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Parametrix construction for a class of subelliptic differential operators. (English) Zbl 0777.35002
The author considers second order partial differential operators of the form \[ L(x,\partial_ x)=A(x,\partial_{x'})-\beta(x)\partial^ 2_{x_ n},\tag{*} \] where \(\beta(x)\) is nonnegative and satisfies the inequality \(\sum_{|\alpha|\leq k}|\partial^ \alpha_{x'}\beta(x)|\geq c_ 0>0\), with \(x'=(x_ 1,\dots,x_{n- 1})\). The operator \(A(x,\partial_{x'})\) is assumed to be an elliptic differential operator in the variables \(x'\), with nonnegative principal symbol, and with coefficients that may depend on \(x_ n\).
The author proves that operators like (*) have parametrices that are pseudo-differential operators with principal symbol equivalent to \[ \bigl[|\xi'|+\sum_{|\alpha|\leq k}|\partial^ \alpha_{x'}\beta(x)\xi^ 2_ n|^{1/(|\alpha|+2)}\bigr]^{-2}. \] From this result, the author derives a sharp regularity result on Sobolev spaces. The strong results proved in this paper required quite delicate techniques, ingeniously combined with some powerful results coming from the Calderon- Zygmund theory.

MSC:
35A08 Fundamental solutions to PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
35S05 Pseudodifferential operators as generalizations of partial differential operators
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46B70 Interpolation between normed linear spaces
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