Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets
Annales de l'Institut Fourier, Tome 31 (1981) no. 1, pp. 157-175.

Soit E⊂R un ensemble fermé de mesure nulle. On démontre une équivalence entre la décomposition de Littlewood-Paley dans Lp par rapport aux intervalles complémentaires de E et les multiplicateurs de Fourier du type de Hörmander-Mihlin et de Marcinkiewicz ayant des singularités sur E. Des propriétés analogues sont étudiées dans R2 pour une réunion de rayons partant de l’origine. Dans ce cas, on considère aussi la fonction maximale par rapport aux rectangles parallèles à ces rayons. On montre notamment que l’opérateur défini par cette fonction maximale est borné dans Lp, 1<p<∞, quand les rayons forment une suite lacunaire itérée.

Let E⊂R be a closed null set. We prove an equivalence between the Littlewood-Paley decomposition in Lp with respect to the complementary intervals of E and Fourier multipliers of Hörmander-Mihlin and Marcinkiewicz type with singularities on E. Similar properties are studied in R2 for a union of rays from the origin. Then there are connections with the maximal function operator with respect to all rectangles parallel to these rays. In particular, this maximal operator is proved to be bounded on Lp, 1<p<∞, when the rays form an iterated lacunary sequence.

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     author = {Sj\"ogren, Peter and Sj\"olin, Per},
     title = {Littlewood-Paley decompositions and {Fourier} multipliers with singularities on certain sets},
     journal = {Annales de l'Institut Fourier},
     pages = {157--175},
     publisher = {Institut Fourier},
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Sjögren, Peter; Sjölin, Per. Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets. Annales de l'Institut Fourier, Tome 31 (1981) no. 1, pp. 157-175. doi : 10.5802/aif.821. https://www.numdam.org/articles/10.5802/aif.821/

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