$ respectively, and $\int \phi(x)\dd x\neq 0$. For any $0<t<\infty$, one defines the frequency localisation operators $P_t$ and $Q_t$ as
\begin{equation}\label{eq:pt_qt}
	 P_t f=\widehat{\phi}(tD) f, \qquad Q_t f=\widehat{\psi}(tD) f.
\end{equation}
In what follows, we shall make use of the following proposition.
\begin{prop}
There exists a constant $C$ such that for any $f\in \BMO$ and any $t>0$
\begin{equation}\label{eq:Q1}
	\norm{Q_t f}_{L^\infty}\leq C \norm{f}_{\BMO}.
\end{equation}
Moreover, for any $f\in \bmo$
\begin{equation}\label{eq:Pt}
	\norm{P_t f}_{L^\infty}\lesssim \brkt{1+\log_+ \frac{1}{t}} \norm{f}_{\bmo}.
\end{equation}
 \end{prop}

\begin{proof}
The statement \eqref{eq:Q1} is a classical result and could be found in \cite{S}*{p. 161}.
So it remains to prove \eqref{eq:Pt}. Without loss of generality we can assume that $\widehat{\phi}$ is equal to one in a neighbourhood of the origin. If not, we can find a compactly supported smooth function function $\widehat{\varphi}$ such that is equal to one on the support of $\widehat{\phi}$. In this way we can write $P_t f=P_t \tilde{P_t} f$, where $\tilde{P_t} f=\widehat{\varphi}(tD) f$, which yields
\[
	\norm{P_t \tilde{P_t} f}_{L^\infty}\leq \int \abs{\phi(x)}\dd x \norm{\tilde{P_t} f}_{L^\infty}.
\]
So, it would be sufficient to prove the result for $\tilde{P_t} f$.

   Observe that for any $0<t<1$
\[
	P_1f(x)-P_tf(x)=\int_t^1 \int  \nabla_\xi \widehat{\phi}(s\xi)\cdot (s\xi) \widehat{f}(\xi)\, e^{ix\cdot \xi}\, \ddd \xi \frac{\dd s}{s}.
\]
If we now set  $\widehat{\Psi}(\xi)=\nabla_\xi\widehat{\phi}(\xi)\cdot \xi$, then  $\widehat{\Psi}$ is a smooth function supported in an annulus around the origin, because $\widehat{\phi}$ is compactly supported { and equal to one in a neighbourhood of the origin}. Then we have that
\[
	P_1f(x)-P_tf(x)=\int_t^1 \widehat{\Psi}(sD) f(x)\frac{\dd s}{s}.
\]
Therefore, using \eqref{eq:bmo} and \eqref{eq:Q1} we obtain
\[
	\norm{P_t f}_{L^\infty}\leq \norm{P_1 f}_{L^\infty}-\log t \norm{f}_{\BMO}\leq (1-\log t)\norm{f}_{\bmo}.
\]
On the other hand, for $t\geq 1$, $\widehat{\phi}(t\xi)$ is supported in a fixed ball independent of $t$. So, we can find an smooth compactly supported  function $\widehat{\Theta}(\xi)$ equal to one in a neighbourhood of the origin such that $P_t f=P_t(\widehat{\Theta}(D)f)$. Thus Minkowskii's inequality and \eqref{eq:bmo} yield
\[
	\norm{P_t f}_{L^\infty}\lesssim \norm{\widehat{\Theta}(D)f}_{L^\infty}\lesssim \norm{f}_{\bmo}.
\]
\end{proof}

\begin{defn}\label{defn: X_w}
For any admissible weight $w$ we shall denote by $X_w$ the space of functions $f\in \BMO$ such that
\begin{equation}\label{eq:X_w}
	\norm{f}_{X_w}:=\norm{f}_{\BMO}+\sup_{t>0} \frac{\norm{P_t f}_{L^\infty}}{w(t)}<+\infty.
\end{equation}
\end{defn}

Note that $X_w$ is nontrivial as it clearly contains $L^\infty$. Also $X_w\subset \BMO$. Moreover, since
\begin{equation}\label{eq:lower_bound}
	\norm{f}_{X_w}\geq \norm{f}_\BMO+\frac{\norm{P_1 f}_{L^\infty}}{w(1)}\thickapprox \norm{f}_{\bmo},
\end{equation}
it follows from \eqref{eq:bmo} that $X_w$ is embedded in  $\bmo$.

Here we shall also observe that for any $f\in \BMO$,
\begin{equation} \label{eq:averages}
	\sup_t \norm{P_tf- c{ \rm Avg}_{B_t(\cdot)} f}_{L^\infty}\lesssim \norm{f}_{\BMO},
\end{equation}
where ${ \rm Avg}_{B_t(x)} f=\frac{1}{\abs{B_t(x)}}\int_{B_t(x)} f(y)\dd y$, $B_t(x)$ is the ball centered at $x$ of radius $t$ and $c=\int \phi(x)\dd x$. The proof of \eqref{eq:averages} is similar to that of \cite{G}*{Prop. 7.1.5.ii}. An easy consequence of this and the fact that $w(t)\geq 1$ is that
\[
	\norm{f}_{X_w}\thickapprox \norm{f}_{\BMO}+\sup_{t>0} \frac{\norm{{ \rm Avg}_{B_t(\cdot)} f}_{L^\infty}}{w(t)}<+\infty.
\]
In this way, we see that the definition of the class $X_w$ does not depend on the different choices of function $\phi$ associated to $P_t$, in the sense that different choices of $\phi$ induce equivalent norms.
Moreover, this expression allows us to rewrite
\begin{equation*}
	\sup_{t>0} \frac{\norm{{ \rm Avg}_{B_t(\cdot)} f}_{L^\infty}}{w(t)}=\sup  \frac{1}{w(r(B))) \abs{B}}\abs{\int_{B} f(x)\dd x},
\end{equation*}
where $r(B)$ is the radius of the ball $B$ and the supremum is taken over the family of all euclidean balls in $\R^n$.
Let us recall that  for $\eta\colon [0,\infty) \rightarrow
[0,\infty)$, the Morrey space $\mathcal{M}^{1,\eta}$ is the class of all locally integrable functions $f$, for which
$$
	\left\|f\right\|_{1,\eta} = \sup \frac{\eta(r(B))}{|B|}\int_{B}\left|f(x)\right|\,\dd x<+\infty,
$$
see  e.g. \cite{MR2833565} and the references therein. Therefore, we have just proved that the class $X_w$ contains the space $\mathcal{M}^{1,\frac{1}{w}}\cap \BMO$.

