1. CMB Online first
 Dang, Pei; Liu, Hua; Qian, Tao

Hilbert Transformation and Representation of the $ax+b$ Group
In this paper we study the Hilbert transformations over
$L^2(\mathbb{R})$
and $L^2(\mathbb{T})$ from
the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$
commutative with the ax+b group we show that the operator is
of the form
$
\lambda I+\eta H,
$
where $I$ and $H$ are the identity operator and Hilbert transformation
respectively, and $\lambda,\eta$ are complex numbers. In the
related literature this result was proved through first invoking
the boundedness result of the operator, proved though a big
machinery.
In our setting the boundedness is a consequence of the boundedness
of the Hilbert transformation. The methodology that we use is
GelfandNaimark's representation of the ax+b group. Furthermore
we prove a similar result on the unit circle. Although there
does not exist a group like ax+b on the unit circle, we construct
a semigroup to play the same symmetry role for the Hilbert transformations
over the circle $L^2(\mathbb{T}).$
Keywords:singular integral, Hilbert transform, the $ax+b$ group Categories:30E25, 44A15, 42A50 

2. CMB 2016 (vol 59 pp. 497)
3. CMB 2005 (vol 48 pp. 370)
 Daly, J. E.; Fridli, S.

Trigonometric Multipliers on $H_{2\pi}$
In this paper we consider multipliers on the real Hardy space
$H_{2\pi}$. It is known that the Marcinkiewicz and the
H\"ormanderMihlin conditions are sufficient for the corresponding
trigonometric multiplier to be bounded on $L_{2\pi}^p$, $1
Keywords:Multipliers, Hardy space Categories:42A45, 42A50, 42A85 
