1. CJM 2018 (vol 70 pp. 1261)
 Fricain, Emmanuel; Hartmann, Andreas; Ross, William T.

Range Spaces of Coanalytic Toeplitz Operators
In this paper we discuss the range of a coanalytic Toeplitz
operator. These range spaces are closely related to de BrangesRovnyak
spaces (in some cases they are equal as sets). In order to understand
its structure, we explore when
the range space decomposes into the range of an associated analytic
Toeplitz operator and an identifiable orthogonal complement.
For certain cases, we compute this orthogonal complement in terms
of the kernel of a certain Toeplitz operator on the Hardy space
where we focus on when this kernel is a model space (backward
shift invariant subspace).
In the spirit of AhernClark, we also discuss the nontangential
boundary behavior in these range spaces. These results give us
further insight into the description of the range of a coanalytic
Toeplitz operator as well as its orthogonal decomposition. Our
AhernClark type results, which are stated in a general abstract
setting, will also have applications to related subHardy Hilbert
spaces of analytic functions such as the de BrangesRovnyak spaces
and the harmonically weighted Dirichlet spaces.
Keywords:Toeplitz operator, Hardy space, range space, de BrangesRovnyak space, boundary behavior, kernel function, nonextreme point, corona pair Categories:30J05, 30H10, 46E22 

2. CJM 2017 (vol 70 pp. 515)
 Chen, Yanni; Hadwin, Don; Liu, Zhe; Nordgren, Eric

A Beurling Theorem for Generalized Hardy Spaces on a Multiply Connected Domain
The object of this paper is to prove a version of the BeurlingHelsonLowdenslager
invariant subspace theorem for operators on certain Banach spaces
of functions on a multiply connected domain in $\mathbb{C}$. The norms
for these spaces are either the usual Lebesgue and Hardy space
norms or certain continuous gauge norms.
In the Hardy space case the expected corollaries include the
characterization of the cyclic vectors as the outer functions
in this context, a demonstration that the set of analytic multiplication
operators is maximal abelian and reflexive, and a determination
of the closed operators that commute with all analytic multiplication
operators.
Keywords:Beurling theorem, invariant subspace, generalized Hardy space, gauge norm, multiply connected domain, Forelli projection, innerouter factorization, affiliated operator Categories:47L10, 30H10 

3. CJM 2017 (vol 70 pp. 721)
 Bao, Guanlong; Göğüş, Nihat Gökhan; Pouliasis, Stamatis

On Dirichlet Spaces with a Class of Superharmonic Weights
In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with
superharmonic weights induced by positive Borel measures $\mu$
on
the open unit disk. We establish the AlexanderTaylorUllman
inequality for $\mathcal{D}_\mu$ spaces and we characterize the cases where
equality occurs.
We define a class of weighted Hardy spaces $H_{\mu}^{2}$ via
the balayage of the measure $\mu$.
We show that $\mathcal{D}_\mu$
is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a
Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the
reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite
sum of point mass measures. We consider the boundary
behavior and innerouter factorization of functions in $\mathcal{D}_\mu$.
We also characterize the boundedness and
compactness of composition operators on $\mathcal{D}_\mu$.
Keywords:Dirichlet space, Hardy space, superharmonic weight Categories:30H10, 31C25, 46E15 

4. CJM 2017 (vol 69 pp. 1219)
5. CJM 2016 (vol 68 pp. 1257)
6. CJM 2015 (vol 67 pp. 1161)
 Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun

Nontangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators
Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights
or in the class of $QC(\mathbb{R}^n)$ weights, and
$L_w:=w^{1}\mathop{\mathrm{div}}(A\nabla)$
the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$,
$n\ge 2$. In this article, the authors establish the nontangential
maximal function characterization
of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for
$p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and
$w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$,
the authors prove that the associated Riesz transform $\nabla L_w^{1/2}$
is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical
Hardy space $H_w^p(\mathbb{R}^n)$.
Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform Categories:42B30, 42B35, 35J70 

7. CJM 2012 (vol 65 pp. 299)
 Grafakos, Loukas; Miyachi, Akihiko; Tomita, Naohito

On Multilinear Fourier Multipliers of Limited Smoothness
In this paper,
we prove certain $L^2$estimate
for multilinear Fourier multiplier operators
with multipliers of limited smoothness.
As a result,
we extend the result of CalderÃ³n and Torchinsky
in the linear theory to the multilinear case.
The sharpness of our results and some
related estimates in Hardy spaces
are also discussed.
Keywords:multilinear Fourier multipliers, HÃ¶rmander multiplier theorem, Hardy spaces Categories:42B15, 42B20 

