Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis

Abstract

We obtain new exact inequalities of the form $$∥x(k)∥_q ⩽ K∥x∥^{α}_p ∥x(r)∥^{1−α}_s$$ for functions defined on the axis $R$ or the semiaxis $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s=1,$$ for functions defined on the axis $R$ in the case where $$r = 2,\; k = 1,\; q ∈ [2,∞),\; p = ∞,\; s= 1,$$ and for functions of constant sign on $R$ or $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s = ∞$$ and in the case where $$r = 2,\; k = 1,\; p ∈ (0,∞),\; q = s = ∞.$$