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

Abstract

We consider a nonvariational degenerate elliptic operator of the kind Lu ≡ Xq i,j=1 aij (x)XiXju where X1, ..., Xq are a system of left invariant, 1-homogeneous, Hörmander’s vector fields on a Carnot group in R n , the matrix {aij} is symmetric, uniformly positive on a bounded domain Ω ⊂ R n and the coefficients satisfy aij ∈ V MOloc (Ω) ∩ L ∞ (Ω). We give a new proof of the interior W2,p X estimates kXiXjukLp(Ω0) + kXiukLp(Ω0) ≤ c n kLukLp(Ω) + kukLp(Ω)o for i, j = 1, 2, ..., q, u ∈ W2,p X (Ω), Ω 0 b Ω and p ∈ (1, ∞), first proved by Bramanti-Brandolini in [3], extending to this context Krylov’ technique, introduced in [15], consisting in estimating the sharp maximal function of XiXju.