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Marcinkiewicz's theorem on operator multipliers of Fourier series

Author(s): Milutin R. Dostanic
Journal: Proc. Amer. Math. Soc. 132 (2004), 391-396.
MSC (2000): Primary 42B15
Posted: June 11, 2003
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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

$\begin{displaymath}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\left\vert \sum\limi... ..._{m}e^{imx}\cdot \Phi_{m}\left( y\right) \right\vert ^{p}dxdy, \end{displaymath}$

$1<p<\infty.$

References:

1.: J. B. Garnett, ``Bounded analytic functions'', Pure and Applied Mathematics, vol. 96, Academic Press, 1981. MR 83g:30037
2.: G. H. Hardy, J. E. Littlewood, and G. Polya, ``Inequalities'', 2 $^{nd.}$ ed., Cambridge University Press, Cambridge, UK, 1952. MR 13:727e
3.: R. Schatten, ``A Theory of cross-spaces'', Ann. Math. Studies, $\mathcal{N}^{o} 26,1950.$ MR 12:186e
4.: A. Zygmund, ``Trigonometric Series'', Izdatelstvo ``Mir'', Moscow, 1965. MR 31:2554


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