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.
"Sufficient conditions for three weight sum inequalities in Lebesgue spaces." Real Anal. Exchange 22 (1) 292 - 317, 1996/1997.