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arXiv · 1511.03536

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

Marco Bramanti·Marisa Toschi
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Abstract

We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, Hörmander's vector fields on a Carnot group in $R^{n}$, where the matrix of coefficients is symmetric, uniformly positive on a bounded domain of $R^{n}$ and the coefficients are bounded, measurable and locally VMO in the domain. We give a new proof of the interior $L^{p}$ estimates on the second order derivatives with respect to the vector fields, first proved by Bramanti-Brandolini in [Rend. Sem. Mat. dell'Univ. e del Politec. di Torino, Vol. 58, 4 (2000), 389-433], extending to this context Krylov' technique, introduced in [Comm. in P.D.E.s, 32 (2007), 453-475], consisting in estimating the sharp maximal function of the second order derivatives.

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Marco Bramanti, Marisa Toschi. 2015-11-11. The sharp maximal function approach to $L^{p}$ estimates for operators structured on Hörmander's vector fields. https://arxiv.org/abs/1511.03536

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