Аннотация:We study norm convolution inequalities in Lebesgue and Lorentz spaces. First, we improve the well-known O’Neil’s inequality for the convolution operators and prove corresponding estimate from below. Second, we obtain Young–O’Neil-type estimate in the Lorentz spaces for the limit value parameters, i.e.,|| K ∗ f||_(L(p,h1)→L(p,h2)). Finally, similar estimates in the weighted Lorentz spaces are presented.