As a consequence of \eqref{eq:Pt}, \eqref{eq:X_w}, \eqref{eq:lower_bound} we have the following alternative  characterisations of $\bmo$:
\begin{thm}\label{cor:bmo} For $w=1+\log_+ \frac{1}{t}$, we have $X_w=\bmo$ and
\[
	\begin{split}
	\norm{f}_\bmo &\thickapprox \norm{f}_\BMO+ \sup_{t>0} \frac{\norm{P_t f}_{L^\infty}}{1+\log_+ \frac{1}{t}}\thickapprox\norm{f}_{\BMO}+\sup  \frac{1}{w(r(B))) \abs{B}}\abs{\int_{B} f(x)\dd x}.
	\end{split}
\]
\end{thm}

\begin{rem}
For $w(t)=1+\log_+1/t$, we have $\mathcal{M}^{1,\frac{1}{w}}\cap \BMO \subset \bmo$ as a consequence of Corollary \ref{cor:bmo}.
\end{rem}
\section{Boundedness of bilinear Paraproducts}\label{paraproducts}
\subsection{Boundedness of constant coefficient paraproducts}

\begin{defn} Let $P_t$ and $Q_t$ be defined as in \eqref{eq:pt_qt}. A bilinear (constant coefficient) paraproduct $\Pi$ on $\R^d$ is a bilinear operator of the form
$$ \Pi(f,g)(x):= \int_0^\infty Q_t f(x)\, P_tg(x) m(t)\frac{\dd t}{t},$$
where $m$ is a bounded measurable function.
\end{defn}

 These paraproducts are of particular interest when dealing with bilinear Coifman-Meyer multipliers (see Section \ref{kato-ponce}).

\begin{thm}\label{thm:main} For any admissible weight $w$ we have
\[
	\norm{\Pi (f,g)}_{\BMO_{\sigma_w}} \leq C \norm{f}_{\BMO}\norm{g}_{X_w}.
\]

\end{thm}

Before proceeding with the proof, we note that as a consequence of  Theorem \ref{thm:main} we have the following result:

\begin{thm}\label{thm:main_thm} For any admissible weight $w$
\[
	\norm{\Pi (f,g)}_{\BMO_{\sigma_w}} \leq C \norm{f}_{\BMO}\norm{g}_{\mathcal{M}^{1,\frac{1}{w}}\cap \BMO}.
\]
For $w=1$,
\[
	\norm{\Pi (f,g)}_{\BMO} \leq C \norm{f}_{\BMO}\norm{g}_{L^\infty(\R^n)}.
\]
For $w=1+\log_+ \frac{1}{t}$,
\[
	\norm{\Pi (f,g)}_{\BMO_{\sigma_w}} \leq C \norm{f}_{\BMO}\norm{g}_{\bmo}.
\]
\end{thm}
\begin{proof}
The strategy consists of applying Theorem \ref{thm:main} to the various weights under consideration.

The first assertion is a direct consequence of the embedding of $\mathcal{M}^{1,\frac{1}{w}}\cap \BMO$ into $X_w$ observed above.

For the second claim, one observes that $w=1$ yields $X_w=L^\infty(\R^n)$ and $\BMO_{\sigma_w}=\BMO$ with equivalent norms in both cases.

For the third claim, Corollary \ref{cor:bmo} implies that $\bmo=X_w$. This concludes the proof of the theorem.
\end{proof}

The proof of Theorem \ref{thm:main} requires a couple of technical results.  The first one is due to Carleson \cite{Carleson} (see also  \cite{G}*{Theorem 7.3.8}).
\begin{thm}\label{tmh:Carleson_thm}
\begin{enumerate}[itemindent=0pt, label=\bfseries (C\arabic*),fullwidth]
\item \label{Carleson:1}For any $b\in \BMO$, the measure $\dd \mu(x,t)$ given by
\[
	\dd \mu(x,t)=\abs{Q_t b(x)}^2\dd x\frac{\dd t}{t},
\]
is a Carleson measure with norm a constant times $\norm{b}_{\BMO}^2$.

\item \label{Carleson:2} Let $\delta,A>0$. Suppose that $\{K_t\}_{t>0}$ are functions in $\R^n\times \R^n$ that satisfy
\[
	\abs{K_t(x,y)}\leq {A t^{-n}}{\brkt{1+\frac{\abs{x-y}}{t}}^{-n-\delta}},
\]
for all $t>0$ and $x,y\in  \R^n$. Let $R_t$ be the linear operator
\[
	R_tf(x)=\int K_t(x,y) f(y)\dd y.
\]
Suppose that $R_t(1)=0$ for all $t>0$ and that there exists a constant $B>0$ such that
\[
	\int_0^\infty \int \abs{R_t f(x)}^2 \dd x \frac{\dd t}{t}\leq B^2 \norm{f}_{L^2(\R^n)}^2,
\]
for all $f\in L^2(\R^n)$, then for any $b\in \BMO$, the measure $\dd \mu(x,t)$ given by
\[
	\dd \mu(x,t)=\abs{R_t b(x)}^2\dd x\frac{\dd t}{t},
\]
is a Carleson measure with norm a constant times $(A+B)^2\norm{b}_{\BMO}^2$.
\end{enumerate}
\end{thm}

We will also need the following proposition, whose proof can be found in \cite{RRS2}*{Proposition 4.11}.

