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

Let $\mathscr{S}_{x}$ be the set of squares centered at $x\in\mathbb{R}^{2}$. The Lebesgue density theorem states that if $f:\mathbb{R}^{2}\to\mathbb{R}$ is Lebesgue summable, then

for $\mathscr{L}^{2}$-almost every $x\in\mathbb{R}^{2}$. This consequence of the Vitali covering theorem fails if $\mathscr{S}_{x}$ is replaced with $\mathscr{R}_{x}$, the set of rectangles centered at $x$ of arbitrary direction and eccentricity. It fails even for some indicator function $f=\mathbbm{1}_{A}$. Indeed, Nikodým [12] defined a set $A\subseteq[0,1]\times[0,1]$ of full measure with the following property: For every $x\in A$, there exists a line $L(x)$ such that $A\cap L(x)=\{x\}$. In fact, Nikodým’s example shows that the Lebesgue density theorem fails if we replace $\mathscr{S}_{x}$ with $\mathscr{R}_{x}^{L}$, the set of rectangles of arbitrary eccentricity, centered at $x$, and one side of which is parallel to $L(x)$. Furthermore [4, Chap. IV Theorem 3.5], replacing $A$ by a nonnegligible subset of $A$, one may choose $L$ to be continuous with respect to $x$. Yet, if $L$ is constant, then the Lebesgue density theorem with respect to $\mathscr{R}_{x}^{L}$ holds, for $p$-summable functions $f$, $1<p\leqslant\infty$, by virtue of a theorem of Zygmund, though the corresponding version of the Vitali covering theorem fails, according to an example of H. Bohr [4, Chap. IV Theorem 1.1]. This raises the question: What regularity condition of $x\mapsto L(x)$ guarantees that the Lebesgue density theorem holds with respect to $\mathscr{R}_{x}^{L}$ for, say, functions that are square summable? The following version is a conjecture reportedly [11] attributed to Zygmund. Assume that $x\mapsto L(x)$ is a Lipschitzian field of lines, with $x\in L(x)$, and $f:\mathbb{R}^{2}\to\mathbb{R}$ is square summable. Is it true that

for $\mathscr{L}^{2}$-almost every $x\in\mathbb{R}^{2}$? Here, $\mathscr{H}^{1}$ is the 1-dimensional Hausdorff measure.

E.M. Stein raised the singular integral variant of this conjecture. Both have received much attention from the harmonic analysis community. To the author’s knowledge, most results recorded so far in the literature, via the maximal function approach, assume some extra regularity property of $L$ – namely, that $L$ be $C^{1}$ together with a hypothesis on the variation of the derivative – see, for instance, [1] and [11].

In this paper we offer a novel approach – a kind of change of variable based on an appropriate fibration and the coarea formula. This allows for treating the case when $L$ is Lipschitzian, without further restriction. We obtain the following lower density bound for indicator functions. Let $x\mapsto L(x)$ be a Lipschitzian field of lines, with $x\in L(x)$, and let $A\subseteq\mathbb{R}^{2}$ be Lebesgue measurable. Then,

for $\mathscr{L}^{2}$-almost every $x\in A$. Incidentally, the following corollary – a nonparallel version of Fubini theorem – seems to be new as well. The set $A$ is Lebesgue negligible if and only if $\mathscr{H}^{1}(A\cap L(x))=0$, for $\mathscr{L}^{2}$-almost every $x\in\mathbb{R}^{2}$.

Our results hold in any dimension and codimension.

