Let be a closed null set. We prove an equivalence between the Littlewood-Paley decomposition in with respect to the complementary intervals of and Fourier multipliers of Hörmander-Mihlin and Marcinkiewicz type with singularities on . Similar properties are studied in 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 , , when the rays form an iterated lacunary sequence.
Soit un ensemble fermé de mesure nulle. On démontre une équivalence entre la décomposition de Littlewood-Paley dans par rapport aux intervalles complémentaires de et les multiplicateurs de Fourier du type de Hörmander-Mihlin et de Marcinkiewicz ayant des singularités sur . Des propriétés analogues sont étudiées dans 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 , , quand les rayons forment une suite lacunaire itérée.
@article{AIF_1981__31_1_157_0, 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}, address = {Grenoble}, volume = {31}, number = {1}, year = {1981}, doi = {10.5802/aif.821}, zbl = {0437.42011}, mrnumber = {82g:42014}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.821/} }
TY - JOUR AU - Sjögren, Peter AU - Sjölin, Per TI - Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets JO - Annales de l'Institut Fourier PY - 1981 SP - 157 EP - 175 VL - 31 IS - 1 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.821/ DO - 10.5802/aif.821 LA - en ID - AIF_1981__31_1_157_0 ER -
%0 Journal Article %A Sjögren, Peter %A Sjölin, Per %T Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets %J Annales de l'Institut Fourier %D 1981 %P 157-175 %V 31 %N 1 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.821/ %R 10.5802/aif.821 %G en %F AIF_1981__31_1_157_0
Sjögren, Peter; Sjölin, Per. Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets. Annales de l'Institut Fourier, Volume 31 (1981) no. 1, pp. 157-175. doi : 10.5802/aif.821. https://aif.centre-mersenne.org/articles/10.5802/aif.821/
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