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# Hilbert Transformation and Representation of the $ax+b$ Group

Published:2017-11-08
Printed: Mar 2018
• 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
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## 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 Gelfand-Naimark'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
 MSC Classifications: 30E25 - Boundary value problems [See also 45Exx] 44A15 - Special transforms (Legendre, Hilbert, etc.) 42A50 - Conjugate functions, conjugate series, singular integrals

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