Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation

De Pauw, Thierry

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1473-1512.

On désigne par $𝔾(n,m)$ la grassmannienne constituée des sous-espaces vectoriels de dimension $m$ dans ${ℝ}^n$, par ${ℒ}^n$ la mesure de Lebesgue dans ${ℝ}^n$, par ${\mathscr{H}}^m$ la mesure de Hausdorff $m$-dimensionnelle dans ${ℝ}^n$ et par $\alpha \left(m\right)={ℒ}^m\left(B(0,1)\right)$ la mesure de Lebesgue de la boule euclidienne unité de ${ℝ}^m$. Nous montrons que si $A\subset {ℝ}^n$ est borélien et $W_0:A\to 𝔾(n,m)$ est lipschitzien, alors

$$\underset{r\to 0^{+}}{lim\; sup}\frac{{\mathscr{H}}^m\left[A\cap B(x,r)\cap (x+W_0\left(x\right))\right]}{\alpha \left(m\right)r^m}\ge \frac{1}{2^n},$$

pour ${ℒ}^n$-presque tout $x\in A$. Il en résulte en particulier que $A$ est ${ℒ}^n$-négligeable si et seulement si ${\mathscr{H}}^m(A\cap (x+W_0\left(x\right))=0$, pour ${ℒ}^n$-presque tout $x\in A$.

Let $𝔾(n,m)$ be the Grassmannian consisting of $m$-dimensional vector subspaces of ${ℝ}^n$, let ${ℒ}^n$ be the Lebesgue measure in ${ℝ}^n$, let ${\mathscr{H}}^m$ be the $m$-dimensional Hausdorff measure in ${ℝ}^n$, and let $\alpha \left(m\right)={ℒ}^m\left(B(0,1)\right)$ be the Lebesgue measure of the Euclidean unit ball of ${ℝ}^m$. We establish that, if $A\subset {ℝ}^n$ is Borel measurable and $W_0:A\to 𝔾(n,m)$ is Lipschitzian, then

$$\underset{r\to 0^{+}}{lim\; sup}\frac{{\mathscr{H}}^m\left[A\cap B(x,r)\cap (x+W_0\left(x\right))\right]}{\alpha \left(m\right)r^m}\ge \frac{1}{2^n},$$

for ${ℒ}^n$-almost every $x\in A$. In particular, it follows that $A$ is ${ℒ}^n$-negligible if and only if ${\mathscr{H}}^m(A\cap (x+W_0\left(x\right))=0$, for ${ℒ}^n$-almost every $x\in A$.

Classification : 28A75, 26B15
Keywords: Lebesgue measure, Nikodým set, negligible set, derivation basis, Zygmund conjecture, Lipschitz differentiation
Mot clés : Mesure de Lebesgue, ensemble de Nikodým, ensemble négligeable, base de derivation, conjecture de Zygmund, differentiation lipschitzienne

[1] Bourgain, J. A remark on the maximal function associated to an analytic vector field, Analysis at Urbana, Vol. I (Urbana, IL, 1986–1987) (London Math. Soc. Lecture Note Ser.), Volume 137, Cambridge Univ. Press, Cambridge, 1989, pp. 111-132 | MR | Zbl

[2] Bruckner, A. M. Differentiation of integrals, Amer. Math. Monthly, Volume 78 (1971) no. 9, part 2, pp. 1-51 | MR | Zbl

[3] Cohn, Donald L. Measure theory, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/Springer, New York, 2013 | DOI | MR

[4] De Pauw, Th. Concentrated, nearly monotonic, epiperimetric measures in Euclidean space, J. Differential Geom., Volume 77 (2007) no. 1, pp. 77-134 http://projecteuclid.org/euclid.jdg/1185550816 | MR | Zbl

[5] Evans, Lawrence C.; Gariepy, Ronald F. Measure theory and fine properties of functions, Textbooks in Math., CRC Press, Boca Raton, FL, 2015

[6] Falconer, K. J. Sets with prescribed projections and Nikodým sets, Proc. London Math. Soc. (3), Volume 53 (1986) no. 1, pp. 48-64 | DOI | Zbl

[7] Federer, Herbert Curvature measures, Trans. Amer. Math. Soc., Volume 93 (1959), pp. 418-491 | DOI | MR | Zbl

[8] Federer, Herbert Geometric measure theory, Grundlehren Math. Wiss., 153, Springer-Verlag New York, Inc., New York, 1969

[9] de Guzmán, Miguel Real variable methods in Fourier analysis, Notas de Matemática, 75, North-Holland Publishing Co., Amsterdam-New York, 1981

[10] Kharazishvili, A. B. Nonmeasurable sets and functions, North-Holland Math. Studies, 195, Elsevier Science B.V., Amsterdam, 2004

[11] Lacey, Michael; Li, Xiaochun On a conjecture of E. M. Stein on the Hilbert transform on vector fields, Mem. Amer. Math. Soc., 205, no. 965, American Mathematical Society, Providence, RI, 2010 | DOI

[12] Nikodým, O. Sur la mesure des ensembles plans dont tous les points sont rectilinéairement accessibles, Fund. Math., Volume 10 (1927), pp. 116-168 | DOI | Zbl

[13] Srivastava, S. M. A course on Borel sets, Graduate Texts in Math., 180, Springer-Verlag, New York, 1998 | DOI