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Zygmund’s program: some partial solutions. (English) Zbl 1080.42019

Summary: 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.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
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References:

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