Corona Decompositions for Parabolic Uniformly Rectifiable Sets

Abstract

We prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular Lip(1,1/2) graphs. Together with our previous work, this allows us to conclude that if $\Sigma \subset {\mathbb {R}}^{n+1}$ is parabolic Ahlfors-David regular, then the following statements are equivalent.

• (1) $\Sigma$ is parabolic uniformly rectifiable.

• (2) $\Sigma$ admits a corona decomposition with respect to regular Lip(1,1/2) graphs.

• (3) $\Sigma$ admits a bilateral corona decomposition with respect to regular Lip(1,1/2) graphs.

• (4) $\Sigma$ is big pieces squared of regular Lip(1,1/2) graphs.

Appendices

Appendix A: Two Lemmas Concerning Approximating Planes

In this appendix we prove two auxiliary and technical lemmas on approximating hyperplanes.

Lemma A.1

Let $\Sigma \subset {\mathbb {R}}^{n+1}$ be a parabolic Ahlfors-David regular set with dyadic cubes ${\mathbb {D}}(\Sigma )$. Then there exists a constant $A=A(n,M)\gg 1$ such that the following holds. Let $Q\in {\mathbb {D}}={\mathbb {D}}(\Sigma )$. Suppose that $(Z_0,\tau _0)\in Q$. Then there exist points $(Z_i,\tau _i)\in Q$ for $i=1,...,n$, such that if we let, for $j=1,...,n$, $L_{j-1}$ be the spatial $(j-1)$-dimensional plane which passes through $Z_0,Z_1,...,Z_{j-1}$, then

$$\begin{aligned} d_p(Z_j,\tau _j,L_{j-1}\times {\mathbb {R}})\ge A^{-1}{{\,\textrm{diam}\,}}(Q) \end{aligned}$$

for $j=1,...,n$. Moreover, the same result holds if Q is replaced by $\lambda Q$ for some $\lambda \ge 1$, with A now depending on $\lambda$ as well.

Proof

We only prove the result for Q as the same proof works for any dilate $\lambda Q$ of Q, $\lambda \ge 1$. We will prove the lemma by induction on j for $j=1,...,n$. Let $L_0:=\{Z_0\}$. Then $L_0$ is a 0-dimensional plane. Assume that $d_p(Z,\tau ,L_{0}\times {\mathbb {R}})\le A^{-1}{{\,\textrm{diam}\,}}(Q)$ for all $(Z,\tau )\in Q$. Then Q can be covered by roughly $A^2$ cubes of size $2A^{-1}{{\,\textrm{diam}\,}}(Q)$. Hence

$$\begin{aligned} ({{\,\textrm{diam}\,}}(Q))^{n+1}\lesssim \sigma (Q)\lesssim A^2(A^{-1}{{\,\textrm{diam}\,}}(Q))^{n+1} \end{aligned}$$

which is impossible if A is large enough. Hence there exists a point $(Z_1,\tau _1)\in Q$ such that $d_p(Z_1,\tau _1,L_{0}\times R)\ge A^{-1}{{\,\textrm{diam}\,}}(Q)$ and the conclusion is true for $j=1$ and we let $L_1$ be the linear span of $Z_0$ and $Z_1$. Let $1\le j<n-1$ and suppose that we have found points $(Z_i,\tau _i)\in Q$ for $i=1,...,j-1$ as stated and that $L_{j-1}$ is the (spatial) $(j-1)$-dimensional plane spanned by $Z_0,Z_1,...,Z_{j-1}$. Assume that $d_p(Z,\tau ,L_{j-1}\times {\mathbb {R}})\le A^{-1}{{\,\textrm{diam}\,}}(Q)$ for all $(Z,\tau )\in Q$. Then Q can be covered by roughly $A^{j-1}A^2$ cubes of size $2A^{-1}{{\,\textrm{diam}\,}}(Q)$ and, again using that $\Sigma$ has the parabolic ADR property, we see that

$$\begin{aligned} 1\lesssim A^{j-1}A^2A^{-n-1} \end{aligned}$$

which again is impossible, as $j<n$, if $A=A(n,M)\gg 1$ is large enough. Hence there exists $(Z_j,\tau _j)\in Q$ such that $d_p(Z_j,\tau _j,L_{j-1}\times {\mathbb {R}})\ge A^{-1}{{\,\textrm{diam}\,}}(Q)$ and we let $L_j$ be the (spatial) j-dimensional plane spanned by $Z_0,Z_1,...,Z_{j}$. The lemma now follows from the induction hypothesis. $\square$