\begin{prop}\label{prop:monster}
Let $F\in H^1$, $v\in L^\infty_{t,x}$, and $T_{t}$ be the convolution operator given by
\[
	T_{t} (f)(x)=\int f(x-y)\dd \nu_t(y),
\]
with $\{\lambda_t\}_t$  being finite measures such that for some $\delta>0$ and for any $t>0$,
\begin{equation*}
	 \int \brkt{1+\frac{\abs{x-y}}{t}}^{-n-\delta} \dd \abs{\lambda_t}(y)\lesssim \brkt{1+\frac{\abs{x}}{t}}^{-n-\delta}.
\end{equation*}
Let $G(t,x)$ be a measurable function on $\R^{n+1}_+$ such that $$\dd \mu_G(t,x)=\abs{G(t,x)}^2\frac{\dd t}{t}\dd x$$ is a Carleson measure with Carleson norm $\norm{\dd \mu_G}_{\mathcal{C}}$. Then
\[
	\abs{\int\int_0^\infty Q_t T_t F(x)\, G(t,x) v(t,x)\frac{\dd t}{t}\dd x}\leq C \norm{F}_{H^1}\norm{\dd \mu_G}_{\mathcal{C}}^{\frac{1}{2}}\norm{v}_{L^\infty_{t,x}}.
\]
\end{prop}
\subsection*{\bf Proof of Theorem \ref{thm:main}}

Let $w$ be an admissible weight.  Suppose that $\widehat{\psi}$ is supported in $0<\alpha\leq \abs{\xi}\leq \beta$ and $\widehat{\psi}$ has its support in $\abs{\xi}\leq \beta$.  Then we can decompose $\phi=\psi_1+\phi_1$ where $\widehat{\psi_1}$ vanishes in $\abs{\xi}\leq \alpha/8$ and $\widehat{\phi_1}$ is supported in $\abs{\xi}\leq \alpha/4$. Then one can  find a smooth real valued radial function $\psi_2$ with spectrum included in an annulus and equal to one in $\alpha/4\leq \abs{\xi}\leq 2\beta$, and another smooth function $\phi_2$ with $\widehat{\phi_2}$ compactly supported and equal to one in $\abs{\xi}\leq 2\beta$ such that we can decompose $\Pi(F,G)$ as
\[
	\Pi(F,G)=\Pi_1(F,G)+\Pi_2(F,G)
\]
where
 \[
 	\Pi_1(F,G)=\int_0^\infty Q_t^{(2)}\brkt{\brkt{Q_t F} \brkt{P_t^{(1)} G}}  m(t)\frac{\dd t}{t},
 \]
 and
  \[
 	\Pi_2(F,G)= \int_0^\infty P_t^{(2)} \brkt{\brkt{Q_t F} \brkt{Q_t^{(1)} G}} m(t)\frac{\dd t}{t},
 \]
 and $P_t^{(j)}$ and $Q_t^{(j)}$ are associated to $\widehat{\phi_j}$ and $\widehat{\psi_j}$ respectively.

 \subsection*{Estimates for $\Pi_2$}
 We shall prove that $\Pi_2: \BMO\times \BMO \to \BMO$. The Cauchy-Schwarz inequality and \ref{Carleson:1} in Theorem \ref{tmh:Carleson_thm} imply that
 $\abs{Q_t F Q_t^{(1)}G }\dd x\frac{\dd t}{t}$ defines a Carleson measure with Carleson norm bounded by a constant multiple of $\norm{F}_\BMO\norm{G}_\BMO$.

For any $H\in H^1$ we can write
\begin{equation}\label{eq:first_term}
	\left<\Pi_2(F,G),H\right>=\int \int_0^\infty {\brkt{Q_t F} \brkt{Q_t^{(1)} G}} \big(P_t^{(2)} H\big)\, m(t)\frac{\dd t}{t}\dd x,
\end{equation}
where we have implicitly used that $P_t^{(2)}(m(t)H)=\big(P_t^{(2)} H\big)\, m(t).$

Using Fefferman-Stein's result \cite{FS} and \eqref{eq:H1} we obtain
\[
	\begin{split}
	\abs{\left<\Pi_2(F,G),H\right>} &\lesssim \norm{F}_{\BMO}\norm{G}_{\BMO} \norm{m}_{L^\infty} \int \sup_{t>0}\sup_{\abs{x-y}<t} \abs{P_t^{(2)} H(y)}\dd x\\
&\lesssim \norm{m}_{L^\infty} \norm{H}_{H^1}  \norm{F}_{\BMO}\norm{G}_{\BMO}.
	\end{split}
\]
   Therefore, we obtain that for any $F,G\in \BMO$ and $H\in H^1(\R^n)$
 \begin{equation}\label{eq:pi_6}
 	\abs{\left<\Pi_2(F,G),H\right>}\lesssim \norm{m}_{L^\infty} \norm{F}_{\BMO}\norm{G}_{\BMO}\norm{H}_{H^1}.
 \end{equation}
\subsection*{Estimates for $\Pi_1$}

For the admissible weight $w$, if we define $v(t,x)=P_t^{(2)}(G)(x)m(t)/w(t)$, by \eqref{eq:X_w} we have
 that
 \begin{equation}\label{eq:v}
 	\norm{v(t,\cdot)}_{L^\infty}\lesssim \norm{m}_{L^\infty}\norm{G}_{X_w}
\end{equation}
and
 \[
 	\left<\Pi_1(F,G),H\right>=\int\int_0^\infty \brkt{Q_t F} \big({Q_t^{(2)}H }\big)w(t) v(t,x)\frac{\dd t}{t}\dd x,
 \]
where once again we have used the fact that $Q_t^{(2)}(m(t)H)=\big(Q_t^{(2)} H\big)\, m(t).$

