Annali di Matematica Pura ed Applicata

, Volume 92, Issue 1, pp 23–28 | Cite as

Remarks on an inequality of Schulenberger and Wilcox

  • Leonard Sarason


J. R. Schulenberger and C. H. Wilcox[1], [2], have proven a coerciveness inequality for a class of nonelliptic first-order partial differential operators of the form Λ = =E −1 AjDj, where Aj (j=1, ..., n) is a constant m × m Hermitian matrix, E=E(x) is uniformly positive definite, bounded, and uniformly differentiable Hermitian m × m matrix, and where the symbol Λ(p, x)=E(x) −1 Ajpj has constant rank for all p ε Rn − {0} and x ε Rn. They prove coerciveness on N(Λ), the orthogonal complement of the null space N(Λ) relative to the inner product
$$\left\langle {u, v} \right\rangle = \int\limits_{R^n } {u(x) \cdot E(x)v(x)dx} $$
. Their proof is rather long, a simpler and shorter proof is given here. This proof leads naturally to a generalization of their results to the case where the Aj's need not be Hermitian.


Differential Operator Null Space Orthogonal Complement Short Proof Hermitian Matrix 
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  1. [1]
    J. R. Schulenberger -C. H. Wilcox,Coerciveness inequalities for nonelliptic systems of partial differential equations, Ann. Mat. Pura Appl.,87 (to appear, 1971).Google Scholar
  2. [2]
    J. R. Schulenberger -C. H. Wilcox,A coerciveness inequality for a class of nonelliptic operators of constant deficit, Technical Summary Report no. 8, Department of Mathematics, University of Denver (October 1970).Google Scholar

Copyright information

© Nicola Zanichelli Editore 1972

Authors and Affiliations

  • Leonard Sarason
    • 1
  1. 1.SeattleU.S.A.

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