Another counterexample to Zygmund’s conjecture

Rey, Guillermo

doi:10.1090/proc/15222

Another counterexample to Zygmund’s conjecture

Author: Guillermo Rey
Journal: Proc. Amer. Math. Soc. 148 (2020), 5269-5275
MSC (2010): Primary 42B25
DOI: https://doi.org/10.1090/proc/15222
Published electronically: September 24, 2020
MathSciNet review: 4163839
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Abstract: We present a simple dyadic construction that yields a new counterexample to Zygmund’s conjecture in differentiation of integrals. Our result recovers Soria’s classical results in dimensions three and four, through a different construction, and gives new ones in all other dimensions.

References

A. Cordoba and R. Fefferman, A geometric proof of the strong maximal theorem, Ann. of Math. (2) 102 (1975), no. 1, 95–100. MR 379785, DOI https://doi.org/10.2307/1970976
Antonio Córdoba, Maximal functions, covering lemmas and Fourier multipliers, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–50. MR 545237
Antonio Córdoba, Maximal functions: a proof of a conjecture of A. Zygmund, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 1, 255–257. MR 513753, DOI https://doi.org/10.1090/S0273-0979-1979-14576-2
Miguel de Guzmán, Differentiation of integrals in $R^{n}$, Lecture Notes in Mathematics, Vol. 481, Springer-Verlag, Berlin-New York, 1975. With appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón. MR 0457661
E. M. Stein, Some problems in harmonic analysis, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–20. MR 545235
Robert Fefferman and Jill Pipher, A covering lemma for rectangles in ${\Bbb R}^n$, Proc. Amer. Math. Soc. 133 (2005), no. 11, 3235–3241. MR 2161145, DOI https://doi.org/10.1090/S0002-9939-05-07902-5

B. Jessen, J. Marcinkiewicz, and A. Zygmund, Note on the differentiability of multiple integrals, Fundamenta Mathematicae, 25(1):217–234, 1935.

Andrei K. Lerner and Fedor Nazarov, Intuitive dyadic calculus: the basics, Expo. Math. 37 (2019), no. 3, 225–265. MR 4007575, DOI https://doi.org/10.1016/j.exmath.2018.01.001

Fernando Soria, Examples and counterexamples to a conjecture in the theory of differentiation of integrals, Ann. of Math. (2) 123 (1986), no. 1, 1–9. MR 825837, DOI https://doi.org/10.2307/1971350

Alexander Stokolos, Zygmund’s program: some partial solutions, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1439–1453 (English, with English and French summaries). MR 2172270

A. Zygmund, A note on the differentiability of integrals, Colloq. Math. 16 (1967), 199–204. MR 210847, DOI https://doi.org/10.4064/cm-16-1-199-204