Thus, taking $H=I_{{1}/{\sigma_w}} h$ with $h\in H^1(\R^n)$ and $\widehat{\psi_3}$ a smooth annulus supported function such that it is equal to one on the support of $\widehat{\psi}_2$,  one can write
\[
	w(t)Q_t^{(2)}H=R_t(Q_t^{(3)} h),
\]
where
\[
	R_tf(x)=\int K_t(x-y) f(y)\dd y,
\]
with
\begin{equation*}
		K_t(z)= w(t)\int \frac{\widehat{\psi}_2(t\xi)}{\sigma_w(\xi)} e^{iz\cdot\xi}\ddd \xi.
\end{equation*}
We claim that for any $N\geq 1$
\begin{equation} \label{eq:Kernel}
	\abs{K_t(z)}\leq A{t^{-n}}{\brkt{1+\frac{\abs{z}}{t}}^{-2N}}.
\end{equation}
Indeed, integration by parts yields that for any $N\geq 1$
\[
	\begin{split}
	K_t(z)&=t^{-n} w(t)\int \frac{\widehat{\psi}_2(\xi)}{\sigma_w(\xi/t)} e^{i t^{-1}z\cdot\xi}\ddd \xi\\
	&=t^{-n} w(t) \brkt{\frac{\abs{z}}{t}}^{-2N}\int (-\Delta)^{N}\left[\frac{\widehat{\psi}_2(\xi)}{\sigma_w(\xi/t)}\right]e^{i t^{-1}z\cdot\xi}\ddd \xi,
	\end{split}
\]
which implies the estimate above since the Leibniz rule, the fact that $\abs{\xi}\approx 1$ and  \ref{eq:sima_inv} in Lemma \ref{lem:regularity_sigma} yield that
\[
	\abs{(-\Delta)^{N}\left[\frac{\widehat{\psi}_2(\xi)}{\sigma_w(\xi/t)}\right]}\lesssim w(t)^ {-1}.
\]

 Thus we can write
 \begin{equation}\label{eq:estimate_pi4}
 	\begin{split}
 		\left<\Pi_1(F,G),H\right>&=\int\int_0^\infty  \brkt{Q_t F} \brkt{R_t {Q_t^{(3)}} h} v(t,x)\frac{\dd t}{t}\dd x\\
 		&=\int\int_0^\infty  \brkt{{Q_t^{(3)}} h} R_t^*\left[{\brkt{Q_t F} v(t,\cdot)}\right]\frac{\dd t}{t}\dd x.
 		\end{split}
 \end{equation}
 Now observe that
 \[
 	R_t^*\brkt{\brkt{Q_t F} v(t,\cdot)}(x)=\int J_t(x,z) F(z)\dd z,
 \]
 where
 \[
 	J_t(x,z)=\int K_t(x-y)v(t,y) \psi_t(y-z)\dd y.
 \]
 Since $\psi$ is a Schwartz function, \eqref{eq:v} holds and $K_t$ satisfies \eqref{eq:Kernel}, it can be shown that for any $N\geq 1$
 \[
 	\abs{J_t(x,z)}\lesssim  t^{-n}\brkt{1+\frac{\abs{x-z}}{t}}^{-2N}\norm{G}_{X_w}\norm{m}_{L^\infty}.
 \]
 On the other hand, for $F=1$, $R_t^*\brkt{\brkt{Q_t F} v(t,\cdot)}(x)=0$. Moreover, Plancherel's theorem, Corollary \ref{cor:equivalence}, and the Minkowski and Young inequalities yield
 \[
 	\begin{split}
 	\int\int_0^\infty &\abs{R_t^*\brkt{\brkt{Q_t F} v(t,\cdot)}(x)}^2 \frac{\dd t}{t}\dd x\thickapprox
 	\int\int_0^\infty \abs{Q_t^{(2)}\brkt{(Q_t F) v(t,\cdot)}(x)}^2 \frac{\dd t}{t}\dd x\\
 	&\leq \int\int_0^\infty \norm{\psi_{2,t}}_{L^1}^2\norm{v(t,\cdot)}_{L^\infty}^2 \abs{Q_t F(x)}^2
 	 \frac{\dd t}{t}\dd x\lesssim \norm{F}_{L^2}^2 \norm{G}_{X_w}^2\norm{m}_{L^\infty}^2.
 		\end{split}
 \]
Finally \ref{Carleson:2} in Theorem \ref{tmh:Carleson_thm} yields that for any $F\in \BMO$
 \[
 	\abs{R_t^*\brkt{\brkt{Q_t F} v(t,\cdot)}(x)}^2 \frac{\dd t}{t}\dd x
 \]
 defines a Carleson measure with Carleson norm bounded by a constant multiple of $$\norm{G}_{X_w}^2\norm{m}_{L^\infty}^2\norm{F}_{\BMO}^2.$$

Now applying Proposition \ref{prop:monster} to the last term in \eqref{eq:estimate_pi4}, with
 $G(t,x)=R_t^*\left[{\brkt{Q_t F} v(t,\cdot)}\right]$, $T_t f=f$ (that is, taking $\lambda_t$ to be the Dirac measure supported at the origin), and $v(t,x)=1$ yield
 \begin{equation}\label{eq:pi_4}
 	\abs{\left<\Pi_1(F,G),I_{\frac{1}{\sigma_w}}h\right>}\lesssim \norm{h}_{H^1}\norm{F}_{\BMO}\norm{G}_{X_w}\norm{m}_{L^\infty}.
 	 \end{equation}
To finish the proof of Theorem \ref{thm:main}, we put \eqref{eq:pi_6} and \eqref{eq:pi_4} together, and use  Proposition \ref{prop:boundedness_H1_BMO} which yields the $H^1$ boundedness of $I_{\frac{1}{\sigma_w}}$. Hence
 \[
 	\begin{split}
 	\Big|\left<\Pi(F,G), I_{\frac{1}{\sigma_w}}h\right>\Big|&\lesssim \norm{F}_{\BMO}\norm{G}_{X_w}\norm{h}_{H^1},
 	\end{split}
 \]
 which implies
 \[
 	\norm{ \Pi(F,G)}_{\BMO_{\sigma_w}}\lesssim \norm{F}_{\BMO}\norm{G}_{X_w},
 \]
 for any $F\in \BMO$, $G\in X_w$. This concludes the proof of Theorem \ref{thm:main}.

