Sufficient conditions for three weight sum inequalities in Lebesgue spaces

Abstract

Conditions (in terms of integrals of the weights) are derived, under which the weighted \(L^q\)-norm of the \(j\)-th order derivative of the function \(u\) can be estimated by the sum of the weighted \(L^r\)-norm of \(u\) and of the weighted \(L^p\)-norm of its \(m\)-th order derivative, \(j\lt m\). All mutual positions of the parameters, \(p,\, q, \, r\) are admissible.