Singular integrals and rectifiable sets in \(R^ n\). Au-delà des graphes lipschitziens.

*(English)*Zbl 0743.49018
Astérisque. 193. Montrouge: Société Mathématique de France, 145 p. (1991).

It is well-known that a \(d\)-dimensional subset of \(R^ n\) is called rectifiable if it is contained in the union of a countable family of \(d\)- dimensional \(C^ 1\) submanifolds, except possibly for a set of Hausdorff measure zero. This monograph is devoted to quantitative versions of the notion of rectifiability which will play a role in the study of the following important problem:

Let \(E\subseteq R^ n\) be a subset with Hausdorff dimension \(d\), \(0<d<n\), where \(d\) is an integer. Assume that the \(d\)-dimensional Hausdorff measure \(H^ d\) is locally finite when restricted to \(E\). Consider the singular integral operators on \(E\) of the form \[ Tf(x)=\hbox{p.v.} \int_ E K(x-y)f(y)dy,\tag{1.1} \] where \(dy\) denotes \(H^ d\mid_ E\), and \(K(x)\) is smooth on \(R^ n\setminus\{0\}\), odd, and satisfies \[ |\bigtriangledown^ j K(x)|\leq C(j)| x|^{-d-j},\quad j=0,1,2,\dots\tag{1.2} \] or \[ \sup_{\varepsilon>0}\int_ E\left|\int_ G K(x- y)f(y)dy\right|^ 2dx<C(K)\int_ E| f|^ 2 dx\tag{1.3} \] for all \(f\in L^ 2(E)\), where \(G:=E\cap\{| x-y|>\varepsilon\}\). A very interesting problem is “What conditions on \(E\) are needed in order that (1.1) defines a bounded operator on \(L^ 2(E)\)”?

It is known that, if \(E\) is a subset of a smooth submanifold which is nice at \(\infty\), or \(E\) is a Lipschitz graph over some \(d\)-plane, then an operator as in (1.1) is bounded on \(L^ 2(E)\). We know also that the rectifiability of \(E\) does not imply the desired boundedness since it is a qualitative condition, while estimates on singular integrals are quantitative. Thus, quantitative versions of the notion of rectifiability are needed in the discussion of the aforementioned problem. As candidates for the notion of quantitative rectifiability there are many known characterizations of rectifiability from which to choose. But the complete relationship between these various candidates is not at all clear. The authors provide some nontrivial equivalence between some of these candidates. The main results are embodied in the next theorem.

Theorem. Let \(E\subseteq R^ n\) be a regular \(d\)-dimensional set in \(R^ n\). The following conditions (C1)-(C7) are equivalent: (C1) if \(K(x)\) is any smooth odd function on \(R^ n\setminus\{0\}\) that satisfies (1.2), then (1.3) holds; (C2) for each smooth odd function \(\psi\) on \(R^ n\) with compact support we have that \[ \sum^ \infty_{k=-\infty}\left|\int_ E\psi_ k(x-y)dy\right|^ 2dx d\delta_ {2^ k}(t) \] is a Carleson measure on \(E\times R_ +\), where \[ \psi_ k(x):=2^{-kd}\psi(2^{-k}x) \] and \(d\delta_ s\) denotes the Dirac mass at \(s\) in \(t\); (C3) \(\beta_ 1(x,t)^ 2t^{-1}dx dt\) is a Carleson measure on \(E\times R_ +\), where \[ \beta_ 1(x,t):=\inf_ P t^{-d}\int_ Q t^{-1}\hbox{dist}(y,P)dy, \] for \(x\in E\), \(t>0\), where \(Q:=E\cap B(x,t)\) and the infimum is taken over all of the \(d\)- plane \(P\), \(B(x,t)\) being the ball with center \(x\) and radius \(t\); (C4) \(E\) admits a Corona decomposition; (C5) \(E\) has very big pieces of bi- Lipschitz images of \(R^ d\) inside \(R^ m\), \(m=\max(n,2d+1)\); (C6) \(E\) has big pieces of Lipschitz images of subsets of \(R^ d\); (C7) there is an \(A_ 1\)-weight \(\omega\) on \(R^ d\) and an \(\omega\)-regular mapping \(z: R^ d\to R^{n+1}\) whose image contains \(E\).