 \subsection{Boundedness of variable coefficient paraproducts}\label{xdependent}
Here we shall investigate the boundedness of certain variable coefficient paraproducts defined by
 $$\PP(f,g)(x):= \int_0^1 Q_t f(x)\, P_tg(x) m(t,x)\frac{\dd t}{t}.$$
 These paraproducts are of particular interest when dealing with bilinear pseudodifferential and Fourier integral operators (see Section \ref{FIOs}). Our main result concerning $\PP(f,g)$ is as follows:
\begin{thm}\label{xdependent version}
 Let $\PP$ be the operator given as above, where $m$ is a bounded measurable function such that
\[
	 \sup_t \norm{m(t,\cdot)}_{L^\infty}+\sup_t \norm{\nabla_x m(t,\cdot)}_{L^\infty}<+\infty.
\]
Let $w$ be an admissible weight such that $\int_0^1 w(t)\dd t<+\infty$. Then
\[
	\norm{\PP(f,g)}_{\BMO_{\sigma_w}} \leq C \norm{f}_{\bmo}\norm{g}_{X_w}.
\]
\end{thm}
 \begin{proof} We shall start by observing that, since $t\in (0,1)$, there exist a Schwartz function $\Theta$ such that $\widehat{\Theta}$ is supported in a small neighbourhood of the origin such that for any $F$,
 \[
 	Q_t\brkt{F}=Q_t\brkt{(1-\widehat{\Theta})(D)F}.
 \]
 In this way, we observe that
 \begin{equation}\label{eq:obs}
 	\PP(F,G)(x)= \PP((1-\widehat{\Theta})(D)F, G).
\end{equation}

For proving the result, one proceeds as in the proof of Theorem \ref{thm:main_thm} and decomposes $\PP(F,G)=\PP_1(F,G)+\PP_2(F,G)$, and pair them with a function $H=I_{1/\sigma_w} h\in H^1_{1/\sigma_w}(\R^n)$.
 Then
 \[
	\left<\PP_2(F,G),H\right>=\int \int_0^1{\brkt{Q_t F} \brkt{Q_t^{(1)} G}} \big(P_t^{(2)} (M_m H)\big)\frac{\dd t}{t}\dd x,
\]
where $M_m(H)(x)= m(t,x) H(x)$. Now if we introduce the operator
\[
	[P_t^{(2)},M_m] H(x)=P_t^{(2)} (M_m H)(x)-m(t,x)P_t^{(2)} (H)(x),
\]
we can write
 \[
 	\begin{split}
	\left<\PP_2(F,G),H\right> &=\int \int_0^1 {\brkt{Q_t F} \brkt{Q_t^{(1)} G}} [P_t^{(2)},M_m] H(x)\frac{\dd t}{t}\dd x+\\
	&+\int \int_0^1 {\brkt{Q_t F} \brkt{Q_t^{(1)} G}}\big(P_t^{(2)} H\big) m(t,\cdot)\frac{\dd t}{t}\dd x = I+II.
	\end{split}
\]
The second term can be dealt with in the same way as \eqref{eq:first_term}, which yields
\[
	\abs{II}\lesssim \sup_t \norm{m(t,\cdot)}_{L^\infty} \int \sup_{0<t<1}\sup_{\abs{x-y}<t} \abs{P_t^{(2)} H(y)}\dd x\lesssim \sup_t \norm{m(t,\cdot)}_{L^\infty} \norm{H}_{H^1}.
\]
For the first term we observe that $[P_t^{(2)},M_m] H(x)(x) =\int K(t,x,y) H(y)\dd y$, with
 \[
 	K(t,x,y)=t^{-n}\phi_2\brkt{\frac{x-y}{t}}\big(m(t,y) - m(t,x)\big)
\]
Observe that by the mean-value theorem, we have that
\[
|m(t,y) - m(t,x)| \lesssim\sup_t \norm{\nabla_x m(t,\cdot)}_{L^\infty}|y-x|
\]
with an implicit constant independent of $t$. Hence, for any $N\geq n+1$
\[
	\abs{K(t,x,y)}\lesssim t \frac{t^{-n}}{(1+\abs{x-y}/t)^N},
\]
which yields that  $[P_t^{(2)},M_m]$ is a bounded operator in $L^1(\R^n)$ with norm bounded by $t$.
This fact and \eqref{eq:Q1} imply that
\[
	\abs{I}\leq \norm{F}_{\BMO}\norm{G}_{\BMO} \int_0^1 \int\abs{[P_t^{(2)},M_m] H(x)}\dd x \frac{\dd t}{t}\leq
	\norm{F}_{\BMO}\norm{G}_{\BMO}\norm{H}_{L^1}.
\]
Then
\begin{equation}\label{eq:step_1}
	\abs{\left<\PP_2(F,G),H\right>}\lesssim \norm{F}_{\BMO}\norm{G}_{X_w}\norm{H}_{H^1}.
\end{equation}
A similar analysis for $\PP_1$ yields
 \[
 	\begin{split}
	\left<\PP_1(F,G),H\right> &=\int \int_0^1 {\brkt{Q_t F} \brkt{P_t^{(1)} G}} [Q_t^{(2)},M_m] H\frac{\dd t}{t}\dd x+\\
	&+\int \int_0^1 {\brkt{Q_t F} \brkt{P_t^{(1)} G}}\big(Q_t^{(2)} H\big) m(t,\cdot)\frac{\dd t}{t}\dd x = A+B.
	\end{split}
\]
Taking $H=I_{1/\sigma_w} h\in H^{1}_{1/\sigma_w}$, proceeding as in the study of \eqref{eq:estimate_pi4} yields
\[
	\abs{B}\lesssim \norm{F}_{\BMO}\norm{G}_{X_w}\norm{h}_{H^1}.
\]
The first term can be written as
\[
	A=\int \int_0^1 \brkt{Q_t F} w(t) [Q_t^{(2)},M_m] H v(t,x)\frac{\dd t}{t}\dd x
\]
where $v(t,x)=\brkt{P_t^{(1)} G}(x)/w(t)$. Observe that $\sup_t \norm{v(t,\cdot)}_{L^\infty}\lesssim \norm{G}_{X_w}$ and also, with a similar argument as above,  $[Q_t^{(2)},M_m]$ is bounded in $L^1$ with norm bounded by a constant multiple of $t$. Then, \eqref{eq:Q1} and the hypothesis that $\int_0^1 w(t)\dd t$ is finite yield
\[
	\abs{A}\lesssim \norm{F}_{\BMO} \norm{G}_{X_w}\norm{H}_{L^1}.
\]
Therefore
\begin{equation}\label{eq:step_2}
		\left<\PP_1(F,G),I_{1/\sigma_w}h\right> \lesssim \norm{F}_{\BMO} \norm{G}_{X_w}\norm{h}_{H^1}
\end{equation}
 Hence, putting together \eqref{eq:step_1} and \eqref{eq:step_2} it follows that
 \[
 	\norm{\PP(F,G)}_{\BMO_{\sigma_w}} \lesssim \norm{F}_{\BMO} \norm{G}_{X_w}.
 \]
 Taking \eqref{eq:obs} into account and using \eqref{eq:bmo}, we have
 \[
 	\norm{\PP(F,G)}_{\BMO_{\sigma_w}} \lesssim \norm{(1-\widehat{\Theta})(D)F}_{\BMO} \norm{G}_{X_w}\lesssim \norm{F}_{\bmo}\norm{G}_{X_w}.
 \]
 \end{proof}

