Geodesic
Mutual estimates of $L^p$-norms, and the Bellman function
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 81-138

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A description of the possible values of the $L^p$-norm of a function is obtained under fixed $L^p$-norms for two other values of $p$ and under a natural multiplicative restriction, like the Muckenhoupt condition. Among special cases of our results, we mention simple interpolation inequalities between two $L^p$-norms, as well as nontrivial ones, such as the Gehring inequality or the reverse Hölder inequality for Mackenhoupt weights. The method of the paper is construction of the true Bellman function for the corresponding extremal problem. Bibl. – 5 titles. 
@article{ZNSL_2008_355_a3,
     author = {V. I. Vasyunin},
     title = {Mutual estimates of $L^p$-norms, and the {Bellman} function},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {81--138},
     publisher = {mathdoc},
     volume = {355},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a3/}
}
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V. I. Vasyunin. Mutual estimates of $L^p$-norms, and the Bellman function. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 81-138. http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a3/