2017/08/09 by David S. Tartakoff, Tartakoff, David S.
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1708.02978
openalex publication_date 2017/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the regularity of Gevrey vectors for H "ormander operators P =\n
sumj=1m Xj2 + X0 + c where the Xj are real vector fields and\nc(x) is a smooth function, all in Gevrey class Gs. The principal\nhypothesis is that P satisfies the subelliptic estimate: for some\n\ε >0, ; \∃ ,C such that
|v
|_
varepsilon2
leq\nC
left(|(Pv, v)| +
|v
|02
right)
qquad
forall v
in C0^
infty. We prove\ndirectly (without the now familiar use of adding a variable t and proving\nsuitable hypoellipticity for Q=-Dt2-P and then, using the hypothesis on the\niterates of P on u, constructiong a homogeneous solution U for Q whose\ntrace on t=0 is just u) that for s\≥ 1, ,Gs(P,\Ω0) \⊂\nGs/\ε(\Ω0); that is,
forall K
Subset
Omega0,
;
exists\nCK:
|Pj u
|L2(K)
leq CKj+1 (2j)!s,
;
forall j
implies\n
forall K'
Subset
Omega0,
;
exists
tilde CK':
,
|D^
ell u
|L2(K')\n
leq
tilde CK'
ell+1
ell!s/
epsilon,
;
forall
ell. In other\nwords, Gevrey growth of derivatives of u as measured by iterates of P\nyields Gevrey regularity for u in a larger Gevrey class. When \ε =1,\nP is elliptic and so we recover the original Kotake-Narasimhan theorem\n( citeKN1962), which has been studied in many other classes, including\nultradistributions ( citeBJ).\n