\begin{cor}\label{cor:bmo_bmo} Under the same conditions as in the previous theorem with $w(t)=1+\log_+ 1/t$ one has
\[
	\norm{\PP (f,g)}_{\BMO_{\sigma_w}} \leq C \norm{f}_{\bmo}\norm{g}_{\bmo}.
\]
\end{cor}
  \section{Application to one dimensional bilinear  Fourier integral operators }\label{FIOs}

	The investigations concerning bilinear Fourier integral operators started in \cite{GP}, followed by \cite{RS}. In \cite{RRS} the authors establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in $S^m_{1,0} (n,2)$ and non-degenerate phase functions on $L^p$ spaces. However it was also shown that many endpoint results fail in dimension one. The following discussion i.e. Theorem \ref{thm:1d_FIO}, can be viewed as one positive result in that direction.

   \begin{defn}\label{defn of hormander symbols}
Let $N\in \N$. A function $\sigma \in \mathcal{C}^{\infty}(\R^n \times\R^{Nn})$ (called the amplitude in this context), belongs to the class $S^{0}_{1,0}(n,N)$, if for all multi-indices $\alpha$ and $\beta$, there exist constants $C_{\alpha,\beta}$ such that
   \begin{align*}
      |\partial_{\Xi}^{\alpha}\partial_{x}^{\beta}\sigma(x,\Xi)| \leq C_{\alpha,\beta} \brkt{1+\abs{\Xi}}^{m-\vert \alpha\vert}, \qquad \text{for all $(x,\Xi)\in \R^n\times \R^{Nn}$}.
   \end{align*}
\end{defn}

\begin{defn}\label{defn:phase} A function $\phase(x,\xi)\in \mathcal{C}^\infty\brkt{\R^n\times (\R^n\setminus \{0\})}$ is called a non-degenerate phase function if it is real-valued, positively homogeneous of degree one in $\xi$ and verifies
\[
	\det {\frac{\d^2 \phase}{\d x \d\xi}}\neq 0,
\]
for $\xi\neq 0$ and $x$ in the spatial support of the amplitude.
\end{defn}
\begin{defn}
A \emph{linear Fourier integral operator} $T^{\phase}_{\sigma}$ is an operator which is defined to act on a Schwartz function $f$ by the formula
      \begin{equation} \label{eq:Linear_FIO}
      	T_\sigma^{\phase}f(x)= \int_{\R^n}  \sigma(x,\xi) \widehat{f}(\xi)e^{i\phase(x,\xi)} \, \ddd\xi.
      \end{equation}
\end{defn}
In what follows, we would need the boundedness of Fourier integral operators on the space $\bmo$. Although the boundedness of these operators on $h^p$ for $p\leq 1$, was established by M. Peloso and S. Secco \cite{PS}, the $\bmo$ boundedness of the operators would not follow directly from their $h^1$ boundedness result and duality. This is because the adjoint of an operator of the type \eqref{eq:Linear_FIO}, although being a general Fourier integral operator with an amplitude in the same class, is not of the same form as \eqref{eq:Linear_FIO}. In particular, the amplitude of this operator would not be compactly supported in both spatial variables, and therefore one can not represent the adjoint as a finite sum of operators of the form \eqref{eq:Linear_FIO} with amplitudes that are compactly supported in $x$.

\begin{thm}\label{eq:thm_bmo} For any linear Fourier integral operator  $T_\sigma^{\phase}$ of the type \eqref{defn RS FIO} with an amplitude $\sigma(x,\xi)\in S^{-(n-1)/2}_{1,0}(\R^n)$ with spatial compact support one  has
\[
	\norm{T_\sigma^{\phase} f}_{\bmo}\leq C\norm{f}_{\bmo}.
\]
\end{thm}
\begin{proof} First we claim that for any $\sigma$ and $\phase$ as above
\begin{equation*}\label{eq:claim}
	\norm{S_{\sigma}^\phase f}_{L^1(\R^n)}\leq \norm{f}_{h^1(\R^n)},
\end{equation*}
where
$
	S_{\sigma}^\phase f(x)=\int \brkt{\int \sigma(y,\xi) e^{-i\phase(y,\xi)+ix\cdot\xi} \ddd \xi}  f(y)\dd y=\int K(x,y) f(y) \dd y.
$

Assuming this claim, since
\[
	r_j (S_{\sigma}^\phase )f(x)=\int \brkt{\int \frac{\xi_j}{\abs{\xi}}(1- \widehat{\Theta }(\xi))\sigma(y,\xi) e^{-i\phase(y,\xi)+ix\cdot\xi} \ddd \xi}  f(y)\dd y,
\]
{with $r_j$ as in \eqref{eq:local_Riesz}}, and since $r_j (S_{\sigma}^\phase )$ is an operator of the same type as $S_{\sigma}^\phase$ with the amplitude $$\frac{\xi_j}{\abs{\xi}}(1- \widehat{\Theta }(\xi))\sigma(y,\xi) \in S^{-(n-1)/2}_{1,0}(\R^n),$$ it follows that for any $j=1,\ldots, n$
$
	\norm{r_j (S_{\sigma}^\phase f) }_{L^1(\R^n)}\leq C \norm{f}_{h^{1}(\R^n)}.
$
\\
Now \eqref{eq:norm_h1} yields $\norm{S_{\sigma}^\phase f }_{h^1(\R^n)}\leq C \norm{f}_{h^{1}(\R^n)}$ . Therefore using duality, since the adjoint of $T^\phase_\sigma$ is $S^\phase_\sigma$, the $\bmo$ boundedness follows.

