Optimal Gevrey Regularity for Certain Sums of Squares in Two Variables

2021/12/11 by Antonio Bove, Bove, Antonio, Marco Mughetti +1
Mathematics · #35A20 #35A27 (secondary) #35B65 #35H10 #35H20 (primary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2112.06001

openalex publication_date 2021/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For q , a integers such that a ≥ 1 , 1 < q , (x, y) ∈ U , U a neighborhood of the origin in ℝ² , we consider the operator D_x² + x^2(q-1) D_y² + y^2a D_y² . Slightly modifying the method of proof of \citemonom we can see that it is Gevrey s₀ hypoelliptic, where s₀^-1 = 1 - a^-1 (q - 1) q^-1 . Here we show that this value is optimal, i.e. that there are solutions to P u = f with f more regular than G^s₀ that are not better than Gevrey s₀ . The above operator reduces to the Métivier operator (\citemetivier81) when a = 1 , q = 2 . We give a description of the characteristic manifold of the operator and of its relation with the Treves conjecture on the real analytic regularity for sums of squares.

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