Zygmund's program: some partial solutions

Stokolos, Alexander

Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1439-1453.

Nous proposons un critère simple pour décider si la fonction maximale associée à une base d’intervalles multidimensionnels, invariante par translation, admet une estimation du type $(1,1)$. Cela nous permet de compléter le programme de Zygmund décrivant les bases d’intervalles multidimensionnels invariantes par translation dans le cas particulier des produits de deux intervalles cubiques. Nous proposons aussi une conjecture qui précise le programme de Zygmund.

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.

Classification : 42B25
Keywords: covering lemmas, maximal functions
Mot clés : lemmes de recouvrement, fonctions maximales

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