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Abstract  The aim of this paper is to give some representation formulas of Riesz and Poisson-Jensen type for super-solutions to a class of hypoelliptic ultraparabolic operators on a homogeneous Lie group . Our results complete the ones obtained in Cinti (Math Scand 100:1–21, 2007). We also provide a suitable theory for -Green functions and for -Green potentials of Radon measures. The proofs mostly rely on the use of appropriate techniques relevant to the Potential Theory for . [ABSTRACT FROM AUTHOR]

Abstract  We establish a covering lemma of Besicovitch type for metric balls in the setting of Hölder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets. For families of measurable functions, we introduce the notions of power decay, critical density and double ball property and with the aid of the covering theorem we show how these notions are related. Next we present an axiomatic procedure to establish Harnack inequality that permits to handle both divergence and non divergence linear equations. [ABSTRACT FROM AUTHOR]

We construct the fundamental solutions $\Gamma$ and $\gamma$ for the non-divergence form operators ${\textstyle\sum_{i,j}} a_{i,j}(x,t)X_iX_j-\partial_t$ and ${\textstyle\sum_{i,j}}a_{i,j}(x)X_iX_j$, where the $X_i$'s are Hörmander vector fields generating a stratified group $\mathbb{G}$ and $(a_{i,j})_{i,j}$ is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of $\Gamma$ and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of $\Gamma$ allow us to construct $\gamma$ using both $t$-saturation and approximation arguments. [ABSTRACT FROM AUTHOR]

The maximum principle on a wide class of unbounded domains is proved for solutions to the partial differential inequality $\Delta_{\mathbb{G}}u+cu\geq 0$, where $c\leq 0$ and $\Delta_{\mathbb{G}}$ is a real sub-Laplacian. A potential theory approach is followed. [ABSTRACT FROM AUTHOR]