Lemma A.2

Let $\Sigma \subset {\mathbb {R}}^{n+1}$ be a parabolic Ahlfors-David regular set with dyadic cubes ${\mathbb {D}}(\Sigma )$. Let $Q\in {\mathbb {D}}={\mathbb {D}}(\Sigma )$ and assume that $P_1$ and $P_2$ are two time-independent hyperplanes such that $d_p(X,t,P_i)\le \epsilon {{\,\textrm{diam}\,}}(Q)$ for all $(X,t)\in Q$ and for $i\in \{1,2\}$. Then

$$\begin{aligned} (i)&\ d_p(Y,s,P_2)\lesssim \epsilon ({{\,\textrm{diam}\,}}(Q)+d_p(Y,s,Q)),\ (Y,s)\in P_1,\nonumber \\ (ii)&\ d_p(Y,s,P_1)\lesssim \epsilon ({{\,\textrm{diam}\,}}(Q)+d_p(Y,s,Q)),\ (Y,s)\in P_2. \end{aligned}$$

(A.1)

In particular, the angle between $P_1$ and $P_2$ is bounded by $\lesssim \epsilon$.

Moreover, (A.1) continues to hold (with slightly different implicit constants) with Q replaced by any “surface ball" $\Delta = \Delta _r(X,t) = C_r(X,t)\cap \Sigma$, where $(X,t) \in \Sigma$, provided that $d_p(X,t,P_i)\le \epsilon {{\,\textrm{diam}\,}}(\Delta )$ for all $(X,t)\in 2\Delta$ and for each $i\in \{1,2\}$.

Proof

It is enough to prove the result for a cube Q: indeed, given a surface ball $\Delta =\Delta _r$ for which $d_p(X,t,P_i)\le \epsilon {{\,\textrm{diam}\,}}(\Delta )$ for all $(X,t)\in 2\Delta$ and for each $i\in \{1,2\}$, we may cover $\Delta$ by a bounded number of disjoint cubes $Q_j$, with ${{\,\textrm{diam}\,}}(Q_j) \approx r$, such that $\cup _j Q_j \subset 2\Delta$, and observe that (A.1) (with $\epsilon$ replaced by $C\epsilon$) may be applied to each $Q_j$.

Fix now a dyadic cube $Q\subset \Sigma$, satisfying the hypotheses of the lemma. Let $(Z_i,\tau _i)\in Q$ for $i=0,1,...,n$, be as in the statement of Lemma A.1 and let $L_{Q}=L_{n-1}$ be the spatial $(n-1)$-dimensional plane which passes through $Z_0,Z_1,...,Z_{n-1}$. Let $P:=L_{Q}\times {\mathbb {R}}$. Consider $(Y,s)\in P_1$. Then

$$\begin{aligned} d_p(Y,s,P_2)&\le d_p(Y,s,P)+d_p(P,P_2)\nonumber \\&\lesssim \max _{i\in \{0,...,n\}} d_p(Y,s,Z_i,\tau _i)+ \max _{i\in \{0,...,n\}} d_p(Z_i,\tau _i,P_2)\nonumber \\&\lesssim \epsilon {{\,\textrm{diam}\,}}(Q)+\max _{i\in \{0,...,n\}} d_p(Y,s,Z_i,\tau _i)\nonumber \\&\lesssim \epsilon ({{\,\textrm{diam}\,}}(Q)+d_p(Y,s,Q)). \end{aligned}$$

(A.2)

This proves the lemma. $\square$

Appendix B: Slice-wise vs. Parabolic Hausdorff Measure

In this appendix we prove the (non-trivial) assertions made in Remark 2.2. Recall $\mu$ is the (global) slice-wise measure defined as

$$\begin{aligned} \mu (E) = \int _{{\mathbb {R}}} {\mathcal {H}}^{n-1}(E_t) \, d{\mathcal {H}}^1(t), \end{aligned}$$

where for $t \in {\mathbb {R}}$, $E_t := E \cap ({\mathbb {R}}^n \times \{t\})$.

