Boundedness of some maximal and multiplicator operators (Q1589189)

Let \(\Omega=\{\sum \varepsilon_i\omega_i : \varepsilon_i=\{0,1\}\}\), for a sequence \(\omega_i>0\), \(\sum_{i=n+1}^{\infty}\omega_i<\omega_n\), \(\sum \omega_i=\pi /2\). The author studies the case when \(\lim \omega_{n+1}/ \omega_n =1/2\), and shows that the set of all rectangles in the directions from \(\Omega\) does not derivate even \(L^\infty\). As an application the following theorem is proved. Theorem. If \(\Omega^*\) is a periodical extension of \(\Omega\) to the circle, let \(P_{\Omega}\) be the intersection of all halfplanes containing the unit circle with boundary tangent to the circle in the points of \(\Omega^*\). Then the Fourier multiplier with the characteristic function of \(P_{\Omega}\) as the symbol is unbounded on \(L^p\), with \(p\neq 2\).