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L2 regularity of measurable solutions of a finite-difference equation of the circle

Published online by Cambridge University Press:  18 October 2004

MICHAEL ROBERT HERMAN
MICHAEL ROBERT HERMAN
Affiliation:
Mathematics Institute, Warwick University, UK and Centre de Mathematiques, Ecole Polytechnique, Plateau de Palaiseau, 91120 Palaiseau, France
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Extract

We show that if $\varphi$ is a lacunary Fourier series and the equation $\psi (x) -\psi (x + \alpha) = \varphi(x), x \bmod 1$ has a measurable solution $\varphi$, then in fact the equation has a solution in L2.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 5 , October 2004 , pp. 1277 - 1281
Copyright
© 2004 Cambridge University Press

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Footnotes

This work of Michel Herman appeared only as a preprint of the Mathematics Institute, University of Warwick, dated May 1976. It was turned into TEX format by Claire Desescures. Minor editorial work was done by Albert Fathi.