The following proposition proves Remark 2.2(vi).

Proposition B.1

There exists a constant c(n) such that $\mu \le c(n){\mathcal {H}}^{n+1}_p$.

Proof

In the proof, for any parabolic cube¹⁵Q, we will write $Q= Q' \times I$, where I is an interval, so that the side length of $Q'$, $\ell (Q')$, and the length of I, $\ell (I)$, satisfy $\ell (I)^2 = \ell (Q')$ and we let $\ell (Q) := \ell (Q')$. Suppose that $E \subset {\mathbb {R}}^{n+1}$, a Borel set, is such that ${\mathcal {H}}^{n+1}_p(E) < \infty$ (otherwise there is nothing to prove). Recall that $H^{n+1}_{p, \delta }(E)$ is a decreasing function in $\delta$. Let $\delta > 0$ be arbitrary. By definition of ${\mathcal {H}}^{n+1}_{p}(E)$ there exists a countable collection of cubes $Q_i$ such that

$$\begin{aligned} \sum _{i} \ell (Q_i)^{n+1} \le 2^{n+1} {\mathcal {H}}^{n+1}_{p}(E) + \delta , \quad E \subseteq \cup _i Q_i \quad \text { and } \quad {{\,\textrm{diam}\,}}(Q_i) \le \delta . \end{aligned}$$

To see this, we use a covering $\{E_i\}$ which nearly minimizes the quantity in the definition of $H^{n+1}_{p,\delta '}(E)$, where $(2\sqrt{n} + \sqrt{2})\delta '= \delta$, then for each i we take $Q_i = C_{r_i}(X_i,t_i)$ with $\ell (Q_i)/2 = r_i = {{\,\textrm{diam}\,}}(E_i)$ and $(X_i,t_i)$ an arbitrary point in $E_i$.

Let ${\mathcal {I}}(t) = \{i: t \in I_i\}$. Then as $E \subset Q_i$ we have $E_t \subseteq \cup _{i \in {\mathcal {I}}(t)} Q'_i$ and ${{\,\textrm{diam}\,}}(Q'_i) \le \delta$. Therefore

$$\begin{aligned} {\mathcal {H}}_\delta ^{n-1}(E_t) \le c'(n) \sum _{i \in {\mathcal {I}}(t)} \ell (Q_i')^{n-1} = c'(n) \sum _{i \in {\mathcal {I}}(t)} \ell (Q_i)^{n-1}, \end{aligned}$$

where $c'(n) := (\sqrt{n})^{n-1}$. Hence

$$\begin{aligned} \int _{{\mathbb {R}}} {\mathcal {H}}^{n-1}_\delta (E_t) \, d{\mathcal {H}}^1(t)&\le c'(n) \int _{{\mathbb {R}}} \sum _{i \in {\mathcal {I}}(t)} \ell (Q_i)^{n-1} \, d{\mathcal {H}}^1(t) \\&\le c'(n)\sum _i \int _{I_i} \ell (Q_i)^{n-1} \, d{\mathcal {H}}^1(t) = c'(n) \sum _i \ell (Q_i)^{n+1} \\&\le c'(n) 2^{n+1} {\mathcal {H}}^{n+1}_{p}(E) + c'(n) \delta . \end{aligned}$$

The result now follows from the monotone convergence theorem as ${\mathcal {H}}^{n-1}_\delta (E_t)$ increases as $\delta$ decreases. $\square$

The following proposition proves Remark 2.2(i) for the measure $\sigma ^{\textbf {s}}$.

