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Abstract <p>Conditions are obtained under which the degenerate operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(P(D)\)</EquationSource> <!--LobJMat2560933Ghazaryan-m1--> </InlineEquation> (polynomial <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(P(\xi)\)</EquationSource> <!--LobJMat2560933Ghazaryan-m2--> </InlineEquation>) will be hypoelliptic. In the case of two-layer degenerate polynomials these conditions are necessary and sufficient and for multilayered ones, these conditions are only sufficient. To impediment this, we first consider the case when polynomial <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(P(\xi)\)</EquationSource> <!--LobJMat2560933Ghazaryan-m3--> </InlineEquation> is two-layer degenerate. It is further shown that in the multi-layer–degenerate case the question comes down to compering the powers of certain sub-polynomials of polynomial <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(P(\xi)\)</EquationSource> <!--LobJMat2560933Ghazaryan-m4--> </InlineEquation> with polynomial <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(P(\xi)\)</EquationSource> <!--LobJMat2560933Ghazaryan-m5--> </InlineEquation>. Therefore, we further study the question of comparing differential operators (polynomials).</p>

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On Multilayered-Degenerate Hypoelliptic Operators (Polynomials)

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • H. Ghazaryan,
  • V. Margaryan

摘要

Abstract

Conditions are obtained under which the degenerate operator \(P(D)\) (polynomial \(P(\xi)\) ) will be hypoelliptic. In the case of two-layer degenerate polynomials these conditions are necessary and sufficient and for multilayered ones, these conditions are only sufficient. To impediment this, we first consider the case when polynomial \(P(\xi)\) is two-layer degenerate. It is further shown that in the multi-layer–degenerate case the question comes down to compering the powers of certain sub-polynomials of polynomial \(P(\xi)\) with polynomial \(P(\xi)\) . Therefore, we further study the question of comparing differential operators (polynomials).