Asymptotical stability of Muckenhoupt weights through Gurov-Reshetnyak classes

Aalto, Daniel; Berkovits, Lauri

doi:10.1090/S0002-9947-2012-05677-7

Asymptotical stability of Muckenhoupt weights through Gurov-Reshetnyak classes
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by Daniel Aalto and Lauri Berkovits PDF

Trans. Amer. Math. Soc. 364 (2012), 6671-6687 Request permission

Abstract:

We show that weights in the Gurov-Reshetnyak class $GR_\varepsilon (\mu )$ satisfy a weak reverse Hölder inequality with an explicit and asymptotically sharp bound for the exponent, thus extending classical results from the Euclidean setting to doubling metric measure spaces. As an application, we study asymptotical behaviour of embeddings between Muckenhoupt classes and reverse Hölder classes.

References

Daniel Aalto and Juha Kinnunen, The discrete maximal operator in metric spaces, J. Anal. Math. 111 (2010), 369–390. MR 2747071, DOI 10.1007/s11854-010-0022-3
D. Aalto. Weak $L^\infty$ and $bmo$ in metric spaces. (arXiv:0910.1207)
B. Bojarski, Remarks on the stability of reverse Hölder inequalities and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 89–94. MR 802470, DOI 10.5186/aasfm.1985.1011
B. Bojarski. On Gurov-Reshetnyak classes. Bounded Mean Oscillation in complex analysis, University of Joensuu, Publications in sciences, no. 14, p. 21-42, Joensuu, 1989.
B. Bojarski, C. Sbordone, and I. Wik, The Muckenhoupt class $A_1(\textbf {R})$, Studia Math. 101 (1992), no. 2, 155–163. MR 1149569, DOI 10.4064/sm-101-2-155-163
Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948
David Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc. 347 (1995), no. 8, 2941–2960. MR 1308005, DOI 10.1090/S0002-9947-1995-1308005-6
Michelangelo Franciosi, Weighted rearrangements and higher integrability results, Studia Math. 92 (1989), no. 2, 131–139. MR 986944, DOI 10.4064/sm-92-2-131-139
Michelangelo Franciosi, On weak reverse integral inequalities for mean oscillations, Proc. Amer. Math. Soc. 113 (1991), no. 1, 105–112. MR 1068122, DOI 10.1090/S0002-9939-1991-1068122-7
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
L. G. Gurov, The stability of Lorentz transformations. Estimates for the derivatives, Dokl. Akad. Nauk SSSR 220 (1975), 273–276 (Russian). MR 0407778
L. G. Gurov and Ju. G. Rešetnjak, A certain analogue of the concept of a function with bounded mean oscillation, Sibirsk. Mat. Ž. 17 (1976), no. 3, 540–546, 716 (Russian). MR 0427565
D. V. Isangulova, A class of mappings with bounded specific oscillation, and the integrability of mappings with bounded distortion on Carnot groups, Sibirsk. Mat. Zh. 48 (2007), no. 2, 313–334 (Russian, with Russian summary); English transl., Siberian Math. J. 48 (2007), no. 2, 249–267. MR 2330061, DOI 10.1007/s11202-007-0025-1
D. V. Isangulova, Local stability of mappings with bounded distortion on Heisenberg groups, Sibirsk. Mat. Zh. 48 (2007), no. 6, 1228–1245 (Russian, with Russian summary); English transl., Siberian Math. J. 48 (2007), no. 6, 984–997. MR 2397505, DOI 10.1007/s11202-007-0101-6
Tadeusz Iwaniec, On $L^{p}$-integrability in PDEs and quasiregular mappings for large exponents, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 2, 301–322. MR 686647, DOI 10.5186/aasfm.1982.0719
