Abstract view
Hilbert Transformation and Representation of the $ax+b$ Group


Pei Dang,
Faculty of Information Technology, Macau University of Science and Technology, Macau, China
Hua Liu,
Department of Mathematics, Tianjin University of Technology and Education, Tianjin 300222, China
Tao Qian,
Department of Mathematics, University of Macau, Macau, China
Abstract
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 $L^2(\mathbb{T}).$
MSC Classifications: 
30E25, 44A15, 42A50 show english descriptions
Boundary value problems [See also 45Exx] Special transforms (Legendre, Hilbert, etc.) Conjugate functions, conjugate series, singular integrals
30E25  Boundary value problems [See also 45Exx] 44A15  Special transforms (Legendre, Hilbert, etc.) 42A50  Conjugate functions, conjugate series, singular integrals