Using the atomic characterisation of $h^1(\R^n)$ as done in \cite{Gol}*{Lemma 4}, it is enough to prove the result for an $h^1(\R)$ atom, which is a  function $a$ supported in a ball $B$, such that $\abs{a}\leq \abs{B}^{-1}$, and $\int a \dd x=0$ if the radius of $B$ is smaller than one, and with no  moment conditions on $a$ in case  the radius of $B$ is bigger than one. In particular, if the radius of $B$ is smaller than one, $a$ is also an $H^1(\R^n)$ atom. Therefore \cite{Sogge}*{Proposition 6.2.6}  implies
\[
	\norm{S_{\sigma}^\phase a}_{L^1(\R^n)}\lesssim 1,
\]
whenever the radius of the support of $a$ is less than one. If the radius is larger than one we proceed as follows. As is observed in \cite{Sogge}*{p. 179}, integration by parts yields that the kernel of  $S_{\sigma}^\phase$, $K(x,y)$ satisfies the estimate $\abs{K(x,y)}\lesssim \abs{x}^{-N}$ for any $N$ when $\abs{x}$ is larger than some fixed constant $r$ depending on $\sigma$ and $\phase$. Therefore,
\[
	\int \abs{S^\phase_\sigma a(x)}\dd x=\brkt{\int_{\abs{x}\leq r}+\int_{\abs{x}\geq r}} \abs{S^\phase_\sigma a(x)}\dd x=I+II.
\]
Now, $II\lesssim \int_{\abs{x}\geq r} \abs{x}^{-n-1}\dd x \int \abs{a(y)}\dd y\lesssim 1$. For $I$, we simply use the compact $x$-support of the domain of integration, the Cauchy-Schwarz inequality and the $L^2$ boundedness of Fourier integral operators of order zero to conclude that $I\lesssim 1$.
\end{proof}

\begin{defn}
A {bilinear oscillatory integral operator} $T^{\phase_1,\phase_2}_{\sigma}$ is an operator which is defined to act on Schwartz functions $f$ and $g$ by the formula
\begin{equation}\label{defn RS FIO}
T^{\phase_1,\phase_2}_{\sigma}(f,g)(x) = \iint \sigma(x,\xi,\eta)\widehat{f}(\xi)\widehat{g}(\eta)e^{i\phase_1(x,\xi)+i \phase_2(x,\eta)} \ddd\xi \ddd\eta.
\end{equation}
\end{defn}

  \begin{thm}\label{thm:1d_FIO} If $\sigma\in S^0_{1,0}(1,2)$ is compactly supported in the spatial variable and $\phase_1,\phase_2$ are non-degenerate phase functions. Then there exists a constant $C$, depending only on a finite number of derivatives of $\sigma$, such that
\begin{equation*}
\|T_\sigma^{\phase_1,\phase_2}(f,g)\|_{\BMO_{\sigma_w}}\leq C \|f\|_{L^\infty}\|g\|_{L^\infty},
\end{equation*}
with $w(t)=1+\log_+ 1/t$.
 \end{thm}
 \begin{proof} It is shown in \cite{RRS2}*{Section 4} that the study of the operator  $T_\sigma^{\phase_1,\phase_2}$ reduces to study an operator of the form
 \[
 	\PP(T^{\phase_1}_{\nu} f,T^{\phase_2}_{\nu} g)+E(f,g),
 \]
 where $\nu(x,\xi)=\chi(x)\mu(\xi)$, $\chi$ is a compactly supported smooth function and $\mu$ is a smooth function such that $1-\mu$ is supported in a neighbourhood of the origin and $E:L^\infty \times L^\infty \to L^\infty$.

Observe that in particular, $\nu \in S^0_{1,0}(1,1)$ with compact support in $x$. Then, Proposition \ref{eq:thm_bmo} implies in particular that $T^{\phase_1}_{\nu}: L^\infty\to \bmo$. So, taking into account Corollary \ref{cor:bmo_bmo}, the result follows by composition of bounded operators and triangle inequality.
 \end{proof}
 \section{Application to bilinear Coifman-Meyer multipliers and Kato-Ponce estimates}\label{kato-ponce}

 In this section, following the method of Coifman-Meyer \cite{CM4} we can also produce a boundedness result for bilinear Coifman-Meyer multipliers. Furthermore, we can apply this to obtain certain Kato-Ponce type estimates, which originated in the seminal paper by T. Kato and G. Ponce \cite{KPo}.

 \begin{thm}\label{thm:H-M} For any $\sigma(\xi,\eta)\in \mathcal{C}^\infty(\R^n\times \R^n\setminus\{0,0\})$ such that
 \begin{equation}\label{eq:H_M}
 	 \abs{\d^\alpha_\xi \d^\beta_\eta \sigma(\xi,\eta)}\lesssim \brkt{\abs{\xi}+\abs{\eta}}^{-\abs{\alpha}-\abs{\beta}}, \quad {\rm for}\, (\xi,\eta)\neq (0,0)
 \end{equation}
 for any $\alpha$ and $\beta$, then the bilinear operator
 \[
 	\sigma(D)(f,g)(x)=\iint \sigma(\xi,\eta) \widehat{f}(\xi)\widehat{g}(\eta) e^{i x(\xi+\eta)}\ddd \xi \ddd \eta
 \]
 satisfies, for any admissible weight $w$
 \[
 	\norm{\sigma(D)(f,g)}_{\BMO_{\sigma_w}}\lesssim \norm{f}_{X_w}\norm{g}_{X_w}.
 \]
 \end{thm}
 \begin{proof} Proceeding as in the proof of Proposition 2 in \cite{CM4}*{p.154}, one decomposes $\sigma=\sigma_1+\sigma_2$ with $\sigma_1(\xi,\eta)$ supported in $\abs{\xi}\geq \abs{\eta}/20$, $\sigma_2(\xi,\eta)$ supported in $10\abs{\xi}\leq \abs{\eta}$ and $\sigma_1,\sigma_2$ satisfying \eqref{eq:H_M}.

