We characterize the pairs of weights[inline-graphic not available: see fulltext] for which the Hardy-Steklov-type operator[inline-graphic not available: see fulltext] applies[inline-graphic not available: see fulltext] into weak-[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext], assuming certain monotonicity conditions on[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext] and[inline-graphic not available: see fulltext].