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
HTML articles powered by AMS MathViewer

by Jason Murcko PDF

Proc. Amer. Math. Soc. 145 (2017), 2139-2153 Request permission

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 $\mathcal {R}$.

References

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, DOI 10.1007/978-3-662-03282-4
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, DOI 10.4310/MRL.2002.v9.n2.a4
Thomas Bagby and William P. Ziemer, Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129–148. MR 344390, DOI 10.1090/S0002-9947-1974-0344390-6
J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257, DOI 10.1007/BF01896376
Michael Christ, Estimates for the $k$-plane transform, Indiana Univ. Math. J. 33 (1984), no. 6, 891–910. MR 763948, DOI 10.1512/iumj.1984.33.33048
Antonio Cordoba, The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977), no. 1, 1–22. MR 447949, DOI 10.2307/2374006
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, DOI 10.1073/pnas.75.3.1060
Virginia Naibo, Bessel capacities and rectangular differentiation in Besov spaces, J. Math. Anal. Appl. 324 (2006), no. 2, 834–840. MR 2265084, DOI 10.1016/j.jmaa.2005.12.071
Otton Nikodým, Sur la mesure des ensembles plans dont tous les points sont rectilinéairement accessibles, Fund. Math. 10 (1927), no. 1, 116–168.
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
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, DOI 10.1070/SM1996v187n07ABEH000148
A. M. Stokolos, On differentiation of integrals with respect to bases of convex sets, Studia Math. 119 (1996), no. 2, 99–108. MR 1391470, DOI 10.4064/sm-119-2-99-108
Terence Tao, The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), no. 2, 363–375. MR 1666558, DOI 10.1215/S0012-7094-99-09610-2
Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540, DOI 10.1007/978-3-0346-0416-1
Thomas Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), no. 3, 651–674. MR 1363209, DOI 10.4171/RMI/188