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Comptes Rendus. Mathématique
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 SJN p (sparse John–Nirenberg space) coincides with L p ,1<p<. This characterization yields the John–Nirenberg inequality by extrapolation and is useful in the theory of commutators.

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

1 O. Domínguez, Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain.
2 M. Milman, Instituto Argentino de Matematica, Buenos Aires, Argentina
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/

[1] Emil Airta; Tuomas Hytönen; Kangwei Li; Henri Martikainen; Tuomas Oikari Off-diagonal estimates for bi-commutators (2020) https://arxiv.org/abs/2005.03548

[2] Sergey Astashkin; Mario Milman Garsia–Rodemich spaces: Local maximal functions and interpolation, Stud. Math., Volume 255 (2020) no. 1, pp. 1-26 | DOI | MR | Zbl

[3] Jesús Bastero; Mario Milman; Francisco J. Ruiz Commutators of the maximal and sharp functions, Proc. Am. Math. Soc., Volume 128 (2000), pp. 65-74 | MR | Zbl

[4] Colin Bennett; Ronald A. DeVore; Robert Sharpley Weak-L and BMO, Ann. Math., Volume 113 (1981), pp. 601-611 | DOI | MR | Zbl

[5] Alexander Brudnyi; Yuri A. Brudnyĭ Multivariate bounded variation functions of Jordan–Wiener type, J. Approx. Theory, Volume 251 (2020), 105346, 70 pages | MR | Zbl

[6] Alexander Brudnyi; Yuri A. Brudnyĭ On the Banach structure of multivariate BV spaces, Diss. Math., Volume 548 (2020), pp. 1-52 | MR | Zbl

[7] Yuri A. Brudnyĭ Spaces defined by means of local approximations, Tr. Mosk. Mat. O.-va, Volume 24 (1971), pp. 69-132 English transl. in Trans. Mosc.. Math. Soc. 24 (1971), p. 73-139 | Zbl

[8] Alberto P. Calderón; Antoni Zygmund On the existence of certain singular integrals, Acta Math., Volume 88 (1952), pp. 85-139 | DOI | MR | Zbl

[9] Sergio Campanato Su un teorema di interpolazione di G. Stampacchia, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 20 (1966), pp. 649-652 | Numdam | MR | Zbl

[10] Ronald R. Coifman; Richard Rochberg; Guido Weiss Factorization theorems for Hardy spaces in several variables, Ann. Math., Volume 103 (1976), pp. 611-635 | DOI | MR | Zbl

[11] Galia Dafni; Tuomas Hytönen; Riikka Korte; Hong Yue The space JN p : nontriviality and duality, J. Funct. Anal., Volume 275 (2018) no. 3, pp. 577-603 | DOI | Zbl

[12] Oscar Domínguez; Mario Milman (in preparation)

[13] Charles Fefferman Characterization of bounded mean oscillation, Bull. Am. Math. Soc., Volume 77 (1970), pp. 587-588 | DOI | MR | Zbl

[14] Charles Fefferman; Elias M. Stein H p spaces of several variables, Acta Math., Volume 129 (1972), pp. 137-193 | DOI | Zbl

[15] Michael Frazier; Björn Jawerth A discrete transform and decompositions of distribution spaces, J. Funct. Anal., Volume 93 (1990) no. 1, pp. 34-170 | DOI | MR | Zbl

[16] Adriano M. Garsia Martingale Inequalities: Seminar Notes on Recent Progress, Mathematics Lecture Notes Series, W. A. Benjamin, Inc., 1973 | Zbl

[17] Adriano M. Garsia; Eugène Rodemich Monotonicity of certain functionals under rearrangements, Ann. Inst. Fourier, Volume 24 (1974) no. 2, pp. 67-116 | DOI | MR | Zbl

[18] Tuomas Hytönen The L p -to-L q boundedness of commutators with applications to the Jacobian operator (2021) https://arxiv.org/abs/1804.11167

[19] Svante Janson Mean oscillation and commutators of singular integral operators, Ark. Mat., Volume 16 (1978), pp. 263-270 | DOI | MR | Zbl

[20] Fritz John; Louis Nirenberg On functions of bounded mean oscillation, Commun. Pure Appl. Math., Volume 14 (1961), pp. 415-426 | DOI | MR | Zbl

[21] Natan Kruglyak Smooth analogues of the Calderón–Zygmund decomposition, quantitative covering theorems and the K-functional for the couple (L q ,W ˙ p k ), Algebra Anal., Volume 8 (1996) no. 4, pp. 110-160 English transl. in St. Petersbg. Math. J. 8 (1997), no. 4, p. 617-649

[22] Natan Kruglyak; Evgeny A. Kuznetsov Sharp integral estimates for the fractional maximal function and interpolation, Ark. Mat., Volume 44 (2006) no. 2, pp. 309-326 | DOI | MR | Zbl

[23] Andrei K. Lerner A simple proof of the A 2 conjecture, Int. Math. Res. Not., Volume 2013 (2013) no. 14, pp. 3159-3170 | DOI | MR | Zbl

[24] Andrei K. Lerner; Fedor Nazarov Intuitive dyadic calculus: the basics, Expo. Math., Volume 37 (2019) no. 3, pp. 225-265 | DOI | MR | Zbl

[25] Mario Milman Marcinkiewicz spaces, Garsia–Rodemich spaces and the scale of John–Nirenberg self improving inequalities, Ann. Acad. Sci. Fenn., Math., Volume 41 (2016) no. 1, pp. 491-501 | DOI | MR | Zbl

[26] Jürgen Moser A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Commun. Pure Appl. Math., Volume 13 (1960), pp. 457-468 | DOI | MR | Zbl

[27] Zeev Nehari On bounded bilinear forms, Ann. Math., Volume 65 (1957), pp. 153-162 | DOI | MR | Zbl

[28] Jaak Peetre New Thoughts on Besov Spaces, Duke University Mathematics Series, Duke University, 1976 | Zbl

[29] Frigyes Riesz Untersuchungen über systeme integrierbarer funktionen, Math. Ann., Volume 69 (1910), pp. 449-497 | DOI | Zbl

[30] Guido Stampacchia (p,λ) -spaces and interpolation, Commun. Pure Appl. Math., Volume 17 (1964), pp. 293-306 | DOI

[31] Guido tampacchia The spaces L (p,λ) ,N (p,λ) and interpolation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 19 (1965), pp. 443-462

[32] Hans Triebel Theory of Function Spaces II, Monographs in Mathematics, 84, Birkhäuser, 1992 | MR | Zbl

[33] Nicholas T. Varopoulos Hardy–Littlewood theory for semigroups, J. Funct. Anal., Volume 63 (1985), pp. 240-260 | DOI | MR | Zbl

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