Marcinkiewicz’s theorem on operator multipliers of Fourier series

Dostanić, Milutin

doi:10.1090/S0002-9939-03-07017-5

Marcinkiewicz’s theorem on operator multipliers of Fourier series
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by Milutin R. Dostanić PDF

Proc. Amer. Math. Soc. 132 (2004), 391-396 Request permission

Abstract:

We give some sufficient conditions on the operators $A_{m}\!\in \!\mathcal {B} \left ( L^{p}\left ( 0,1\right ) \right )$ which for each $\Phi _{m}\in L^{p}\left ( 0,1\right )$ imply the inequality \[ \int \limits _{0}^{1}\int \limits _{0}^{2\pi }\left | \sum \limits _{m}e^{imx}\cdot A_{m}\Phi _{m}\left ( y\right ) \right | ^{p}dxdy\leq c_{p}^{p}\int \limits _{0}^{1}\int \limits _{0}^{2\pi }\left | \sum \limits _{m}e^{imx}\cdot \Phi _{m}\left ( y\right ) \right | ^{p}dxdy, \] $1<p<\infty .$

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