Density and differentiation theorems for sparse sets

Over the next couple of weeks, I will be posting short expositions of various parts of an upcoming paper by Malabika Pramanik and myself on maximal estimates associated with sparse sets in {\mathbb R}. I’ll start by explaining some of the questions that motivated us to do this work. We first learned about them from Nir Lev. We are grateful to him for the many conversations we had at the Fields Institute and for pointing us to references that would otherwise be very hard to find.

The following question was raised and investigated by Vincenzo Aversa and David Preiss in the 1980s and 90s: to what extent can the Lebesgue density theorem be viewed as “canonical” in {\mathbb R}, in the sense that any other density theorem that takes into account the affine structure of the reals must follow from the Lebesgue density theorem?

Let’s make this more precise. For the purpose of this post, we will say that family {\cal S} of measurable subsets of {\mathbb R} has the density property if for every measurable set E \subset {\mathbb R} we have

\lim_{S \in {\cal S}, diam ( S \cup \{ 0  \} ) \to 0 } \frac{ |(x+S) \cap E  | }{ |S| } = 1 for a.e. x\in E.

This is slightly different from standard terminology, but there should be no danger of confusion, as we will not use any other density properties here. We write x+S= \{ x+y:\  y\in S \}.

The Lebesgue density theorem states that the collection of intervals \{ (-r,r): \ r>0 \} has this property. It also implies that collections such as \{(0,r):\ r>0\} or \{(\frac{r}{2},r):\ r>0\} have it, just because the intervals in question occupy a positive and bounded from below proportion of (-r,r).

But that does not exhaust all examples. For instance, consider the family \{ I_n \}_{n=1}^\infty, where I_n=( \frac{ n }{ (n+1)! } , \frac{ 1 }{ n! } ). We have |I_n|=\frac{1}{(n+1)!} and diam ( I_n \cup \{ 0 \} )= \frac{ 1 }{ n! }, hence the Lebesgue argument no longer works. Nonetheless, this collection does have the density property, by the hearts density theorem of Preiss and Aversa-Preiss.

Note, however, that the collection in the last example is not closed under scaling x \to r x, $r>0$. Aversa and Preiss have in fact proved that if a family of intervals is invariant under such scaling and has the density property, then its density property must follow from the Lebesgue theorem in the manner described above.

On the other hand, if we consider more general sets than intervals, then it turns out that there are indeed scaling-invariant density theorems that are independent of the Lebesgue theorem. This was announced by Aversa and Preiss in 1987; the proof (via a probabilistic construction) was published in a 1995 preprint.

Theorem 1 (Aversa-Preiss): There is a sequence \{ S_n \} of compact sets of positive measure such that |S_n|\to 0 and:

  • 0 is a Lebesgue density point for {\mathbb R } \setminus \bigcup S_n, and in particular we have \lim_{ n\to\infty } \frac{ |S_n| }{ diam (S_n \cup \{ 0 \} ) }=0;

  • the family \{rS_n:\ r>0, n\in {\mathbb N} \} has the density property.

The analogous question for L^p differentiation theorems turned out to be much more difficult.

We will say that {\cal S} differentiates L^p_{loc} ( {\mathbb R} ) for some 1\leq p\leq\infty if for every f\in L^p_{loc} ( {\mathbb R} ) we have

\lim_{ S\in {\cal S}, diam (S\cup \{ 0 \} )\to 0 } \frac{ 1 }{ |S| } \int_{x+S} f( y  ) dy = f(x) for a.e. x\in E.

For example, the Lebesgue differentiation theorem states that the collection \{ (-r,r): r>0\} differentiates L^1_{loc}( {\mathbb R }).

The differentiation property is formally stronger than the density property, by letting f range over characteristic functions of measurable sets. However, there is no automatic implication in the other direction.

The following theorem was conjectured by Aversa and Preiss in 1995, and proved very recently by Malabika Pramanik and myself (paper in preparation).

Theorem 2. There is a sequence [1,2]\supset S_1\supset S_2\supset\dots of compact sets of positive measure with |S_n| \to 0 such that {\cal S} =\{ rS_n:\ r>0, n=1,2,\dots \} differentiates L^2( {\mathbb R}). More explicitly, for every f \in L^2 we have

\lim_{r\to 0} \sup_{n} \frac{ 1 }{ r|S_n| } \int_{ x+rS_n } f(y)dy = f(x) for a.e. x\in {\mathbb R}.

Our construction of S_n, like that of Aversa and Preiss, is probabilistic. We prove that the sequence S_n can be chosen so that the maximal operator associated with it is bounded on appropriate L^p spaces. This in particular implies the differentiation theorem.

The exact statement of the maximal estimate, and some of the ideas from the proof, will follow in the next installment.

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