A higher-dimensional version of Peter Jones’ travelling salesman theorem is included in the equivalence of (C3) and (C7) when \(d>1\) at least for regular sets. An amusing feature of the methods used in the monograph is the role played by singular integral operators, which provide a bridge for passing between various geometrical conditions. Some related open problems are also indicated.

Let \(E\subseteq R^ n\) be a subset with Hausdorff dimension \(d\), \(0<d<n\), where \(d\) is an integer. Assume that the \(d\)-dimensional Hausdorff measure \(H^ d\) is locally finite when restricted to \(E\). Consider the singular integral operators on \(E\) of the form \[ Tf(x)=\hbox{p.v.} \int_ E K(x-y)f(y)dy,\tag{1.1} \] where \(dy\) denotes \(H^ d\mid_ E\), and \(K(x)\) is smooth on \(R^ n\setminus\{0\}\), odd, and satisfies \[ |\bigtriangledown^ j K(x)|\leq C(j)| x|^{-d-j},\quad j=0,1,2,\dots\tag{1.2} \] or \[ \sup_{\varepsilon>0}\int_ E\left|\int_ G K(x- y)f(y)dy\right|^ 2dx<C(K)\int_ E| f|^ 2 dx\tag{1.3} \] for all \(f\in L^ 2(E)\), where \(G:=E\cap\{| x-y|>\varepsilon\}\). A very interesting problem is “What conditions on \(E\) are needed in order that (1.1) defines a bounded operator on \(L^ 2(E)\)”?

It is known that, if \(E\) is a subset of a smooth submanifold which is nice at \(\infty\), or \(E\) is a Lipschitz graph over some \(d\)-plane, then an operator as in (1.1) is bounded on \(L^ 2(E)\). We know also that the rectifiability of \(E\) does not imply the desired boundedness since it is a qualitative condition, while estimates on singular integrals are quantitative. Thus, quantitative versions of the notion of rectifiability are needed in the discussion of the aforementioned problem. As candidates for the notion of quantitative rectifiability there are many known characterizations of rectifiability from which to choose. But the complete relationship between these various candidates is not at all clear. The authors provide some nontrivial equivalence between some of these candidates. The main results are embodied in the next theorem.

Theorem. Let \(E\subseteq R^ n\) be a regular \(d\)-dimensional set in \(R^ n\). The following conditions (C1)-(C7) are equivalent: (C1) if \(K(x)\) is any smooth odd function on \(R^ n\setminus\{0\}\) that satisfies (1.2), then (1.3) holds; (C2) for each smooth odd function \(\psi\) on \(R^ n\) with compact support we have that \[ \sum^ \infty_{k=-\infty}\left|\int_ E\psi_ k(x-y)dy\right|^ 2dx d\delta_ {2^ k}(t) \] is a Carleson measure on \(E\times R_ +\), where \[ \psi_ k(x):=2^{-kd}\psi(2^{-k}x) \] and \(d\delta_ s\) denotes the Dirac mass at \(s\) in \(t\); (C3) \(\beta_ 1(x,t)^ 2t^{-1}dx dt\) is a Carleson measure on \(E\times R_ +\), where \[ \beta_ 1(x,t):=\inf_ P t^{-d}\int_ Q t^{-1}\hbox{dist}(y,P)dy, \] for \(x\in E\), \(t>0\), where \(Q:=E\cap B(x,t)\) and the infimum is taken over all of the \(d\)- plane \(P\), \(B(x,t)\) being the ball with center \(x\) and radius \(t\); (C4) \(E\) admits a Corona decomposition; (C5) \(E\) has very big pieces of bi- Lipschitz images of \(R^ d\) inside \(R^ m\), \(m=\max(n,2d+1)\); (C6) \(E\) has big pieces of Lipschitz images of subsets of \(R^ d\); (C7) there is an \(A_ 1\)-weight \(\omega\) on \(R^ d\) and an \(\omega\)-regular mapping \(z: R^ d\to R^{n+1}\) whose image contains \(E\).

A higher-dimensional version of Peter Jones’ travelling salesman theorem is included in the equivalence of (C3) and (C7) when \(d>1\) at least for regular sets. An amusing feature of the methods used in the monograph is the role played by singular integral operators, which provide a bridge for passing between various geometrical conditions. Some related open problems are also indicated.

Reviewer: Yang En Hao (Guangzhou)

##### MSC:

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |