2023/12/23 by Stefano Biagi, Biagi, Stefano, Marco Bramanti +1
Mathematics · #35B45 #35B65 (primary) #35H10 #35J70 #42B25 (secondary) #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2312.15367
openalex publication_date 2023/12/23 · openalex created_date 2023/12/29 · openalex updated_date 2026/08/01
We consider a class of nonvariational degenerate elliptic operators of the kind Lu=∑i,j=1maij( x) XiXju where \ aij( x) \ i,j=1m is a symmetric uniformly positive matrix of bounded measurable functions defined in the whole ℝn (n>m), possibly discontinuos but satisfying a VMO assumption, and X1,...,Xm are real smooth vector fields satisfying Hörmander rank condition in the whole ℝn and 1-homogeneous w.r.t. a family of nonisotropic dilations. We do not assume that the vector fields are left invariant w.r.t. an underlying Lie group of translations. We prove global WX2,p a-priori estimates, for every p∈( 1,∞) , of the kind: \Vert u\Vert_WX2,p(ℝn)≤ c\ \Vert Lu\Vert Lp( ℝn) +\Vert u\Vert Lp( ℝn) \ for every u∈ WX2,p( ℝn) . We also prove higher order estimates and corresponding regularity results: if aij∈ WXk,∞( ℝn) , u∈ WX2,p( ℝn) , Lu∈ WXk,p( ℝn) , then u∈ WXk+2,p( ℝn) and \Vert u\Vert_WXk+2,p(ℝn)≤ c\ \Vert Lu\Vert_WXk,p(ℝn)+\Vert u\VertLp(ℝn )\ .