# Differential operators of the first order with degenerate principal symbols

• Volume: 27, Issue: 1, page 147-161
• ISSN: 0137-6934

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## Abstract

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Let there be given a differential operator on ${ℝ}^{n}$ of the form $D={\sum }_{i,j=1}^{n}{a}_{ij}·{x}_{j}\partial /\partial {x}_{i}+\mu$, where $A=\left({a}_{ij}\right)$ is a real matrix and μ is a complex number. We study the following question: To what extent the mapping $D:{S}^{\text{'}}\left({ℝ}^{n}\right)\to {S}^{\text{'}}\left({ℝ}^{n}\right)$ is surjective? We shall give some conditions on A and μ which assure the surjectivity of D.

## How to cite

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Felix, Rainer. "Differential operators of the first order with degenerate principal symbols." Banach Center Publications 27.1 (1992): 147-161. <http://eudml.org/doc/262773>.

@article{Felix1992,
abstract = {Let there be given a differential operator on $ℝ^n$ of the form $D = ∑^\{n\}_\{i,j=1\} a_\{ij\}·x_j ∂/∂x_i + μ$, where $A = (a_\{ij\})$ is a real matrix and μ is a complex number. We study the following question: To what extent the mapping $D :S^\{\prime \}(ℝ^n) → S^\{\prime \}(ℝ^n)$ is surjective? We shall give some conditions on A and μ which assure the surjectivity of D.},
author = {Felix, Rainer},
journal = {Banach Center Publications},
keywords = {Schwartz space},
language = {eng},
number = {1},
pages = {147-161},
title = {Differential operators of the first order with degenerate principal symbols},
url = {http://eudml.org/doc/262773},
volume = {27},
year = {1992},
}

TY - JOUR
AU - Felix, Rainer
TI - Differential operators of the first order with degenerate principal symbols
JO - Banach Center Publications
PY - 1992
VL - 27
IS - 1
SP - 147
EP - 161
AB - Let there be given a differential operator on $ℝ^n$ of the form $D = ∑^{n}_{i,j=1} a_{ij}·x_j ∂/∂x_i + μ$, where $A = (a_{ij})$ is a real matrix and μ is a complex number. We study the following question: To what extent the mapping $D :S^{\prime }(ℝ^n) → S^{\prime }(ℝ^n)$ is surjective? We shall give some conditions on A and μ which assure the surjectivity of D.
LA - eng
KW - Schwartz space
UR - http://eudml.org/doc/262773
ER -

## References

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1. [1] R. Felix, Solvability of differential equations with linear coefficients of nilpotent type, Proc. Amer. Math. Soc. 94 (1985), 161-166. Zbl0541.35010
2. [2] R. Felix, Zentrale Distributionen auf Exponentialgruppen, J. Reine Angew. Math. 389 (1988), 133-156.
3. [3] G. B. Folland, Real Analysis. Modern Techniques and Their Applications, Wiley, New York 1984. Zbl0549.28001
4. [4] L. Hörmander, On the division of distributions by polynomials, Ark. Mat. 3 (1958), 555-568. Zbl0131.11903
5. [5] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, Berlin 1983. Zbl0521.35001
6. [6] D. Müller and F. Ricci, Analysis of second order differential operators on Heisenberg groups II, preprint. Zbl0790.43011
7. [7] H. H. Schaefer, Topological Vector Spaces, 5th printing, Springer, New York 1986.
8. [8] L. Schwartz, Théorie des distributions, Hermann, Paris 1966.

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