The sharp maximal function approach to L p estimates for operators structured on Hörmander’s vector fields

• Published in: Revista matemática complutense 2016-01, Vol.29 (3), p.531-557
• Main Authors: Bramanti, Marco, Toschi, Marisa
• Format: Article
• Language: English
• Subjects: Fields (mathematics); Mathematical analysis; Matrix methods
• ISSN: 1139-1138; 1988-2807
• DOI: 10.1007/s13163-016-0206-1

We consider a nonvariational degenerate elliptic operator of the kind L u ≡ ∑ i , j = 1 q a i j ( x ) X i X j u where X 1 , … , X q are a system of left invariant, 1-homogeneous, Hörmander’s vector fields on a Carnot group in R n , the matrix a i j is symmetric, uniformly positive on a bounded domain Ω ⊂ R n and the coefficients satisfy a i j ∈ V M O l o c Ω ∩ L ∞ Ω . We give a new proof of the interior W X 2 , p estimates X i X j u L p Ω ′ + X i u L p Ω ′ ≤ c L u L p Ω + u L p Ω for i , j = 1 , 2 , … , q , u ∈ W X 2 , p Ω , Ω ′ ⋐ Ω and p ∈ 1 , ∞ , first proved by Bramanti–Brandolini in (Rend. Sem. Mat. dell’Univ. e del Politec. di Torino, 58:389–433, 2000), extending to this context Krylov’ technique, introduced in (Comm. PDEs, 32, 453–475, 2007), consisting in estimating the sharp maximal function of X i X j u .