Estimates of the Lebesgue function of Fourier sums over trigonometric polynomials orthogonal with a weight not belonging to the spaces $L^r$ $(r>1)$

A two-sided pointwise estimate is obtained for the Lebesgue function of Fourier sums with respect to trigonometric polynomials orthogonal with a $2\pi$-periodic weight that differs from the function $1/|\sin(\tau/2)|$ by some factor slowly changing at zero. The weight under consideration does not belong to the space $L^r$ for any $r>1$. A similar result for polynomials orthogonal on the interval $[-1,1]$ is obtained in the form of a corollary.