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In Mattila's book Fourier Analysis and Hausdorff Dimension, Mattila presents a result of Barcelo, Bennett, Carbery, and Rogers about convergence of solutions of the Schrodinger equation to the initial data. In this paper, the authors show that if the initial data lies in an appropriate Sobolev space, then the solutions converge to the initial data except on a set of points whose Hausdorff dimension is determined by the Sobolev space in which the initial data lies.

Are there any similar results for convergence of Fourier series? Given a subset of [0,1] and a function in an appropriate Sobolev space on [0,1], are there any known results about the Hausdorff dimension of the set for which pointwise convergence of the Fourier series fails?

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  • $\begingroup$ Yes, there are many results along these lines. You could take a look at Carleson's own book on exceptional sets, and Zygmund has some discussion of this too in his book. $\endgroup$
    – Christian Remling
    Commented Jun 21, 2016 at 23:38
  • $\begingroup$ This doesn't directly answer your question but there is a result of Katznelson which states that for any measure 0 set there is a continuous function with a Fourier series diverging on it (see: Y. Katznelson, "Sur les ensembles de divergence des séries trigonométriques", Studia Mathematica 26: 301–304, 1966). $\endgroup$
    – Mark Lewko
    Commented Jun 22, 2016 at 0:03

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