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\(L^p\)-bounded pseudodifferential operators and regularity for multi-quasi-elliptic equations. (English) Zbl 1082.35175

The authors present some new results on the \(L^p\)-interior regularity of the solutions of the equation with smooth coefficients \[ Pu= \sum_{\alpha\in Q} a_\alpha(x) D^\alpha_x u= f, \] where the Newton polyhedron of the coefficients \(Q\) is assumed complete, and the operator \(P\) is multi-quasi-elliptic, in the sense that \[ \Biggl| \sum_{\alpha\in Q} a(x)\xi^\alpha\Biggr|\leq C\Biggl(\sum_{\alpha\in Q} \xi^{2\alpha}\Biggr)^{1/2}, \] cf. [S. Gindikin and L. R. Volevich, The method of Newton’s polyhedron in the theory of partial differential equations. Mathematics and Its Applications. Soviet Series. 86. Dordrecht: Kluwer Academic Publishers (1992; Zbl 0779.35001)].
In particular, the authors obtain that \(Pu= f\in L^p\), \(1< p<\infty\), implies \(D^\alpha u\in L^p\) for every \(\alpha\in Q\). These results are deduced from a pseudodifferential calculus of the type of R. Beals, L. Hörmander, for a general class of operators including \(P\) and its parametrix \(P'\). Basic point is a theorem of \(L^p\)-boundedness for pseudodifferential operators, new with respect to the existing literature.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
42B15 Multipliers for harmonic analysis in several variables
35A17 Parametrices in context of PDEs

Citations:

Zbl 0779.35001
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