Differentiation of Besov spaces and the Nikodym maximal operator

Murcko, Jason

doi:10.1090/proc/13396

Differentiation of Besov spaces and the Nikodym maximal operator

Author: Jason Murcko
Journal: Proc. Amer. Math. Soc. 145 (2017), 2139-2153
MSC (2010): Primary 42B25, 42B35
DOI: https://doi.org/10.1090/proc/13396
Published electronically: December 30, 2016
MathSciNet review: 3611327
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study several questions related to differentiation of integrals for Besov spaces relative to the basis $\mathcal {R}$ of arbitrarily oriented rectangular parallelepipeds in $\mathbb{R}^{d}$ , $d \geq 2$ . We improve on positive and negative differentiation results of Aimar, Forzani, and Naibo and on capacitary and dimensional bounds for exceptional sets of Naibo. Our main tool in obtaining these improvements involves showing that bounds for the Nikodym maximal operator can be used to deduce boundedness properties of the local maximal operator associated to $\mathcal {R}$ .

[1] David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441
[2] Hugo Aimar, Liliana Forzani, and Virginia Naibo, Rectangular differentiation of integrals of Besov functions, Math. Res. Lett. 9 (2002), no. 2-3, 173–189. MR 1909636, https://doi.org/10.4310/MRL.2002.v9.n2.a4
[3] Thomas Bagby and William P. Ziemer, Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129–148. MR 344390, https://doi.org/10.1090/S0002-9947-1974-0344390-6
[4] J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257, https://doi.org/10.1007/BF01896376
[5] Michael Christ, Estimates for the 𝑘-plane transform, Indiana Univ. Math. J. 33 (1984), no. 6, 891–910. MR 763948, https://doi.org/10.1512/iumj.1984.33.33048
[6] Antonio Cordoba, The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977), no. 1, 1–22. MR 447949, https://doi.org/10.2307/2374006
[7] A. Nagel, E. M. Stein, and S. Wainger, Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 3, 1060–1062. MR 466470, https://doi.org/10.1073/pnas.75.3.1060
[8] Virginia Naibo, Bessel capacities and rectangular differentiation in Besov spaces, J. Math. Anal. Appl. 324 (2006), no. 2, 834–840. MR 2265084, https://doi.org/10.1016/j.jmaa.2005.12.071
[9] Otton Nikodým, Sur la mesure des ensembles plans dont tous les points sont rectilinéairement accessibles, Fund. Math. 10 (1927), no. 1, 116-168.
[10] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
[11] A. M. Stokolos, On the differentiation of integrals by bases that do not possess the density property, Mat. Sb. 187 (1996), no. 7, 113–138 (Russian, with Russian summary); English transl., Sb. Math. 187 (1996), no. 7, 1061–1085. MR 1404191, https://doi.org/10.1070/SM1996v187n07ABEH000148
[12] A. M. Stokolos, On differentiation of integrals with respect to bases of convex sets, Studia Math. 119 (1996), no. 2, 99–108. MR 1391470, https://doi.org/10.4064/sm-119-2-99-108
[13] Terence Tao, The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), no. 2, 363–375. MR 1666558, https://doi.org/10.1215/S0012-7094-99-09610-2
[14] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540
[15] Thomas Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), no. 3, 651–674. MR 1363209, https://doi.org/10.4171/RMI/188