Fourier coefficients of continuous functions and a class of multipliers

Kislyakov, Serguei V.. "Fourier coefficients of continuous functions and a class of multipliers." Annales de l'institut Fourier 38.2 (1988): 147-183. <http://eudml.org/doc/74798>.

@article{Kislyakov1988,
abstract = {If $x$ is a bounded function on $\{\bf Z\}$, the multiplier with symbol $x$ (denoted by $M_ x)$ is defined by $(M_ xf)\{\hat\{\ \}\}=x\hat\{f\}$, $f\in L^ 2(\{\bf T\})$. We give some conditions on $x$ ensuring the “interpolation inequality” $\Vert M_ xf\Vert _\{L^ p\}\le C\Vert f\Vert ^\{\alpha \}_\{L^ 1\}\Vert M_ xf\Vert _\{L^ q\}^\{1-\alpha \}$ (here $1&lt; p&lt; q$ and $\alpha =\alpha (p,q,x)$ is between 0 and 1). In most cases considered $M_ x$ fails to have stronger $L^ 1$-regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E\subset \{\bf Z\}$ every positive sequence in $\ell ^ 2(E)$ can be majorized by the sequence $\lbrace $$\vert \hat\{f\}(n)\vert \rbrace _\{n\in E\}$ for some continuous funtion $f$ with spectrum in $E$.},
author = {Kislyakov, Serguei V.},
journal = {Annales de l'institut Fourier},
keywords = {multiplier; interpolation inequality; -regularity properties},
language = {eng},
number = {2},
pages = {147-183},
publisher = {Association des Annales de l'Institut Fourier},
title = {Fourier coefficients of continuous functions and a class of multipliers},
url = {http://eudml.org/doc/74798},
volume = {38},
year = {1988},
}

TY - JOUR
AU - Kislyakov, Serguei V.
TI - Fourier coefficients of continuous functions and a class of multipliers
JO - Annales de l'institut Fourier
PY - 1988
PB - Association des Annales de l'Institut Fourier
VL - 38
IS - 2
SP - 147
EP - 183
AB - If $x$ is a bounded function on ${\bf Z}$, the multiplier with symbol $x$ (denoted by $M_ x)$ is defined by $(M_ xf){\hat{\ }}=x\hat{f}$, $f\in L^ 2({\bf T})$. We give some conditions on $x$ ensuring the “interpolation inequality” $\Vert M_ xf\Vert _{L^ p}\le C\Vert f\Vert ^{\alpha }_{L^ 1}\Vert M_ xf\Vert _{L^ q}^{1-\alpha }$ (here $1&lt; p&lt; q$ and $\alpha =\alpha (p,q,x)$ is between 0 and 1). In most cases considered $M_ x$ fails to have stronger $L^ 1$-regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E\subset {\bf Z}$ every positive sequence in $\ell ^ 2(E)$ can be majorized by the sequence $\lbrace $$\vert \hat{f}(n)\vert \rbrace _{n\in E}$ for some continuous funtion $f$ with spectrum in $E$.
LA - eng
KW - multiplier; interpolation inequality; -regularity properties
UR - http://eudml.org/doc/74798
ER -