Dyadic cubes

In mathematics, the dyadic cubes are a collection of cubes in Rⁿ of different sizes or scales such that the set of cubes of each scale partition Rⁿ and each cube in one scale may be written as a union of cubes of a smaller scale. These are frequently used in mathematics (particularly harmonic analysis) as a way of discretizing objects in order to make computations or analysis easier. For example, to study an arbitrary subset of A of Euclidean space, one may instead replace it by a union of dyadic cubes of a particular size that cover the set. One can consider this set as a pixelized version of the original set, and as smaller cubes are used one gets a clearer image of the set A. Most notable appearances of dyadic cubes include the Whitney extension theorem and the Calderón–Zygmund lemma.

Dyadic cubes in Euclidean space

In Euclidean space, dyadic cubes may be constructed as follows: for each integer k let Δ_k be the set of cubes in Rⁿ of sidelength 2^−k and corners in the set

2^{{-k}}{\mathbf {Z}}^{n}=\left\{2^{{-k}}(v_{1},\dots ,v_{n}):v_{j}\in {\mathbf {Z}}\right\}

and let Δ be the union of all the Δ_k.

The most important features of these cubes are the following:

For each integer k, Δ_k partitions Rⁿ.
All cubes in Δ_k have the same sidelength, namely 2^−k.
If the interiors of two cubes Q and R in Δ_k have nonempty intersection, then either Q is contained in R or R is contained in Q.
Each Q in Δ_k may be written as a union of 2ⁿ cubes in Δ_k+1 with disjoint interiors.

We use the word "partition" somewhat loosely: for although their union is all of Rⁿ, the cubes in Δ_k can overlap at their boundaries. These overlaps, however, have zero Lebesgue measure, and so in most applications this slightly weaker form of partition is no hindrance.

It may also seem odd that larger k corresponds to smaller cubes. One can think of k as the degree of magnification. In practice, however, letting Δ_k be the set of cubes of sidelength 2^k or 2^−k is a matter of preference or convenience.

The one-third trick

One disadvantage to dyadic cubes in Euclidean space is that they rely too much on the specific position of the cubes. For example, for the dyadic cubes Δ described above, it is not possible to contain an arbitrary ball inside some Q in Δ (consider, for example, the unit ball centered at zero). Alternatively, there may be such a cube that contains the ball, but the sizes of the ball and cube are very different. Because of this caveat, it is sometimes useful to work with two or more collections of dyadic cubes simultaneously.

Definition

The following is known as the one-third trick:^[1]

Let Δ_k be the dyadic cubes of scale k as above. Define

\Delta _{k}^{\alpha }=\{Q+\alpha :Q\in \Delta _{k}\}.

This is the set of dyadic cubes in Δ_k translated by the vector α. For each such α, let Δ^α be the union of the Δ_k^α over k.

There is a universal constant C > 0 such that for any ball B with radius r < 1/3, there is α in {0,1/3}ⁿ and a cube Q in Δ^α containing B whose diameter is no more than Cr.
More generally, if B is a ball with any radius r > 0, there is α in {0, 1/3, 4/3, 4²/3, ...}ⁿ and a cube Q in Δ^α containing B whose diameter is no more than Cr.

An example application

The appeal of the one-third trick is that one can first prove dyadic versions of a theorem and then deduce "non-dyadic" theorems from those. For example, recall the Hardy-Littlewood Maximal function

Mf(x)=\sup _{r>0}{\frac {1}{|B(x,r)|}}\int _{B(x,r)}|f(y)|dy

where f is a locally integrable function and |B(x, r)| denotes the measure of the ball B(x, r). The Hardy–Littlewood maximal inequality states that for an integrable function f,

\left|\{x\in {\mathbf {R}}^{{n}}:Mf(x)>\lambda \}\right|\leq {\frac {C_{n}}{\lambda }}\|f\|_{{1}}

for λ > 0 where C_n is some constant depending only on dimension.

This theorem is typically proven using the Vitali Covering Lemma. However, one can avoid using this lemma by proving the above inequality first for the dyadic maximal functions

M_{{\Delta ^{{\alpha }}}}f(x)=\sup _{{x\in Q\in \Delta ^{{\alpha }}}}{\frac {1}{|Q|}}\int _{{Q}}|f(x)|dx.

The proof is similar to the proof of the original theorem, however the properties of the dyadic cubes rid us of the need to use the Vitali covering lemma. We may then deduce the original inequality by using the one-third trick.

Dyadic cubes in metric spaces

Analogues of dyadic cubes may be constructed in some metric spaces.^[2] In particular, let X be a metric space with metric d that supports a doubling measure µ, that is, a measure such that for x ∈ X and r > 0, one has:

\mu (B(x,2r))\leq C\mu (B(x,r))

where C > 0 is a universal constant independent of the choice of x and r.

If X supports such a measure, then there exist collections of sets Δ_k such that they (and their union Δ) satisfy the following:

For each integer k, Δ_k partitions X, in the sense that

\mu \left(X\backslash \bigcup \nolimits _{{Q\in \Delta _{{k}}}}Q\right)=0.

All sets Q in Δ_k have roughly the same size. More specifically, each such Q has a center z_Q such that

B(z_{{Q}},c_{{1}}\delta ^{{k}})\subseteq Q\subseteq B(z_{{Q}},c_{{2}}\delta ^{{k}})

where c₁, c₂, and δ are positive constants depending only on the doubling constant C of the measure µ and independent of Q.

Each Q in Δ_k is contained in a unique set R in Δ_k−1.
There are constants constant C₃, η > 0 depending only on µ such that for all k and t > 0,

\mu \left(\left\{x\in Q:d(x,X\backslash Q)\leq t\delta ^{k}\right\}\right)\leq C_{3}t^{\eta }\mu (Q).

These conditions are very similar to the properties for the usual Euclidean cubes described earlier. The last condition says that the area near the boundary of a "cube" Q in Δ is small, which is a property taken for granted in the Euclidean case although is very important for extending results from harmonic analysis to the metric space setting.

References

↑ Okikiolu, Kate (1992). "Characterization of subsets of rectifiable curves in Rⁿ". J. London Math. Soc. (2). 46 (2): 336–348.
↑ Christ, Michael (1990). "A T(b) theorem with remarks on analytic capacity and the Cauchy integral". Colloq. Math. 60/61 (2): 601–628.

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