Książka - szczegóły

We prove sufficiency of conditions on pairs of measures μ and ν, defined respectively on the interior and the boundary of a bounded Lipschitz domain Ω in d-dimensional Euclidean space, which ensure that, if u is the solution of the Dirichlet problem.
Δu = 0 in Ω,
$u|_{∂Ω} = f$,
with f belonging to a reasonable test class, then
$(∫_{Ω} |∇u|^{q} dμ) ^{1/q} ≤ (∫_{∂Ω} |f|^{p} dν)^{1/p}$,
where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on $ℝ₊^{d+1}$. As in that case we attack the problem by means of Littlewood-Paley theory. However, the lack of translation invariance forces us to use a general result of Wilson, which must then be translated into the setting of homogeneous spaces. We also consider what can be proved when a strictly elliptic divergence form operator replaces the Laplacian.