Proposition B.2

Let $E \subset {\mathbb {R}}^{n+1}$ be closed. There exists $c(n) > 0$ such that the following holds. If there is a constant c such that

$$\begin{aligned} c^{-1}r^{n+1} \le \mu (C_r(X,t) \cap E) \le cr^{n+1}, \quad \forall (X,t) \in E, r > 0, \end{aligned}$$

then

$$\begin{aligned} c(n)^{-1} c^{-1}r^{n+1} \le {\mathcal {H}}^{n+1}_p(C_r(X,t) \cap E) \le c(n)c r^{n+1}, \quad \forall (X,t) \in E, r > 0. \end{aligned}$$

Proof

The lower bound follows directly from the previous proposition. To prove the other inequality fix $(X,t) \in E, r > 0$, let $\delta \in (0,r)$ be arbitrary and let $r_\delta$ be such that ${{\,\textrm{diam}\,}}(C_{5r_\delta }(0)) = \delta$. Clearly,

$$\begin{aligned} C_r(X,t) \cap E \subseteq \bigcup _{(Y,s) \in C_r(X,t) \cap E}C_{r_\delta }(Y,s) \cap E. \end{aligned}$$

By the 5r-covering lemma (see e.g., [41, Theorem 2.1]) there exists a countable collection of $(Y_i,s_i)\in E \cap C_r(X,t)$ such that the cubes $C_{r_\delta }(Y_i,s_i)$ are disjoint and

$$\begin{aligned} C_r(X,t) \cap E \subset \cup _i C_{5r_\delta }(Y_i,s_i) \cap E. \end{aligned}$$

It follows that

$$\begin{aligned} {\mathcal {H}}^{n+1}_{p,\delta }(C_r(X,t) \cap E)&\lesssim \sum _i {{\,\textrm{diam}\,}}(C_{5r_\delta }(Y_i,s_i))^{n+1} \lesssim c\sum _i \mu (C_{r_\delta }(Y_i,s_i) \cap E)^{n+1} \\&\lesssim \mu (C_{2r}(X,t) \cap E) \lesssim c r^{n+1}, \end{aligned}$$

where now $\lesssim$ means that constants only depend on n, and where we have used that ${{\,\textrm{diam}\,}}(C_{r_\delta }(Y_i,s_i))< \delta < r$ in the second-to-last inequality. Letting $\delta \rightarrow 0^+$ yields the upper bound. $\square$

On the other hand, ${\mathcal {H}}^{n+1}_p \not \ll \mu$, even when the set is p-ADR with respect to ${\mathcal {H}}^{n+1}_p$, as the following example shows.

Example B.3

Let $E= [C_{1-1/2n}]^n \times C_{3/4}$, where $C_s$ denotes a (generalized) s-dimensional Cantor-type set in ${\mathbb {R}}$. Then E is p-ADR with respect to ${\mathcal {H}}^{n+1}_p$, but $\mu (E) = 0$. The fact that $\mu (E) = 0$ follows from the fact that ${\mathcal {H}}^1(C_{3/4}) = 0$. Moreover, one can verify that E is p-ADR with respect to the measure ${\mathcal {H}}^{n-1/2} \times {\mathcal {H}}^{3/4}$ and hence p-ADR with respect to ${\mathcal {H}}^{n+1}_p$.

We will require the following definition.

Definition B.4

Let $E\subset {\mathbb {R}}^{n+1}$ be a p-ADR set (with respect to ${\mathcal {H}}^{n+1}_p$) and let ${\mathcal {S}}$ be a collection of p-ADR sets, with uniform control on the p-ADR constant. We say E is big pieces of ${\mathcal {S}}$, written E is $BP({\mathcal {S}})$, if there exists $\theta > 0$ such that for every $(X,t) \in E$ and $r \in (0,{{\,\textrm{diam}\,}}(E))$ there exists $\Gamma \in {\mathcal {S}}$ with

$$\begin{aligned} {\mathcal {H}}^{n+1}_p(E \cap C_r(X,t) \cap \Gamma ) \ge \theta {\mathcal {H}}^{n+1}_p(E \cap C_r(X,t)). \end{aligned}$$

We say E is big pieces squared of ${\mathcal {S}}$, written E is $BP^2({\mathcal {S}})$ if there exists a constant M such that E is $BP(BP({\mathcal {S}},M))$, where $BP({\mathcal {S}},M)$ is the collection of all p-ADR sets with p-ADR constant less than M which are $BP({\mathcal {S}})$.

