Celebratio Mathematica — Calderón

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Works connected to Ivan Georgievich Petrovsky

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A. P. Calderón: “Algebras of singular integral operators,” pp. 393–395 in Trudy Mezhdunarodnogo Kongressa Matematikov [Proceedings of the International Congress of Mathematicians] (Moscow, 16–26 August 1966), vol. 1. Edited by I. G. Petrovsky. Mir (Moscow), 1968. Also published in Singular integrals (1967). MR 0248563 Zbl 0187.39401 incollection

In dealing with differential operator with non-smooth coefficients one is led to consider pseudo-differential operators and singular integral operators with non-smooth symbols and kernels. Such pseudo-differential operators, however, do not have desirable algebraic properties. This becomes evident if one observes that in particular differential operators cannot be composed freely unless one assumes their coefficients to be infinitely differentiable. The situation is different with singular integral operators. Here it is possible to construct a theory, which is fairly general as far as regularity assumptions is concerned, preserving at the same time an algebra structure and a relatively simple approximate functional calculus. In what follows we will give a brief description.