Estimates for fundamental solutions of second-order subelliptic differential operators

Author:
Michael Christ

Journal:
Proc. Amer. Math. Soc. **105** (1989), 166-172

MSC:
Primary 35H05; Secondary 32F20, 58G05

MathSciNet review:
953002

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Abstract: A simple proof is given of pointwise estimates of C. Fefferman and A. Sánchez-Calle for the fundamental solution of a subelliptic, second-order partial differential operator with nonnegative characteristic form, based on a rescaling argument.

**[C]**Michael Christ,*Regularity properties of the \overline∂_{𝑏} equation on weakly pseudoconvex CR manifolds of dimension 3*, J. Amer. Math. Soc.**1**(1988), no. 3, 587–646. MR**928903**, 10.1090/S0894-0347-1988-0928903-2**[FP]**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. MR**730094****[FS]**Charles L. Fefferman and Antonio Sánchez-Calle,*Fundamental solutions for second order subelliptic operators*, Ann. of Math. (2)**124**(1986), no. 2, 247–272. MR**855295**, 10.2307/1971278**[M]**M. Machedon,*Estimates of the parametrix of the Kohn Laplacian on certain weakly pseudoconvex domains*, preprint.**[NSW]**Alexander Nagel, Elias M. Stein, and Stephen Wainger,*Balls and metrics defined by vector fields. I. Basic properties*, Acta Math.**155**(1985), no. 1-2, 103–147. MR**793239**, 10.1007/BF02392539**[OR]**O. A. Oleĭnik,*On hypoellipticity of second order equations*, Partial differential equations (Proc. Sympos. Pure Math., Vol. LXXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 145–151. MR**0470454**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1989-0953002-6

Article copyright:
© Copyright 1989
American Mathematical Society