Resumen

MARCO BRAMANTI; TOSCHI, MARISA

Santa Fe

Congreso; LXIV Reunion anual de la Unión Matematica Argentina; 2015

In 1993 Chiarenza-Frasca-Longo cite{Chiarenza-Frasca-Longo} proved $W^{2,p}$ estimates for nonvariational elliptic operators $a_{ij}left( xight) u_{x_{i}x_{j}}$ with $a_{ij}in L^{infty}cap VMO$, introducing a new technique based on representation formulas by means of the fundamental solution of the constant coefficient operator, and commutators of singular integrals with $BMO$ functions. This technique, by now classic, has been extended to several contexts, for instance parabolic operators (see cite{Bramanti-Cerutti}) and nonvariational operators structured on H"{o}rmander´s vector fields (see cite{Bramanti-Brandolini}). In 2007 Krylov (cite{Krylov pap}) introduced a differerent technique to prove similar andmore general results for elliptic and parabolic operators, based on theemph{pointwise} estimate of the emph{sharp maximal function of }%$u_{x_{i}x_{j}}$, that is $left( u_{x_{i}x_{j}}ight) ^{#}$. The idea of this work is to see if Krylov´ technique can be extended also to the context of linear degenerate equations structured on H"{o}rmander´s vector fields. We give a positive answers for an operator of the kind of $a_{ij}(x)X_{i}X_{j}u$ on Carnot groups, giving an alternative, shorter proof of the results obtained in cite{Bramanti-Brandolini}.