##
**Wavelets and singular integrals on curves and surfaces.**
*(English)*
Zbl 0764.42019

Lecture Notes in Mathematics. 1465. Berlin etc.: Springer-Verlag. x, 109 p. (1992).

This text covers the state of the art of singular integral operators on curves and surfaces.

Its first part explains the construction of multiresolution analyses and orthogonal wavelet bases and follows closely Y. Meyer’s exposition in “Ondelettes et opérateurs I” (1991; Zbl 0694.41037).

The second part treats necessary and sufficient conditions for the boundedness of singular integral operators in the form of the \(T(b)\)- theorem. Its main objects are a standard kernel \(K(x,y) \leq C| x- y| ^{-n}\) defined for \(x, y \in R^ n, x\neq y\), with some smoothness in \(x\) and \(y\), and bounded functions \(b\) satisfying \(| Q| ^{-1} | \int _ Q b(y)dy| \geq \delta > 0\) for all dyadic cubes, so-called (special) “paraaccretive functions”. Then the \(T(b)\)-theorem states – modulo technical conditions which guarantee that the kernel defines a bounded operator on some space of test functions – that the operator \(Tf(x) = \int _{R^ n} K(x,y)f(y) dy \) is bounded on \(L^ p(R^ n), 1 < p < \infty\), if and only if there exist two paraaccretive functions \(b_ 1, b_ 2\) so that \(T(b_ 1) \in\text{BMO}(R^ n)\) and \(T^ t(b_ 2) \in\text{BMO}(R^ n)\) for the transpose of \(T\). The notes give a new and relatively short proof of this theorem that is based on wavelet techniques. It proceeds in three steps: (1) The construction of a Riesz basis for \(L^ 2(R^ n)\) which resembles the classical Haar basis, but is adjusted to \(b_ 1, b_ 2\). (2) Estimates for the infinite matrix of \(T\) in this special basis, and (3) Treatment of paraproducts.

One of the most important applications of the \(T(b)\)-theorem is the boundedness of the Cauchy-integral on Lipschitz curves, i.e., the operator \(C_ A f(x) = p.v. \int (x -y +i A(x) - iA(y))^{-1} f(y) dy\) for \(f \in L^ 2(R)\), where \(A:R \to R\) is a Lipschitz function.

The third part treats extensions of the Cauchy integral to higherdimensional objects. Given a \(k\)-dimensional “surface” \(S \subseteq R^ n, k < n\), a reasonable kernel \(K(x)\) on \(R^ n \setminus \{0\}\), and a positive Radon measure \(\mu \) supported on \(S\), the question is under which geometric conditions on \(S\) (and to a lesser extent on \(K\) and \(\mu \)) the operator \(Tf(x) = \int _ S K(x-y)f(y) d\mu (y) \) is bounded on \(L^ 2(R^ n, \mu )\). Since \(S\) is not assumed to be smooth, this question requires a deeper study of the geometry of surfaces. Roughly speaking, the problem is how to recognize whether a given surface contains big pieces of Lipschitz graphs. This part of the book is based on very recent and partially still unpublished work of the author, P. Jones, S. Semmes and others and is devoted to a deep, at times fairly technical investigation of this problem. These results, however, are too complicated to be described here.

Its first part explains the construction of multiresolution analyses and orthogonal wavelet bases and follows closely Y. Meyer’s exposition in “Ondelettes et opérateurs I” (1991; Zbl 0694.41037).

The second part treats necessary and sufficient conditions for the boundedness of singular integral operators in the form of the \(T(b)\)- theorem. Its main objects are a standard kernel \(K(x,y) \leq C| x- y| ^{-n}\) defined for \(x, y \in R^ n, x\neq y\), with some smoothness in \(x\) and \(y\), and bounded functions \(b\) satisfying \(| Q| ^{-1} | \int _ Q b(y)dy| \geq \delta > 0\) for all dyadic cubes, so-called (special) “paraaccretive functions”. Then the \(T(b)\)-theorem states – modulo technical conditions which guarantee that the kernel defines a bounded operator on some space of test functions – that the operator \(Tf(x) = \int _{R^ n} K(x,y)f(y) dy \) is bounded on \(L^ p(R^ n), 1 < p < \infty\), if and only if there exist two paraaccretive functions \(b_ 1, b_ 2\) so that \(T(b_ 1) \in\text{BMO}(R^ n)\) and \(T^ t(b_ 2) \in\text{BMO}(R^ n)\) for the transpose of \(T\). The notes give a new and relatively short proof of this theorem that is based on wavelet techniques. It proceeds in three steps: (1) The construction of a Riesz basis for \(L^ 2(R^ n)\) which resembles the classical Haar basis, but is adjusted to \(b_ 1, b_ 2\). (2) Estimates for the infinite matrix of \(T\) in this special basis, and (3) Treatment of paraproducts.

One of the most important applications of the \(T(b)\)-theorem is the boundedness of the Cauchy-integral on Lipschitz curves, i.e., the operator \(C_ A f(x) = p.v. \int (x -y +i A(x) - iA(y))^{-1} f(y) dy\) for \(f \in L^ 2(R)\), where \(A:R \to R\) is a Lipschitz function.

The third part treats extensions of the Cauchy integral to higherdimensional objects. Given a \(k\)-dimensional “surface” \(S \subseteq R^ n, k < n\), a reasonable kernel \(K(x)\) on \(R^ n \setminus \{0\}\), and a positive Radon measure \(\mu \) supported on \(S\), the question is under which geometric conditions on \(S\) (and to a lesser extent on \(K\) and \(\mu \)) the operator \(Tf(x) = \int _ S K(x-y)f(y) d\mu (y) \) is bounded on \(L^ 2(R^ n, \mu )\). Since \(S\) is not assumed to be smooth, this question requires a deeper study of the geometry of surfaces. Roughly speaking, the problem is how to recognize whether a given surface contains big pieces of Lipschitz graphs. This part of the book is based on very recent and partially still unpublished work of the author, P. Jones, S. Semmes and others and is devoted to a deep, at times fairly technical investigation of this problem. These results, however, are too complicated to be described here.

Reviewer: K.Gröchenig (Stoors)

### MSC:

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

42B25 | Maximal functions, Littlewood-Paley theory |