Functional analysis
Sparse Brudnyi and John–Nirenberg Spaces
Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 1059-1069.

A generalization of the theory of Y. Brudnyi [7], and A. and Y. Brudnyi [5, 6], is presented. Our construction connects Brudnyi’s theory, which relies on local polynomial approximation, with new results on sparse domination. In particular, we find an analogue of the maximal theorem for the fractional maximal function, solving a problem proposed by Kruglyak–Kuznetsov. Our spaces shed light on the structure of the John–Nirenberg spaces. We show that $SJ{N}_{p}$ (sparse John–Nirenberg space) coincides with ${L}^{p},1 This characterization yields the John–Nirenberg inequality by extrapolation and is useful in the theory of commutators.

Accepted:
Published online:
DOI: 10.5802/crmath.252
Classification: 42B35,  42B25,  46E30,  46E35
Óscar Domínguez 1; Mario Milman 2

2 M. Milman, Instituto Argentino de Matematica, Buenos Aires, Argentina
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Óscar Domínguez; Mario Milman. Sparse Brudnyi and John–Nirenberg Spaces. Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 1059-1069. doi : 10.5802/crmath.252. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.252/

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