Summary.
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.
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Received: March 22, 2001, revised version: August 21, 2001.
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Kałamajska, A. A remark on Kolmogorov‚s theorem. Aequ. math. 64, 165–169 (2002). https://doi.org/10.1007/s00010-002-8039-z
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DOI: https://doi.org/10.1007/s00010-002-8039-z