Juha Kinnunen, A stability result on Muckenhoupt’s weights, Publ. Mat. 42 (1998), no. 1, 153–163. MR 1628162, DOI 10.5565/PUBLMAT_{4}2198_{0}8
A. A. Korenovskiĭ, The inverse Hölder inequality, the Muckenhoupt condition, and equimeasurable permutations of functions, Dokl. Akad. Nauk 323 (1992), no. 2, 229–232 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 45 (1992), no. 2, 301–304 (1993). MR 1191537
A. A. Korenovskiĭ, The connection between mean oscillations and exact exponents of summability of functions, Mat. Sb. 181 (1990), no. 12, 1721–1727 (Russian); English transl., Math. USSR-Sb. 71 (1992), no. 2, 561–567. MR 1099524, DOI 10.1070/SM1992v071n02ABEH001409
A. A. Korenovskyy, A. K. Lerner, and A. M. Stokolos, A note on the Gurov-Reshetnyak condition, Math. Res. Lett. 9 (2002), no. 5-6, 579–583. MR 1906061, DOI 10.4310/MRL.2002.v9.n5.a1
Michael Brian Korey, Ideal weights: asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation, J. Fourier Anal. Appl. 4 (1998), no. 4-5, 491–519. MR 1658636, DOI 10.1007/BF02498222
Riikka Korte and Outi Elina Kansanen, Strong $A_\infty$-weights are $A_\infty$-weights on metric spaces, Rev. Mat. Iberoam. 27 (2011), no. 1, 335–354. MR 2815740, DOI 10.4171/rmi/638
O. E. Kansanen. The Gehring lemma in metric spaces. (arXiv:0704.3916)
N. A. Malaksiano, On exact inclusions of Gehring classes in Muckenhoupt classes, Mat. Zametki 70 (2001), no. 5, 742–750 (Russian, with Russian summary); English transl., Math. Notes 70 (2001), no. 5-6, 673–681. MR 1882347, DOI 10.1023/A:1012983028054
Joaquim Martín and Mario Milman, Gehring’s lemma for nondoubling measures, Michigan Math. J. 47 (2000), no. 3, 559–573. MR 1813544, DOI 10.1307/mmj/1030132591
J. Mateu, P. Mattila, A. Nicolau, and J. Orobitg, BMO for nondoubling measures, Duke Math. J. 102 (2000), no. 3, 533–565. MR 1756109, DOI 10.1215/S0012-7094-00-10238-4
Themis Mitsis, Embedding $B_\infty$ into Muckenhoupt classes, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1057–1061. MR 2117206, DOI 10.1090/S0002-9939-04-07803-7
Benjamin Muckenhoupt, Weighted reverse weak type inequalities for the Hardy-Littlewood maximal function, Pacific J. Math. 117 (1985), no. 2, 371–377. MR 779926
C. J. Neugebauer. The precise range of indices for the $RH_r$- and $A_p$-classes. (arXiv:9809162)
Joan Orobitg and Carlos Pérez, $A_p$ weights for nondoubling measures in $\textbf {R}^n$ and applications, Trans. Amer. Math. Soc. 354 (2002), no. 5, 2013–2033. MR 1881028, DOI 10.1090/S0002-9947-02-02922-7
Anastasios Politis, Sharp results on the relation between weight spaces and BMO, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–The University of Chicago. MR 2716561
Yu. G. Reshetnyak. Stability estimates in Liouville’s theorem, and the Lp-integrability of the derivatives of quasi-conformal mappings. Siberian Math. J. 17 (1976), 653-674.
Jan-Olov Strömberg and Alberto Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, vol. 1381, Springer-Verlag, Berlin, 1989. MR 1011673, DOI 10.1007/BFb0091154
Ingemar Wik, Note on a theorem by Reshetnyak-Gurov, Studia Math. 86 (1987), no. 3, 287–290. MR 917054, DOI 10.4064/sm-86-3-287-290
Ingemar Wik, Reverse Hölder inequalities with constant close to $1$, Ricerche Mat. 39 (1990), no. 1, 151–157. MR 1101311