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Acta Mathematica Hungarica
Author:
Rostom Getsadze
Summary
Let
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$\big\{\varphi_k(x)\big\}_{k=1}^\infty$
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and
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$\big\{\psi_l(y)\big\}_{l=1}^\infty$
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be arbitrary orthonormal systems (ONS) on
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$[0,1]$
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that satisfy the conditions (5)
where
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\begin{document}
$M_1$
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and
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$M_2$
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are positive constants. Let
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$A$
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be a Lebesgue measurable subset of
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${[0,1]}^2$
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such that
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$S^{\varphi,\psi}(f,x,y)< \infty$
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, for a.e.\
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$(x,y)\in A$
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for every Lebesgue integrable function
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$f$
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on
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${[0,1]}^2$
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, where
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$S^{\varphi,\psi}$
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is the
Sunouchi operator with respect to the product system
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$\big\{\varphi_k(x) \psi_l(y)$
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, $k, l=1,2,\dots\big\}$. We study the
following problem: How large may the measure of
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$A$
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be? We prove that for each such system we have \documentclass{aastex}
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$$\mu_2A \le 1-\frac{1}{M_1^2 M_2^2}$$
\end{document} (for the
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$d$
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-fold product systems we have
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$\mu_d A \le 1-\frac{1}{M_1^2 M_2^2\dots M_d^2}$
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,
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$d\ge 2$
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). This estimate is sharp in the class of all such product systems.
