Marco Bramanti

Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities

In this work the authors deal with linear second order partial differential operators of the following type

Singular integrals in nonhomogeneous spaces: L 2 and L p continuity from Holder estimates

H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}-\sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)

Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality

where

Symmetrization in parabolic neumann problems

X_{1},X_{2},\ldots,X_{q}

To Understand, i.e., to Know How to Explain

is a system of real Hormanders vector fields in some bounded domain

To Understand, i.e., to Know How to Apply

\Omega\subseteq\mathbb{R}^{n}

Negations and Indirect Proofs

,

Functions and Set Theoretic Arguments

A=\left\{ a_{ij}\left( t,x\right) \right\} _{i,j=1}^{q}

Propositions and Properties (Level B)

is a real symmetric uniformly positive definite matrix such that

Formulae and Indices (Level A)

\lambda^{-1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x) \xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\text{}\forall\xi\in\mathbb{R}^{q}, x \in\Omega,t\in(T_{1},T_{2})