Let $A$ be a subset of Euclidean space $\mathbb{R}^{n}$, $n\geqslant 2$, and let $\mathscr{L}^{n}$ be the Lebesgue outer measure. We start by considering the following weak question: Can one tell whether $A$ is Lebesgue negligible from the knowledge only of its trace on each member of some given collection of “lower dimensional” subsets $\Gamma_{i}\subseteq\mathbb{R}^{n}$, $i\in I$. One expects that if $A\cap\Gamma_{i}$ is “negligible in the dimension of $\Gamma_{i}$”, for each $i\in I$, then $\mathscr{L}^{n}(A)=0$. Of course, a necessary condition is that the sets $\Gamma_{i}$ cover almost all of $A$, i.e., $\mathscr{L}^{n}(A\setminus\cup_{i\in I}\Gamma_{i})=0$. Consider, for instance, $n=2$, $I=\mathbb{R}$, and $\Gamma_{t}=\{t\}\times\mathbb{R}$, for $t\in\mathbb{R}$, the collection of all vertical lines in the plane. It is not true in general that if $A\subseteq\mathbb{R}^{2}$ and $A\cap\Gamma_{t}$ is a singleton, for each $t\in\mathbb{R}$, then $\mathscr{L}^{2}(A)=0$. There exist, indeed, functions $f:\mathbb{R}\to\mathbb{R}$ whose graph $A$ satisfies $\mathscr{L}^{2}(A)>0$ – see, e.g., [10, Chapter 2 Theorem 4] for an example due to Sierpiński. In order to rule out such examples, we will henceforth assume that $A\subseteq\mathbb{R}^{n}$ is Borel measurable. In that case, the theorem of Fubini, together with the invariance of the Lebesgue measure under orthogonal transformations, imply the following. Given an integer $1\leqslant m\leqslant n-1$, if $(\Gamma_{i})_{i\in I}$ is the collection of all $m$-dimensional affine subspaces of $\mathbb{R}^{n}$ of some fixed direction, and if $\mathscr{H}^{m}(A\cap\Gamma_{i})=0$ for all $i\in I$, then $\mathscr{L}^{n}(A)=0$. Here, $\mathscr{H}^{m}$ denotes the $m$-dimensional Hausdorff measure. A special feature of this collection $(\Gamma_{i})_{i\in I}$ is that it partitions $\mathbb{R}^{n}$, its members being the level sets $f^{-1}\{y\}$, $y\in\mathbb{R}^{n-m}$, of a “nice map” $f:\mathbb{R}^{n}\to\mathbb{R}^{n-m}$, indeed, an orthogonal projection. This is an occurrence of the following more general situation when $f$ and its leaves $f^{-1}\{y\}$ are allowed to be nonlinear. The coarea formula due to Federer [8] asserts that if $f:\mathbb{R}^{n}\to\mathbb{R}^{n-m}$ is Lipschitzian and $A\subseteq\mathbb{R}^{n}$ is Borel measurable, then

Thus, if the Jacobian coarea factor $Jf$ is positive, $\mathscr{L}^{n}$-almost everywhere in $A$, then the collection $\left(f^{-1}\{y\}\right)_{y\in\mathbb{R}^{n-m}}$ is suitable for detecting whether or not $A$ is Lebesgue null. At $\mathscr{L}^{n}$-almost all $x\in\mathbb{R}^{n}$, the map $f$ is differentiable, by virtue of Rademacher’s theorem, and

see [6, Chapter 3 §4] and [9, 3.2.1 and 3.2.11].

In this paper, we focus on the case when $\Gamma_{i}$, $i\in I$, are affine subspaces of $\mathbb{R}^{n}$, but not necessarily members of a partition of the ambient space. Specifically, we assume that with each $x\in\mathbb{R}^{n}$ is associated an $m$-dimensional affine subspace $\mathbf{W}(x)$ of $\mathbb{R}^{n}$ containing $x$. Given a Borel set $A\subseteq\mathbb{R}^{n}$, the question whether

has a negative answer, in view of Nikodým’s set $A\subseteq\mathbb{R}^{2}$ evoked in the previous section. Indeed, corresponding to this set $A$, there exists a field of lines $x\mapsto\mathbf{W}(x)$ such that $A\cap\mathbf{W}(x)=\{x\}$, for all $x\in A$. In this context, a selection theorem due to von Neumann implies that (possibly considering a smaller, non Lebesgue null, Borel subset of $A$) the correspondence $x\mapsto\mathbf{W}(x)$ can be chosen to be Borel measurable (see 3.19) and, in turn, it can be chosen to be continuous, according to Lusin’s theorem. Nonetheless, when $\mathbf{W}$ is Lipschitzian, the situation improves, as illustrated in our theorem below; $\mathbb{G}(n,m)$ is the Grassmannian manifold consisting of $m$-dimensional vector subspaces of $\mathbb{R}^{n}$.

