Wavelets and operators. II: Calderón-Zygmund operators. (Ondelettes et opérateurs. II: Opérateurs de Calderón-Zygmund.) (French) Zbl 0745.42011

Meyer, Yves

Actualités Mathématiques. Paris: Hermann, Éditeurs des Sciences et des Arts. xii, p. 217-381 (1990).

The main contents of this volume are the studies of Calderón-Zygmund operators with the aid of the theory of ondelettes (= wavelets). [For Part I (1990) see Zbl 0694.41037 and Part III (1991) the following review Zbl 0745.42012]
The first chapter, Chapter VII, consists of the definition of Calderón-Zygmund operators and their fundamental properties. Let \(T\) be a linear continuous operator from \({\mathcal D}(\mathbb R^ n)\) to \({\mathcal D}'(\mathbb R^ n)\) with distribution kernel \(S\). The restriction of \(S\) to \(\Omega=\{(x,y)\in \mathbb R^ n\times\mathbb R^ n: x\neq y\}\) is denoted by \(K\). It is said that \(T\) is an operator of Calderón-Zygmund if the following conditions are satisfied:
(1) there exists a constant \(C_ 0\) such that \(K(x,y)\) is a locally integrable function satisfying \(| K(x,y)|\leq C_ 0| x- y|^{-n}\),
(2) there exist an exponent \(\gamma\in]0,1]\) and a constant \(C_ 1\) such that if \((x,y)\in\Omega\) and \(| x'-x|\leq{1\over 2}| x- y|\), then \[ | K(x',y)-K(x,y)|\leq C_ 1| x'-x|^ \gamma| x-y|^{-n-\gamma}, \] (3) analogously if \(| y'- y|\leq{1\over 2}| x-y|\) then \[ | K(x,y')-K(x,y)|\leq C_ 1| y'-y|^ \gamma| x-y|^{-n-\gamma}, \] (4) \(T\) is extended to a linear continuous operator in \(L^ 2(\mathbb R^ n)\).
It is shown that Calderón-Zygmund operators are bounded mappings from \(L^ p(\mathbb R^ n)\) to itself for any \(p\in]1,\infty[\), from the Hardy space \(H^ 1(\mathbb R^ n)\) to \(L^ 1(\mathbb R^ n)\), and from \(L^ \infty(\mathbb R^ n)\) to the space \(\text{BMO}(\mathbb R^ n)\) of functions of bounded mean oscillations. Since \(1\in L^ \infty(\mathbb R^ n)\), \(T(1)\) and \(T^*(1)\) are defined. It is shown that \(T\) is extended to a linear continuous operator in \(H^ 1(\mathbb R^ n)\) (resp. \(\text{BMO}(\mathbb R^ n))\) if and only if \(T^*(1)=0\) (resp. \(T(1)=0)\). The author then gives piecewise estimates of Calderón-Zygmund operators and the reconstruction of these operators by their kernels which are extensions of the corresponding results for classical singular integral operators of Calderón-Zygmund type.
Chapter VIII is devoted to the proof of the \(T(1)\) theorem of David and Journé and its applications to the study of algebras of Calderón-Zygmund operators. Let \(V\) be a topological vector space such that \({\mathcal D}(\mathbb R^ n)\subset V\subset L^ 2(\mathbb R^ n)\subset V'\subset{\mathcal D}'(\mathbb R^ n)\), e.g. \(V=C^ 1_ 0(\mathbb{R}^ n)\). A linear continuous operator \(T:V\to V'\) is said to be weakly continuous on \(L^ 2(\mathbb R^ n)\) if there exist a constant \(C\) and an integer \(q\) such that for any ball \(B\subset\mathbb R^ n\) and for any pair \(f,g\in V\) with support contained in \(B\) \[ |\langle Tf,g\rangle|\leq CN^ B_ q(f)N^ B_ q(g), \] where \(N^ B_ q(f)=R^{n/2}\sum_{|\alpha|\leq q} R^{|\alpha|}\|\partial^ \alpha f\|_ \infty\) and \(R\) is the radius of \(B\). A linear continuous operator \(T: V\to V'\) is said to be associated with a singular integral if there exist an exponent \(\gamma\in ]0,1]\), two constants \(C_ 0\), \(C_ 1\) and a function \(K:\Omega\to\mathbb C\) such that (1), (2) and (3) hold, and \[ Tf(x)=\int K(x,y)f(y)\,dy \] for any \(f\in V\) and \(x\) not belonging to the support of \(f\). For such an operator \(T(1)\) is defined modulo a constant. The following \(T(1)\) theorem is due to David and Journé.
Let \(T: V\to V'\) be a linear continuous operator associated with a singular integral. Then a necessary and sufficient condition in order that \(T\) is extended to a continuous operator in \(L^ 2(\mathbb R^ n)\) is that the following three conditions are satisfied:
