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Real analysis and operator theory. (English) Zbl 0591.47041

Pseudodifferential operators and applications, Proc. Symp., Notre Dame/Indiana 1984, Proc. Symp. Pure Math. 43, 219-235 (1985).
[For the entire collection see Zbl 0562.00004.]
In his Helsinki adress, A. P. Calderón proposed a systematic study of the third generation of singular integral operators and explained the role played by some operators of this class in complex analysis and in elliptic P.D.E.’s in Lipschitz domains. An operator of the third generation is given as a weakly defined linear operator T mapping test functions into distributions whose distributional kernel satisfies some smoothness and size estimates away from the diagonal. Those estimates are reminiscent of the ones in the case of the Riesz transforms: off the diagonal the distributional kernel coincides with an ordinary function K(x,y) with the following estimates
(1) \(| K(x,y)| \leq C| x-y|^{-n}\)
(2) \(| K(x',y)-K(x,y)| \leq C| x'-x|^{\delta}| x- y|^{-n-\delta}\) \((0<\delta \leq 1\) is a fixed exponent and (2) should hold for \(| x'-x| \leq 1/2| x-y|)\)
(3) \(| K(x,y')-K(x,y)| \leq C| y'-y|^{\delta}| x- y|^{-n-\delta}\) for \(| y'-y| \leq 1/2| x-y|.\)
Major break-throughs have been made by G. David and J.-L. Journé [Ann. Math., II. Ser. 120, 371-397 (1984; Zbl 0567.47025)] who completely solved the fundamental problem of giving a necessary and sufficient condition for such an operator T to be bounded on \(L^ 2({\mathbb{R}}^ n;dx)\). Soon afterwards P. G. Lemarié [Ann. Inst. Fourier 35, No.4, 175-187 (1985; Zbl 0555.47032)] could give a sufficient condition for the continuity of T on homogeneous Besov spaces \(B^ s_{p,q}\) when \(0<s<\delta\), \(1\leq p\leq +\infty\), \(1\leq q\leq +\infty\).

MSC:

47Gxx Integral, integro-differential, and pseudodifferential operators
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
47B38 Linear operators on function spaces (general)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47L10 Algebras of operators on Banach spaces and other topological linear spaces
22E30 Analysis on real and complex Lie groups