Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets

Sjögren, Peter, and Sjölin, Per. "Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets." Annales de l'institut Fourier 31.1 (1981): 157-175. <http://eudml.org/doc/74479>.

@article{Sjögren1981,
abstract = {Let $E\subset \{\bf R\}$ be a closed null set. We prove an equivalence between the Littlewood-Paley decomposition in $L^p$ 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 $\{\bf R\}^2$ 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 $L^p$, $1&lt; p&lt; \infty $, when the rays form an iterated lacunary sequence.},
author = {Sjögren, Peter, Sjölin, Per},
journal = {Annales de l'institut Fourier},
keywords = {Lp spaces; Fourier multipliers; Littlewood-Paley decompositions; maximal functions},
language = {eng},
number = {1},
pages = {157-175},
publisher = {Association des Annales de l'Institut Fourier},
title = {Littlewood-Paley decompositions and Fourier multipliers with singularities on certain sets},
url = {http://eudml.org/doc/74479},
volume = {31},
year = {1981},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 1
SP - 157
EP - 175
AB - Let $E\subset {\bf R}$ be a closed null set. We prove an equivalence between the Littlewood-Paley decomposition in $L^p$ 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 ${\bf R}^2$ 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 $L^p$, $1&lt; p&lt; \infty $, when the rays form an iterated lacunary sequence.
LA - eng
KW - Lp spaces; Fourier multipliers; Littlewood-Paley decompositions; maximal functions
UR - http://eudml.org/doc/74479
ER -