Proposition B.5

Let ${\mathcal {S}}$ be a collection of closed subsets of ${\mathbb {R}}^{n+1}$, which are uniformly p-ADR with respect to $\mu$, that is, there exists $M > 1$ such that for all $\Gamma \in {\mathcal {S}}$ it holds

$$\begin{aligned} M^{-1} r^{n+1} \le \mu (\Gamma \cap C_r(X,t)) \le M r^{n+1}, \quad \forall (X,t) \in \Gamma , r > 0. \end{aligned}$$

If E is p-ADR (with respect to ${\mathcal {H}}^{n+1}_p$) and E is $BP({\mathcal {S}})$, then E is p-ADR with respect to $\mu$ with constant depending only on M, n, the ADR constant of E, and the constant $\theta$ in the definition of $BP({\mathcal {S}})$. In particular, $\mu |_E \approx {\mathcal {H}}^{n+1}_p|_E$, with implicit constant depending only on M, n, the p-ADR constant of E, and the constant $\theta$ in the definition of $BP({\mathcal {S}})$.

Proof

Let $(X,t) \in E$ and $r \in (0,{{\,\textrm{diam}\,}}(E))$. By Proposition B.1

$$\begin{aligned} \mu (C_r(X,t) \cap E) \le c(n) H_p^{n+1}(E \cap C_r(X,t) )\lesssim r^{n+1}, \end{aligned}$$

where the implicit constant depends on the p-ADR constant of E and n. To prove the lower bound, we note that for $\Gamma \in {\mathcal {S}}$, ${\mathcal {H}}^{n+1}_p|_\Gamma \approx \mu |_\Gamma$, with implicit constant depending on M since any set which is p-ADR with respect to $\mu$ is p-ADR with respect to ${\mathcal {H}}^{n+1}_p$ (by Proposition B.2). Then using that E is p-ADR and E is $BP({\mathcal {S}})$ there exists $\Gamma \in {\mathcal {S}}$ such that

$$\begin{aligned} r^{n+1} \approx {\mathcal {H}}^{n+1}(E \cap C_r(X,t))&\le \theta ^{-1} {\mathcal {H}}^{n+1}(E \cap C_r(X,t) \cap \Gamma ) \\&\approx _M \theta ^{-1} \mu (E \cap C_r(X,t) \cap \Gamma ) \\&\lesssim \theta ^{-1} \mu (E \cap C_r(X,t)) . \end{aligned}$$

This proves the proposition. $\square$

The following corollary finishes the proof of the assertion in Remark 2.2(v).

Corollary B.6

If E is parabolic UR then $\mu |_E \approx {\mathcal {H}}^{n+1}_p|_E$, that is, $\sigma ^s \approx \sigma$. Here the implicit constants depend on dimension and the parabolic UR constants for E.

Proof

We have shown in Theorem 3.1 that if E is parabolic UR, then E admits a corona decomposition with respect to regular Lip(1,1/2) graphs, with (uniform) control on the Lip(1,1/2) constant in terms of the parabolic UR constants for E. Then it follows from [6, Theorem 1.1] that E is $BP^2({\mathcal {S}})$, where ${\mathcal {S}}$ is a collection of Lip(1,1/2) graphs with uniform control on the Lip(1,1/2) constant. In particular, ${\mathcal {S}}$ is a collection of sets which are uniformly p-ADR with respect to $\mu$. Applying Proposition B.5 twice yields the corollary. $\square$