As should be apparent from the discussion above, one difficulty stands with the fact that the affine $m$-planes $\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$ may not be disjointed. Nevertheless, they locally are, in the following sense. Given $x_{0}\in A$ there exist a neighborhood $U$ of $x_{0}$ and Lipschitzian maps $\mathbf{w}_{1},\ldots,\mathbf{w}_{m}:U\to\mathbb{R}^{n}$ so that $\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x)$ is an orthonomal frame spanning $\mathbf{W}_{0}(x)$, for $x\in U$, and the map $\Phi:(V_{x_{0}}\cap U)\times\mathbb{R}^{m}\to\mathbb{R}^{n}:(\xi,t)\mapsto\xi+\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(\xi)$ is a lipeomorphism of a neighborhood of $x_{0}$ onto its image – here, $V_{x_{0}}=x_{0}+V$ and $V\in\mathbb{G}(n,n-m)$ is close to $\mathbf{W}_{0}(x_{0})^{\perp}$. This, and applications of Fubini’s theorem, yield $(B)\Rightarrow(A)$ in the theorem above (see 8.2).

However, we aim at obtaining a quantative version of this result, that the change of variable just described does not seem to provide. A natural route is to reduce the problem to applying the coarea formula, by spreading out the $\mathbf{W}(x)$’s in a disjointed way, in a higher dimensional space – i.e., adding a variable $u\in\mathbf{W}(x)$ to the given $x\in\mathbb{R}^{n}$ and considering $\mathbf{W}(x)$ as a fiber above the base space $\mathbb{R}^{n}$. We thus define

where $E\subseteq\mathbb{R}^{n}$ is Borel measurable. This set is $(n+m)$-rectifiable, owing to the Lipschitz continuity of $\mathbf{W}$. It is convenient to assume that $\mathscr{L}^{n}(E)<\infty$, so that

$B\subseteq\mathbb{R}^{n}$, is a locally finite Borel measure 3.16. Now, $\Sigma$ was precisely set up so that, for each $x\in E$,

where $\pi_{1}$ and $\pi_{2}$ denote the projections of $\mathbb{R}^{n}\times\mathbb{R}^{n}$ to the $x$ and $u$ variable, respectively. Abbreviating $\Sigma_{B}=\Sigma\cap\pi_{2}^{-1}(B)$, the coarea formula yields

A simple calculation 5.4 shows that $J_{\Sigma}\pi_{2}>0$, $\mathscr{H}^{n+m}$-almost everywhere on $\Sigma_{B}$. Since also

the implication $(A)\Rightarrow(C)$ above should now be clear: Letting $B=A$ and $E=\mathbf{B}(0,R)$, one infers from hypothesis (A) and (4) that $\mathscr{H}^{n+m}(\Sigma_{B})=0$, whence, $\phi_{\mathbf{B}(0,R)}(A)=0$, by (3), and, in turn, conclusion (C) ensues from (2).

In order to establish that $(B)\Rightarrow(A)$ by using this new change of variable, we need to observe 5.2 that $J_{\Sigma}\pi_{1}>0$, $\mathscr{H}^{n+m}$-almost everywhere, and, ideally, to show that $\mathscr{H}^{m}\left(\Sigma_{B}\cap\pi_{2}^{-1}\{u\}\right)>0$, for almost every $u\in B$. This last part involves some difficulty. To understand this, we let $m=n-1$, in order to keep the notations short. Now $u\in\mathbf{W}(x)$ iff $u-x\in\mathbf{W}_{0}(x)$ iff $\langle\mathbf{v}_{0}(x),x-u\rangle=0$, where $\mathbf{v}_{0}(x)\in\mathbf{W}_{0}(x)^{\perp}$ is, say, a unit vector. Abbreviating $g_{u}(x)=\langle\mathbf{v}_{0}(x),x-u\rangle$, we infer that

