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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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##### References:
 [1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $$N$$-body Schrödinger operators , Mathematical Notes, vol. 29, Princeton University Press, Princeton, NJ, 1982. · Zbl 0503.35001 [2] R. Beals, $$L\spp$$ and Hölder estimates for pseudodifferential operators: sufficient conditions , Ann. Inst. Fourier (Grenoble) 29 (1979), no. 3, vii, 239-260. · Zbl 0387.35065 · doi:10.5802/aif.760 · numdam:AIF_1979__29_3_239_0 · eudml:74422 [3] R. Beals, Weighted distribution spaces and pseudodifferential operators , J. Analyse Math. 39 (1981), 131-187. · Zbl 0474.35089 · doi:10.1007/BF02803334 [4] A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions , Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 33-49. · Zbl 0195.41103 [5] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes , Springer-Verlag, Berlin, 1971. · Zbl 0224.43006 · doi:10.1007/BFb0058946 [6] C. L. Fefferman, The uncertainty principle , Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129-206. · Zbl 0526.35080 · doi:10.1090/S0273-0979-1983-15154-6 [7] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems , Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 590-606. · Zbl 0503.35071 [8] C. L. Fefferman and A. Sanchez-Calle, Fundamental solutions for second order subelliptic operators , Ann. of Math. (2) 124 (1986), no. 2, 247-272. JSTOR: · Zbl 0613.35002 · doi:10.2307/1971278 · links.jstor.org [9] L. Hormander, The Weyl calculus of pseudodifferential operators , Comm. Pure Appl. Math. 32 (1979), no. 3, 360-444. [10] A. Nagel and E. M. Stein, Some new classes of pseudodifferential operators , Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 2, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 159-169. · Zbl 0427.35081 [11] A. Nagel and E. M. Stein, Lectures on pseudodifferential operators: regularity theorems and applications to nonelliptic problems , Mathematical Notes, vol. 24, Princeton University Press, Princeton, N.J., 1979. · Zbl 0415.47025 [12] O. A. Oleinik and E. Radkevitch, Second Order Equations with Nonnegative Characteristic Form , Amer. Math. Soc., Providence, R.I., 1973. [13] L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups , Acta Math. 137 (1976), no. 3-4, 247-320. · Zbl 0346.35030 · doi:10.1007/BF02392419 [14] H. F. Smith, The subelliptic oblique derivative problem , Comm. Partial Differential Equations 15 (1990), no. 1, 97-137. · Zbl 0711.35157 · doi:10.1080/03605309908820679
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