Kernel operators with variable intervals of integration in lebesgue spaces and applications

New criteria of Lp - Lq boundedness of Hardy-Steklov type operator (1.1) with both increasing on (0,∞) boundary functions a(x) and b(x) are obtained for 1< p < q < ∞ and 0 < q < p < ∞, p > 1. This result is applied for two-weighted Lp - Lq characterization of the corresponding geometric Steklov operator (1.3) and other related problems.