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Spaces of Lorentz Multipliers

Published online by Cambridge University Press:  20 November 2018

Kathryn E. Hare  and
Enji Sato
Kathryn E. Hare
Affiliation:
Dept. of Pure Mathematics Univ. of Waterloo Waterloo, Ont. N2L 3G1, email: kehare@uwaterloo.ca
Enji Sato
Affiliation:
Dept. of Mathematical Sciences Faculty of Science Yamagata University Yamagata 990-8560 Japan, email: esato@sci.kj.yamagata-u.ac.jp
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Abstract

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We study when the spaces of Lorentz multipliers from ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ are distinct. Our main interest is the case when $s\,<\,t$, the Lorentz-improving multipliers. We prove, for example, that the space of multipliers which map ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ is different from those mapping ${{L}^{r,v}}\,\to \,{{L}^{r,u}}$ if either $r\,=\,p$ or ${p}'$ and $1/s\,-\,1/t\,\ne \,1/u\,-\,1/v$, or $r\,\ne \,p$ or ${p}'$. These results are obtained by making careful estimates of the Lorentz multiplier norms of certain linear combinations of Fejer or Dirichlet kernels. For the case when the first indices are different the linear combination we analyze is in the spirit of a Rudin-Shapiro polynomial.

Keywords

43A2242A4546E30multipliersconvolution operatorsLorentz spacesLorentz-improving multipliers
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 3 , 01 June 2001 , pp. 565 - 591
Copyright
Copyright © Canadian Mathematical Society 2001

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