Luca Capogna

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The classical embedding theorem of Sobolev W (R) ↪→ L(R) was originally proved in the case 1 < p < n, with q = np n−p . It was only in the late fifties that by means of an elegant integral inequality Gagliardo and Nirenberg were independently able to obtain the limiting case p = 1 and prove that: (∗) W (R) ↪→ L n n−1 (R) (see, e.g., [S]). Meanwhile, in his(More)
We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of the L∞ variational problem { inf ||∇0u||L∞(Ω), u = g ∈ Lip(∂Ω) on ∂Ω, where Ω ⊂G is an open subset of a Carnot group, ∇0u denotes the horizontal gradient of u : Ω → R, and the Lipschitz class is defined in relation to the(More)
In 1977 B. Dahlberg [7] proved his celebrated theorem stating that for a bounded Lipschitz domain in R harmonic and surface measure are mutually absolutely continuous and, furthermore, the Radon-Nikodym derivative of harmonic measure with respect to surface measure satisfies a reverse Hölder inequality. The aim of this note is to announce a similar theorem(More)