The problem remains that two of the nonlinear $m$ sets $E\cap g_{u}^{-1}\{0\}$ and $E\cap g_{u^{\prime}}^{-1}\{0\}$ may intersect, thereby preventing another application of the coarea formula when looking out for their lower bound. Yet, we already know that

where $\mathscr{Z}_{E}\mathbf{W}$ is a Radon-Nikodým derivative (see 5.6) and also that $(\mathscr{Z}_{E}\mathbf{W})(u)$ is comparable to $\mathscr{H}^{m}\left(E\cap g_{u}^{-1}\{0\}\right)$ (see 5.8). Adding an extra variable $y$ to the fibered space $\Sigma$, 5.10, we improve on this by showing that

where the last equality defines $\mathscr{Y}_{E}\mathbf{W}$, and $\boldsymbol{\eta}(n,m)>0$ is a lower bound for the coarea Jacobian factor of the restriction to the fibered space of the projection $(x,u,y)\mapsto(x,y)$ (see 5.12 and 5.16). We are reduced to showing that $\mathscr{Y}_{E}\mathbf{W}>0$, almost everywhere. The reason why this holds is the following. Fix a Borel set $Z\subseteq\mathbb{R}^{n}$, $x_{0}\in\mathbb{R}^{n}$ and $r>0$. Let $\mathbf{C}_{\mathbf{W}}(x_{0},r)$ be the cylindrical box consisting of those $x\in\mathbb{R}^{n}$ such that $\left|P_{\mathbf{W}_{0}(x_{0})}(x-x_{0})\right|\leqslant r$ and $\left|P_{\mathbf{W}_{0}(x_{0})^{\perp}}(x-x_{0})\right|\leqslant r$. We want to find a lower bound for

To this end, we fix $z\in\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)$ and we let $V_{z}=\mathbb{R}^{n}\cap\{x_{0}+z+s\mathbf{v}_{0}(x_{0}):-r\leqslant s\leqslant r\}$ denote the corresponding vertical line segment. According to Fubini’s theorem we are reduced to estimating

By virtue of Vitali’s covering theorem, we can find a disjointed family of line segments $I_{1},I_{2},\ldots$, covering almost all $V_{z}$, such that the above integral nearly equals

where the first near equality follows from the coarea formula, and the second one because $\left|\nabla g_{u_{k}}\right|\cong 1$ at small scales (see 3.12) and the “nonlinear horizontal stripes” $g_{u_{k}}^{-1}(I_{k})$ are nearly pairwise disjoint. Verification of these claims takes up sections 6 and 7. Now, we reach a contradiction if $Z=\mathbb{R}^{n}\cap\{\mathscr{Y}_{E}\mathbf{W}=0\}$ is assumed to have $\mathscr{L}^{n}(Z)>0$, and $x_{0}$ is a point of density of $Z$.

Along this path, we have, in fact, achieved a quantitative version of these observations. Indeed, assuming that $\limsup_{r\to 0^{+}}\frac{\mathscr{H}^{m}(A\cap\mathbf{B}(x,r)\cap\mathbf{W}(x))}{\boldsymbol{\alpha}(m)r^{m}}<\frac{\boldsymbol{\eta}(n,m)}{2^{m}}$, for $\mathscr{L}^{n}$ positively many $x\in A$, we start by choosing $E\subseteq A$ with the same property, whose diameter is small enough for the various estimates above to hold, and we refer to Egoroff’s theorem to select $\varepsilon>0$ so that the set

has positive $\mathscr{L}^{n}$ measure. Thus, for $0<\hat{\varepsilon}\ll\varepsilon$, there exists $x_{0}\in Z$ and there exists $r>0$ as small as we please so that

where the symbol $\lesssim$ means that we loose a multiplicative factor $\lambda>1$ as close as we wish to 1, according to the scale $r$. The above contradiction proves our main result 8.4, namely,

I extend my warm thanks to Jean-Christophe Léger for carefully reading an early version of the manuscript.