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Zygmund's program: some partial solutions

Alexander Stokolos (2005)

Annales de l’institut Fourier

We present a simple criterion to decide whether the maximal function associated with a translation invariant basis of multidimensional intervals satisfies a weak type ( 1 , 1 ) estimate. This allows us to complete Zygmund’s program of the description of the translation invariant bases of multidimensional intervals in the particular case of products of two cubic intervals. As a conjecture, we suggest a more precise version of Zygmund’s program.

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