 Let $\widehat{\psi}$ be a smooth Schwartz function supported in $4\leq 5 \abs{\xi}\leq 6$ and such that
 $$\int_0^\infty \abs{\widehat{\psi}(\xi)}^2\frac{\dd t}{t}=1, \qquad \mathrm{for}\,\,\, \xi\neq 0.$$
 Let $\widehat{\phi}$  be a smooth compactly supported Schwartz function such that is equal to $1$ if $\abs{\xi}\leq 30$. Then an argument based on integration by parts shows that for any positive real number $N$
one can write
 \[
 	\sigma_1(D)(f,g)(x)=\iint  \Pi_{u,v}(f,g)(x)\frac{\dd u \dd v}{(1+\abs{u}^2+\abs{v}^2)^N},
 \]
 where
 \[
 	\Pi_{u,v}(f,g)(x)=\int_0^\infty Q_t^u(f) P_t^v(g) m(t,u,v) \frac{\dd t}{t},
 \]
$\sup_{t,u,v} \abs{m(t,u,v)}<+\infty$ and $$Q_t^u(f)=\widehat{\psi^u}(tD) f,\quad P_t^u(f)=\widehat{\phi^v}(tD) g$$ with $\psi^u(x)=\psi(x+u)$ and $\phi^v(x)=\phi(x+v)$. Another integration by parts argument revelas that, for any $\delta>0$,
\[
	\abs{\psi^u(x)}\lesssim \frac{(1+\abs{u})^{n+\delta}}{(1+\abs{x})^{n+\delta}}\quad {\rm and} \quad \abs{\phi^v(x)}\lesssim \frac{(1+\abs{v})^{n+\delta}}{(1+\abs{x})^{n+\delta}}.
\]
So one can apply Theorem \ref{thm:main_thm} to deduce
\[
	\norm{\Pi_{u,v}(f,g)}_{\BMO_{\sigma_w}}\lesssim P(u,v) \norm{f}_{X_w}\norm{g}_{X_w},
\]
with $P(u,v)$ a polynomial expression in $\abs{u}$ and $\abs{v}$, independent on $f,g$. Then, taking $N$ large enough yields
\[
	\norm{\sigma_1(D)(f,g)}_{\BMO_{\sigma_w}}\lesssim \norm{f}_{X_w}\norm{g}_{X_w}.
\]
For $\sigma_2(D)$ the roles of $\xi$ and $\eta$ are reversed, but by the symmetry of the estimates, a similar argument to the previous one yields
\[
	\norm{\sigma_2(D)(f,g)}_{\BMO_{\sigma_w}}\lesssim \norm{f}_{X_w}\norm{g}_{X_w}.
\]
Then, the result follows by putting these last two estimates together.
    \end{proof}

\begin{rem}\label{eq:final_remark}  One can weaken the assumption \eqref{eq:H_M} on the symbol by requiring only finite number of derivatives of the symbol. Tracing the proofs of theorems \ref{thm:main}  and \ref{thm:H-M}, it turns out that we would need at least $4n+1$ derivatives. In light of the results of \cite{GT} and \cite{MT,Tomita}, there seems to be some room for improvement of the number of derivatives.
\end{rem}

\begin{cor} If $\sigma(\xi,\eta)\in \mathcal{C}^\infty(\R^n\times \R^n\setminus\{0,0\})$ satisfying \eqref{eq:H_M}, then
\[
	\norm{\sigma(D)(f,g)}_{\BMO_{\sigma_w}}\lesssim \norm{f}_{\bmo}\norm{g}_{\bmo}.
\]
with $w=1+\log_+ 1/t$ and
\[
		\norm{\sigma(D)(f,g)}_{\BMO}\lesssim \norm{f}_{L^\infty}\norm{g}_{L^\infty}.
\]
\end{cor}

\begin{cor} For any $f,g\in \bmo$
\[
	\norm{f g}_{\BMO_{\sigma_w}}\lesssim \norm{f}_{\bmo}\norm{g}_{\bmo},
\]
with $w=1+\log_+ 1/t$.
\end{cor}
\begin{proof} This follows from the previous corollary by taking $\sigma(\xi,\eta)=1$.
\end{proof}
As a consequence of Theorem \ref{thm:H-M} and Remark \ref{eq:final_remark}  we obtain the following endpoint Kato-Ponce type inequality, which is a variant of \cite{GMN}*{Theorem 3}.

\begin{thm}\label{thm:kato_ponce} If $s >4n+1$ and $w$ is an admissible weight, then for every $f, g \in \mathcal{S}$
\begin{equation*}
\norm{D^s(fg)}_{\BMO_{\sigma_w}}  \lesssim \norm{D^{s}f}_{X_w} \norm{g}_{X_w} + \norm{f}_{X_w} \norm{D^{s}g}_{X_w},
\end{equation*}
where $D^s$ denotes the operator defined for $h \in \mathcal{S}$ as
$
\widehat{D^sh}(\xi):= |\xi|^s \hat{h}(\xi)$ for all $\xi \in \R^n$.
In particular, for $w=1+\log_+ 1/t$
\begin{equation*}
\norm{D^s(fg)}_{\BMO_{\sigma_w}}  \lesssim \norm{D^{s}f}_{\bmo} \norm{g}_{\bmo} + \norm{f}_{\bmo} \norm{D^{s}g}_{\bmo}.
\end{equation*}
\end{thm}
\begin{proof} Following the approach in \cite{GOh}, it is shown in \cite{GMN} that the bilinear mapping $(f,g) \mapsto D^s(fg)$ can be decomposed into the sum of three bilinear multipliers
\begin{equation*}
D^s(fg) = \sigma_{1,s}(D)(D^s f,g) + \sigma_{2,s}(D)(f,D^s g) + \sigma_{3,s}(D)(f,D^sg),
\end{equation*}
where, for $s > 4n+1$, these have symbols satisfying \eqref{eq:H_M} {for $\abs{\alpha}+\abs{\beta}\leq 4n+1$}. Then the result follows from Theorem \ref{thm:H-M} and Remark \ref{eq:final_remark}.

The last assertion follows by taking the particular weight $w=1+\log_+ 1/t$ and applying Corollary \ref{cor:bmo}.
\end{proof}

\begin{rem} Observe that for $w=1$, the previous theorem recovers \cite{GMN}*{Theorem 3} albeit with a larger amount of derivatives required in \eqref{eq:H_M}.
 \end{rem}

\end{document}

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