On a multiplicative inequality for derived functions (Q1812844)

The author considers the space \(L_ \infty\) of all essentially bounded functions on \([0,1]\) and introduces a subspace \(H^{r,s}\). This subspace consists of all functions with essentially bounded derivatives of order at most \(r\) and with a Lipschitz condition in \(L_ \infty\) for the highest order derivatives involving a logarithmic factor to the power \(s\) in the denominator. The following estimate is obtained: \(\| x\|^{\alpha_ 0+ \alpha_ 1+\alpha_ 2}_{H^{r,s}}\leq c\| x^{\alpha_ 0}(\dot x)^{\alpha_ 1}(\ddot x)^{\alpha_ 2}\|_{L_ \infty}\) under certain conditions on \(x\) and with \(r\), \(s\) determined by the \(\alpha\)'s.