8. CJM 2011 (vol 64 pp. 1329)
9. CJM 2010 (vol 62 pp. 961)
 Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan

Multiplicative Isometries and Isometric ZeroDivisors
For some Banach spaces of analytic functions in the unit disk
(weighted Bergman spaces, Bloch space, Dirichlettype spaces), the
isometric pointwise multipliers are found to be unimodular constants.
As a consequence, it is shown that none of those spaces have isometric
zerodivisors. Isometric coefficient multipliers are also
investigated.
Keywords:Banach spaces of analytic functions, Hardy spaces, Bergman spaces, Bloch space, Dirichlet space, Dirichlettype spaces, pointwise multipliers, coefficient multipliers, isometries, isometric zerodivisors Categories:30H05, 46E15 

10. CJM 2009 (vol 62 pp. 439)
 Sundhäll, Marcus; Tchoundja, Edgar

On Hankel Forms of Higher Weights: The Case of Hardy Spaces
In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by SundhÃ¤ll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.
Keywords:Hankel forms, Schattenâvon Neumann classes, Bergman spaces, Hardy spaces, Besov spaces, transvectant, unitary representations, MÃ¶bius group Categories:32A25, 32A35, 32A37, 47B35 

11. CJM 2009 (vol 62 pp. 202)
12. CJM 2003 (vol 55 pp. 1231)
 Havin, Victor; Mashreghi, Javad

Admissible Majorants for Model Subspaces of $H^2$, Part I: Slow Winding of the Generating Inner Function
A model subspace $K_\Theta$ of the Hardy space $H^2 = H^2
(\mathbb{C}_+)$ for the upper half plane $\mathbb{C}_+$ is
$H^2(\mathbb{C}_+) \ominus \Theta H^2(\mathbb{C}_+)$ where $\Theta$
is an inner function in $\mathbb{C}_+$. A function $\omega \colon
\mathbb{R}\mapsto[0,\infty)$ is called an admissible
majorant for $K_\Theta$ if there exists an $f \in K_\Theta$, $f
\not\equiv 0$, $f(x)\leq \omega(x)$ almost everywhere on
$\mathbb{R}$. For some (mainly meromorphic) $\Theta$'s some parts
of Adm $\Theta$ (the set of all admissible majorants for
$K_\Theta$) are explicitly described. These descriptions depend on
the rate of growth of $\arg \Theta$ along $\mathbb{R}$. This paper
is about slowly growing arguments (slower than $x$). Our results
exhibit the dependence of Adm $B$ on the geometry of the zeros of
the Blaschke product $B$. A complete description of Adm $B$ is
obtained for $B$'s with purely imaginary (``vertical'') zeros. We
show that in this case a unique minimal admissible majorant exists.
Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant Categories:30D55, 47A15 

13. CJM 2003 (vol 55 pp. 1264)
 Havin, Victor; Mashreghi, Javad

Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function
This paper is a continuation of Part I [6]. We consider the model
subspaces $K_\Theta=H^2\ominus\Theta H^2$ of the Hardy space $H^2$
generated by an inner function $\Theta$ in the upper half plane. Our
main object is the class of admissible majorants for $K_\Theta$,
denoted by Adm $\Theta$ and consisting of all functions $\omega$
defined on $\mathbb{R}$ such that there exists an $f \ne 0$, $f \in
K_\Theta$ satisfying $f(x)\leq\omega(x)$ almost everywhere on
$\mathbb{R}$. Firstly, using some simple Hilbert transform techniques,
we obtain a general multiplier theorem applicable to any $K_\Theta$
generated by a meromorphic inner function. In contrast with
[6], we consider the generating functions $\Theta$ such that
the unit vector $\Theta(x)$ winds up fast as $x$ grows from $\infty$
to $\infty$. In particular, we consider $\Theta=B$ where $B$ is a
Blaschke product with ``horizontal'' zeros, i.e., almost
uniformly distributed in a strip parallel to and separated from $\mathbb{R}$.
It is shown, among other things, that for any such $B$, any even
$\omega$ decreasing on $(0,\infty)$ with a finite logarithmic integral
is in Adm $B$ (unlike the ``vertical'' case treated in [6]),
thus generalizing (with a new proof) a classical result related to
Adm $\exp(i\sigma z)$, $\sigma>0$. Some oscillating $\omega$'s in
Adm $B$ are also described. Our theme is related to the
BeurlingMalliavin multiplier theorem devoted to Adm $\exp(i\sigma z)$,
$\sigma>0$, and to de Branges' space $\mathcal{H}(E)$.
Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant Categories:30D55, 47A15 

14. CJM 1998 (vol 50 pp. 897)