The sharp maximal function approach to $$L^{p}$$ L p estimates for operators structured on Hörmander?s vector fields

MARCO BRAMANTI; MARISA TOSCHI

REVISTA MATEMATICA COMPLUTENSE

UNIV COMPLUTENSE MADRID

Lugar: Obsrvación: La editorial es Springer y no UCM; Año: 2016 vol. 29 p. 531 - 557

1139-1138

egin{abstract}We consider a nonvariational degenerate elliptic operator of the kind%[Luequivsum_{i,j=1}^{q}a_{ij}(x)X_{i}X_{j}u]where $X_{1},...,X_{q}$ are a system of left invariant, $1$-homogeneous,H"{o}rmander´s vector fields on a Carnot group in $mathbb{R}^{n}$, thematrix $left{ a_{ij}ight} $ is symmetric, uniformly positive on abounded domain $Omegasubsetmathbb{R}^{n}$ and the coefficients satisfy%[a_{ij}in VMO_{loc}left( Omegaight) cap L^{infty}left(Omegaight) .]We give a new proof of the interior $W_{X}^{2,p}$ estimates%[leftVert X_{i}X_{j}uightVert _{L^{p}left( Omega^{prime}ight)}+leftVert X_{i}uightVert _{L^{p}left( Omega^{prime}ight) }leqcleft{ leftVert LuightVert _{L^{p}left( Omegaight) }+leftVertuightVert _{L^{p}left( Omegaight) }ight}]for $i,j=1,2,...,q$, $uin W_{X}^{2,p}left( Omegaight) ,$ $Omega^{prime}SubsetOmega$ and $pinleft( 1,inftyight) $, first proved byBramanti-Brandolini in cite{bb1}, extending to this context Krylov´technique, introduced in cite{K1}, consisting in estimating the sharp maximalfunction of $X_{i}X_{j}u.$end{abstract}