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By an easy “trick” taken from the caloric polynomial theory, we prove the existence of a basis of the Euclidean topology whose elements are resolutive sets of the heat equation. This result can be used to construct, with a very elementary approach, the Perron solution of the caloric Dirichlet problem on arbitrary bounded open subsets of the Euclidean space-time.

Let H be a linear second order partial differential operator with non-negative characteristic form in a strip S ⊂ R^N ×R. We assume that H as a fundamental solution, smooth out of its poles and bounded from above and from below by Gaussian kernels modeled on subriemannian doubling distances in R^N. Under these assumptions we show that H endows S with a structure of β-harmonic space. This allows us to study boundary value problems for L with a Perron-Wiener-Brelot-Bauer method, and to obtain pointwise regularity estimates at the boundary in terms of Wiener series modeled on the Gaussian kernels. Our analysis includes the proof of a scale invariant Harnack inequality for nonnegative solutions. We also show an application to the real hypersurphaces of C^{n+1} with given Levi-curvature.

Let L be a real sub-Laplacian on a stratified Lie group G. In this note we present some results on the Dirichlet problem for L with L^p-boundary data, on domains which are contractible with respect to the natural dilations of G. One of the main difficulties we overcome is the presence of non-regular boundary points for the usual Dirichlet problem for L. A potential theoretical approach is followed.

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In this note we present mean value characterizations of subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution ¡. These characterizations are based on suitable average operators on the level sets of ¡. Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach and Reade. The results presented here generalize and carry forward former results of the authors in [6, 8].