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The \(L^ 2\)-boundedness of pseudodifferential operators. (English) Zbl 0651.35089

The author proves several new results of \(L^ 2\)-boundedness for pseudo differential operators a(x,D) with symbol a(x,\(\xi)\) in the classes \(S^ m_{\rho,\delta}\) of Hörmander assuming estimates for a finite number only of derivatives \(D_ x^{\alpha}D^{\beta}_{\xi}a(x,\xi)\). In particular the theorem of A. P. Calderon and R. Vaillancourt [Proc. Nat. Acad. Sci. USA 69, 1185-1187 (1972; Zbl 0244.35074)] is recaptured by using only elementary tools, as integration by parts, Fourier transform and Parseval formula.
Reviewer: L.Rodino

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
47Gxx Integral, integro-differential, and pseudodifferential operators

Citations:

Zbl 0244.35074
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