ИСТИНА

The talk is a survey of resluts obtained in collaboration with G.V.Kalachev on the existence of maximizers of the functional f-> ||f*k||_r, where f is a function of norm 1 in Lp(R^n), the convolution kernel k is a given function in Lq, and the Lebesgue exponents p,q,r are related by Young's condition 1/p+1/q=1+1/r. If p,q,r lie strictly between 1 and infinity, the existence without further assumption is out result of 2019. The recent addition is the existence result in the case of k lying in weak Lq. Here some extra assumptions are needed, for example, it suffices to have k lying in a Lorentz space L_{q,s} with s<infinity.