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Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations

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In this note we state weighted Poincaré inequalities associated with a family of vector fields satisfying Hörmander rank condition. Then, applications are given to relative isoperimetric inequalities and to local regularity (Harnack's inequality) for a class of degenerate elliptic equations with measurable coefficients.

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Additional information

The first author thanks the Institute for Advaced Study in Princeton for its hospitality during the preparation of the manuscript.

The first author was partially supported by MURST, Italy (40% and 60%) and GNAFA of CNR, Italy.

The second and third authors were partially supported by NSF Grants DMS93-15963 and 93-02991.

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Franchi, B., Lu, G. & Wheeden, R.L. Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations. Potential Anal 4, 361–375 (1995).

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Key words

  • Hörmander vector fields
  • Poincaré inequality
  • relative isoperimetric inequality
  • Harnack's inequality