(a) \(T(1)\) belongs to \(\text{BMO}(\mathbb R^ n)\),
(b) \(^ tT(1)\) belongs to \(\text{BMO}(\mathbb R^ n)\),
(c) \(T\) is weakly continuous on \(L^ 2(\mathbb R^ n)\).
The necessity part is a direct consequence of the previous chapter. The sufficiency part is established as follows. First consider the case \(T(1)=^ tT(1)=0\). In this case some estimates of \(\tau(\lambda,\lambda')=\langle T(\psi_ \lambda),\psi_{\lambda'}\rangle\) are obtained, where \(\psi_ \lambda\), \(\lambda\in\Lambda\), are ondelettes of class \(C^ 1\) and with compact support. Then, the desired result is proved with the aid of Schur’s lemma on infinite matrices. In the general case two operators \(R\) and \(S\) of Calderón-Zygmund are constructed so that \[ R(1)=T(1),\;^ tR(1)=0,\;S(1)=0,\;^ tS(1)=^ tT(1). \] Then \(N=T-R-S\) is an operator associated with a singular integral which satisfies the conditions of the special case. Hence, \(R\), \(S\), \(N\) are all continuous in \(L^ 2(\mathbb R^ n)\), and so is \(T\).
Chapter IX is devoted to examples of Calderón-Zygmund operators. A necessary and sufficient condition in order that an operator \(T:{\mathcal S}(\mathbb R^ n)\to{\mathcal S}'(\mathbb R^ n)\) belongs to some class of pseudodifferential operators is given in terms of the matrix with elements \(\langle T(\psi_ \lambda),\psi_{\lambda'}\rangle\), \(\lambda\in \Lambda\), \(\lambda'\in\Lambda\). It is shown that the commutators of operators of pointwise multiplication by Lipschitz continuous functions and pseudodifferential operators of order 1 are operators of Calderón-Zygmund. The \(L^ 2\) continuity of the operator \(\text{v.p.}(A(x)-A(y))^ k/(x-y)^{k+1}\) is studied, where \(A\) is a Lipschitz continuous function.
Chapter \(X\) is concerned with conditions in order that operators of Calderón-Zygmund are continuous in the class of Hölder continuous functions, Besov spaces and Sobolev spaces.
The main topic of Chapter XI is the \(T(b)\) theorem which is a generalization of the \(T(1)\) theorem of Chapter VIII. Let \(b\in L^ \infty(\mathbb R^ n)\) be a function satisfying \(\text{Re}\, b(x)\geq 1\). Then, for a given multiresolution analysis \(V_ j\), \(-\infty<j<\infty\), one can construct ondelettes \(\tilde\psi_ \lambda\), \(\lambda\in\Lambda\), so that \(\tilde\psi_ \lambda\in V_{j+1}\) for \(\lambda\in\Lambda_ j\), and \(\{\tilde\psi_ \lambda\}\) are biorthogonal with respect to the bilinear form \(B(f,g)=\int_{\mathbb R^ n}f(x)g(x)b(x)\,dx\). Let \(T:{\mathcal D}(\mathbb R^ n)\to{\mathcal D}'(\mathbb R^ n)\) be a linear continuous operator such that the restriction of its distribution kernel to \(\Omega\) is \(b(x)K(x,y)b(y)\), where \(K(x,y)\) is a function satisfying (1), (2), (3). Then \(T(1)\), \(^ tT(1)\) can be defined, and one says that \(T(1)\in\text{BMO}_ b\) if \(\sum_{Q(\lambda)\subset Q}|\langle T(1),\tilde\psi_ \lambda\rangle|^ 2\leq C| Q|\). Then, the \(T(b)\) theorem asserts that \(T\) is extended to a linear continuous operator in \(L^ 2(\mathbb R^ n)\) if and only if \(T\) is weakly continuous and \(T(1)\in\text{BMO}_ b\), \(^ tT(1)\in\text{BMO}_ b\).

Reviewer: H. Tanabe (Toyonaka)

MSC:

42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
42-02	Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47G10	Integral operators
47G30	Pseudodifferential operators

Keywords:

Calderón-Zygmund operators; ondelettes; wavelets; BMO; functions of bounded mean oscillations; singular integral operators; \(T(1)\) theorem; pseudodifferential operators; commutators; Lipschitz continuous function; Hölder continuous functions; Besov spaces; Sobolev spaces; \(T(b)\) theorem; multiresolution analysis

Citations:

Zbl 0745.42012; Zbl 0694.41037