Geodesic
On Holder's inequality in Lebesgue spaces with variable order of summability
Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 472-482

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In this paper, we introduce a new version of the definition of a quasi-norm (in particular, a norm) in Lebesgue spaces with variable order of summability. Using it, we prove an analogue of Hölder's inequality for such spaces, which is more general and more precise than those known earlier. 
@article{CMFD_2021_67_3_a4,
     author = {V. I. Burenkov and T. V. Tararykova},
     title = {On {Holder's} inequality in {Lebesgue} spaces with variable order of summability},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {472--482},
     publisher = {mathdoc},
     volume = {67},
     number = {3},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a4/}
}
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V. I. Burenkov; T. V. Tararykova. On Holder's inequality in Lebesgue spaces with variable order of summability. Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 472-482. http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a4/