# Necessary condition for measures which are $\left({L}^{q},{L}^{p}\right)$ multipliers

Bérenger Akon Kpata[1]; Ibrahim Fofana[1]; Konin Koua[1]

• [1] UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire
• Volume: 16, Issue: 2, page 339-353
• ISSN: 1259-1734

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## Abstract

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Let $G$ be a locally compact group and $\rho$ the left Haar measure on $G$. Given a non-negative Radon measure $\mu$, we establish a necessary condition on the pairs $\left(q,\phantom{\rule{4pt}{0ex}}p\right)$ for which $\mu$ is a multiplier from ${L}^{q}\left(G,\phantom{\rule{4pt}{0ex}}\rho \right)$ to ${L}^{p}\left(G,\phantom{\rule{4pt}{0ex}}\rho \right)$. Applied to ${ℝ}^{n}$, our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7].When $G$ is the circle group, we obtain a generalization of a condition stated by Oberlin [15] and improve on it in some cases.

## How to cite

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Kpata, Bérenger Akon, Fofana, Ibrahim, and Koua, Konin. "Necessary condition for measures which are $(L^{q},L^{p})$ multipliers." Annales mathématiques Blaise Pascal 16.2 (2009): 339-353. <http://eudml.org/doc/10584>.

@article{Kpata2009,
abstract = {Let $G$ be a locally compact group and $\rho$ the left Haar measure on $G$. Given a non-negative Radon measure $\mu$, we establish a necessary condition on the pairs $\left( q,\ p\right)$ for which $\mu$ is a multiplier from $L^\{q\}\left( G,\ \rho \right)$ to $L^\{p\}\left( G,\ \rho \right)$. Applied to $\mathbb\{R\}^\{n\}$, our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7].When $G$ is the circle group, we obtain a generalization of a condition stated by Oberlin [15] and improve on it in some cases.},
affiliation = {UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire; UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire; UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22 Côte d’Ivoire},
author = {Kpata, Bérenger Akon, Fofana, Ibrahim, Koua, Konin},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Cantor-Lebesgue measure; $L^\{q\}$-improving measure; non-negative Radon measure; Cantor–Lebesgue measure; Radon measure; multiplier},
language = {eng},
month = {7},
number = {2},
pages = {339-353},
publisher = {Annales mathématiques Blaise Pascal},
title = {Necessary condition for measures which are $(L^\{q\},L^\{p\})$ multipliers},
url = {http://eudml.org/doc/10584},
volume = {16},
year = {2009},
}

TY - JOUR
AU - Kpata, Bérenger Akon
AU - Fofana, Ibrahim
AU - Koua, Konin
TI - Necessary condition for measures which are $(L^{q},L^{p})$ multipliers
JO - Annales mathématiques Blaise Pascal
DA - 2009/7//
PB - Annales mathématiques Blaise Pascal
VL - 16
IS - 2
SP - 339
EP - 353
AB - Let $G$ be a locally compact group and $\rho$ the left Haar measure on $G$. Given a non-negative Radon measure $\mu$, we establish a necessary condition on the pairs $\left( q,\ p\right)$ for which $\mu$ is a multiplier from $L^{q}\left( G,\ \rho \right)$ to $L^{p}\left( G,\ \rho \right)$. Applied to $\mathbb{R}^{n}$, our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7].When $G$ is the circle group, we obtain a generalization of a condition stated by Oberlin [15] and improve on it in some cases.
LA - eng
KW - Cantor-Lebesgue measure; $L^{q}$-improving measure; non-negative Radon measure; Cantor–Lebesgue measure; Radon measure; multiplier
UR - http://eudml.org/doc/10584
ER -

## References

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