A remark on Kolmogorov‚s theorem

We deduce the pointwise one-dimensional multiplicative inequalities \( |f^{(k)}(x)| \le CMf(x)^{1-k/m}Mf^{(m)}(x)^{k/m} \), where Mh is the Hardy-Littlewood maximal function of h from a classical theorem of Kolmogorov. This implies that in the one-dimensional case the Gagliardo-Nirenberg inequalities in weighted L p spaces with Muckenhoupt‚s weights are consequences of Kolmogorov's theorem.