Additive inequalities for the norms of intermediate derivatives (Q5937508)

Let \(f\) be a function with absolutely continuous derivative \(f^{ (n-1)}\) and \(f^{(n)}\in L_r(a,b)\). Then it is well-known that for \(0\leq k<n\), \(1 \leq p\), \(q,r\leq +\infty\) there exist constants \(A,B\) such that for \(f\) we have \[ \|f^{(k)} \|_{L_q (a,b)}\leq A\|f|_{L_p (a,b)}+ B\|f^{(n)} \|_{L_r(a,b)}. \tag{1} \] Let \(\inf B\) denote the infimum of all \(B\) for which (1) is valid for all \(f\in L_r(a,b)\). Then \(\inf B=0\). But the analogously defined \(\inf A\) is strictly positive. In the paper the authors give a general scheme for obtaining inequalities (1) and give a formula determining \(\inf A\).